Strategy Complexity of Reachability in Countable Stochastic 2-Player Games

Abstract

We study countably infinite stochastic 2-player games with reachability objectives. Our results provide a complete picture of the memory requirements of $\varepsilon$-optimal (resp. optimal) strategies. These results depend on the size of the players’ action sets and on whether one requires strategies that are uniform (i.e., independent of the start state). Our main result is that $\varepsilon$-optimal (resp. optimal) Maximizer strategies requires infinite memory if Minimizer is allowed infinite action sets. This lower bound holds even under very strong restrictions. Even in the special case of infinitely branching turn-based reachability games, even if all states allow an almost surely winning Maximizer strategy, strategies with a step counter plus finite private memory are still useless. Regarding uniformity, we show that for Maximizer there need not exist memoryless (i.e., positional) uniformly $\varepsilon$-optimal strategies even in the special case of finite action sets or in finitely branching turn-based games. On the other hand, in games with finite action sets, there always exists a uniformly $\varepsilon$-optimal Maximizer strategy that uses just one bit of public memory.

1 Introduction

We study 2-player zero-sum stochastic games on countably¹ infinite graphs. This section outlines the background and our contribution. Formal definitions of games, strategies, memory, etc., are given in Sect. 2.

Stochastic games were first introduced by Shapley in his seminal 1953 work [51], and model dynamic interactions in which the environment responds randomly to players’ actions. Shapley’s games were generalized by [24] and [35] to allow infinite state and action sets and non-termination. They play a central role in the solution of many problems in economics, see [4, 27, 45, 52, 53], evolutionary biology, e.g., [48], and computer science, see [1, 2, 9, 15, 43, 52, 54] among others.

In general concurrent games, in each state both Maximizer and Minimizer independently choose an action and the next state is determined according to a pre-defined distribution that depends on the chosen pair of actions. Turn-based games (also called switching-control games) are a subclass where each state is owned by some player and only this player gets to choose an action. These games were studied first in the 1980s and 90s in [14, 18, 19, 55, 56] but have recently received much attention by computer scientists, for instance in [5, 9, 13, 25, 30]. An even more special case of stochastic games are Markov Decision Processes (MDPs): MDPs are turn-based games where all controlled states are Maximizer states. Since Minimizer is passive, they are also called games against nature.

In order to get the strongest results, we will show that our lower bound results hold even for the special subclass of turn-based games while our upper bounds hold even for general games.

A strategy for a player is a function that, given a history of a play, determines the next action of the player. Objectives are defined via functions that assign numerical rewards to plays, and the Maximizer (resp. Minimizer) aim to maximize (resp. minimize) the expected reward. A central result in zero-sum 2-player stochastic games with finite action sets is the existence of a value for the large class of Borel measurable objectives [38, 41] (i.e., that $\sup _{ Max}\inf _{ Min} = { value} = \inf _{ Min}\sup _{ Max}$ over Maximizer/Minimizer strategies). In particular, this implies the existence of $\varepsilon$-optimal strategies for every $\varepsilon >0$ and either player, i.e., strategies that enforce that the outcome of a game is $\varepsilon$-close to its value, regardless of the behavior of the other player. Optimal strategies ($\varepsilon$-optimal for $\varepsilon =0$) need not exist in general, but their properties have been studied in those cases where they do exist, for example in [30, 33, 36, 47].

The nature of good strategies in stochastic games – that is $\varepsilon$-optimality vs. optimality, and their memory requirements – is relevant in computer science [11, 30, 33], in particular, in the sense of computability [36]. It is also recognized as a central notion in branches of mathematics and economics, especially operations research[40], probability theory [20], game theory[22, 37, 40] and economic theory [3, 4, 29].

The simplest type of strategy bases its decisions only on the current state, and not on the history of the play. Such strategies are called memoryless or positional.² By default, we assume that strategies can use randomization (i.e., use mixed actions), while the subclass of deterministic (pure) strategies are limited to choosing a single pure action at each state. Memoryless randomized (MR) strategies choose a mixed action at each state, while memoryless deterministic (MD) strategies choose a pure action at each state, both independently of the history.

More complex strategies might use some finite amount of memory. The strategy chooses an action depending only on the current state and the current memory mode. The memory mode can be updated in every round according to the current state, the observed chosen actions and the next state. We assume perfect-information games, so the actions and states are observable at the end of every round. In general, for strategies that are not deterministic but use randomization, this memory update may also be randomized. Therefore, in the case of games, a player does not necessarily know for sure the current memory mode of the other player. It may be advantageous for a player to keep his memory mode hidden from the other player. We distinguish between public memory, where the strategies’ memory mode is public knowledge, and private memory, which is hidden from the opponent. A step counter is an infinite memory device corresponding to a discrete clock that is incremented after every round. We consider this to be a type of public memory, because the update is deterministic and the memory mode can be deduced by the opponent. Strategies that use only a step counter are called Markov strategies. Combinations of the above are possible, e.g., a strategy that uses a step counter and an additional finite public/private general purpose memory. The amount/type of memory and randomization required for a good ($\varepsilon$-optimal, resp. optimal) strategy for a given objective is also called its strategy complexity.

1.1 The Reachability Objective

With a reachability objective, a play is defined as winning for Maximizer iff it visits a defined target state (or a set of target states) at least once. Thus Maximizer aims to maximize the probability that the target is reached. Dually, Minimizer aims to minimize the probability of reaching the target. So, from Minimizer’s point of view, this is the dual safety objective of avoiding the target.

Reachability is arguably the simplest objective in games on graphs. It can trivially be encoded into the usual reward-based objectives, i.e., every play that reaches the target gets reward 1 and all other plays get reward 0. Moreover, it can be encoded into many other objectives including Büchi, Parity and average-payoff conditions, by turning the target vertex into a good (for the new objective) sink.

Despite their apparent simplicity, reachability games are not trivial. While both players have optimal MD strategies in finite-state turn-based reachability games [14]; see also [36, Proposition 5.6.c, Proposition 5.7.c], this does not carry over to finite-state concurrent reachability games. A counterexample where Maximizer has no optimal strategy is the Hide-or-Run game [17, Example 1], also see [16, 35].

In countably infinite reachability games, Maximizer does not have an optimal strategy even if the game is turn-based, in fact not even in countably infinite MDPs that are finitely branching [31]. On the other hand, [46, Proposition A] shows that Maximizer has $\varepsilon$-optimal MD strategies in countably infinite MDPs. Better yet, the MD strategies can be made uniform, i.e., independent of the start state.³ This led to the question whether Ornstein’s results can be generalized from MDPs to countably infinite stochastic games.[49], Corollary 3.9, proved the following.

Proposition 1

Maximizer has $\varepsilon$-optimal memoryless (MR) strategies in countably infinite concurrent reachability games with finite action sets.

Table 1 The strategy complexity of Maximizer for the reachability objective

Table 2 The strategy complexity of Minimizer for the reachability objective

However, these MR strategies are not uniform, i.e., they depend on the start state. In fact, [44] showed that there cannot exist any uniformly $\varepsilon$-optimal memoryless Maximizer strategies in countably infinite concurrent reachability games with finite action sets. Their counterexample is called the Big Match on $\mathbb {N}$ which, in turn, is inspired by the Big Match [7, 24, 26, 52]. Several fundamental questions remained open:

Q1.:

Does the negative result of [44] still hold in the special case of countable turn-based (finitely branching) reachability games?

Q2.:

If uniformly $\varepsilon$-optimal Maximizer strategies cannot be memoryless, how much memory do they need?

Q3.:

Does the positive result of Secchi (Proposition 1 above) still hold if the restriction to finite action sets is dropped? The question is meaningful, since concurrent games where only one player has countably infinite action sets are still determined [21, Theorem 11] though not if both players have infinite action sets, unless one imposes other restrictions. Moreover, what about infinitely branching turn-based reachability games? How much memory do good Maximizer strategies need in these cases?

1.2 Our Contribution

Our results, summarized in Tables 1 and 2, provide a comprehensive view on the strategy complexity of (uniformly) $\varepsilon$-optimal strategies for reachability (and also about optimal strategies when they exist).

Our first result strengthens the negative result of [44] to the turn-based case.

First Lower-Bound result (Q1)

(Theorem 7) There exists a finitely branching turn-based version of the Big Match on $\mathbb {N}$ where Maximizer still does not have any uniformly $\varepsilon$-optimal MR strategy.

Our second result solves the open question about uniformly $\varepsilon$-optimal Maximizer strategies. While uniformly $\varepsilon$-optimal Maximizer strategies cannot be memoryless, 1 bit of memory is enough.

Main Upper-Bound result (Q2)

(Theorem 6) In concurrent games with finite action sets and reachability objective, for any $\varepsilon >0$, Maximizer has a uniformly $\varepsilon$-optimal public-memory 1-bit strategy. This strategy can be chosen as deterministic if the game is turn-based and finitely branching.

Our main contribution (Theorem 2) addresses Q3. It determines the strategy complexity of Maximizer in infinitely branching reachability games. Our result is a strong lower bound, and we present the path towards it by disproving a sequence of hopeful conjectures towards upper bounds.

Hope 1

In turn-based reachability games, Maximizer has $\varepsilon$-optimal MD strategies.

This is motivated by the fact that the property holds if the game is finitely branching [36, Proposition 5.7.c] and Lemma 5 or if it is just an MDP as in [46].

One might even have hoped for uniformly $\varepsilon$-optimal MD strategies, i.e., strategies that do not depend on the start state of the game, but this hope was crushed by the answer to Q1.

Let us mention a concern about Hope 1 as stated (i.e., disregarding uniformity). Consider any turn-based reachability game that is finitely branching, and let $x \in [0,1]$ be the value of the game. The proof of Proposition 1 actually shows that for every $\varepsilon > 0$, Maximizer has both a strategy and a time horizon $n \in \mathbb {N}$ such that for all Minimizer strategies, the game visits the target state with probability at least $x-\varepsilon$ within the first n steps of the game. There is no hope that such a guarantee on the time horizon can be given in infinitely branching games. Indeed, consider the infinitely many states $f_0, f_1, f_2, \ldots$, where $f_0$ is the target state and for $i>0$ state $f_i$ leads to $f_{i-1}$ regardless of the players’ actions, and an additional Minimizer state, u, where Minimizer chooses, by her action, one of the $f_i$ as successor state. In this game, starting from u, Maximizer wins with probability 1 (he is passive in this game). Minimizer cannot avoid losing, but her strategy determines when $f_0$ is visited. This shows that a proof of Hope 1 would require different methods.

In case Hope 1 turns out to be false, there are various plausible weaker versions. Let us briefly discuss their motivation.

Hope 2

Hope 1 is true if MD is replaced by MR.

This is motivated by Proposition 1, i.e., that in concurrent games with finite action sets for both players, Maximizer has $\varepsilon$-optimal MR strategies. In fact, [21, Theorem 12.3] implies that this holds even under the weaker assumption that just Minimizer has finite action sets (while Maximizer is allowed infinite action sets).

Hope 3

Hope 1 is true if Maximizer has an optimal strategy.

This is motivated by the fact that in MDPs with Büchi objective (i.e., the player tries to visit a set of target states infinitely often), if the player has an optimal strategy, he also has an MD optimal strategy. The same is not true for $\varepsilon$-optimal strategies as shown in [31]. This example shows that although optimal strategies do not always exist, if they do exist, they may be simpler.

Hope 4

Hope 1 is true if Maximizer has an almost surely winning strategy, i.e., a strategy that guarantees him to visit the target state with probability 1.

This is weaker than Hope 3, because an almost surely winning strategy is necessarily optimal.

In a turn-based game, let us associate to each state $s$ its value, which is the value of the game when started in s. We call a controlled step $s\rightarrow s^{\prime }$ value-decreasing (resp., value-increasing), if the value of $s^{\prime }$ is smaller (resp., larger) than the value of $s$. It is easy to see that Maximizer cannot do value-increasing steps and Minimizer cannot do value-decreasing steps, but the opposite is possible in general.

Hope 5

Hope 1 is true if Maximizer does not have value-decreasing steps.

Hope 6

Hope 1 is true if Minimizer does not have value-increasing steps.

Hopes 5 and 6 are motivated by the fact that sometimes the absence of Maximizer value-decreasing steps or the absence of Minimizer value-increasing steps implies the existence of optimal Maximizer strategies and then Hope 3 might apply. For example, in finitely branching turn-based reachability games, the absence of Maximizer value-decreasing steps or the absence of Minimizer value-increasing steps implies the existence of optimal Maximizer strategies, and they can be chosen MD [30, Theorem 5].

Hope 7

Hope 1 is true for games with acyclic game graph.

This is motivated, e.g., by the fact that in safety MDPs (where the only active player tries to avoid a particular state f) with acyclic game graph and infinite action sets the player has $\varepsilon$-optimal MD strategies [33, Corollary 26]. The same does not hold without the acyclicity assumption [31, Theorem 3].

Hope 8

Hope 1 is true if Maximizer can additionally use a step counter to choose his actions.

This is weaker than Hope 7, because by using a step counter Maximizer effectively makes the game graph acyclic. However, the reverse does not hold. Not every acyclic game graph has an implicit step counter.

Hope 9

In turn-based reachability games, Maximizer has $\varepsilon$-optimal strategies that use only finite memory.

This is motivated, e.g., by the fact that in MDPs with acyclic game graph and Büchi objective, the player has $\varepsilon$-optimal deterministic strategies that require only 1 bit of memory, but no $\varepsilon$-optimal MR strategies [32].

It might be advantageous for Maximizer to keep his memory mode private. This motivates the following final weakening of Hope 9.

Hope 10

In turn-based reachability games, Maximizer has $\varepsilon$-optimal strategies that use only private finite memory.

The main contribution of this paper is to crush all these hopes. That is, Hope 1 is false, even if all weakenings proposed in Hopes 2-10 are imposed at the same time. Specifically, we show the following theorem (stated in more detail as Theorem 15 later on).

Theorem 2

There is a turn-based reachability game (necessarily, by Proposition 1, with infinite action sets for Minimizer) with the following properties:

• 1. for every Maximizer state, Maximizer has at most two actions to choose from;

• 2. for every state Maximizer has a strategy to visit the target state with probability 1, regardless of Minimizer’s strategy;

• 3. for every Maximizer strategy that uses only a step counter and private finite memory and randomization, for every $\varepsilon > 0$, Minimizer has a strategy so that the target state is visited with probability at most $\varepsilon$.

This lower bound trivially carries over to concurrent stochastic games with infinite Minimizer action sets, and for all Borel objectives that subsume reachability, e.g., Büchi, co-Büchi, Parity, average-reward and total-reward.

To put this result into perspective, we show in Sect. 8 that it is crucial that Minimizer can use infinite branching (resp. infinite actions sets) infinitely often. If the game is restricted such that Minimizer can use infinite actions sets only finitely often in any play then Maximizer still has uniformly $\varepsilon$-optimal public 1-bit strategies.

While optimal Maximizer strategies need not exist in general, it is still relevant to study the case where they do exist. If Minimizer can use infinite branching (resp. infinite actions sets) then Theorem 2 shows that Maximizer needs infinite memory even in that case. However, optimal Maximizer strategies in finitely branching turn-based games can be chosen to use a step counter plus 1 bit of public memory, while just a step counter is not enough; cf. Sect. 9.

Finally, in Sect. 10, we determine the strategy complexity of Minimizer for all cases. In particular, Minimizer has uniformly $\varepsilon$-optimal memoryless strategies in turn-based games that are infinitely branching but acyclic.

2 Preliminaries and Notations

A probability distribution over a countable set $S$ is a function $f:S\rightarrow [0,1]$ with $\sum _{s\in S}f(s)=1$. Let $\texttt{supp}(f) \overset{{\textrm{def}}}{=}\{s\mid f(s) >0\}$ denote the support of f. We write $\mathcal {D}(S)$ for the set of all probability distributions over $S$.

We study perfect-information 2-player stochastic games between the two players Maximizer (also denoted as $\Box$) and Minimizer (also denoted as $\Diamond$).

2.1 2-Player Concurrent Stochastic Games

A 2-player concurrent game ${\mathcal {G}}$ is played on a countable set of states $S$. For each state $s\in S$ there are nonempty countable action sets $A(s)$ and $B(s)$ for Maximizer and Minimizer, respectively. Let $Z \overset{{\textrm{def}}}{=}\{(s,a,b) \mid s\in S, a \in A(s), b \in B(s)\}$. For every triple $(s,a,b) \in Z$ there is a distribution $p(s,a,b) \in \mathcal {D}(S)$ over successor states. We call a state $s \in S$ a sink state, or absorbing, if $p(s,a,b) = s$ for all $a \in A(s)$ and $b \in B(s)$. The set of plays from an initial state $s_0$ is given by the infinite sequences in $Z^\omega$ where the first triple contains $s_0$. The game from $s_0$ is played in stages $\mathbb {N}=\{0,1,2,\dots \}$. At every stage $t \in \mathbb {N}$, the play is in some state $s_t$. Maximizer chooses an action $a_t \in A(s_t)$ and Minimizer chooses an action $b_t \in B(s_t)$. The next state $s_{t+1}$ is then chosen according to the distribution $p(s_t,a_t,b_t)$. (Since we just consider the reachability objective here, we don’t define a reward function.)

2.2 2-Player Turn-based Stochastic Games

A special subclass of concurrent stochastic games are turn-based games, where in each round either Maximizer or Minimizer is passive (i.e., has just a single action to play). Turn-based games are often represented in a form that explicitly separates local decisions into Maximizer-controlled ones, Minimizer-controlled ones, and random decisions. Thus one describes the turn-based game as ${\mathcal {G}}=(S,(S_\Box ,S_\Diamond ,S_\ocircle ),{\longrightarrow },P)$ where the countable set of states $S$ is partitioned into the set $S_\Box$ of states controlled by Maximizer ($\Box$), the set $S_\Diamond$ of states controlled by Minimizer ($\Diamond$) and random states $S_\ocircle$. The relation $\mathord {{\longrightarrow }}\subseteq S\times S$ is the transition relation. We write $s{\longrightarrow }s^{\prime }$ if $(s,s^{\prime })\in {\longrightarrow }$, and we assume that each state $s$ has a successor state $s^{\prime }$ with $s{\longrightarrow }s^{\prime }$. The probability function $P:S_\ocircle \rightarrow \mathcal {D}(S)$ assigns to each random state $s\in S_\ocircle$ a probability distribution over its successor states.

The game ${\mathcal {G}}$ is called finitely branching if each state has only finitely many successors; otherwise, it is infinitely branching. A game is acyclic if the underlying graph $(S,{\longrightarrow })$ is acyclic. Let $\odot \in \{\Box ,\Diamond \}$. At each stage t, if the game is in state $s_t \in S_\odot$ then player $\odot$ chooses a successor state $s_{t+1}$ with $s_t{\longrightarrow } s_{t+1}$; otherwise the game is in a random state $s_t\in S_\ocircle$ and proceeds randomly to $s_{t+1}$ with probability $P(s_t)(s_{t+1})$. If $S_\odot =\emptyset$, we say that player $\odot$ is passive, and the game is a Markov decision process (MDP). A Markov chain is an MDP where both players are passive.

2.3 Strategies and Probability Measures

The set of histories at stage n, with $n\in \mathbb {N}$, is denoted by $H_n$. That is, $H_0 \overset{{\textrm{def}}}{=}S$ and $H_n \overset{{\textrm{def}}}{=}Z^n \times S$ for all $n>0$. Let $H \overset{{\textrm{def}}}{=}\bigcup _{n \in \mathbb {N}} H_n$ be the set of all histories. For each history $h = (s_0,a_0,b_0) \cdots (s_{n-1},a_{n-1},b_{n-1}) s_{n} \in H_n$, let $s_h \overset{{\textrm{def}}}{=}s_{n}$ denote the final state in h. In the special case of turn-based games, the history can be represented by the sequence of states $s_0s_1\cdots s_h$, where $s_i{\longrightarrow } s_{i+1}$ for all $i\in \mathbb {N}$. We say that a history $h \in H$ visits the set of states $T\subseteq S$ at stage t if $s_t \in T$.

A mixed action for Maximizer (resp. Minimizer) in state $s$ is a distribution over $A(s)$ (resp. $B(s)$). A strategy for Maximizer (resp. Minimizer) is a function $\sigma$ (resp. $\pi$) that to each history $h \in H$ assigns a mixed action $\sigma (h) \in \mathcal {D}(A(s_h))$ (resp. $\pi (h) \in \mathcal {D}(B(s_h))$ for Minimizer). For turn-based games this means instead a distribution $\sigma (h) \in \mathcal {D}(S)$ over successor states if $s_h \in S_\Box$ (and similarly for Minimizer with $s_h \in S_\Diamond$). Let $\Sigma$ (resp. $\Pi$) denote the set of strategies for Maximizer (resp. Minimizer).

An initial state $s_0$ and a pair of strategies $\sigma , \pi$ for Maximizer and Minimizer induce a probability measure on sets of plays. We write ${{\mathcal {P}}}_{{\mathcal {G}},s_0,\sigma ,\pi }({{{\mathfrak {R}}}})$ for the probability of a measurable set of plays ${{\mathfrak {R}}}$ starting from $s_0$. More generally, if $f: Z^\omega \rightarrow \mathbb {R}$ is a measurable reward function on plays then we write ${\mathcal {E}}_{{\mathcal {G}},s_0,\sigma ,\pi }(f)$ for the expected reward w.r.t. f and ${{\mathcal {P}}}_{{\mathcal {G}},s_0,\sigma ,\pi }$. The case of measurable sets of plays ${{\mathfrak {R}}}$ is subsumed by this, since we can choose f as the indicator function of ${{\mathfrak {R}}}$. These measures are initially defined for the cylinder sets and extended to the sigma algebra by Carathéodory’s unique extension theorem [6].

2.4 Objectives

We consider the reachability objective for Maximizer. Given a set $T\subseteq S$ of states, the reachability objective $\texttt{Reach}(T)$ is the set of plays that visit $T$ at least once. From Minimizer’s point of view, this is the dual safety objective $\texttt{Safety}(T) \overset{{\textrm{def}}}{=}Z^\omega \setminus \texttt{Reach}(T)$ of plays that never visit T. Maximizer (resp. Minimizer) attempts to maximize (resp. minimize) the probability of $\texttt{Reach}(T)$.

For any subset of states $R\subseteq S$, let $\texttt{Reach}_{R}(T)$ denote the objective of visiting $T$ while remaining in R before visiting $T$. For $X \subseteq \mathbb {N}$, let $\texttt{Reach}_{X}(T)$ denote the objective of visiting $T$ in some number of rounds $n \in X$. For $n \in \mathbb {N}$ let $\texttt{Reach}_{n}(T) \overset{{\textrm{def}}}{=}\texttt{Reach}_{{\{k \mid k \le n\}}}(T)$ denote the objective of reaching $T$ in at most n rounds.

2.5 Value and Optimality

For a game ${\mathcal {G}}$, initial state $s_0$ and objective ${{\mathfrak {R}}}$ the lower value is defined as

$$\begin{aligned} \alpha (s_0) \overset{{\textrm{def}}}{=}\sup _{\sigma \in \Sigma } \inf _{\pi \in \Pi } {{\mathcal {P}}}_{{\mathcal {G}},s_0,\sigma ,\pi }({{{\mathfrak {R}}}}) \end{aligned}$$

Similarly, the upper value is defined as

$$\begin{aligned} \beta (s_0) \overset{{\textrm{def}}}{=}\inf _{\pi \in \Pi } \sup _{\sigma \in \Sigma } {{\mathcal {P}}}_{{\mathcal {G}},s_0,\sigma ,\pi }({{{\mathfrak {R}}}}) \end{aligned}$$

The inequality $\alpha (s_0) \le \beta (s_0)$ trivially holds. If $\alpha (s_0) = \beta (s_0)$, then this quantity is called the value of the game, denoted by ${\texttt{val}_{{\mathcal {G}},{{\mathfrak {R}}}}(s_0)}$. Reachability objectives, like all Borel objectives, have value [38]. For $\varepsilon >0$, a strategy $\sigma \in \Sigma$ from $s_0$ for Maximizer is called $\varepsilon$-optimal if $\forall \pi \in \Pi .\, {{\mathcal {P}}}_{{\mathcal {G}},s_0,\sigma ,\pi }({{{\mathfrak {R}}}}) \ge {\texttt{val}_{{\mathcal {G}},{{\mathfrak {R}}}}(s_0)} - \varepsilon$. Similarly, a strategy $\pi \in \Pi$ from $s_0$ for Minimizer is called $\varepsilon$-optimal if $\forall \sigma \in \Sigma .\, {{\mathcal {P}}}_{{\mathcal {G}},s_0,\sigma ,\pi }({{{\mathfrak {R}}}}) \le {\texttt{val}_{{\mathcal {G}},{{\mathfrak {R}}}}(s_0)} + \varepsilon$. If a strategy is 0-optimal we simply call it optimal.

2.6 Memory-based Strategies

A memory-based strategy $\sigma$ of Maximizer is a strategy that can be described by a tuple $(\textsf{M}, \textsf{m}_0, \sigma _\alpha , \sigma _\textsf{m})$ where $\textsf{M}$ is the set of memory modes, $\textsf{m}_0 \in \textsf{M}$ is the initial memory mode, and the functions $\sigma _\alpha$ and $\sigma _\textsf{m}$ describe how actions are chosen and memory modes updated; see below. A play according to $\sigma$ generates a random sequence of memory states $\textsf{m}_0, \dots , \textsf{m}_t, \textsf{m}_{t+1}, \dots$ from a given set of memory modes $\textsf{M}$, where $\textsf{m}_t$ is the memory mode at stage t. The strategy $\sigma$ selects the action at stage t according to a distribution that depends only on the current state $s_t$ and the memory $\textsf{m}_t$. Maximizer’s action $a_t$ is chosen via a distribution $\sigma _\alpha (s_t, \textsf{m}_t) \in \mathcal {D}(A(s_t))$. (Minimizer’s action is $b_t$). The next memory mode $\textsf{m}_{t+1}$ of Maximizer is chosen according to a distribution $\sigma _m(s_t, a_t, b_t, s_{t+1}) \in \mathcal {D}(\textsf{M})$ that depends on the chosen actions and the observed outcome. The memory is private if the other player cannot see the memory mode. Otherwise, it is public.

Let $\sigma [\textsf{m}]$ denote the memory-based strategy $\sigma$ that starts in memory mode $\textsf{m}$. In cases where the time is relevant and the strategy has access to the time (by using a step counter) $\sigma [\textsf{m}](t)$ denotes the strategy $\sigma$ in memory mode $\textsf{m}$ at time t.

A finite-memory strategy is one where $|\textsf{M}| < \infty$. A k-memory strategy is a memory-based strategy with at most k memory modes, i.e., $|\textsf{M}| \le k$. A 2-memory strategy is also called a 1-bit strategy. A strategy is memoryless (also called positional) if $|\textsf{M}|=1$. A strategy is called Markov if it uses only a step counter but no additional memory. A strategy is deterministic (also called pure) if the distributions for the action and memory update are Dirac. Otherwise, it is called randomized (or mixed). Memoryless randomized strategies are also called MR and memoryless deterministic strategies are also called MD. Similarly, randomized (resp. deterministic) finite-memory strategies are also called FR (resp. FD).

A finite-memory strategy $\sigma$ is called uniformly $\varepsilon$-optimal for an objective ${{\mathfrak {R}}}$ iff $\forall s\in S.\forall \pi .\, {{\mathcal {P}}}_{{\mathcal {G}},s,\sigma [\textsf{m}_0],\pi }({{{\mathfrak {R}}}}) \ge {\texttt{val}_{{\mathcal {G}},{{\mathfrak {R}}}}(s)} - \varepsilon$, i.e., the strategy performs well from every state.

The definitions above carry over directly to the simpler turn-based games where we have chosen/observed transitions instead of actions.

3 Uniform Strategies in Concurrent Games

Fig. 1

The Concurrent Big Match on $\mathbb {Z}$; see Definition 1. On the right is a depiction of the game graph; on the left we see how joint actions from state $c_i$ are resolved

First, we consider the concurrent version of the Big Match on the integers, in the formulation of [23].

Definition 1

(Concurrent Big Match on $\mathbb {Z}$) This game is shown in Fig. 1. The state space is $\{c_i \mid i \in \mathbb {Z}\} \cup \{\text {win},\text {lose}\}$, where the states $\text {win}$ and $\text {lose}$ are absorbing. Both players have the action set $\{0,1\}$ at each state. If Maximizer chooses action 1 in $c_i$ then the game is decided in this round: If Minimizer chooses 0 (resp. 1) then the game goes to $\text {lose}$ (resp. $\text {win}$). If Maximizer chooses action 0 in $c_i$ and Minimizer chooses action 0 (resp. 1) then the game goes to $c_{i-1}$ (resp. $c_{i+1}$). Maximizer wins iff state $\text {win}$ is reached or $\liminf \{i \mid c_i \text{ visited }\} = - \infty$.

Theorem 3

([23, Theorem 1.1]) In the concurrent Big Match on $\mathbb {Z}$, shown in Fig. 1, every state $c_i$ has value 1/2. An optimal strategy for Minimizer is to toss a fair coin at every stage. Maximizer has no optimal strategy, but for any start state $c_x$ and any positive integer N, he can win with probability $\ge N/(2N+2)$ by choosing action 1 with probability $1/(n+1)^2$ whenever the current state is $c_i$ with $i=x+N-n$ for some $n \ge 0$.

The concurrent Big Match on $\mathbb {Z}$ is not a reachability game, due to its particular winning condition. However, the following slightly modified version (played on $\mathbb {N}$) is a reachability game.

Definition 2

(Concurrent Big Match on $\mathbb {N}$) This game is shown in Fig. 2. The state space is $\{c_i \mid i \in \mathbb {N}\} \cup \{\text {lose}\}$ where $\text {lose}$ and $c_0$ are absorbing. Both players have the action set $\{0,1\}$ at each state. If Maximizer chooses action 1 in $c_i$ then the game is decided in this round: If Minimizer chooses 0 (resp. 1) then the game goes to $\text {lose}$ (resp. $c_0$). If Maximizer chooses action 0 in $c_i$ and Minimizer chooses action 0 (resp. 1) then the game goes to $c_{i-1}$ (resp. $c_{i+1}$).

Maximizer wins iff $c_0$ is reached, i.e., we have the reachability objective $\texttt{Reach}(\{c_0\})$.

The following theorem summarizes results on the concurrent Big Match on $\mathbb {N}$ by combining results from [23] and [44].

Fig. 2

The Concurrent Big Match on $\mathbb {N}$; see Definition 2. On the right is a depiction of the game graph; on the left we see how joint actions from state $c_i$ are resolved

Theorem 4

Denote by ${\mathcal {G}}$ the concurrent Big Match game on $\mathbb {N}$, as shown in Fig. 2, and let $x\in \mathbb {N}$. Then,

• 1. ${\texttt{val}_{{\mathcal {G}}}(c_x)} = (x+2)/(2x+2) \ge 1/2$.

• 2. For every start state $c_x$ and $N\ge 0$, Maximizer can win with probability $\ge N/(2N+2)$ by choosing action 1 with probability $1/(n+1)^2$ whenever the current state is $c_i$ with $i=x+N-n$ for some $n \ge 0$.

• 3. For any $\varepsilon < 1/2$ there is no uniformly $\varepsilon$-optimal memoryless (MR) strategy for Maximizer. Every MR Maximizer strategy $\sigma$ attains arbitrarily little from $c_x$ as $x \rightarrow \infty$. Formally, $\limsup _{x \rightarrow \infty }\inf _{\pi }{{\mathcal {P}}}_{{\mathcal {G}},c_x,\sigma ,\pi }(\texttt{Reach}(\{c_0\}))=0.$

Proof

Item 1 follows directly from [23, Proposition 5.1].

Item 2 follows from Theorem 3, since it is easier for Maximizer to win in the game of Definition 2 than in the game of Definition 1.

Towards item 3, we follow the proof of [44, Lemma 4]. Let $\sigma$ be an MR Maximizer strategy and f(x) the probability that $\sigma$ picks action 1 at state $c_x$. There are two cases.

In the first case $\sum _{x \ge 1} f(x) < \infty$. Let $\pi$ be the strategy of Minimizer that always picks action 1. For all $x \ge 1$ we have

$$\begin{aligned} {{\mathcal {P}}}_{{\mathcal {G}},c_x,\sigma ,\pi }(\texttt{Reach}(\{c_0\})) \quad \le \quad&\ f(x) + (1-f(x))f(x+1) \\&+ (1-f(x))(1-f(x+1))f(x+2) \\&+ \dots \ \le \sum _{k=x}^\infty f(k) < \infty . \end{aligned}$$

Thus $\limsup _{x \rightarrow \infty }{{\mathcal {P}}}_{{\mathcal {G}},c_x,\sigma ,\pi }(\texttt{Reach}(\{c_0\}))=0$.

In the second case $\sum _{x \ge 1} f(x) = \infty$. Let $\pi$ be the strategy of Minimizer that always picks action 0. For all $x \ge 1$ we have

$$\begin{aligned} {{\mathcal {P}}}_{{\mathcal {G}},c_x,\sigma ,\pi }(\texttt{Reach}(\{c_0\}))&=(1-f(x))(1-f(x-1))\cdots (1-f(1)) \\&=\prod _{k=1}^x (1-f(k))\\&\le \frac{1}{1+ \sum _{k=1}^x f(k)} \end{aligned}$$

For the final inequality we refer the reader to Proposition 31 in Appendix A. Thus $\limsup _{x \rightarrow \infty }{{\mathcal {P}}}_{{\mathcal {G}},c_x,\sigma ,\pi }(\texttt{Reach}(\{c_0\}))=0$.

Since, by item 1, ${\texttt{val}_{{\mathcal {G}}}(c_x)} = (x+2)/(2x+2) \ge 1/2$ for every $x \ge 0$, the MR strategy $\sigma$ cannot be uniformly $\varepsilon$-optimal for any $\varepsilon < 1/2$. $\square$

While uniformly $\varepsilon$-optimal Maximizer strategies cannot be memoryless, we show that they can be chosen with just 1 bit of public memory in Theorem 6.

First we need an auxiliary lemma that is essentially known; see, e.g., [39, Sect. 7.7], and [21] Theorem 12.1. We extend it slightly to fit our purposes, i.e., for the proof of Theorem 6 below.

Lemma 5

Consider a concurrent game with countable state space S, finite action sets for Minimizer at every state and unrestricted (possibly infinite) action sets for Maximizer.

For the reachability objective $\texttt{Reach}(T)$, for every finite set $S_0 \subseteq S$ of initial states, and for every $\varepsilon >0$, there exists a memoryless strategy $\sigma$ and a finite set of states $R \subseteq S$ such that for all $s_0 \in S_0$

$$\begin{aligned} \inf _{\pi \in \Pi } {{\mathcal {P}}}_{s_0,\sigma ,\pi }(\texttt{Reach}_{R}(T)) \ \ge \ {\texttt{val}_{\texttt{Reach}(T)}(s_0)} - \varepsilon \,, \end{aligned}$$

where $\texttt{Reach}_{R}(T)$ denotes the objective of visiting $T$ while remaining in R before visiting T. If the game is turn-based and finitely branching at Minimizer-controlled states, there is a deterministic (i.e., MD) such strategy $\sigma$.

Proof

Since Minimizer’s action sets are finite, by [21, Theorem 11.1], the game has a value. Moreover, using the finiteness of Minimizer’s action sets again, it follows from [21, Theorem 12.1] that for all $s \in S$

$$\begin{aligned} \lim _{n \rightarrow \infty } {\texttt{val}_{\texttt{Reach}_{n}(T)}(s)} \ = \ {\texttt{val}_{\texttt{Reach}(T)}(s)}\,, \end{aligned}$$

(1)

where $\texttt{Reach}_{n}(T)$ denotes the objective of visiting $T$ within at most n rounds of the game.

To achieve the uniformity (across the set $S_0$ of initial states) required by the statement of the lemma, we add a fresh “random” state (i.e., a state in which each player has only a single action available) that branches uniformly at random to a state in $S_0$. Call this state ${\hat{s}}_0$. The value of ${\hat{s}}_0$ is the arithmetic average of the values of the states in $S_0$. It follows that every $(\varepsilon /|S_0|)$-optimal memoryless strategy for Maximizer in ${\hat{s}}_0$ must be $\varepsilon$-optimal in every state in $S_0$. So it suffices to prove the statement of the lemma under the assumption that $S_0$ is a singleton, say $S_0 = \{s_0\}$.

Fix $\varepsilon >0$ and let $\varepsilon ^{\prime } \overset{{\textrm{def}}}{=}\varepsilon /4$. By Eq. (1) there is a number n such that ${\texttt{val}_{\texttt{Reach}_{n}(T)}(s_0)} = {\texttt{val}_{\texttt{Reach}(T)}(s_0)} - \varepsilon ^{\prime }$. Let $\sigma$ be a Maximizer strategy such that

$$\begin{aligned} \inf _{\pi \in \Pi } {{\mathcal {P}}}_{s_0,\sigma ,\pi }(\texttt{Reach}_{n}(T)) \ \ge \ {\texttt{val}_{\texttt{Reach}(T)}(s_0)} - 2\varepsilon ^{\prime }\,. \end{aligned}$$

(2)

For each m with $0 \le m \le n$ we will inductively construct a finite subset $H^{\prime }_m \subseteq H_m$ of the m-step histories of plays from $s_0$ that are compatible with $\sigma$ such that, for every Minimizer strategy $\pi$, the plays in $H_m^{\prime }Z^\omega$ have probability $\ge 1 - \frac{m}{n} \varepsilon ^{\prime }$, where the event $H_m^{\prime }Z^\omega$ is defined as the set of continuations of the m-step histories in $H^{\prime }_m$. Formally,

$$\begin{aligned} \inf _{\pi \in \Pi } {{\mathcal {P}}}_{s_0,\sigma ,\pi }(H_m^{\prime }Z^\omega ) \ \ge \ 1 - \frac{m}{n} \varepsilon ^{\prime } \end{aligned}$$

(3)

The base case of $m=0$ is trivial. Now we show the inductive step from m to $m+1$. For any of the finitely many histories $h \in H^{\prime }_m$ ending in some state $s$, consider the chosen mixed actions $a \in \mathcal {D}(A(s))$ and $b \in \mathcal {D}(B(s))$ by Maximizer and Minimizer, respectively. Since B(s) is finite, b has finite support. However, the distribution a can have infinite support. We fix a sufficiently large finite subset $A^{\prime }$ of the support of a that has probability mass $\ge 1-\frac{\varepsilon ^{\prime }}{2n}$. Consider the set $\gamma (s)$ of possible successor states of s. Since the size of the support of b is upper bounded by the finite number $|B(s)|$ independently of $\pi$, we can pick a finite subset $\gamma ^{\prime }(s) \subseteq \gamma (s)$ sufficiently large such that both Maximizer’s chosen action is inside $A^{\prime }$ and the chosen successor state is inside $\gamma ^{\prime }(s)$ with probability $\ge 1 - \frac{1}{n} \varepsilon ^{\prime }$. We then define $H^{\prime }_{m+1}$ as the finitely many one-round extensions of histories in $H^{\prime }_m$ with Maximizer action in $A^{\prime }$ and successor state in $\gamma ^{\prime }(s)$. Using the induction hypothesis and the properties above, we obtain that

$$\begin{aligned} \inf _{\pi \in \Pi } {{\mathcal {P}}}_{s_0,\sigma ,\pi }(H^{\prime }_{m+1}Z^\omega )&\ge \ \inf _{\pi \in \Pi } {{\mathcal {P}}}_{s_0,\sigma ,\pi }(H^{\prime }_{m}Z^\omega ) \left( 1-\frac{1}{n}\varepsilon ^{\prime }\right) \\&\ge \ \left( 1-\frac{m}{n}\varepsilon ^{\prime }\right) \left( 1-\frac{1}{n}\varepsilon ^{\prime }\right) \\&= \ 1 - \frac{m+1}{n}\varepsilon ^{\prime } + \frac{m}{n^2} (\varepsilon ^{\prime })^2\\&\ge \ 1 - \frac{m+1}{n} \varepsilon ^{\prime }. \end{aligned}$$

This completes the induction step, and thus we obtain (3).

For every $0 \le m \le n$ let $R_m$ be the finite set of states that are visited during the first m steps of the histories in $H_m^{\prime }$. Then $R \overset{{\textrm{def}}}{=}R_n$ is a finite set of states. It follows that

$$\begin{aligned}&\inf _{\pi \in \Pi } {{\mathcal {P}}}_{s_0,\sigma ,\pi }(\texttt{Reach}_{R}(T))\\&\quad \ge \inf _{\pi \in \Pi } {{\mathcal {P}}}_{s_0,\sigma ,\pi }(H^{\prime }_n Z^\omega \cap \texttt{Reach}_{R}(T))\\&\quad \ge \inf _{\pi \in \Pi } {{\mathcal {P}}}_{s_0,\sigma ,\pi }(H'_n Z^\omega \cap \texttt{Reach}_{n}(T))&\text{ set } \text{ incl. }\\&\quad = \inf _{\pi \in \Pi } ({{\mathcal {P}}}_{s_0,\sigma ,\pi }(\texttt{Reach}_{n}(T)) - {{\mathcal {P}}}_{s_0,\sigma ,\pi }(\overline{H^{\prime }_n Z^\omega } \cap \texttt{Reach}_{n}(T)))\\&\quad \ge \inf _{\pi \in \Pi } ({{\mathcal {P}}}_{s_0,\sigma ,\pi }(\texttt{Reach}_{n}(T)) - {{\mathcal {P}}}_{s_0,\sigma ,\pi }(\overline{H^{\prime }_n Z^\omega }))\\&\quad \ge \inf _{\pi \in \Pi } ({{\mathcal {P}}}_{s_0,\sigma ,\pi }(\texttt{Reach}_{n}(T))) - \sup _{\pi \in \Pi }({{\mathcal {P}}}_{s_0,\sigma ,\pi }(\overline{H^{\prime }_n Z^\omega }))\\&\quad = \inf _{\pi \in \Pi } ({{\mathcal {P}}}_{s_0,\sigma ,\pi }(\texttt{Reach}_{n}(T))) - (1-\inf _{\pi \in \Pi } {{\mathcal {P}}}_{s_0,\sigma ,\pi }(H^{\prime }_n Z^\omega ))\\&\quad \ge \inf _{\pi \in \Pi } {{\mathcal {P}}}_{s_0,\sigma ,\pi }(\texttt{Reach}_{n}(T)) - \varepsilon ^{\prime }&\hbox {by }(3)\\&\quad \ge {\texttt{val}_{\texttt{Reach}(T)}(s_0)} - 3\varepsilon ^{\prime }.&\hbox {by } (2) \end{aligned}$$

Note that the restriction on the time horizon, n, has been lifted here. In particular, the above implies that

$$\begin{aligned} {\texttt{val}_{\texttt{Reach}_{R}(T)}(s_0)} \ge {\texttt{val}_{\texttt{Reach}(T)}(s_0)} - 3\varepsilon ^{\prime }. \end{aligned}$$

(4)

The restriction of the objective to the (finitely many) states in R means that we have effectively another reachability game. It is known [49, Corollary 3.9] that for concurrent games with finite action sets and reachability objective, Maximizer has a memoryless $\varepsilon ^{\prime }$-optimal strategy. In turn-based games with finitely many states, he even has an MD optimal strategy  [14]. So Maximizer has a memoryless (in the turn-based case: MD) strategy $\sigma ^{\prime }$ such that

$$\begin{aligned}&\inf _{\pi \in \Pi } {{\mathcal {P}}}_{s_0,\sigma ^{\prime },\pi }(\texttt{Reach}_{R}(T))\\&\ge {\texttt{val}_{\texttt{Reach}_{R}(T)}(s_0)} - \varepsilon ^{\prime }&\hbox {by}\,\varepsilon ^{\prime }\,\hbox {-optimality of}\,\sigma ^{\prime } \\&\ge {\texttt{val}_{\texttt{Reach}(T)}(s_0)} - 4\varepsilon ^{\prime }&\hbox {by }(4)\\&={\texttt{val}_{\texttt{Reach}(T)}(s_0)} - \varepsilon . \end{aligned}$$

$\square$

Theorem 6

For any concurrent game with finite action sets and reachability objective, for any $\varepsilon >0$, Maximizer has a uniformly $\varepsilon$-optimal public 1-bit strategy. If the game is turn-based and finitely branching, Maximizer has a deterministic such strategy.

Proof

Denote the game by ${\hat{{\mathcal {G}}}}$, over state space S. Let $\varepsilon >0$. We show how to construct the uniformly $\varepsilon$-optimal public 1-bit strategy. It is convenient to describe the 1-bit strategy in ${\hat{{\mathcal {G}}}}$ in terms of a memoryless strategy in a derived game ${\mathcal {G}}$ with state space $S \times \{0,1\}$, where the second component (0 or 1) reflects the current memory mode of Maximizer. Accordingly, we think of the state space of ${\mathcal {G}}$ as organized in two layers, the “inner” and the “outer” layer, with the memory mode being 0 and 1, respectively. In each state (s, j) of ${\mathcal {G}}$ (where $j \in \{0,1\}$ denotes the layer), Maximizer can choose the layer, $j^{\prime } \in \{0,1\}$, of the successor state $(s^{\prime },j^{\prime })$, possibly depending on $s^{\prime }$. This is exactly analogous to Maximizer using 1 bit of memory. In these terms, our goal is to construct, for the layered game ${\mathcal {G}}$, a memoryless strategy for Maximizer. From this one can naturally extract a public 1-bit strategy for Maximizer in the original game ${\hat{{\mathcal {G}}}}$. Upon reaching the target, the memory mode is irrelevant, so for notational simplicity we denote the objective as $\texttt{Reach}(T)$, also in the layered game ${\mathcal {G}}$ (instead of $\texttt{Reach}(T \times \{0,1\})$). The current state of the layered game is known to both players (to Minimizer in particular); this corresponds to the (1-bit) memory being public in the original game ${\hat{{\mathcal {G}}}}$: at each point in the game, Minimizer knows the distribution of actions that Maximizer is about to play. Notice that the values of states (s, 0) and (s, 1) in ${\mathcal {G}}$ are equal to the value of s in ${\hat{{\mathcal {G}}}}$; this is because the definition of value does not impose restrictions on the memory of strategies and so the players could, in ${\hat{{\mathcal {G}}}}$, simulate the two layers of ${\mathcal {G}}$ in their memory if that were advantageous.

In general Maximizer does not have uniformly $\varepsilon$-optimal memoryless strategies in reachability games; cf. Theorem 4. So our construction will exploit the special structure in the layered game, namely, the symmetry of the two layers. The memoryless Maximizer strategy we construct will be $\varepsilon$-optimal from each state (s, 0) in the inner layer, but not necessarily from the states in the outer layer.

As building blocks we use the non-uniformly $\varepsilon$-optimal memoryless strategies that we get from Lemma 5; in the turn-based finitely branching case they are even MD. We combine them by “plastering” the state space (of the layered game). This is inspired by the construction in [46]; see [34, Sect. 3.2] for a recent description.⁴

In the general concurrent case, a memoryless strategy prescribes for each state (s, i) a probability distribution over Maximizer’s actions. We define a memoryless strategy by successively fixing such distributions in more and more states. Technically, one can fix a state s by replacing the actions A(s) available to Maximizer by a single action which is a convex combination over A(s). Visually, we “plaster” the whole state space by the fixings. This is in general an infinite (but countable) process; it defines a memoryless strategy for Maximizer in the limit.

In the turn-based and finitely branching case, an MD strategy prescribes one outgoing transition for each Maximizer state. Accordingly, fixing a Maximizer state means restricting the outgoing transitions to a single such outgoing transition. The plastering proceeds similarly as in the concurrent case; it defines an MD strategy for Maximizer in the limit.

Put the states of ${\hat{{\mathcal {G}}}}$ in some order, i.e., $s_1, s_2, \ldots$ with $S = \{s_1, s_2, \ldots \}$. The plastering proceeds in rounds. In round $i \ge 1$ we fix the states in $S_i[0] \times \{0\}$ and in $S_i[1] \times \{1\}$, where $S_1[0], S_2[0], \ldots \subseteq S$ are pairwise disjoint and $S_1[1], S_2[1], \ldots \subseteq S$ are pairwise disjoint; see Fig. 3 for an example of sets $S_i[0]$ and $S_i[1]$ in a two-layer game. Define $F_i[0] \overset{{\textrm{def}}}{=}\bigcup _{j \le i} S_j[0]$ and $F_i[1] \overset{{\textrm{def}}}{=}\bigcup _{j \le i} S_j[1]$. So $F_i[0] \times \{0\}$ and $F_i[1] \times \{1\}$ are the states that have been fixed by the end of round i. We will keep an invariant $F_i[0] \subseteq F_i[1] \subseteq F_{i+1}[0]$.

Let ${\mathcal {G}}_i$ be the game obtained from ${\mathcal {G}}$ after the fixings of the first $i-1$ rounds (with ${\mathcal {G}}_1 = {\mathcal {G}}$). Define

$$\begin{aligned}S_i[0] \overset{{\textrm{def}}}{=}(\{s_i\} \cup S_{i-1}[1]) \setminus F_{i-1}[0]\end{aligned}$$

(and $S_1[0] \overset{{\textrm{def}}}{=}\{s_1\}$), the set of states to be fixed in round i. In particular, round i guarantees that states (s, 0) whose “outer sibling” (s, 1) has been fixed previously are also fixed, ensuring $F_{i-1}[1] \subseteq F_i[0]$. It follows from the invariant above that $\bigcup _{j=1}^\infty S_j[0] = \bigcup _{j=1}^\infty S_j[1] = S$. The set $S_i[1]$ will be defined below.

Fig. 3

Example of sets $S_i[0]$ and $S_i[1]$ in the two layers of ${\mathcal {G}}$, where the outer and inner layers are $S\times \{1\}$ and $S \times \{0\}$, respectively

In round i we fix the states in $(S_i[0] \times \{0\}) \cup (S_i[1] \times \{1\})$ in such a way that

• (A) starting from any (s, 0) with $s \in S_i[0]$, the (infimum over all Minimizer strategies $\pi$) probability of reaching T using only fixed states is not much less than the value ${\texttt{val}_{{\mathcal {G}}_i,\texttt{Reach}(T)}((s,0))}$; and

• (B) for all states $(s,0) \in S \times \{0\}$ in the inner layer, the value ${\texttt{val}_{{\mathcal {G}}_{i+1},\texttt{Reach}(T)}((s,0))}$ is almost as high as ${\texttt{val}_{{\mathcal {G}}_{i},\texttt{Reach}(T)}((s,0))}$.

The purpose of goal (A) is to guarantee good progress towards the target when starting from any state (s, 0) in $S_i[0] \times \{0\}$. The purpose of goal (B) is to avoid fixings that would cause damage to the values of other states in the inner layer.

We want to define the fixings in round i. First we define an auxiliary game $\bar{{\mathcal {G}}}_i$ with state space ${\bar{S}}_i \overset{{\textrm{def}}}{=}(F_i[0] \times \{0,1\}) \cup (S \setminus F_i[0])$. Game $\bar{{\mathcal {G}}}_i$ is obtained from ${\mathcal {G}}_i$ by collapsing, for all $s \in S \setminus F_i[0]$, the siblings (s, 0), (s, 1) (neither of which have been fixed yet) to a single state s. See Fig. 4. The game $\bar{{\mathcal {G}}}_i$ inherits the fixings from ${\mathcal {G}}_i$. The values remain equal; in particular, for $s \in S \setminus F_i[0]$, the values of (s, 0) and (s, 1) in ${\mathcal {G}}_i$ and the value of s in $\bar{{\mathcal {G}}}_i$ are all equal.

Fig. 4

Example of a game $\bar{{\mathcal {G}}}_3$

Let $\varepsilon _i > 0$. We apply Lemma 5 to $\bar{{\mathcal {G}}}_i$ with set of initial states $S_i[0] \times \{0\}$. So Maximizer has a memoryless strategy $\sigma _i$ for $\bar{{\mathcal {G}}}_i$ and a finite set of states $R \subseteq {\bar{S}}_i$ so that for all $s \in S_i[0]$ we have $\inf _\pi {{\mathcal {P}}}_{\bar{{\mathcal {G}}}_i, (s,0),\sigma _i,\pi }(\texttt{Reach}_{R}(T)) \ge {\texttt{val}_{{\mathcal {G}}_i,\texttt{Reach}(T)}((s,0))} - \varepsilon _i$.

Now we carry the strategy $\sigma _i$ from $\bar{{\mathcal {G}}}_i$ to ${\mathcal {G}}_i$ by suitably adapting it (see below). Then we obtain ${\mathcal {G}}_{i+1}$ from ${\mathcal {G}}_i$ by fixing (the adapted version of) $\sigma _i$ in ${\mathcal {G}}_i$.

The adaption of $\sigma _i$ to ${\mathcal {G}}_i$ is by treating states $s \in S \setminus F_i[0]$ in $\bar{{\mathcal {G}}}_i$ as states in the outer layer (s, 1) of ${\mathcal {G}}_i$, as follows. Every transition that in $\bar{{\mathcal {G}}}_i$ goes from a state $(s,j) \in F_i[0] \times \{0,1\}$ to a state $s^{\prime } \in S \setminus F_i[0]$ is redirected so that in ${\mathcal {G}}_i$ it goes from (s, j) to $(s^{\prime },1)$. Similarly, every transition that in $\bar{{\mathcal {G}}}_i$ goes from a state $s^{\prime } \in S \setminus F_i[0]$ to a state $(s,j) \in F_i[0] \times \{0,1\}$ goes in ${\mathcal {G}}_i$ from $(s^{\prime },1)$ to (s, j). Finally, every transition that in $\bar{{\mathcal {G}}}_i$ goes from a state $s^{\prime } \in S \setminus F_i[0]$ to another state $t^{\prime } \in S \setminus F_i[0]$ goes in ${\mathcal {G}}_i$ from $(s^{\prime },1)$ to $(t^{\prime },1)$.

Accordingly, define $S_i[1] \overset{{\textrm{def}}}{=}(S_i[0] \setminus F_{i-1}[1]) \cup ((S \setminus F_i[0]) \cap R)$ (this ensures that $F_i[0] \subseteq F_i[1]$), and obtain ${\mathcal {G}}_{i+1}$ from ${\mathcal {G}}_i$ by fixing the adapted version of $\sigma _i$ in $(S_i[0] \times \{0\}) \cup (S_i[1] \times \{1\})$. This yields, for all $s \in S_i[0]$,

$$\begin{aligned} \begin{aligned} \inf _{\sigma ,\pi } {{\mathcal {P}}}_{{\mathcal {G}}_{i+1}, (s,0),\sigma ,\pi } (\texttt{Reach}_{(F_i[0] \times \{0\}) \cup (F_i[1] \times \{1\})}(T)) \ge {\texttt{val}_{{\mathcal {G}}_i,\texttt{Reach}(T)}((s,0))} - \varepsilon _i, \end{aligned} \end{aligned}$$

(5)

achieving goal (A) above. Notice that the fixings in ${\mathcal {G}}_{i+1}$ “lock in” a good attainment from $S_i[0] \times \{0\}$, regardless of the Maximizer strategy $\sigma$. Now we extend (5) to achieve goal (B) from above: for all $s \in S$ we have

$$\begin{aligned} {\texttt{val}_{{\mathcal {G}}_{i+1},\texttt{Reach}(T)}((s,0))} \ \ge \ {\texttt{val}_{{\mathcal {G}}_i,\texttt{Reach}(T)}((s,0))} - \varepsilon _i\,. \end{aligned}$$

(6)

Indeed, consider any Maximizer strategy $\sigma$ in ${\mathcal {G}}_i$ from any (s, 0). Without loss of generality we can assume that $\sigma$ is such that the play enters the outer layer only (if at all) after having entered $F_i[0] \times \{0\}$. Now change $\sigma$ to a strategy $\sigma ^{\prime }$ in ${\mathcal {G}}_{i+1}$ so that as soon as $F_i[0] \times \{0\}$ is entered, $\sigma ^{\prime }$ respects the fixings (and plays arbitrarily afterwards). By (5) this decreases the (infimum over Minimizer strategies $\pi$) probability by at most $\varepsilon _i$. Thus,

$$\begin{aligned} \inf _\pi {{\mathcal {P}}}_{{\mathcal {G}}_{i+1}, (s,0),\sigma ^{\prime },\pi }(\texttt{Reach}(T)) ~\ge ~ \inf _\pi {{\mathcal {P}}}_{{\mathcal {G}}_{i}, (s,0),\sigma ,\pi }(\texttt{Reach}(T)) - \varepsilon _i\,. \end{aligned}$$

Taking the supremum over strategies $\sigma$ in ${\mathcal {G}}_i$ yields (6).

For any $\varepsilon > 0$ choose $\varepsilon _i \overset{{\textrm{def}}}{=}2^{-i} \varepsilon$; thus, $\sum _{i \ge 1} \varepsilon _i = \varepsilon$. Let $\sigma$ be the memoryless strategy that respects all fixings in all ${\mathcal {G}}_i$. Then, by (6), for all $s \in S$ we have

$$\begin{aligned} \inf _\pi {{\mathcal {P}}}_{{\mathcal {G}}, (s,0),\sigma ,\pi }(\texttt{Reach}(T)) \ \ge \ {\texttt{val}_{{\mathcal {G}},\texttt{Reach}(T)}((s,0))} - \sum _{i=1}^\infty \varepsilon _i\,, \end{aligned}$$

so $\sigma$ is $\varepsilon$-optimal in ${\mathcal {G}}$ from all (s, 0). Hence, the corresponding public 1-bit memory strategy (with initial memory mode 0, corresponding to the inner layer) is uniformly $\varepsilon$-optimal in ${\hat{{\mathcal {G}}}}$. $\square$

4 Uniform Strategies in Turn-based Games

Theorem 4 in the previous section shows that Maximizer has no uniformly $\varepsilon$-optimal memoryless strategies in concurrent reachability games with finite action sets (since the concurrent Big Match game on $\mathbb {N}$ is a counterexample). Here we strengthen this negative result by showing that it holds even for the subclass of finitely branching turn-based reachability games.

To this end, we define a finitely branching turn-based game in Definition 3 that is very similar to the concurrent Big Match on $\mathbb {N}$, as shown in Fig. 2. The difference is that at each $c_i$ Maximizer has to announce his mixed choice of actions first, rather than concurrently with Minimizer. Note that Maximizer only announces his distribution over the actions $\{0,1\}$, not any particular action. Since a good Maximizer strategy needs to work even if Minimizer knows it in advance, this makes no difference with respect to the attainment of memoryless Maximizer strategies. Another slight difference is that Maximizer is restricted to choosing distributions with only rational probabilities where the probability of picking action 1 is of the form 1/k for some $k \in \mathbb {N}$. However, since we know that there exist good Maximizer strategies of this form (cf. Theorem 4), it is not a significant restriction.

Fig. 5

Turn-based Big Match on $\mathbb {N}$

Definition 3

(Turn-based Big Match on $\mathbb {N}$) This game is shown in Fig. 5. Maximizer controls the set $\{c_i \mid i \in \mathbb {N}\} \cup \{c_{i,j} \mid i,j \in \mathbb {N}\} \cup \{\text {lose}\}$ of states, whereas Minimizer controls only the states in $\{d_{i,j} \mid i,j \in \mathbb {N}\}$. The remaining set $\{r_{i,j}^0, r_{i,j}^1 \mid i,j \in \mathbb {N}\}$ of states are random. For all $i,j \in \mathbb {N}$, there are the following transitions

$$\begin{aligned}c_i {\longrightarrow }c_{i,1} \qquad \quad c_{i,j} {\longrightarrow }c_{i,j+1} \qquad \quad c_{i,j} {\longrightarrow }d_{i,j} \\ d_{i,j} {\longrightarrow }r_{i,j}^0 \qquad \quad d_{i,j} {\longrightarrow }r_{i,j}^1 \end{aligned}$$

and $\text {lose}{\longrightarrow }\text {lose}$. Intuitively, by going from $c_i$ to $d_{i,j}$, Maximizer chooses action 1 with probability 1/j and action 0 with probability $1-1/j$. Minimizer chooses actions 0 or 1 by going from $d_{i,j}$ to $r_{i,j}^0$ or $r_{i,j}^1$, respectively. The probabilistic function is defined by

$$\begin{aligned}P(r_{i,j}^0)(\text {lose}) = 1/j \qquad&\qquad P(r_{i,j}^0)(c_{i-1}) = 1-1/j \\ P(r_{i,j}^1)(c_0) = 1/j\qquad&\qquad P(r_{i,j}^1)(c_{i+1}) = 1-1/j\end{aligned}$$

where $i,j \in \mathbb {N}$. The objective is $\texttt{Reach}(\{c_0\})$.

This finitely branching turn-based game mimics the behavior of the game in Definition 2.

Theorem 7

Consider the turn-based Big Match game ${\mathcal {G}}$ on $\mathbb {N}$ from Definition 3 and let $x\in \mathbb {N}$.

• 1. For every start state $c_x$ and $N\ge 0$, Maximizer can win with probability $\ge N/(2N+2)$ by choosing the transitions $c_i {\longrightarrow }\dots d_{i,j}$ where $j = (n+1)^2$ whenever he is in state $c_i$ with $i=x+N-n$ for some $n \ge 0$.

  In particular, ${\texttt{val}_{{\mathcal {G}}}(c_x)} \ge 1/2$.

• 2. For any $\varepsilon < 1/2$ there does not exist any uniformly $\varepsilon$-optimal memoryless (MR) strategy for Maximizer.

  Every MR Maximizer strategy $\sigma$ attains arbitrarily little from $c_x$ as $x \rightarrow \infty$. Formally, $\limsup _{x \rightarrow \infty }\inf _{\pi }{{\mathcal {P}}}_{{\mathcal {G}},c_x,\sigma ,\pi }(\texttt{Reach}(\{c_0\}))=0.$

Proof

Let ${\mathcal {G}}^{\prime }$ be the concurrent game from Definition 2.

Towards item 1, consider the concurrent game ${\mathcal {G}}^{\prime }$ and the turn-based game ${\mathcal {G}}$ from Definition 3. Let $c_x$ be our start state. After fixing the Maximizer strategy from Theorem 4(2) in ${\mathcal {G}}^{\prime }$, we obtain an MDP ${\mathcal {M}}^{\prime }$ from Minimizer’s point of view. Similarly, after fixing the strategy described above in ${\mathcal {G}}$, we obtain an MDP ${\mathcal {M}}$. Then ${\mathcal {M}}^{\prime }$ and ${\mathcal {M}}$ are almost isomorphic (apart from linear chains of steps $c_{i,j}$ ${\longrightarrow }c_{i,j+1} \dots$ in ${\mathcal {M}}^{\prime }$), and thus the infimum of the chance of winning, over all Minimizer strategies are the same. Therefore the result follows from Theorem 4(2).

Towards item 2, note that every Maximizer MR strategy $\sigma$ in ${\mathcal {G}}$ corresponds to a Maximizer MR strategy $\sigma ^{\prime }$ in ${\mathcal {G}}^{\prime }$. First, forever staying in states $c_{i,j}$ is losing, since the target is never reached. Thus, without restriction, we assume that $\sigma$ almost surely moves from $c_i$ to some $d_{i,j}$ eventually. Let $p_{i,j}$ be the probability that $\sigma$ moves from $c_i$ to $d_{i,j}$. Thus the corresponding strategy $\sigma ^{\prime }$ in ${\mathcal {G}}^{\prime }$ in $c_i$ plays action 1 with probability $\sum _j p_{i,j}(1/j)$ and action 0 otherwise. Again the MDPs resulting from fixing the respective strategies in ${\mathcal {G}}$ and ${\mathcal {G}}^{\prime }$ are (almost) isomorphic, and thus the result follows from Theorem 4(3). $\square$

In the rest of this section we briefly describe an alternative construction of a turn-based finitely branching reachability game without uniformly $\varepsilon$-optimal memoryless Maximizer strategies, i.e., Theorem 8 is a different proof of the same result as in Theorem 7. In the direct construction in Definition 3, Maximizer had many alternatives in the states $c_i$ (by going to some state $d_{i,j}$ for some $j \ge 1$). However, Theorem 6 shows that a deterministic 1-bit strategy suffices for Maximizer. Thus, it suffices for Maximizer to have just two alternatives, corresponding to the two memory modes of the 1-bit strategy. The following definition uses this observation to construct an alternative counterexample.

Definition 4

Consider the concurrent reachability game from Definition 2 and let $\varepsilon = 1/4$. By Theorem 6, Maximizer has a uniform $\varepsilon$-optimal 1-bit strategy $\hat{\sigma }$. Let $p_{i,0}$ (resp. $p_{i,1}$) be the probability that $\hat{\sigma }$ picks action 1 at state $c_i$ when in memory mode 0 (resp. memory mode 1).

We construct a turn-based reachability game ${\mathcal {G}}$ with branching degree two where Maximizer can pick randomized actions according to these probabilities $p_{i,0}, p_{i,1}$, but nothing else. Let ${\mathcal {G}}=(S,(S_\Box ,S_\Diamond ,S_\ocircle ),{\longrightarrow },P)$ where $S_\Box = \{c_i \mid i \in \mathbb {N}\} \cup \{\text {lose}\}$, $S_\Diamond = \{d_{i,0}, d_{i,1} \mid i \in \mathbb {N}\}$, and $S_\ocircle = \{r_{i,0,0}, r_{i,0,1}, r_{i,1,0}, r_{i,1,1} \mid i \in \mathbb {N}\}$. We have controlled transitions $c_i {\longrightarrow }d_{i,j}$, $d_{i,j} {\longrightarrow }r_{i,j,k}$ for all $i \in \mathbb {N}$ and $j,k \in \{0,1\}$ and $\text {lose}{\longrightarrow }\text {lose}$. Intuitively, by going from $c_i$ to $d_{i,j}$, Maximizer chooses action 1 with probability $p_{i,j}$ and action 0 otherwise. Minimizer chooses action k by going from $d_{i,j}$ to $r_{i,j,k}$. The random transitions are defined by $P(r_{i,j,0})(\text {lose}) = p_{i,j}$, $P(r_{i,j,0})(c_{i-1}) = 1-p_{i,j}$, $P(r_{i,j,1})(c_0) = p_{i,j}$, $P(r_{i,j,1})(c_{i+1}) = 1-p_{i,j}$.

The objective is $\texttt{Reach}(\{c_0\})$.

Theorem 8

Consider the turn-based reachability game ${\mathcal {G}}$ of branching degree two from Definition 4 and let $x\in \mathbb {N}$.

• 1. ${\texttt{val}_{{\mathcal {G}}}(c_x)} \ge 1/4$.

• 2. There does not exist any uniformly $\varepsilon$-optimal memoryless (MR) strategy for Maximizer.

  Every MR Maximizer strategy $\sigma$ attains arbitrarily little from $c_x$ as $x \rightarrow \infty$. Formally, $\limsup _{x \rightarrow \infty }\inf _{\pi }{{\mathcal {P}}}_{{\mathcal {G}},c_x,\sigma ,\pi }(\texttt{Reach}(\{c_0\}))=0.$

Proof

Let ${\mathcal {G}}^{\prime }$ be the concurrent game from Definition 2. We have ${\texttt{val}_{{\mathcal {G}}^{\prime }}(c_x)} \ge 1/2$ by Theorem 4. Consider the (1/4)-optimal 1-bit Maximizer strategy $\hat{\sigma }$ used in ${\mathcal {G}}^{\prime }$ in Definition 4. We can define a corresponding 1-bit Maximizer strategy $\sigma$ in ${\mathcal {G}}$. In every state $c_i$, it picks the move $c_i {\longrightarrow }d_{i,j}$ whenever its memory mode is j, and it updates its memory in the same way as $\hat{\sigma }$. Then

$$\begin{aligned}{\texttt{val}_{{\mathcal {G}}}(c_x)}&\ge \inf _\pi {{\mathcal {P}}}_{{\mathcal {G}},c_x,\sigma ,\pi }(\texttt{Reach}(\{c_0\}))\\&= \inf _\pi {{\mathcal {P}}}_{{\mathcal {G}}^{\prime },c_x,\hat{\sigma },\pi }(\texttt{Reach}(\{c_0\}))\\&\ge {\texttt{val}_{{\mathcal {G}}^{\prime }}(c_x)} - 1/4 \ge 1/4. \end{aligned}$$

For item 2 the argument is exactly the same as in Theorem 7(2). $\square$

5 No Memoryless Strategies for Reachability in Infinitely Branching Games

Fig. 6

The scheme of games defined in Definition 5 and in Definition 6, where in the former the teal-colored boxes are replaced with a line connecting u directly to the $c_i$, whereas in the latter such a teal-colored box is replaced with a Delay gadget $D_i$ illustrated in Fig. 7 (Color figure online)

In finitely branching turn-based stochastic 2-player games with reachability objectives, Maximizer has $\varepsilon$-optimal MD strategies (Lemma 5). We go on to show that this does not carry over to infinitely branching turn-based reachability games. In this case, there are not even $\varepsilon$-optimal MR strategies, i.e., good Maximizer strategies need memory. The reason for this is the infinite branching of Minimizer. Infinite branching of Maximizer states and random states does not make a difference in the case of reachability objectives. Each infinitely branching Maximizer state can be encoded into a gadget containing an infinite sequence of binary branching Maximizer states, where the sequence must eventually be left, because the target state is not on the sequence. Similarly, each infinitely branching random state can be encoded into a gadget containing an infinite sequence of binary branching random states, where the sequence is left eventually almost surely. Such an encoding is not possible for infinitely branching Minimizer states, because Minimizer could choose to stay inside the gadget forever, and spuriously win the game. (Strictly speaking, such encodings do not preserve path lengths. However, we show in Sect. 6 that a step counter does not help Maximizer anyway.)

Definition 5

Consider the finitely branching turn-based reachability game from Definition 3. We construct an infinitely branching game ${\mathcal {G}}$ by adding a new Minimizer-controlled initial state u, Minimizer-transitions $u {\longrightarrow }c_i$ for all $i \in \mathbb {N}$ and $\text {lose}{\longrightarrow }u$. See Fig. 6 for a scheme of this game. The objective is still $\texttt{Reach}(\{c_0\})$.

Theorem 9

Let ${\mathcal {G}}$ be the infinitely branching turn-based reachability game from Definition 5.

• 1. All states in ${\mathcal {G}}$ are almost surely winning. I.e., for every state s there exists a Maximizer strategy $\sigma$ such that $\inf _{\pi }{{\mathcal {P}}}_{{\mathcal {G}},s,\sigma ,\pi }(\texttt{Reach}(\{c_0\}))=1$.

• 2. For each MR Maximizer strategy $\sigma$ we have

  $$\begin{aligned}\inf _{\pi }{{\mathcal {P}}}_{{\mathcal {G}},u,\sigma ,\pi }(\texttt{Reach}(\{c_0\}))=0.\end{aligned}$$

  I.e., for any $\varepsilon < 1$ there does not exist any $\varepsilon$-optimal MR Maximizer strategy $\sigma$ from state u.

Proof

Towards item 1, first note that by playing

$$\begin{aligned}c_i {\longrightarrow }c_{i,1} {\longrightarrow }d_{i,1}\end{aligned}$$

Maximizer can enforce that he either wins (if Minimizer goes to $r_{i,1}^1$) or the game returns to state u (via state $\text {lose}$ if Minimizer goes to $r_{i,1}^0$). Thus it suffices to show that Maximizer can win almost surely from state u. We construct a suitable strategy $\sigma$ (which is not MR). By Theorem 7, ${\texttt{val}_{{\mathcal {G}}}(c_x)} \ge 1/2$ for every x. Moreover, the subgraph between any $c_x$ and a return to u (which goes via a losing state and is thus to be avoided by Maximizer) is finitely branching. Thus there exists a strategy $\sigma _x$ and a finite horizon $h_x$ such that $\inf _\pi {{\mathcal {P}}}_{{\mathcal {G}},c_x,\sigma _x,\pi }(\texttt{Reach}_{h_x}(\{c_0\})) \ge 1/4$. Then $\sigma$ plays from u as follows. If Minimizer moves $u {\longrightarrow }c_x$ then first play $\sigma _x$ for $h_x$ steps, unless $c_0$ or u are reached first. Then play to reach u again, i.e., the next time that the play reaches a state $c_i$ play $c_i {\longrightarrow }c_{i,1} {\longrightarrow }d_{i,1}$ (thus either Maximizer wins or the play returns to u). So after every visit to u the Maximizer strategy $\sigma$ wins with probability $\ge 1/4$ before seeing u again, and otherwise the play returns to u. Thus $\inf _{\pi }{{\mathcal {P}}}_{{\mathcal {G}},s,\sigma ,\pi }(\texttt{Reach}(\{c_0\})) \ge 1 - (1/4)^\infty = 1$. This proves item 1.

Claim 10

Suppose that for each MR Maximizer strategy $\sigma$ and every $\varepsilon > 0$ there is a Minimizer strategy $\pi (\sigma ,\varepsilon )$ such that, from u, the probability of visiting $c_0$ before revisiting u is at most $\varepsilon$. Then item  is true.

Proof of the claim

Suppose the precondition of the claim. Let $\sigma$ be an MR Maximizer strategy. Let $\pi$ be the Minimizer strategy that, after the i-th visit to u, continues to play $\pi (\sigma ,\varepsilon \cdot 2^{-i})$ until the next visit to u. It follows that ${{\mathcal {P}}}_{{\mathcal {G}},u,\sigma ,\pi }(\texttt{Reach}(\{c_0\})) \le \sum _{i \ge 1} \varepsilon \cdot 2^{-i} = \varepsilon$. $\square$

It remains to prove the precondition of the claim. Let $\sigma$ be an MR Maximizer strategy, and let $\varepsilon > 0$. By Theorem 7(2), there are $i \in \mathbb {N}$ and a Minimizer strategy $\pi$ such that ${{\mathcal {P}}}_{{\mathcal {G}}^{\prime },c_i,\sigma ,\pi }(\texttt{Reach}(\{c_0\})) \le \varepsilon$, where ${\mathcal {G}}^{\prime }$ is the finitely branching subgame of ${\mathcal {G}}$ from Definition 3. The Minimizer strategy $\pi (\sigma ,\varepsilon )$ that, in ${\mathcal {G}}$, from u goes to $c_i$ and then plays $\pi$ has the required property. $\square$

Fig. 7

Delay gadget $D_i$, used in Fig. 6

6 No Markov Strategies for Reachability in Infinitely Branching Games

Recall that Markov strategies are strategies that use just a step counter as memory. We strengthen the result from the previous section by modifying the game so that even Markov strategies are useless for Maximizer. The modification of the game allows Minimizer to cause an arbitrary but finite delay before any state $c_i$ is entered.

Definition 6

Consider the infinitely branching turn-based reachability game from Definition 5. We modify it as follows. For each $i \in \mathbb {N}$ we add a Minimizer-controlled state $b_i$ and redirect all transitions going into $c_i$ to go into $b_i$ instead. Each $b_i$ is infinitely branching: for each $j \in \mathbb {N}$ we add a random state $b_{i,j}$ and a transition $b_i {\longrightarrow }b_{i,j}$. We add further states so that the game moves (deterministically, via a chain of random states) from $b_{i,j}$ to $c_i$ in exactly j steps. See Figs. 6 and 7 for a depiction of this game. The objective is still $\texttt{Reach}(\{c_0\})$.

Theorem 11

Let ${\mathcal {G}}$ be the infinitely branching turn-based reachability game from Definition 6.

• 1. All states in ${\mathcal {G}}$ are almost surely winning. I.e., for every state s there exists a Maximizer strategy $\sigma$ such that $\inf _{\pi }{{\mathcal {P}}}_{{\mathcal {G}},s,\sigma ,\pi }(\texttt{Reach}(\{c_0\}))=1$.

• 2. For every Markov Maximizer strategy $\sigma$ it holds that $\inf _{\pi }{{\mathcal {P}}}_{{\mathcal {G}},u,\sigma ,\pi }(\texttt{Reach}(\{c_0\}))=0$. I.e., no Markov Maximizer strategy is $\varepsilon$-optimal from state u for any $\varepsilon < 1$.

Proof

Item 1 follows from Theorem 9(1), as the modification in Definition 6 only allows Minimizer to cause finite delays.

Towards item 2, the idea of the proof is that for every Markov Maximizer strategy $\sigma$, Minimizer can cause delays that make $\sigma$ behave in the way it would after a long time. This way, Minimizer turns $\sigma$ approximately to an MR-strategy, which is useless by Theorem 9(2).

In more detail, fix any Markov Maximizer strategy $\sigma$. As in the proof of Theorem 7(2), we can assume that whenever the game is in $c_i$, the strategy $\sigma$ almost surely moves eventually to some $d_{i,j}$. Let $p_{i,j,t}$ be the probability that strategy $\sigma$, when it is in $c_i$ at time t, moves to $d_{i,j}$. Thus, a corresponding Maximizer strategy in the (concurrent) Big Match, when it is in $c_i$ at time t, picks action 1 with probability $f(i,t) \overset{{\textrm{def}}}{=}\sum _j p_{i,j,t}(1/j)$; cf. the proof of Theorem 7(2). For each $i \in \mathbb {N}$, let f(i) be an accumulation point of $f(i,1), f(i,2), \ldots$; e.g., take $f(i) \overset{{\textrm{def}}}{=}\liminf _t f(i,t)$. We have that

$$\begin{aligned} \forall {i\in \mathbb {N}}\ \forall {t_0 \in \mathbb {N}}\ \forall {\varepsilon > 0}\ \exists {t\ge t_0} :&\quad f(i,t) \le f(i)+\varepsilon \end{aligned}$$

(7)

$$\begin{aligned} \forall {i\in \mathbb {N}}\ \forall {t_0 \in \mathbb {N}}\ \forall {\varepsilon > 0}\ \exists {t\ge t_0} :&\quad f(i,t) \ge f(i)-\varepsilon \end{aligned}$$

(8)

Similarly to the proof of Theorem 9(2) (see Claim 10 therein), it suffices to show that after each visit to u, Minimizer can make the probability of visiting $c_0$ before seeing u again arbitrarily small. Let $\varepsilon > 0$. We show that Minimizer has a strategy $\pi$ to make this probability at most $\varepsilon$.

Consider the first case where $\sum _{i \ge 1} f(i) < \infty$. Then there is $i_0 \in \mathbb {N}$ such that $\sum _{i \ge i_0} f(i) \le \varepsilon /2$. In u, strategy $\pi$ moves to $b_{i_0}$. Whenever the game is in some $b_i$, strategy $\pi$ moves to some $b_{i,j}$ so that the game will arrive in $c_i$ at a time t that satisfies $f(i,t) \le f(i) + 2^{-i} \cdot \varepsilon /2$; such t exists due to (7). In $c_i$ Maximizer (using $\sigma )$ moves (eventually) to some $d_{i,j}$. Then $\pi$ always chooses “action 1”; i.e., $\pi$ moves to $r_{i,j}^1$. In this way, the play, restricted to states $c_i$, is either of the form $c_{i_0}, c_{i_0+1}, \ldots$ (Maximizer loses) or of the form $c_{i_0}, c_{i_0+1}, \ldots , c_{i_0+k}, c_0$ (Maximizer wins). The probability of the latter can be bounded similarly to the proof of Theorem 4(3); i.e., we have

$$\begin{aligned} {{\mathcal {P}}}_{{\mathcal {G}},u,\sigma ,\pi }(\texttt{Reach}(\{c_0\})) ~\le ~ \sum _{i=i_0}^\infty f(i) + 2^{-i} \cdot \frac{\varepsilon }{2} ~\le ~ \frac{\varepsilon }{2} + \frac{\varepsilon }{2} ~=~ \varepsilon . \end{aligned}$$

Now consider the second case where $\sum _{i \ge 1} f(i) = \infty$. Then there is $i_0 \in \mathbb {N}$ such that $\sum _{i=1}^{i_0} f(i) \ge \frac{1}{\varepsilon }$. In u, strategy $\pi$ moves to $b_{i_0}$. Whenever the game is in some $b_i$, strategy $\pi$ moves to some $b_{i,j}$ so that the game will arrive in $c_i$ at a time t that satisfies $f(i,t) \ge f(i) - 2^{-i}$; such t exists due to (8). In $c_i$ Maximizer (using $\sigma )$ moves (eventually) to some $d_{i,j}$. Then $\pi$ always chooses “action 0”; i.e., $\pi$ moves to $r_{i,j}^0$. In this way, the play, restricted to states $c_i$, is either of the form $c_{i_0}, c_{i_0-1}, \ldots , c_0$ (Maximizer wins) or of the form $c_{i_0}, c_{i_0-1}, \ldots , c_{i_0-k}$ (for some $k < i_0$), followed by $\text {lose}, u$ (Maximizer does not reach $c_0$ before revisiting u). The probability of the former can be bounded similarly to the proof of Theorem 4(3); i.e., the probability that the play reaches $c_0$ before u is upper-bounded by

$$\begin{aligned} \prod _{i=1}^{i_0}(1 - \max \{f(i) - 2^{-i}, 0\}) \quad&\le \quad \frac{1}{1+\sum _{i=1}^{i_0} (f(i) - 2^{-i})}&\text {by Proposition}~(31)\\&\le \quad \frac{1}{\sum _{i=1}^{i_0} f(i)} \; \le \ \varepsilon . \end{aligned}$$

$\square$

7 Good Strategies for Reachability Require Infinite Memory

Fig. 8

The scheme of the nested construction in Definition 7

We show that even finite private memory, in addition to a step counter, is useless for Maximizer in infinitely branching reachability games. To this end, we define a nested version of the game of Definition 6, where the memory requirements increase unboundedly with the nesting depth.

Definition 7

Let ${\mathcal {G}}$ be the game from Definition 6. We inductively define the k-nested game ${\mathcal {G}}_k$ as follows (see Fig. 8). For the base case, let ${\mathcal {G}}_1 \overset{{\textrm{def}}}{=}{\mathcal {G}}$.

For every $i\ge 1$ let ${\mathcal {G}}_k^i$ be a fresh copy of ${\mathcal {G}}_k$ and let $u^{k,i}$ (resp. $c_0^{k,i}$) be the initial state u (resp. the target state $c_0$) in ${\mathcal {G}}_k^i$. For every $k \ge 1$ we construct ${\mathcal {G}}_{k+1}$ by modifying ${\mathcal {G}}$ as follows. The idea is that at every state $c_i$ Maximizer first needs to win the subgame ${\mathcal {G}}_k^i$ before continuing in the game ${\mathcal {G}}_{k+1}$, but Minimizer can choose at which state $s$ in ${\mathcal {G}}_k^i$ the subgame is entered.

We make the state $c_i$ Minimizer-controlled and replace the previous Maximizer transition $c_i {\longrightarrow }c_{i,1}$ by Minimizer transitions $c_i {\longrightarrow }s$ for every state $s$ in ${\mathcal {G}}_k^i$. Moreover, we add the transitions $c_0^{k,i} {\longrightarrow }\text {win}^{k,i} {\longrightarrow }c_{i,1}$. (The new state $\text {win}^{k,i}$ is not strictly needed. It just indicates that Maximizer has won and exited the subgame ${\mathcal {G}}_k^i$.) Note that also in ${\mathcal {G}}_{k+1}$ Minimizer can introduce arbitrary delays between states $b_i$ and $c_i$.

The objective in ${\mathcal {G}}_{k+1}$ is still $\texttt{Reach}(\{c_0\})$.

Lemma 12

For any $k \ge 1$ let ${\mathcal {G}}_k$ be the infinitely branching turn-based reachability game from Definition 7.

• 1. All states in ${\mathcal {G}}_k$ are almost surely winning. I.e., for every state $s$ there exists a Maximizer strategy $\sigma$ such that

  $$\begin{aligned} \inf _{\pi }{{\mathcal {P}}}_{{\mathcal {G}}_k,s,\sigma ,\pi }(\texttt{Reach}(\{c_0\}))=1. \end{aligned}$$

• 2. For each Maximizer strategy $\sigma$ with a step counter plus a private finite memory with $\le k$ modes

  $$\begin{aligned}\inf _{\pi }{{\mathcal {P}}}_{{\mathcal {G}}_k,u,\sigma ,\pi }(\texttt{Reach}(\{c_0\}))=0.\end{aligned}$$

  I.e., for any $\varepsilon < 1$ there does not exist any $\varepsilon$-optimal step counter plus k memory mode Maximizer strategy $\sigma$ from state u in ${\mathcal {G}}_k$.

Proof

We show Item 1 by induction on k.

In the base case of $k=1$ we have ${\mathcal {G}}_1 = {\mathcal {G}}$ from Definition 6, and thus the result holds by Theorem 11(Item 1).

Induction step $k {\longrightarrow }k+1$. For every state $s$ in ${\mathcal {G}}_{k+1}$ outside of any subgame, let $\sigma ^{\prime }(s)$ be the almost surely winning Maximizer strategy from $s$ in the non-nested game ${\mathcal {G}}_1$, obtained from above. By the induction hypothesis, for any state $s$ in a subgame ${\mathcal {G}}_k^i$ there exists a Maximizer strategy $\sigma _k^i(s)$ from $s$ that almost surely wins this subgame ${\mathcal {G}}_k^i$.

We now construct a Maximizer strategy $\sigma (s)$ from any state $s$ in ${\mathcal {G}}_{k+1}$. If $s$ is not in any strict subgame then $\sigma (s)$ plays like $\sigma ^{\prime }(s)$ outside of the subgames. Whenever a subgame ${\mathcal {G}}_k^i$ is entered at some state x then it plays like $\sigma _k^i(x)$ until $\text {win}^{k,i}$ is reached (which happens eventually almost surely by the definition of $\sigma _k^i(x)$) and the play exits the subgame, and then it continues with the outer strategy $\sigma ^{\prime }(s)$. Similarly, if the start state $s$ is in some subgame ${\mathcal {G}}_k^i$ then it first plays $\sigma _k^i(s)$ until $\text {win}^{k,i}$ is reached (which happens eventually almost surely by the definition of $\sigma _k^i(s)$) and the play exits the subgame, and then it continues like the strategy $\sigma ^{\prime }(c_{i,1})$ described above. Then $\sigma (s)$ wins almost surely, since the strategies $\sigma ^{\prime }$ and $\sigma _k^i$ win almost surely.

Towards Item 2, we show, by induction on k, the following slightly stronger property. For each Maximizer strategy $\sigma$ in ${\mathcal {G}}_k$ with a step counter plus a private finite memory with $\le k$ modes, for every $\delta >0$ there exists a Minimizer strategy $\pi$ that upper-bounds Maximizer’s attainment by $\delta$ regardless of Maximizer’s initial memory mode and the starting time. Formally,

$$\begin{aligned} \forall \delta >0\,\exists \pi \, \forall \textsf{m}\, \forall t\ {{\mathcal {P}}}_{{\mathcal {G}}_{k},u,\sigma [\textsf{m}](t),\pi }(\texttt{Reach}(\{c_0\})) \le \delta \end{aligned}$$

(9)

For the base case $k=1$ we have ${\mathcal {G}}_1 = {\mathcal {G}}$ from Definition 6. Since $k=1$, Maximizer has only one memory mode. Moreover, Minimizer’s strategy from the proof of Theorem 11(2) works regardless of the starting time t (since it just chooses delays in the states $b_i$ to satisfy (7) and (8)). Thus we obtain (9).

Induction step $k {\longrightarrow }k+1$. Consider the game ${\mathcal {G}}_{k+1}$ and a fixed Maximizer strategy $\sigma$ with a step counter plus $(k+1)$ private memory modes from state u. Let $\{0,1,\dots ,k\}$ denote the $k+1$ private memory modes of $\sigma$.

From every state $c_i$ in ${\mathcal {G}}_{k+1}$ we enter the subgame ${\mathcal {G}}_k^i$, at some state chosen by Minimizer. When (and if) Maximizer wins this subgame then we are in state $\text {win}^{k,i}$ and the game ${\mathcal {G}}_{k+1}$ continues with Maximizer’s choice at $c_{i,1}$, etc.

Consider the state $c_i$, visited at some time t with some Maximizer memory mode $\textsf{m}$. Let $\pi ^{\prime }$ be some Minimizer strategy. Then let $\alpha (i,\textsf{m},t,\pi ^{\prime })$ be the probability that Maximizer will play action “1” (w.r.t. the encoded concurrent game) in the next round in ${\mathcal {G}}_{k+1}$ after winning the subgame ${\mathcal {G}}_k^i$ (i.e., after reaching $\text {win}^{k,i}$), or loses the subgame (never reaches $\text {win}^{k,i}$). So $\alpha (i,\textsf{m},t,\pi ^{\prime })$ is the probability of losing the subgame ${\mathcal {G}}_k^i$ plus $\sum _j (1/j)\cdot p_j$, where $p_j$ is the probability of winning the subgame and then directly going to $d_{i,j}$ (i.e., in the same round, without seeing any other state $c_i$ before). To formally capture this notion of the “same round”, we let $C \overset{{\textrm{def}}}{=}\{c_i \mid i \in \mathbb {N}\}$ and define

$$\begin{aligned} \alpha (i,\textsf{m},t,\pi ^{\prime }) ~\overset{{\textrm{def}}}{=}~&{{\mathcal {P}}}_{{\mathcal {G}}_{k+1},c_i,\sigma [\textsf{m}](t),\pi ^{\prime }}(\lnot \texttt{Reach}(\{\text {win}^{k,i}\}))\\&\quad + \sum _j (1/j) {{\mathcal {P}}}_{{\mathcal {G}}_{k+1},c_i,\sigma [\textsf{m}](t),\pi ^{\prime }}(d_{i,j}\ \textsf{beforeagain}\ C) \end{aligned}$$

where $(d_{i,j}\ \textsf{beforeagain}\ C)$ denotes the set of plays that visit state $d_{i,j}$ before visiting any state in C again, i.e., any visit to C at the start state (here $c_i$) does not count. The probability $\alpha (i,\textsf{m},t,\pi ^{\prime })$ depends on i (since we are looking at state $c_i$), on Maximizer’s private memory mode $\textsf{m}\in \{0,1,\dots ,k\}$ at state $c_i$, at the time $t\in \mathbb {N}$ when we are at $c_i$ and on Minimizer’s strategy $\pi ^{\prime }$. Let $\alpha (i,\textsf{m},t) \overset{{\textrm{def}}}{=}\sup _{\pi ^{\prime }} \alpha (i,\textsf{m},t,\pi ^{\prime })$ be the supremum over all Minimizer strategies. Let $\alpha (i,t) \overset{{\textrm{def}}}{=}\min _{\textsf{m}\in \{0,\dots ,k\}} \alpha (i,\textsf{m},t)$ the minimum over all memory modes. Intuitively, when entering the subgame ${\mathcal {G}}_k^i$ from $c_i$ at time t, for each Maximizer memory mode $\textsf{m}$, for each $\varepsilon >0$, Minimizer has a strategy to make the probability of Maximizer playing action “1” after winning the subgame (or else losing the subgame) at least $\alpha (i,t)-\varepsilon$. Let $\alpha (i)$ be the maximal accumulation point of the infinite sequence $\alpha (i,1), \alpha (i,2), \dots$, i.e., $\alpha (i) \overset{{\textrm{def}}}{=}\limsup _t \alpha (i,t)$. We have:

$$\begin{aligned} \forall \,i\in \mathbb {N}\ \forall \,\varepsilon > 0\ \forall \,t_0 \in \mathbb {N}\ \exists \,t\ge t_0 :&\quad \alpha (i,t) \ge \alpha (i)-\varepsilon \end{aligned}$$

(10)

$$\begin{aligned} \forall \,i\in \mathbb {N}\ \forall \,\varepsilon > 0\ \exists \,t_0 \in \mathbb {N}\ \forall \,t\ge t_0 :&\quad \alpha (i,t) \le \alpha (i)+\varepsilon \end{aligned}$$

(11)

Now there are two cases.

In the first case, $\sum _i \alpha (i)$ diverges. Intuitively, since we have $\alpha (i) = \limsup _t \alpha (i,t)$, Minimizer chooses the delays at the states $b_i$ in order to make $\alpha (i,t)$ “large”, i.e., close to $\alpha (i)$, by using (10).

Analogously to Claim 10, it suffices, for every $\varepsilon >0$ to construct a Minimizer strategy $\pi$ in ${\mathcal {G}}_{k+1}$ from state u that makes the probability of visiting $c_0$ before revisiting u (denoted as the event “${{c_0}\ \textsf{beforeagain}\ u}$”) at most $\varepsilon$.

We construct a Minimizer strategy $\pi$ that plays as follows. First $\pi$ picks the transition $u {\longrightarrow }b_{i_0}$ for a sufficiently high $i_0 \in \mathbb {N}$, to be determined. At every state $d_{i,j}$, outside of the subgames, $\pi$ always plays action “0”, i.e., $d_{i,j} {\longrightarrow }r_{i,j}^0$. This implies that the state $c_i$ is not visited again, unless the state u is re-visited first. (If Maximizer plays action “1” then he loses this round and the game goes back to u. If Maximizer plays “0” then the game goes down to $c_{i-1}$.)

At every state $b_i$ at each time $t^{\prime }$ the strategy $\pi$ picks a delay such that the game arrives at $c_i$ at a time t such that

$$\begin{aligned} \alpha (i,t) \ge \alpha (i)- \frac{1}{4} 2^{-i}. \end{aligned}$$

(12)

This is possible by Eq. (10). Let $T_i$ be the set of the times t that satisfy (12). Minimizer’s strategy ensures that the states $c_i$ are only reached at times $t \in T_i$.

From state $c_i$, at each time t, for each memory mode $\textsf{m}$ there exists a Minimizer strategy $\pi (i,\textsf{m},t)$ such that we have $\alpha (i,\textsf{m},t,\pi (i,\textsf{m},t)) \ge \alpha (i,\textsf{m},t) - \frac{1}{4} 2^{-i}$, by the definition of $\alpha (i,\textsf{m},t) \overset{{\textrm{def}}}{=}\sup _{\pi ^{\prime }} \alpha (i,\textsf{m},t,\pi ^{\prime })$.

Since the Maximizer memory mode $\textsf{m}$ is private, Minimizer does not know it. So our Minimizer strategy $\pi$ hedges her bets over all possible $\textsf{m}\in \{0,\dots ,k\}$ and plays each strategy $\pi (i,\textsf{m},t)$ with equal probability $\frac{1}{k+1}$. It follows that for every i and time $t \in T_i$ chosen according to Eq. (12), after (and if) winning the subgame ${\mathcal {G}}_k^i$, Maximizer plays action “1” with probability

$$\begin{aligned} \begin{aligned} \alpha (i,\textsf{m},t,\pi )&~\ge ~ \frac{1}{k+1} \left( \alpha (i,t) - \frac{1}{4} 2^{-i}\right) \\&~\ge ~ \frac{1}{k+1} \left( \alpha (i) - \frac{1}{4} 2^{-i} - \frac{1}{4} 2^{-i}\right) = \frac{1}{k+1} \left( \alpha (i) - \frac{1}{2} 2^{-i}\right) . \end{aligned} \end{aligned}$$

(13)

Since, outside of subgames, Minimizer always plays action “0” and each state $c_i$ is visited at most once (unless state u is re-visited), we get for every starting memory mode $\textsf{m}^{\prime }$ and starting time $t^{\prime } \in T_{i_0}$ that the probability of visiting $c_0$ before revisiting u is upper-bounded, i.e.,

$$\begin{aligned} {{\mathcal {P}}}_{{\mathcal {G}}_{k+1},c_{i_0},\sigma [\textsf{m}^{\prime }](t^{\prime }),\pi }({{c_0}\ \textsf{beforeagain}\ u}) ~\le ~ \prod _{i=1}^{i_0} (1 - \min _{\textsf{m}}\inf _{t \in T_i} \alpha (i,\textsf{m},t,\pi )). \end{aligned}$$

Since $\sum _i \alpha (i)$ diverges, it follows from (13) that

$$\begin{aligned} \sum _{i=1}^\infty \min _{\textsf{m}}\inf _{t \in T_i} \alpha (i,\textsf{m},t,\pi ) \ge \left( \frac{1}{k+1}\sum _i \alpha (i)\right) - \frac{1}{2k+2} \end{aligned}$$

also diverges. From Proposition 32 we obtain

$$\begin{aligned} \prod _{i=1}^\infty (1 - \min _{\textsf{m}}\inf _{t \in T_i} \alpha (i,\textsf{m},t,\pi )) = 0 \end{aligned}$$

and hence

$$\begin{aligned} \lim _{i_0 \rightarrow \infty } \prod _{i=1}^{i_0} (1 - \min _{\textsf{m}}\inf _{t \in T_i} \alpha (i,\textsf{m},t,\pi ))= 0. \end{aligned}$$

Thus, for every $\varepsilon >0$, there exists a sufficiently large $i_0$ such that for all $t^{\prime } \in T_{i_0}$ and all memory modes $\textsf{m}^{\prime }$

$$\begin{aligned} {{\mathcal {P}}}_{{\mathcal {G}}_{k+1},c_{i_0},\sigma [\textsf{m}^{\prime }](t^{\prime }),\pi }({{c_0}\ \textsf{beforeagain}\ u}) \le \varepsilon . \end{aligned}$$

Recall that at state u the Minimizer strategy $\pi$ picks the transition $u {\longrightarrow }b_{i_0}$, i.e., the number $i_0$ is Minimizer’s choice. Moreover, the delay gadget $D_{i_0}$ ensures that state $c_{i_0}$ is entered at some time $t^{\prime } \in T_{i_0}$, regardless of the time t at state u. Hence, for all $t \in \mathbb {N}$ and all memory modes $\textsf{m}$ we have

$$\begin{aligned} {{\mathcal {P}}}_{{\mathcal {G}}_{k+1},u,\sigma [\textsf{m}](t),\pi }({{c_0}\ \textsf{beforeagain}\ u}) \le \varepsilon . \end{aligned}$$

Analogously to Claim 10, we can pick $i_0^l$ and $\varepsilon ^l = \delta \cdot 2^{-l}$ after the l-th visit to u such that for all $t \in \mathbb {N}$ and all $\textsf{m}$ we have

$$\begin{aligned} {{\mathcal {P}}}_{{\mathcal {G}}_{k+1},u,\sigma [\textsf{m}](t),\pi }(\texttt{Reach}(\{c_0\})) \le \sum _{l=1}^\infty \varepsilon _l = \delta \end{aligned}$$

as required.

Now we consider the second case where $\sum _i \alpha (i)$ converges. We construct a Minimizer strategy $\pi$ in ${\mathcal {G}}_{k+1}$ that plays as follows. First $\pi$ picks the transition $u {\longrightarrow }b_{i_0}$ for a sufficiently high $i_0 \in \mathbb {N}$, to be determined. At every state $d_{i,j}$, outside of the subgames, $\pi$ always plays action “1”, i.e., $d_{i,j} {\longrightarrow }r_{i,j}^1$. This implies that each state $c_i$ with $i \ge i_0$ is visited at most once, and states $c_i$ with $0< i < i_0$ are never visited. (If Maximizer plays action “1” then he immediately wins at $c_0$ and if he plays “0” then the games goes up to $c_{i+1}$.)

By Eq. (11), for every $i \ge 1$ there exists a time $t_i \in \mathbb {N}$ such that

$$\begin{aligned} \forall t \ge t_i.\ \alpha (i,t) \le \alpha (i) + \frac{1}{4} 2^{-i}. \end{aligned}$$

(14)

Let $T_i = \{t \in \mathbb {N}\mid t \ge t_i\}$ be the set of the times t that satisfy (14). At each state $b_i$ Minimizer’s strategy $\pi$ delays sufficiently long such that $c_i$ is reached at a time $t \in T_i$. This is possible for every arrival time $t^{\prime }$ at $b_i$.

Consider a state $s$ in the subgame ${\mathcal {G}}_k^i$ that is reached at some time t when Maximizer’s strategy is in some memory mode $\textsf{m}$, and let $\pi ^{\prime }$ be a Minimizer strategy. Let $\beta (s,i,\textsf{m},t,\pi ^{\prime })$ be the probability that Maximizer will play action “1” (w.r.t. the encoded concurrent game) in the next round in ${\mathcal {G}}_{k+1}$ after winning the subgame ${\mathcal {G}}_k^i$ (i.e., after reaching $\text {win}^{k,i}$), or loses the subgame (never reaches $\text {win}^{k,i}$). So $\beta (s,i,\textsf{m},t,\pi ^{\prime })$ is the probability, from state $s$, of losing the subgame ${\mathcal {G}}_k^i$ plus $\sum _j (1/j)\cdot p_j$, where $p_j$ is the probability of winning the subgame and then directly going to $d_{i,j}$ (without visiting any other state $c_i$ in between). Recall that $C = \{c_i \mid i \in \mathbb {N}\}$. We let

$$\begin{aligned} \beta (s,i,\textsf{m},t,\pi ^{\prime }) ~\overset{{\textrm{def}}}{=}~&{{\mathcal {P}}}_{{\mathcal {G}}_{k+1},s,\sigma [\textsf{m}](t),\pi ^{\prime }}(\lnot \texttt{Reach}(\{\text {win}^{k,i}\}))\\&\quad + \sum _j (1/j) {{\mathcal {P}}}_{{\mathcal {G}}_{k+1},s,\sigma [\textsf{m}](t),\pi ^{\prime }}(d_{i,j}\ \textsf{beforeagain}\ C) \end{aligned}$$

Let $\beta (s,i,\textsf{m},t) \overset{{\textrm{def}}}{=}\sup _{\pi ^{\prime }} \beta (s,i,\textsf{m},t,\pi ^{\prime })$ be the supremum over all Minimizer strategies. Let

$$\begin{aligned} \beta (s,i,t) \overset{{\textrm{def}}}{=}\min _{\textsf{m}\in \{0,\dots ,k\}} \beta (s,i,\textsf{m},t) \end{aligned}$$

be the minimum over all memory modes.

Claim 13

For all states $s$ in ${\mathcal {G}}_k^i$ and times t we have $\beta (s,i,t+1) \le (k+1)\alpha (i,t)$.

Proof

Consider the situation where we are at state $c_i$ at time t and the Maximizer strategy $\sigma$ is in some memory mode $\textsf{m}$ (unknown to Minimizer). Then some particular Minimizer strategy $\hat{\pi }$ could play $c_i {\longrightarrow }s$ to arrive at $s$ in one step at time $t+1$. Meanwhile, $\sigma$ can update its memory to some other mode $\textsf{m}^{\prime }$ (or a distribution over memory modes), still unknown to Minimizer. Then $\hat{\pi }$ hedges her bets by guessing Maximizer’s memory mode $\textsf{m}^{\prime }$. For each of the $k+1$ possible modes $\textsf{m}^{\prime }$, the Minimizer strategy $\hat{\pi }$ plays, with probability $\frac{1}{k+1}$, an $\varepsilon$-optimal strategy to maximize the probability that Maximizer plays action “1” after winning ${\mathcal {G}}_k^i$ (or loses the subgame). Thus

$$\begin{aligned}\alpha (i,t)&= \min _\textsf{m}\sup _{\pi ^{\prime }} \alpha (i,\textsf{m},t,\pi ^{\prime })\\&\ge \min _\textsf{m}\alpha (i,\textsf{m},t,\hat{\pi })\\&\ge \min _{\textsf{m}^{\prime }} \frac{1}{k+1} \beta (s,i,\textsf{m}^{\prime },t+1,\hat{\pi })\\&\ge \frac{1}{k+1} (\min _{\textsf{m}^{\prime }} \sup _{\pi ^{\prime }} \beta (s,i,\textsf{m}^{\prime },t+1,\pi ^{\prime }) - \varepsilon )\\&= \frac{1}{k+1} (\beta (s,i,t+1) - \varepsilon ). \end{aligned}$$

Since this holds for every $\varepsilon >0$, the claim follows. $\square$

By Claim 13, for every $s$ in ${\mathcal {G}}_k^i$ and time $t+1$ there exists at least one memory mode $\textsf{m}(s,t+1)$ (a mode $\textsf{m}$ where the minimum $\beta (s,i,t+1) = \min _{\textsf{m}\in \{0,\dots ,k\}} \beta (s,i,\textsf{m},t+1)$ is realized) such that if $\sigma$ enters $s$ at time $t+1$ in mode $\textsf{m}(s,t+1)$ then after winning ${\mathcal {G}}_k^i$ (if at all) Maximizer plays action “1” with a “small” probability $\le (k+1)\alpha (i,t)$. Crucially, this property holds for the $\sup$ over the Minimizer strategies and thus for every Minimizer strategy inside the subgame ${\mathcal {G}}_k^i$. In particular it holds for the Minimizer strategy $\pi$ that we will construct. We call $\textsf{m}(s,t+1)$ the forbidden memory mode for state $s$ at time $t+1$.

Above we have defined our Minimizer strategy $\pi$ so that it adds sufficient delays in the states $b_i$ such that $c_i$ is only visited at times $t \ge t_i$. This implies that states $s$ in ${\mathcal {G}}_k^i$ are only visited at times $t+1$ where $t \ge t_i$. Since for these times Eq. (14) is satisfied, we obtain

$$\begin{aligned} \forall t \ge t_i.\ \beta (s,i,\textsf{m}(s,t+1),t+1) \le (k+1)(\alpha (i) + \frac{1}{4}2^{-i}). \end{aligned}$$

(15)

Let $\sigma ^{\prime }$ be a restriction of $\sigma$ that, inside the subgame ${\mathcal {G}}_k^i$, is never in the forbidden memory mode $\textsf{m}(s,t+1)$ at state $s$ at time $t+1$, or else concedes defeat.

Claim 14

Consider a step counter plus $(k+1)$ mode Maximizer strategy $\sigma ^{\prime }$ in ${\mathcal {G}}_k^i$ that is never in the forbidden memory mode $\textsf{m}(s,t+1)$ at state $s$ at time $t+1$. Then there exists a step counter plus k mode Maximizer strategy $\sigma ^{\prime \prime }$ in ${\mathcal {G}}_k^i$ that performs equally well as $\sigma ^{\prime }$ against any Minimizer strategy.

Proof

The strategy $\sigma ^{\prime }$ has $k+1$ memory modes $\{0,\dots ,k\}$, plus the step counter. We will construct the strategy $\sigma ^{\prime \prime }$ to only have k memory modes $\{0,\dots ,k-1\}$, plus the step counter. The strategy $\sigma ^{\prime \prime }$ can directly imitate the behavior of $\sigma ^{\prime }$ as follows. Suppose that $\sigma ^{\prime }$ enters memory mode k at some state $s$ and time $t+1$. From our assumption that $\sigma ^{\prime }$ never enters the forbidden memory mode it follows that $k \ne \textsf{m}(s,t+1)$. In this situation $\sigma ^{\prime \prime }$ enters memory mode $\textsf{m}(s,t+1)$ instead. Whenever $\sigma ^{\prime \prime }$ is in memory mode $\textsf{m}(s,t+1)$ at some state $s$ and time $t+1$ then it plays like $\sigma ^{\prime }$ at state $s$ in memory mode k. By the condition on the behavior of $\sigma ^{\prime }$ there is no confusion, $\sigma ^{\prime \prime }$ just uses the memory modes $\{0,\dots ,k-1\}$ and it still imitates the behavior of $\sigma$. $\square$

By Claim 14, the Maximizer strategy $\sigma ^{\prime }$ is equivalent to a strategy with just a step counter and k memory modes. By the induction hypothesis (9), for this restricted Maximizer strategy $\sigma ^{\prime }$ there exists a Minimizer strategy $\pi _i$ in ${\mathcal {G}}_k^i$ such that

$$\begin{aligned} \forall \textsf{m}\ \forall t\ {{\mathcal {P}}}_{{\mathcal {G}}_k^i,u^{k,i},\sigma ^{\prime }[\textsf{m}](t),\pi _i}(\texttt{Reach}(\{c_0^{k,i}\})) \le \delta \cdot 2^{-(i+1)}. \end{aligned}$$

(16)

We are now ready to construct Minimizer’s strategy $\pi$ in ${\mathcal {G}}_{k+1}$. At every state $d_{i,j}$, outside of the subgames, $\pi$ always plays action “1”, i.e., $d_{i,j} {\longrightarrow }r_{i,j}^1$. This implies that each state $c_i$ is visited at most once. At the states $b_i$, Minimizer chooses the delays such that $c_i$ is reached at a time $t \ge t_i$, as described above. From each $c_i$ Minimizer goes to state $u^{k,i}$ of the subgame ${\mathcal {G}}_k^i$. Inside each subgame ${\mathcal {G}}_k^i$ Minimizer plays like $\pi _i$. By (16), $\pi _i$ performs well in ${\mathcal {G}}_k^i$ (regardless of the initial memory mode and time) if Maximizer limits himself to $\sigma ^{\prime }$.

Now we show that $\pi$ performs well in ${\mathcal {G}}_{k+1}$. Since $\pi$ first picks the transition $u {\longrightarrow }b_{i_0}$ and then always plays action “1” outside of the subgames, it follows that each subgame ${\mathcal {G}}_k^i$ with $i \ge i_0$ is played at most once, and subgames ${\mathcal {G}}_k^i$ with $i < i_0$ are never played. For each subgame ${\mathcal {G}}_k^i$, let $\text {Forb}_i$ be the set of plays where Maximizer enters a forbidden memory mode (for the current state and time) at least once.⁵

From Eq. (16) we obtain that Maximizer loses ${\mathcal {G}}_k^i$ (and thus ${\mathcal {G}}_{k+1}$) with high probability if he never enters a forbidden memory mode.

$$\begin{aligned} \begin{aligned} \max _{\textsf{m}}\sup _{t}&\; {{\mathcal {P}}}_{{\mathcal {G}}_{k+1},c_i,\sigma [\textsf{m}](t),\pi }(\texttt{Reach}(\{c_0\}) \cap \overline{\text {Forb}_i})\\&\le \max _{\textsf{m}}\sup _{t} {{\mathcal {P}}}_{{\mathcal {G}}_k^i,u^{k,i},\sigma ^{\prime }[\textsf{m}](t),\pi _i}(\texttt{Reach}(\{c_0^{k,i}\}))\\&\le \delta \cdot 2^{-(i+1)}. \end{aligned} \end{aligned}$$

(17)

On the other hand, we can show that if Maximizer does enter a forbidden memory mode (for the current state and time) in ${\mathcal {G}}_k^i$ then his chance of playing action “1” (and thus winning ${\mathcal {G}}_{k+1}$ in that round) after (and if) winning the subgame ${\mathcal {G}}_k^i$ is small. This holds for every Minimizer’s strategy inside ${\mathcal {G}}_k^i$ and thus in particular this holds for our chosen Minimizer strategy $\pi _i$.

Recall that $\pi$ ensures that states in ${\mathcal {G}}_k^i$ are only reached at times $t+1$ where $t \ge t_i$, and thus Eq. (15) applies. Hence, at $c_i$, Maximizer’s chance of satisfying $\text {Forb}_i$ and still winning the game in this round (without going to $c_{i+1}$ and the next subgame $G_k^{i+1}$) is upper bounded by $(k+1)(\alpha (i) + \frac{1}{4}2^{-i})$. For all $\textsf{m}$ and all $t \ge t_i$ we have

$$\begin{aligned} \begin{aligned} {{\mathcal {P}}}_{{\mathcal {G}}_{k+1},c_i,\sigma [\textsf{m}](t),\pi }(\texttt{Reach}(\{c_0\}) \cap \text {Forb}_i \cap \lnot \texttt{Reach}(\{c_{i+1}\}))\\ \le (k+1)(\alpha (i) + \frac{1}{4}2^{-i}) \end{aligned} \end{aligned}$$

(18)

Since in our current case $\sum _i \alpha (i)$ converges, it follows that $\sum _i (k+1)(\alpha (i) + \frac{1}{4}2^{-i})$ also converges, and thus there exists a sufficiently large $i_0 \in \mathbb {N}$ such that

$$\begin{aligned} \sum _{i \ge i_0} (k+1)(\alpha (i) + \frac{1}{4}2^{-i}) \le \delta /2 \end{aligned}$$

(19)

Let

$$\begin{aligned} { nFo}(i,\textsf{m}^{\prime },t^{\prime }) ~\overset{{\textrm{def}}}{=}~ {{\mathcal {P}}}_{{\mathcal {G}}_{k+1},c_i,\sigma [\textsf{m}^{\prime }](t^{\prime }),\pi }(\texttt{Reach}(\{c_0\}) \cap \overline{\text {Forb}_i}) \end{aligned}$$

and

$$\begin{aligned} { Fo}(i,\textsf{m}^{\prime },t^{\prime }) ~\overset{{\textrm{def}}}{=}~ {{\mathcal {P}}}_{{\mathcal {G}}_{k+1},c_i,\sigma [\textsf{m}^{\prime }](t^{\prime }),\pi }(\texttt{Reach}(\{c_0\}) \cap \text {Forb}_i \cap \lnot \texttt{Reach}(\{c_{i+1}\})) \end{aligned}$$

Then from (17), (18) and (19) we obtain that for every initial memory mode $\textsf{m}$ and time t,

$$\begin{aligned} {{\mathcal {P}}}_{{\mathcal {G}}_{k+1},u,\sigma [\textsf{m}](t),\pi }(\texttt{Reach}(\{c_0\})&\le \sum _{i \ge i_0} \max _{\textsf{m}^{\prime }}\sup _{t^{\prime }\ge t_i} { nFo}(i,\textsf{m}^{\prime },t^{\prime }) + \sum _{i \ge i_0} \max _{\textsf{m}^{\prime }}\sup _{t^{\prime } \ge t_i} { Fo}(i,\textsf{m}^{\prime },t^{\prime }) \\&\le \sum _{i \ge i_0} \delta \cdot 2^{-(i+1)} + \sum _{i \ge i_0} (k+1)(\alpha (i) + \frac{1}{4}2^{-i})\\&\le \delta /2 + \delta /2 = \delta \ \end{aligned}$$

$\square$

In order to show that Maximizer needs infinite memory, in addition to a step counter, we combine all the nested games ${\mathcal {G}}_k$ into a single game.

Definition 8

For all $k \ge 1$ consider the nested games ${\mathcal {G}}_k$ from Definition 7 with initial state $u^k$ and target state $c_0^k$, respectively. We construct a game ${\mathcal {G}}$ with initial state $s_0$, target state f, Minimizer-controlled transitions $s_0 {\longrightarrow }u_k$ for all k, and Maximizer controlled transitions $c_0^k {\longrightarrow }f$ for all k. The objective in ${\mathcal {G}}$ is $\texttt{Reach}(\{f\})$.

The following theorem is the formal version of Theorem 2 from the introduction.

Theorem 15

Let ${\mathcal {G}}$ be the infinitely branching turn-based reachability game from Definition 8.

• 1. All states in ${\mathcal {G}}$ are almost surely winning. I.e., for every state $s$ there exists a Maximizer strategy $\sigma$ such that $\inf _{\pi }{{\mathcal {P}}}_{{\mathcal {G}},s,\sigma ,\pi }(\texttt{Reach}(\{f\}))=1$.

• 2. For each Maximizer strategy $\sigma$ with a step counter plus a private finite memory we have

  $$\begin{aligned} \inf _{\pi }{{\mathcal {P}}}_{{\mathcal {G}},s_0,\sigma ,\pi }(\texttt{Reach}(\{f\}))=0. \end{aligned}$$

  I.e., for any $\varepsilon < 1$ there does not exist any $\varepsilon$-optimal step counter plus finite private memory Maximizer strategy $\sigma$ from state $s_0$ in ${\mathcal {G}}$.

Proof

Towards Item 1, every state in ${\mathcal {G}}_k$ is almost surely winning by Lemma 12(1). Thus, after the first step $s_0 {\longrightarrow }u^k$ into some game ${\mathcal {G}}_k$, Maximizer just needs to play the respective almost surely winning strategy in ${\mathcal {G}}_k$.

Towards Item 2, consider a Maximizer strategy $\sigma$ with a step counter and a finite memory with some number of modes $k \in \mathbb {N}$. Then, by Lemma 12(2), for every $\delta >0$, Minimizer can choose a first step $s_0 {\longrightarrow }u^k$ into ${\mathcal {G}}_k$ and a strategy in ${\mathcal {G}}_k$ that upper-bounds Maximizer’s attainment to $\le \delta$. $\square$

Remark 1

Theorem 15 has implications even for games with finite action sets (resp. finitely branching turn-based games).

Consider a finitely branching game where the states are labeled with rewards in $\{-1,1\}$. The objective of Maximizer is to ensure that the $\liminf$ of the seen rewards is $\ge 0$. (Equivalently that states with reward $-1$ are visited only finitely often. This is also called a co-Büchi objective in [31]). The infinitely branching reachability game of Theorem 15 can be encoded into a finitely branching $\liminf$ game, and thus the lower bound of Theorem 15 carries over. One just replaces every infinite Minimizer branching $s\rightarrow s_i$ for $i \in \mathbb {N}$ by a Minimizer-controlled gadget $s\rightarrow s^{\prime }_1 \rightarrow s^{\prime }_2 \dots$ and $s^{\prime }_i \rightarrow s_i$ with new states $s^{\prime }_i$ that have reward 1. Minimizer cannot stay in states $s^{\prime }_i$ forever, since their rewards of 1 makes this winning for Maximizer. Thus the finitely branching gadgets faithfully encode Minimizer’s original infinitely branching choice. Finally, the target state $c_0$ is given reward 1 and a self-loop, and all other states are given reward $-1$. Thus the $\liminf \ge 0$ objective in the new game corresponds to the reachability objective to reach state $c_0$ in the original game. The only problem with this construction is that the new gadgets incur extra steps, i.e., the step counters in the two games do not coincide. Thus the property that a step counter does not help Maximizer does not follow immediately from Theorem 15 if taken as a black box. However, the delay gadgets $D_i$ in Definition 7 (and their finitely branching encoding in the new game) still ensure that the step counter does not help Maximizer. I.e., the proof of the lower bound for the finitely branching $\liminf \ge 0$ game is nearly identical to the proof of Theorem 15.

In the new finitely branching game above, the objective to reach $c_0$ also coincides with the objective to attain a high expected $\liminf$ of the rewards. Thus $\varepsilon$-optimal (resp. optimal) strategies to maximize the expected $\liminf$ also require infinite memory.

Finally, if one flips the signs of the rewards of all transitions in the game above, then the objective to reach $c_0$ coincides with the objective to minimize the expected $\limsup$ of the rewards. Thus $\varepsilon$-optimal (resp. optimal) strategies to minimize the expected $\limsup$ also require infinite memory. This solves the open question in Sect. 5 of [50].

8 Infinitely Branching but only Finitely Often

In Theorem 15 we showed that already for turn-based reachability games, $\varepsilon$-optimal strategies for Maximizer require infinite memory. The lower bound construction crucially uses that Minimizer’s action set is infinite. If Minimizer has only finite action sets then Maximizer’s $\varepsilon$-optimal strategies can be simpler, namely MR for concurrent games and MD for turn-based games (cf. Table 1).

However, the connection between Minimizer’s infinite action sets and Maximizer’s need for infinite memory is not as direct as it might seem. In this section we consider the restricted setting of concurrent games where Maximizer has finite action sets and Minimizer can use an infinite action set at most finitely often in any play (see also Definition 10 below).

We show in Theorem 20 that in this case, Maximizer still has uniformly $\varepsilon$-optimal 1-bit strategies. Moreover, we show that this upper bound is tight in the sense that even if Minimizer can use an infinite action set only once, $\varepsilon$-optimal Maximizer strategies still require 1 bit of memory.

We start with the lower bound. The following theorem shows that, even in turn-based reachability games, if Minimizer can use infinite branching just once, MR strategies cannot be $\varepsilon$-optimal for Maximizer for any $\varepsilon < 1/2$.

Definition 9

Consider the Turn-based Big Match on $\mathbb {N}$ (Definition 3 on page 19), and add a new infinitely branching Minimizer-controlled initial state u and Minimizer-transitions $u {\longrightarrow }c_x$ for all $x \in \mathbb {N}$.

Theorem 16

There exists a turn-based game ${\mathcal {G}}$ as in Definition 9 with initial state u and reachability objective $\texttt{Reach}(\{c_0\})$ such that

• 1. All states except u are finitely branching, and all plays from u use infinite branching exactly once.

• 2. ${\texttt{val}_{{\mathcal {G}}}(u)} \ge 1/2$.

• 3. $\inf _{\pi }{{\mathcal {P}}}_{{\mathcal {G}},u,\sigma ,\pi }(\texttt{Reach}(\{c_0\}))=0$ holds for every Maximizer MR strategy $\sigma$.

Proof

Item 1 holds by the construction in Definition 9, since u is the only state with infinite branching and plays cannot return to u.

Towards Item 2., Theorem 7(Item 1) yields ${\texttt{val}_{{\mathcal {G}}}(c_x)} \ge 1/2$ for every $x \in \mathbb {N}$, and thus ${\texttt{val}_{{\mathcal {G}}}(u)} \ge 1/2$.

Towards Item 3., by Theorem 7(Item 2), for every MR Maximizer strategy $\sigma$ we have $\limsup _{x \rightarrow \infty }\inf _{\pi }{{\mathcal {P}}}_{{\mathcal {G}},c_x,\sigma ,\pi }(\texttt{Reach}(\{c_0\}))=0$. Since in our game the Minimizer strategy $\pi$ gets to pick the transition $u \rightarrow c_x$ for an arbitrary $x \in \mathbb {N}$, $\inf _{\pi }{{\mathcal {P}}}_{{\mathcal {G}},u,\sigma ,\pi }(\texttt{Reach}(\{c_0\}))=0$. $\square$

Towards the upper bound, we first define the case where Minimizer can use infinite action sets only finitely often in any play.

Definition 10

Let ${\mathcal {G}}$ be a concurrent game on a countable set of states $S$ and $S^\infty \overset{{\textrm{def}}}{=}\{s\in S\mid |B(s)|=\infty \}$ the states with an infinite Minimizer action set. Let $S^{\prime } \subseteq S$ be the subset of states $s$ such that every play from $s$ (under any strategies) visits $S^\infty$ only finitely often.

Different plays from the same start state can have different numbers of visits to $S^\infty$. Even if this number is finite for every play, there is no uniform finite upper bound. Thus the condition of Definition 10 on plays does not imply a finite bound for the start state. However, we show that an ordinal bound exists.

We introduce a ranking function $I: S^{\prime } \rightarrow {\mathbb {O}}$ so that I(x) is an upper bound on the number of possible visits to $S^\infty$, including the current state x. This is based on a classic result on well-founded relations. Recall that a binary relation $E\subseteq S\times S$ is well-founded if every non-empty subset $X\subseteq S$ has a minimal element w.r.t. E.

Theorem 17

([28, Theorem 2.27]) If $E\subseteq S\times S$ is well-founded then there exists a unique function $\rho : V \rightarrow {\mathbb {O}}$ such that for all $x \in V$

$$\begin{aligned} \rho (x) = \sup \{\rho (y)+1 \mid y E x\}. \end{aligned}$$

In particular, yEx implies $\rho (y) < \rho (x)$. Moreover, if V is countable then $\sup \rho (V)$ is a countable ordinal.

Definition 11

(Ranking function I) Let ${\mathcal {G}}$ be a concurrent reachability game on a countable set of states $S$, and let $S^{\prime }, S^\infty \subseteq S$ be as in Definition 10. Let $\mathord {\rightarrow } \subseteq S\times S$ be the induced game graph, i.e., $x \mathord {\rightarrow } y \iff \exists a,b.\, y \in \texttt{supp}(p(x,a,b))$.

Let $V \overset{{\textrm{def}}}{=}S^{\prime } \cap S^\infty$ and $E \subseteq V \times V$ be the reversal of the closure of $\rightarrow$ over states in $S^{\prime } \setminus S^\infty$, i.e., $(y,x) \in E \iff \exists k\ge 0, z_1,\dots ,z_k \in S^{\prime } \setminus S^\infty \ : x \rightarrow z_1 \rightarrow \dots \rightarrow z_k \rightarrow y$.

From the definition of $S^{\prime }$ we obtain that E is a well-founded relation on V. By Theorem 17, there exists a unique function $\rho : V \rightarrow {\mathbb {O}}$ such that for all $x \in V$

$$\begin{aligned} \rho (x) = \sup \{\rho (y)+1 \mid y E x\}. \end{aligned}$$

(20)

By convention, $\sup \emptyset =0$. We first define our ranking function $I: V \rightarrow {\mathbb {O}}$ only on the set V by $I(x) \overset{{\textrm{def}}}{=}\rho (x)+1$. Intuitively, I(x) is the upper bound on the number of visits to $S^\infty$, including the current state x. We then extend the function I from V to $S^{\prime }$ as follows. For every $x \in S^{\prime } \setminus V$ let

$$\begin{aligned} I(x) \overset{{\textrm{def}}}{=}\sup \{\rho (y) \mid y \in V \wedge x \rightarrow ^+ y\}, \end{aligned}$$

where $\rightarrow ^+$ is the transitive closure of $\rightarrow$. Since we assume $\sup \emptyset =0$, the states x that cannot reach $S^\infty$ satisfy $I(x)=0$.

Lemma 18

The ranking function $I:S^{\prime }\rightarrow {\mathbb {O}}$ satisfies the following properties.

$$\begin{aligned} x \mathord {\rightarrow } y \ \text{ implies }\ I(x) \ge I(y) \end{aligned}$$

(21)

$$\begin{aligned} x \in S^\infty \cap S^{\prime } \ \wedge \ x \mathord {\rightarrow } y \ \text{ implies }\ I(x) > I(y) \end{aligned}$$

(22)

and $\gamma ({\mathcal {G}}) \overset{{\textrm{def}}}{=}\sup I(S^{\prime })$ is a countable ordinal.

Proof

Equations (21) and (22) follow directly from the definition of function I in Definition 11. Since $S$ and $S^{\prime }$ are countable, $\sup I(S^{\prime })$ is a countable ordinal by Theorem 17. $\square$

It follows from Lemma 18 that states in $S^{\prime }$ can be part of cycles, but not part of any cycle that contains a state from $S^\infty$. E.g., in the game in Fig. 8 we have $S^{\prime } = \{c_0\}$, i.e., none of the states are in $S^{\prime }$ except for the target.

Now we show that 1 bit of public memory is sufficient for Maximizer, provided that Minimizer can use infinite action sets only finitely often in any play. I.e., for every $\varepsilon >0$, Maximizer has a public 1-bit strategy for reachability that is uniformly $\varepsilon$-optimal from $S^{\prime }$.

First we need a slight generalization of the reachability objective.

Definition 12

(Weighted reachability) Let ${\mathcal {G}}$ be a concurrent game on $S$ and $T\subseteq S$. Let $f: T\rightarrow [0,1]$ be a reward function. We lift f to plays $f: Z^\omega \rightarrow [0,1]$ as follows. If a play $h \in Z^\omega$ never visits $T$ then $f(h) \overset{{\textrm{def}}}{=}0$. Otherwise, let $f(h) \overset{{\textrm{def}}}{=}f(s)$ where $s$ is the first state in $T$ that is visited by h. Let ${{\mathcal {W}}}_{f}$ denote the weighted reachability objective, i.e., to maximize the expected payoff w.r.t. function f.

Weighted reachability generalizes reachability (just let $f(s)=1$ for all $s\in T$). Now we generalize Theorem 6 to weighted reachability.

Theorem 19

For any concurrent game with finite action sets and weighted reachability objective, for any $\varepsilon >0$, Maximizer has a uniformly $\varepsilon$-optimal public 1-bit strategy. If the game is turn-based and finitely branching, Maximizer has a deterministic such strategy.

Proof

We can encode weighted reachability into ordinary reachability. Given a concurrent game ${\mathcal {G}}$ with target set $T$ and weighted reachability objective ${{\mathcal {W}}}_{f}$, we construct a modified game ${\mathcal {G}}^{\prime }$ with target set $\{t\}$, where t is a new state, as follows. From every state $s\in T$, regardless of the chosen actions, the game goes to t with probability $f(s)$ and to a special new sink state $\bot$ with probability $1-f(s)$. Then ${{\mathcal {W}}}_{f}$ in ${\mathcal {G}}$ coincides with $\texttt{Reach}(\{t\})$ in ${\mathcal {G}}^{\prime }$, i.e., ${\mathcal {E}}_{{\mathcal {G}},s,\sigma ,\pi }(f)= {{\mathcal {P}}}_{{\mathcal {G}}^{\prime },s,\sigma ,\pi }(\texttt{Reach}(\{t\}))$. The result follows from Theorem 6, since the 1-bit strategy can be carried from ${\mathcal {G}}^{\prime }$ to ${\mathcal {G}}$. $\square$

Theorem 20

Let ${\mathcal {G}}$ be a concurrent game with finite Maximizer action sets on a countable set of states $S$ with reachability objective $\texttt{Reach}(T)$ and $S^{\prime } \subseteq S$ as in Definition 10.

For every $\varepsilon >0$, Maximizer has a public 1-bit strategy that is uniformly $\varepsilon$-optimal from every state in $S^{\prime }$. If the game is turn-based then Maximizer has a deterministic such strategy.

Proof

Let $I: S^{\prime } \rightarrow {\mathbb {O}}$ be the ranking function from Definition 11. For every ordinal $\alpha \in {\mathbb {O}}$ let $S_\alpha \overset{{\textrm{def}}}{=}\{s\in S^{\prime } \mid I(s)=\alpha \}$. We have $S^{\prime } = \bigcup _{\alpha \le \gamma ({\mathcal {G}})} S_\alpha$ for the countable ordinal $\gamma ({\mathcal {G}})$ by Lemma 18. Let $S_{< \alpha } \overset{{\textrm{def}}}{=}\bigcup _{\beta < \alpha } S_\beta$ and $S_{\le \alpha } \overset{{\textrm{def}}}{=}\bigcup _{\beta \le \alpha } S_\beta$. We can assume without restriction that the states in $T$ are absorbing and thus $T\subseteq S_0$.

Since $\gamma ({\mathcal {G}})$ is a countable ordinal, the set $\{\alpha \in {\mathbb {O}}\mid \alpha \le \gamma ({\mathcal {G}})\}$ is countable and thus we can pick an injection $g: \{\alpha \in {\mathbb {O}}\mid \alpha \le \gamma ({\mathcal {G}})\} \rightarrow \mathbb {N}$. Let $\varepsilon _\alpha \overset{{\textrm{def}}}{=}\varepsilon \cdot 2^{-g(\alpha )}$ for every $\alpha \le \gamma ({\mathcal {G}})$.

For every ordinal $\alpha \le \gamma ({\mathcal {G}})$ we consider a restricted subgame ${\mathcal {G}}_\alpha$ of ${\mathcal {G}}$ that is played on the subspace $S_{\le \alpha }$. The objective of ${\mathcal {G}}_\alpha$ is a weighted reachability objective, defined relative to a reward function $f_\alpha$ like in Definition 12. Let $T_\alpha \overset{{\textrm{def}}}{=}S_{< \alpha } \cup T$ be a target set. We consider the weighted reachability objective ${{\mathcal {W}}}_{f_\alpha }$ where $f_\alpha : T_\alpha \rightarrow [0,1]$ with $f_\alpha (s) \overset{{\textrm{def}}}{=}{\texttt{val}_{{\mathcal {G}},\texttt{Reach}(T)}(s)}$.

For every $\alpha \le \gamma ({\mathcal {G}})$ and $s\in S_\alpha$ we show that

$$\begin{aligned} {\texttt{val}_{{\mathcal {G}}_\alpha ,{{\mathcal {W}}}_{f_\alpha }}(s)} = {\texttt{val}_{{\mathcal {G}},\texttt{Reach}(T)}(s)} \end{aligned}$$

(23)

If $\alpha =0$ then the equality (23) holds trivially, since $T_0 = T$ and $f_0(s)=1$ for every $s\in T$.

Now we consider the case of ${\alpha >0}$. For the $\le$ inequality of (23), first assume towards a contradiction that ${\texttt{val}_{{\mathcal {G}}_\alpha ,{{\mathcal {W}}}_{f_\alpha }}(s)} > {\texttt{val}_{{\mathcal {G}},\texttt{Reach}(T)}(s)}$ for some state $s\in S_\alpha$. Let

$$\begin{aligned}\varepsilon ^{\prime } \overset{{\textrm{def}}}{=}({\texttt{val}_{{\mathcal {G}}_\alpha ,{{\mathcal {W}}}_{f_\alpha }}(s)} - {\texttt{val}_{{\mathcal {G}},\texttt{Reach}(T)}(s)})/3 > 0\end{aligned}$$

and $\sigma$ an $\varepsilon ^{\prime }$-optimal Maximizer strategy from $s$ for ${{\mathcal {W}}}_{f_\alpha }$ in ${\mathcal {G}}_\alpha$. We construct a Maximizer strategy $\sigma ^{\prime }$ in ${\mathcal {G}}$ from $s$ as follows. Initially, $\sigma ^{\prime }$ plays like $\sigma$. Then upon reaching some state $s^{\prime }$ in $T_\alpha$ it switches to an $\varepsilon ^{\prime }$-optimal strategy for $\texttt{Reach}(T)$ from $s^{\prime }$. Hence, we get that ${{\mathcal {P}}}_{{\mathcal {G}},s,\sigma ^{\prime },\pi }(\texttt{Reach}(T)) \ge {{\texttt{val}_{{\mathcal {G}}_\alpha ,{{\mathcal {W}}}_{f_\alpha }}(s)} - 2\varepsilon ^{\prime }} > {\texttt{val}_{{\mathcal {G}},\texttt{Reach}(T)}(s)}$, a contradiction. Therefore ${\texttt{val}_{{\mathcal {G}}_\alpha ,{{\mathcal {W}}}_{f_\alpha }}(s)} \le {\texttt{val}_{{\mathcal {G}},\texttt{Reach}(T)}(s)}$.

Towards the $\ge$ inequality of (23), consider an $\varepsilon ^{\prime }$-optimal strategy $\sigma$ from $s$ for $\texttt{Reach}(T)$ in ${\mathcal {G}}$ and apply it in ${\mathcal {G}}_\alpha$. For any Minimizer strategy $\pi$ from $s$, let ${{\mathfrak {R}}}^{s,\sigma ,\pi }$ be the set of plays from $s$ consistent with $\sigma ,\pi$. Let ${{\mathfrak {R}}}^{s,\sigma ,\pi }_{s^{\prime }} \subseteq {{\mathfrak {R}}}^{s,\sigma ,\pi }$ be the subset of plays where $s^{\prime }$ is the first visited state with $I(s^{\prime }) < \alpha$. Since $s\in S_\alpha$ and $\alpha >0$ but $T\subseteq S_0$, every play from $s$ that reaches $T$ must first visit some state $s^{\prime } \in S_{< \alpha }$. Thus these subsets ${{\mathfrak {R}}}^{s,\sigma ,\pi }_{s^{\prime }}$ are a disjoint partition of ${{\mathfrak {R}}}^{s,\sigma ,\pi }$, i.e.,

$$\begin{aligned} {{\mathfrak {R}}}^{s,\sigma ,\pi } = \biguplus _{s^{\prime } \in S_{< \alpha }} {{\mathfrak {R}}}^{s,\sigma ,\pi }_{s^{\prime }} \end{aligned}$$

(24)

Then

$$\begin{aligned}&{\texttt{val}_{{\mathcal {G}}_\alpha ,{{\mathcal {W}}}_{f_\alpha }}(s)} \\&\ge \inf _\pi {\mathcal {E}}_{{\mathcal {G}}_\alpha ,s,\sigma ,\pi }(f_\alpha )&\text{ def. } \text{ of } \text{ value } \\&= \inf _\pi \sum _{s^{\prime } \in S_{< \alpha }} {{\mathcal {P}}}_{{\mathcal {G}},s,\sigma ,\pi }({{\mathfrak {R}}}^{s,\sigma ,\pi }_{s^{\prime }})\cdot f_\alpha (s^{\prime })&\text{ by } (24) \\&= \inf _\pi \sum _{s^{\prime } \in S_{< \alpha }} {{\mathcal {P}}}_{{\mathcal {G}},s,\sigma ,\pi }({{\mathfrak {R}}}^{s,\sigma ,\pi }_{s^{\prime }})\cdot {\texttt{val}_{{\mathcal {G}},\texttt{Reach}(T)}(s^{\prime })}&\text{ def. } \text{ of } f_\alpha \\&\ge \inf _\pi \sum _{s^{\prime } \in S_{< \alpha }} {{\mathcal {P}}}_{{\mathcal {G}},s,\sigma ,\pi }({{\mathfrak {R}}}^{s,\sigma ,\pi }_{s^{\prime }} \cap \texttt{Reach}(T))&\pi \text{ can } \text{ restart } \text{ at } s^{\prime } \\&= \inf _\pi {{\mathcal {P}}}_{{\mathcal {G}},s,\sigma ,\pi }(\texttt{Reach}(T))&\text{ by } (24) \\&\ge {\texttt{val}_{{\mathcal {G}},\texttt{Reach}(T)}(s)} - \varepsilon ^{\prime }.&\text{ def. } \text{ of } \sigma \end{aligned}$$

Since the above holds for every $\varepsilon ^{\prime }>0$, it follows that ${\texttt{val}_{{\mathcal {G}}_\alpha ,{{\mathcal {W}}}_{f_\alpha }}(s)} \ge {\texttt{val}_{{\mathcal {G}},\texttt{Reach}(T)}(s)}$ and we obtain (23).

We now define Maximizer’s public 1-bit strategy $\sigma$ on $S^{\prime }$ in ${\mathcal {G}}$. It uses two memory modes $\{0,1\}$ and $\sigma [\textsf{m}]$ denotes $\sigma$ with current memory mode $\textsf{m}$. The strategy $\sigma$ starts in memory mode 0, i.e., $\sigma = \sigma [0]$ (cf. “Memory-based Strategies” in Sect. 2).

First we consider a slightly modified weighted reachability objective ${{\mathcal {W}}}_{f^{\prime }_\alpha }$ on ${\mathcal {G}}_\alpha$ where $f^{\prime }_\alpha : T_\alpha \cup S^\infty \rightarrow [0,1]$ and $f^{\prime }_\alpha (s) \overset{{\textrm{def}}}{=}{\texttt{val}_{{\mathcal {G}},\texttt{Reach}(T)}(s)}$. This game effectively ends when a state in $S^\infty$ (with an infinite Minimizer action set) is visited, unlike for the ${{\mathcal {W}}}_{f_\alpha }$ objective where the game only ends in the following step when it inevitably (by definition of the ranking function) visits a state in $S_{< \alpha }$. Thus effectively the game ${\mathcal {G}}_\alpha$ with objective ${{\mathcal {W}}}_{f^{\prime }_\alpha }$ has only finite action sets, since it stops before infinite action sets can be used. Therefore, by Theorem 19, there exists a uniformly $(\epsilon _\alpha /2)$-optimal public 1-bit strategy $\sigma ^{\prime }_\alpha$ for Maximizer on ${\mathcal {G}}_\alpha$ with objective ${{\mathcal {W}}}_{f^{\prime }_\alpha }$. We now extend $\sigma ^{\prime }_\alpha$ to a uniformly $\epsilon _\alpha$-optimal public 1-bit strategy $\sigma _\alpha$ for Maximizer on ${\mathcal {G}}_\alpha$ with objective ${{\mathcal {W}}}_{f_\alpha }$. It suffices for Maximizer to play $(\epsilon _\alpha /2)$-optimal in all states $S_\alpha \cap S^\infty$ w.r.t. the one-shot game with reward function $f_\alpha$, regardless of the current memory mode. (These one-shot games with infinite Minimizer action sets and finite Maximizer action sets have a value by [21, Theorem 3], and thus Maximizer can play $\epsilon _\alpha /2$-optimally.) After this one-shot game, the index of any successor state will always be $<\alpha$, by definition of the ranking function.

In the special case of turn-based games, $\sigma ^{\prime }_\alpha$ can be chosen as deterministic by Theorem 19. Moreover, Maximizer is then passive in the one-shot games from states in $S^\infty$, since these states belong to Minimizer who has an infinitely branching choice there. Thus, in turn-based games, $\sigma _\alpha$ is deterministic as well.

The Maximizer strategy $\sigma =\sigma [0]$ starts with memory mode 0. In every state $s$ with $I(s) = \alpha$ the strategy $\sigma$ plays like $\sigma _\alpha$. Whenever we make a step $s\rightarrow s^{\prime }$ with $I(s^{\prime }) < I(s)$ then its sets the memory mode to 0 again. (It is impossible that $I(s^{\prime }) > I(s)$ by the definition of I.) Since all the $\sigma _\alpha$ are public 1-bit strategies, so is $\sigma$. In the special case of turn-based games, the $\sigma _\alpha$ are deterministic and thus also $\sigma$ is deterministic.

We now show by induction on $\alpha$ (for every $\alpha \le \gamma ({\mathcal {G}})$) that $\sigma$ is uniformly $\varepsilon ^{\prime }_\alpha$-optimal in ${\mathcal {G}}$ for objective $\texttt{Reach}(T)$ from every state $s\in S_{\le \alpha }$, where $\varepsilon ^{\prime }_\alpha \overset{{\textrm{def}}}{=}\sum _{\beta \le \alpha } \varepsilon _\beta$.

In the base case of $\alpha =0$ we have ${\mathcal {G}}={\mathcal {G}}_0$ on $S_0$, $\varepsilon ^{\prime }_0 = \varepsilon _0$ and $\sigma = \sigma _0$. Since $T_0 = T\subseteq S_0$, the objectives ${{\mathcal {W}}}_{f_0}$ and $\texttt{Reach}(T)$ coincide. Formally, for any $\sigma ^{\prime },\pi ^{\prime }$, we have

$$\begin{aligned} {\mathcal {E}}_{{\mathcal {G}}_0,s,\sigma ^{\prime },\pi ^{\prime }}(f_0)= {{\mathcal {P}}}_{{\mathcal {G}}_0,s,\sigma ^{\prime },\pi ^{\prime }}(\texttt{Reach}(T)) \end{aligned}$$

(25)

By our construction above, $\sigma _0$ is a uniformly $\epsilon _0$-optimal public 1-bit strategy for Maximizer on ${\mathcal {G}}_0$ with objective ${{\mathcal {W}}}_{f_0}$. Thus, for every $s\in S_0$ we have

$$\begin{aligned}&\inf _\pi {{\mathcal {P}}}_{{\mathcal {G}},s,\sigma ,\pi }(\texttt{Reach}(T)) \\&= \inf _\pi {{\mathcal {P}}}_{{\mathcal {G}}_0,s,\sigma _0,\pi }(\texttt{Reach}(T))&{\mathcal {G}}={\mathcal {G}}_0 \text{ and } \sigma =\sigma _0 \\&= \inf _\pi {\mathcal {E}}_{{\mathcal {G}}_0,s,\sigma _0,\pi }(f_0)&\text{ by } (25) \\&\ge {\texttt{val}_{{\mathcal {G}}_0,{{\mathcal {W}}}_{f_0}}(s)} - \varepsilon _0&\varepsilon _0\text{-optimality } \text{ of } \sigma _0 \\&= {\texttt{val}_{{\mathcal {G}},\texttt{Reach}(T)}(s)} - \varepsilon ^{\prime }_0&\text{ by } (23)\hbox { and }\varepsilon ^{\prime }_0 = \varepsilon _0 \end{aligned}$$

For the induction step let $\alpha >0$. If $s\in S_{< \alpha }$ then the claim holds by induction hypothesis. Now let $s\in S_\alpha$ and $\pi$ be an arbitrary Minimizer strategy. Let ${{\mathfrak {R}}}$ be the set of induced plays from $s$ under $\sigma$ and $\pi$ and ${{\mathfrak {R}}}_{s^{\prime }} \subseteq {{\mathfrak {R}}}$ be the subset of plays where $s^{\prime }$ is the first visited state with $I(s^{\prime }) < \alpha$. Recall that $\sigma$ is a 1-bit strategy with two memory modes $\{0,1\}$. For $\textsf{m}\in \{0,1\}$, $\sigma [\textsf{m}]$ denotes the strategy $\sigma$ with current memory mode $\textsf{m}$. Therefore, $\sigma [\textsf{m}]$ can be applied to start at any state, since it does not depend on the history. The initial memory mode is 0, i.e., $\sigma =\sigma [0]$.

$$\begin{aligned}&{{\mathcal {P}}}_{{\mathcal {G}},s,\sigma ,\pi }(\texttt{Reach}(T))\\&\quad {= {{\mathcal {P}}}_{{\mathcal {G}},s,\sigma [0],\pi }(\texttt{Reach}(T))}\\&\quad \ge \sum _{s^{\prime } \in S_{< \alpha }} {{\mathcal {P}}}_{{\mathcal {G}},s,\sigma [0],\pi }({{\mathfrak {R}}}_{s^{\prime }})\cdot \inf _{\pi ^{\prime }}{{\mathcal {P}}}_{{\mathcal {G}},s^{\prime },{\sigma [0]},\pi ^{\prime }}(\texttt{Reach}(T)) \\&\quad {\text{(the } \text{ memory } \text{ mode } \text{ of } \sigma \hbox { is set to }0\hbox { at }s^{\prime }, \hbox {since }I(s^{\prime })<\alpha )} \\&\quad \ge \sum _{s^{\prime } \in S_{< \alpha }} {{\mathcal {P}}}_{{\mathcal {G}},s,\sigma [0],\pi }({{\mathfrak {R}}}_{s^{\prime }})\cdot ({\texttt{val}_{{\mathcal {G}},\texttt{Reach}(T)}(s^{\prime })} - \varepsilon ^{\prime }_{I(s^{\prime })})&\text{ ind. } \text{ hyp. } \\&\quad = \sum _{s^{\prime } \in S_{< \alpha }} {{\mathcal {P}}}_{{\mathcal {G}},s,\sigma [0],\pi }({{\mathfrak {R}}}_{s^{\prime }})\cdot f_{\alpha }(s^{\prime }) - \sum _{s^{\prime } \in S_{< \alpha }} {{\mathcal {P}}}_{{\mathcal {G}},s,\sigma [0],\pi }({{\mathfrak {R}}}_{s^{\prime }})\cdot \varepsilon ^{\prime }_{I(s^{\prime })}&\text{ def. } f_\alpha \\&\quad \ge {\mathcal {E}}_{{\mathcal {G}}_\alpha ,s,\sigma _\alpha ,\pi }(f_\alpha ) - \sup _{s^{\prime } \in S_{< \alpha }} \varepsilon ^{\prime }_{I(s^{\prime })} \\&\quad \ge {\texttt{val}_{{\mathcal {G}}_\alpha ,{{\mathcal {W}}}_{f_\alpha }}(s)} - \varepsilon _\alpha - \sup _{s^{\prime } \in S_{< \alpha }} \varepsilon ^{\prime }_{I(s^{\prime })}&\sigma _\alpha \hbox { is } \varepsilon _\alpha \hbox {-optimal} \\&\quad = {\texttt{val}_{{\mathcal {G}},\texttt{Reach}(T)}(s)} - \varepsilon _\alpha - \sup _{s^{\prime } \in S_{< \alpha }} \varepsilon ^{\prime }_{I(s^{\prime })}&\hbox {by } (23) \\&\quad = {\texttt{val}_{{\mathcal {G}},\texttt{Reach}(T)}(s)} - (\varepsilon _\alpha + \sup _{s^{\prime } \in S_{< \alpha }} \varepsilon ^{\prime }_{I(s^{\prime })}) \\&\quad \ge {\texttt{val}_{{\mathcal {G}},\texttt{Reach}(T)}(s)} - \varepsilon ^{\prime }_\alpha \end{aligned}$$

Therefore, for every $s\in S^{\prime } = S_{\le \gamma ({\mathcal {G}})}$ our strategy $\sigma$ is $\varepsilon ^{\prime }_{\gamma ({\mathcal {G}})}$-optimal. Moreover, $\varepsilon ^{\prime }_{\gamma ({\mathcal {G}})} = \sum _{\beta \le \gamma ({\mathcal {G}})} \varepsilon _\beta = \sum _{\beta \le \gamma ({\mathcal {G}})} \varepsilon \cdot 2^{-g(\beta )} \le \varepsilon$, since $g: \{\alpha \in {\mathbb {O}}\mid \alpha \le \gamma ({\mathcal {G}})\} \rightarrow \mathbb {N}$ is injective. Thus $\sigma$ is uniformly $\varepsilon$-optimal from $S^{\prime }$ in ${\mathcal {G}}$. $\square$

9 Optimal Maximizer Strategies

In finite turn-based reachability games, there always exist optimal Maximizer strategies, and even optimal memoryless deterministic ones [14, 36, Proposition 5.6.c, Proposition 5.7.c]. This does not carry over to finite concurrent reachability games. E.g., in the snowball game (aka Hide-or-Run game) described in [17, Example 1] and [16, 35], Maximizer does not have any optimal strategy. However, it was recently shown by [8] that, in finite concurrent games with finite action sets, optimal Maximizer strategies, if they exist, can be chosen as memoryless randomized.

In countably infinite reachability games, optimal strategies for Maximizer need not exist in general even if the game is turn-based, in fact not even in countably infinite MDPs that are finitely branching [31, 46].

In this section we study the memory requirements of optimal Maximizer strategies under the condition that such an optimal strategy exists.

If we allow infinite action sets for Minimizer (resp. infinite Minimizer branching in turn-based games) then optimal (and even almost surely winning) Maximizer strategies require infinite memory by Theorem 15. Thus, in the rest of this section, we consider games with finite action sets (resp. turn-based games where the players are finitely branching).

9.1 Turn-Based Games

Here we consider turn-based reachability games where the players have only finitely many choices at each controlled state (i.e., finite action sets). It turns out that the memory requirements of optimal Maximizer strategies, if they exist, also depend on whether random states are infinitely branching or finitely branching, i.e., on whether these distributions have finite support.

If we allow infinite branching at random states, then optimal Maximizer strategies require infinite memory, even with a step counter, by the following example. (A weaker result, without considering the step counter, was shown in [36, Prop. 5.7.b].)

Definition 13

Let ${\mathcal {G}}$ be the following turn-based reachability game depicted in Fig. 9, where Maximizer and Minimizer have only finite branching (i.e., finite action sets), with initial state $s_0$ and target state t. State $s_0$ is a random state and the distribution $p(s_0)$ over its infinitely many successor states is defined as $p(s_0)(s_i^{\prime }) = \frac{1}{2^i}$ for all $i \ge 1$. Further, for every $i\ge 1$ there is a Minimizer-controlled state $s_i^{\prime }$ and a Maximizer-controlled state $s_i^{\prime \prime }$. In $s_i^{\prime }$ Minimizer chooses between moving to state $s_1^{\prime \prime }$ or (via a random state) to the target with probability $1 - \frac{1}{2^i}$ and to a losing sink with probability $\frac{1}{2^i}$. At $s_i^{\prime \prime }$ Maximizer chooses between moving to state $s_{i+1}^{\prime \prime }$ or (via a random state) to target t with probability $1 - \frac{1}{2^i}$ and to a losing sink with probability $\frac{1}{2^i}$.

Fig. 9

The game ${\mathcal {G}}$ from Definition 13. Choices depicted as double arrows (in red) immediately end the game after one round (Color figure online)

Proposition 21

There exists a turn-based reachability game ${\mathcal {G}}$ where Maximizer and Minimizer have only finite branching (i.e., finite action sets) with initial state $s_0$ and objective $\texttt{Reach}(\{t\})$ as in Definition 13, such that

• 1. Maximizer has an optimal strategy from $s_0$.

• 2. Every randomized Maximizer strategy from $s_0$ that uses only a step counter and finite private memory is not optimal.

Proof

We have ${\texttt{val}_{{\mathcal {G}}}(s_i^{\prime \prime } )} = 1$ for all i, and so ${\texttt{val}_{{\mathcal {G}}}(s_i^{\prime })} = 1 - \frac{1}{2^i}$ for all i, and so ${\texttt{val}_{{\mathcal {G}}}(s_0)} = \sum _{i=1}^\infty \frac{1}{2^i} \cdot (1 - \frac{1}{2^i})$. (The latter series equals $\frac{2}{3}$, but that will not be needed.) It follows that the only optimal Minimizer strategy is the one that chooses the red option at any state $s_i^{\prime }$ (where $i \ge 1$). Note that Maximizer does not make any choices if Minimizer plays her optimal strategy.

Towards Item 1, Maximizer’s optimal strategy $\sigma$ from $s_0$ is defined as follows. In plays where the state $s_1^{\prime \prime }$ is not reached, Maximizer does not make any decisions. If $s_1^{\prime \prime }$ is reached, Maximizer considers the history of this play: If Minimizer chose the move from $s_i^{\prime }$ to $s_1^{\prime \prime }$ for some $i\ge 1$, then Maximizer chooses moves (via states $s_2^{\prime \prime } , \ldots , s_{i-1}^{''}$) to state $s_i^{\prime \prime }$ for the same i, and at state $s_i^{\prime \prime }$ he chooses the red option (end the game and win with probability $1-2^{-i}$. Note that in this way Maximizer takes the action that Minimizer refused to take (although it would have been optimal for her) at $s_i^{\prime }$. With this Maximizer strategy $\sigma$, for every Minimizer strategy $\pi$, the probability to reach t equals

$$\begin{aligned} {{\mathcal {P}}}_{{\mathcal {G}},s_0,\sigma ,\pi }(\texttt{Reach}(\{t\})) = \sum _{i=1}^\infty \frac{1}{2^i} \cdot \left( 1 - \frac{1}{2^i}\right) = {\texttt{val}_{{\mathcal {G}}}(s_0)} \end{aligned}$$

meaning that $\sigma$ is optimal.

Towards Item 2, we note that the step counter from $s_0$ is implicit in the states of ${\mathcal {G}}$ (except in the target t and the losing sink state), and thus superfluous for Maximizer strategies. Hence it suffices to prove the property for Maximizer strategies with finite memory. Let $\sigma$ be an FR Maximizer strategy with finitely many memory modes $\{1,\dots ,k\}$. At state $s_1^{\prime \prime }$ this strategy $\sigma$ can base its decision only on the current memory mode $\textsf{m}\in \{1,\dots ,k\}$. Let $X(\textsf{m}) \overset{{\textrm{def}}}{=}\inf _\pi {{\mathcal {P}}}_{{\mathcal {G}},s_1^{\prime \prime } ,\sigma [\textsf{m}],\pi }(\texttt{Reach}(\{t\}))$ be the probability of reaching the target if $\sigma$ is in mode $\textsf{m}$ at state $s_1^{\prime \prime }$. (From state $s_1^{\prime \prime }$ only Maximizer plays, thus Minimizer has no influence.) Since $X(\textsf{m}) < 1$ and the memory is finite, we have $Y \overset{{\textrm{def}}}{=}\max _{\textsf{m}\in \{1,\dots ,k\}} X(\textsf{m}) < 1$. There exists a number i sufficiently large such that $Y < 1 - \frac{1}{2^i}$. Let $\pi$ be a Minimizer strategy from $s_0$ that takes the blue option from $s_i^{\prime }$ to $s_1^{\prime \prime }$, but chooses the red option in all states $s_j^{\prime }$ with $j \ne i$. Then we have

$$\begin{aligned} {{\mathcal {P}}}_{{\mathcal {G}},s_0,\sigma ,\pi }(\texttt{Reach}(\{t\})) \ &\le \ \frac{1}{2^i} Y + \sum _{j \ne i} \frac{1}{2^j} \cdot \left( 1 - \frac{1}{2^j}\right) \\ \ &< \ \sum _{j=1}^\infty \frac{1}{2^j} \cdot \left( 1 - \frac{1}{2^j}\right) \ = \ {\texttt{val}_{{\mathcal {G}}}(s_0)} \end{aligned}$$

and thus $\sigma$ is not optimal. $\square$

Note that the counterexample in Definition 13 and Proposition 21 has some particular properties. Even though the players have finite action sets, the random state $s_0$ is infinitely branching. Moreover, while $s_0$ admits an optimal Maximizer strategy, the same does not hold for all states in the game, e.g., the states $s_i^{\prime \prime }$ have value 1, but do not admit any optimal Maximizer strategy.

The following theorem shows that if we impose any such extra condition on the game (i.e., even all random states are finitely branching, or all states admit an optimal Maximizer strategy) then the memory requirements of optimal Maximizer strategies are somewhat lower. In these cases, just a step counter and 1 bit of public memory are sufficient.

Theorem 22

Let ${\mathcal {G}}$ be a turn-based reachability game with finite action sets with initial state $s_0$ and objective $\texttt{Reach}(\{t\})$ such that at least one of the following two conditions is satisfied:

• (A) ${\mathcal {G}}$ is finitely branching (at all states, including the random states), or

• (B) Every state in ${\mathcal {G}}$ admits an optimal Maximizer strategy.

Suppose that Maximizer has an optimal strategy $\sigma$, i.e., ${{\mathcal {P}}}_{{\mathcal {G}},s_0,\sigma ,\pi }(\texttt{Reach}(\{t\})) \ge {\texttt{val}_{{\mathcal {G}}}(s_0)}$ holds for all Minimizer strategies $\pi$. Then Maximizer also has a deterministic such strategy that uses 1 bit of public memory and a step counter.

In the proof of Theorem 22 we will use the following version of the optional stopping theorem.

Theorem 23

(Optional Stopping Theorem) Suppose $X_0, X_1, \ldots$ is a submartingale adapted to a filtration $\mathcal {F}_0, \mathcal {F}_1, \ldots$; i.e., $X_n \le {\mathcal {E}}(X_{n+1} \mid \mathcal {F}_n)$ for all n. Suppose further that there is $c \in \mathbb {R}$ with $|X_n| \le c$ almost surely for all n. Then the limit $X_\infty \overset{{\textrm{def}}}{=}\lim _{n \rightarrow \infty } X_n$ exists almost surely. Let $\tau _1, \tau _2$ be stopping times with $\tau _1 \le \tau _2$ almost surely (where $\tau _1 = \infty$ and $\tau _2 = \infty$ may have positive probability). Then we have $X_{\tau _1} \le {\mathcal {E}}(X_{\tau _2} \mid \mathcal {F}_{\tau _1})$ almost surely.

Proof

The proof is immediate from [42, Proposition IV-5-24, Corollary IV-2-25]. $\square$

Notice that if, in addition to the other preconditions of Theorem 23, the submartingale $X_0, X_1, \ldots$ is a martingale, i.e., $X_n = {\mathcal {E}}(X_{n+1} \mid \mathcal {F}_n)$ for all n, then it follows, by considering $Y_n \overset{{\textrm{def}}}{=}-X_n$ for all n, that we have $X_{\tau _1} = {\mathcal {E}}(X_{\tau _2} \mid \mathcal {F}_{\tau _1})$ almost surely.

For the proof of Theorem 22 we also use [30, Theorem 5(2)], slightly generalized as the following lemma.

Lemma 24

Let ${\mathcal {G}}$ be a turn-based reachability game, such that Minimizer has finite action sets and Minimizer does not have any value-increasing transitions; i.e., for all transitions $s {\longrightarrow } s^{\prime }$ with $s \in S_\Diamond$ we have ${\texttt{val}_{{\mathcal {G}}}(s)} = {\texttt{val}_{{\mathcal {G}}}(s^{\prime })}$.

Then there exists some MD Maximizer strategy that is optimal from every state that admits an optimal strategy.

Proof

First we consider the special case where ${\mathcal {G}}$ is finitely branching (i.e., at every state, not just at the Minimizer-controlled states). The statement then follows from [30, Thm. 5(2)], but there it is stated only for a single initial state that admits an optimal strategy. Therefore, denote by $S_\textit{opt}\subseteq S$ the set of states that admit an optimal strategy. Add a fresh random state, say $s_0$, such that the support of $P(s_0)$ equals $S_\textit{opt}$. This might require infinite branching, but one can easily encode infinite branching of random states into finite branching in the case of reachability objectives, using a “ladder” gadget of fresh intermediate finitely branching random states. Since every state in $S_\textit{opt}$ admits an optimal strategy, the new state $s_0$ admits an optimal strategy. The mentioned result [30, Thm. 5(2)] applied to $s_0$ gives an MD Maximizer strategy $\sigma$ that is optimal starting from $s_0$. But then $\sigma$ must be optimal from every state in $S_\textit{opt}$.

The above result can be generalized to allow infinitely branching random states by the same encoding as above, using a “ladder” of fresh intermediate finitely branching random states. Similarly, infinitely branching Maximizer states can also be encoded into a “ladder” of fresh intermediate finitely branching Maximizer states. This encoding gives Maximizer the additional option to remain on the ladder forever, but this is not a problem. Since the target is not on the ladder, staying on the ladder forever would be losing for Maximizer. Finally, since we are dealing with MD strategies, the strategies can be carried back from the finitely branching game that uses the “ladder” gadget encoding to the infinitely branching original game. (The same would not hold for Markov strategies in general, since the encoding does not preserve path lengths. Also it is not possible to encode infinite Minimizer branching in this way, because Minimizer could spuriously win by staying on the ladder gadget forever.) $\square$

Proof of Theorem 22

Denote by $\bar{{\mathcal {G}}}$ the game obtained from ${\mathcal {G}}$ by deleting Minimizer’s value-increasing transitions, i.e., those transitions $s {\longrightarrow } s^{\prime }$ with $s \in S_\Diamond$ and ${\texttt{val}_{{\mathcal {G}}}(s)} < {\texttt{val}_{{\mathcal {G}}}(s^{\prime })}$. As ${\mathcal {G}}$ has finite Minimizer action sets, each Minimizer state still has at least one outgoing transition in $\bar{{\mathcal {G}}}$, and all states have the same value in ${\mathcal {G}}$ and $\bar{{\mathcal {G}}}$, and all states that admit an optimal Maximizer strategy in ${\mathcal {G}}$ admit an optimal Maximizer strategy in $\bar{{\mathcal {G}}}$ and vice versa. Denote by $S_\textit{opt}\subseteq S$ the set of states that admit an optimal Maximizer strategy. By Lemma 24, there exists an MD Maximizer strategy $\bar{\sigma }$ in $\bar{{\mathcal {G}}}$ that is optimal from every state in $S_\textit{opt}$. Thus, for any $s \in S_\Box \cap S_\textit{opt}$, the transition $s {\longrightarrow } s^{\prime }$ that $\bar{\sigma }$ prescribes preserves the value, i.e., $\texttt{val}(s) = \texttt{val}(s^{\prime })$, and $s^{\prime } \in S_\textit{opt}$. By assumption, $s_0 \in S_\textit{opt}$. Thus, $\bar{\sigma }$ is optimal from $s_0$ in $\bar{{\mathcal {G}}}$. Hence, $\bar{\sigma }$ is optimal from $s_0$ in ${\mathcal {G}}$ if Minimizer never chooses a value-increasing transition $s {\longrightarrow } s^{\prime }$ with $s \in S_\Diamond$ and $\texttt{val}(s) < \texttt{val}(s^{\prime })$. We view such a transition as a gift from Minimizer of size $\texttt{val}(s^{\prime }) - \texttt{val}(s) >0$.

Let us now sketch a first draft of a Maximizer strategy that is optimal from $s_0$ in ${\mathcal {G}}$.

• Play the strategy $\bar{\sigma }$ until Minimizer gives a gift of, say, $\varepsilon >0$. Use the 1 bit of public memory to record the fact that a gift has been given.

• Then play an $\varepsilon$-optimal MD strategy, which exists by Lemma 5.

The problem with this draft strategy is that storing the size of Minimizer’s gift $\varepsilon$ appears to require infinite memory, not just 1 bit, because $\varepsilon$ could be arbitrarily small. Moreover, the different $\varepsilon$-optimal MD strategies might prescribe different choices for different $\varepsilon$.

Therefore, we use the step counter to deduce a lower bound on any nonzero gift that Minimizer may have given up to that point in time.

Let R(i) be the set of states that could be reached from $s_0$ with nonzero probability under any pair of strategies within $\le i$ steps.

Under condition (A), R(i) is finite for every $i \in \mathbb {N}$, because ${\mathcal {G}}$ is finitely branching. Here we just define the finite set $S(i) \overset{{\textrm{def}}}{=}R(i)$. (Under condition (B), $S(i)$ will be a subset instead.)

Under condition (B), even though both players have finite action sets, random states are still allowed to be infinitely branching. Thus R(i) could be infinite. However, since the players have finite action sets, for every time $i \ge 0$ and $\delta >0$, there exists a finite subset of states $S(i,\delta ) \subseteq R(i)$ such that under any pair of strategies $\sigma ,\pi$ from $s_0$, the probability of ever being outside $S(i,\delta )$ at any time $t \le i$ is upper-bounded by $\delta$. I.e.,

$$\begin{aligned} \forall \sigma ,\pi \ {{\mathcal {P}}}_{{\mathcal {G}},s_0,\sigma ,\pi }(\texttt{Reach}_{i}(S\setminus S(i,\delta ))) \le \delta . \end{aligned}$$

(26)

Since only random states can be infinitely branching, $S(i,\delta )$ can easily be defined by cutting infinite tails off distributions, e.g., losing $\le \delta \cdot 2^{-(t+1)}$ in the t-th round. Additionally, we can define these sets such that they are monotone increasing in i. That is, $S(i,\delta ) \subseteq S(i+1,\delta )$ for all $i \in \mathbb {N}$. We then define $S(i) \overset{{\textrm{def}}}{=}S(i,2^{-i})$, and these sets are also monotone increasing in i.

Let $\varepsilon _i >0$ denote the size of the smallest nonzero gift that Minimizer can give from any state inside $S(i)$, i.e.,

$$\begin{aligned} \varepsilon _i \overset{{\textrm{def}}}{=}\min \{(\texttt{val}(s^{\prime }) - \texttt{val}(s)) > 0 \mid s\rightarrow s^{\prime }\ \wedge \ s\in S(i) \cap S_\Diamond \}. \end{aligned}$$

We have $\varepsilon _i >0$, because $S(i)$ is finite and Minimizer has finite action sets. Moreover, the $\varepsilon _i$ are monotone decreasing in i, because the sets $S(i)$ are monotone increasing.

Under condition (A), $\varepsilon _i$ is a lower bound on all possible Minimizer gifts until time i, while under condition (B) it is only a lower bound on most of Minimizer’s possible gifts until time i (namely on those originating from a state in the subset $S(i)$).

However, we will show that, under condition (B), it is safe for Maximizer to ignore gifts from Minimizer if gifts are given only finitely often. (This does not hold under condition (A).) So our Maximizer strategy will ignore gifts from Minimizer at time i if the gift originates from a state outside $S(i)$. Indeed, except for a nullset of plays, Minimizer cannot give a gift at infinitely many times i at states outside of $S(i)$, because it is so unlikely to be outside $S(i)$ at time i. Consider an arbitrary pair of strategies $\sigma , \pi$ and let ${{\mathfrak {R}}}\subseteq s_0S^\omega$ be the set of plays $s_0 s_1 \cdots s_i s_{i+1} \dots$ from $s_0$ where $s_i \notin S(i)$ for infinitely many $i \in \mathbb {N}$.

Claim 25

$\forall \sigma ,\pi \ {{\mathcal {P}}}_{{\mathcal {G}},s_0,\sigma ,\pi }({{\mathfrak {R}}}) = 0$.

Proof

Consider a number $k \in \mathbb {N}$. For every play $s_0 s_1 \cdots s_i s_{i+1} \dots$ in ${{\mathfrak {R}}}$ there exists a number $k^{\prime } > k$ such that $s_{k^{\prime }} \notin S(k^{\prime })$. Let ${{\mathfrak {R}}}_{k^{\prime }} \subseteq {{\mathfrak {R}}}$ be the subset of plays where $k^{\prime }$ is the smallest number $> k$ where $s_{k^{\prime }} \notin S(k^{\prime })$. Then ${{\mathfrak {R}}}$ can be partitioned as ${{\mathfrak {R}}}= \uplus _{k^{\prime } > k} {{\mathfrak {R}}}_{k^{\prime }}$. However, by $S(k^{\prime }) = S(k^{\prime },2^{-k^{\prime }})$ and (26), we have ${{\mathcal {P}}}_{{\mathcal {G}},s_0,\sigma ,\pi }({{\mathfrak {R}}}_{k^{\prime }}) \le 2^{-k^{\prime }}$ and thus ${{\mathcal {P}}}_{{\mathcal {G}},s_0,\sigma ,\pi }({{\mathfrak {R}}}) \le \sum _{k^{\prime } >k} 2^{-k^{\prime }} \le 2^{-k}$. Since this holds for every $k \in \mathbb {N}$, the result follows. $\square$

Another problem with the draft strategy is that the $\varepsilon$-optimal MD strategy from Lemma 5 is $\varepsilon$-optimal only from a finite set of initial states (uniformly $\varepsilon$-optimal memoryless strategies do not always exist; see Theorem 7).

Therefore, we partition time into infinitely many finite phases $\Phi _1 = \{1, \ldots , t_1\}, \Phi _2 = \{t_1+1, \ldots , t_2\}, \Phi _3 = \{t_2+1,\ldots , t_3\}$, etc., and refer to $\Phi _1, \Phi _3, \ldots$ as odd and to $\Phi _2, \Phi _4, \ldots$ as even phases. The length of the phases is determined inductively; see below. Let $S_i$ with $S(t_{i-1}+1) \subseteq S_i \subseteq R(t_{i-1}+1)$ be a sufficiently large finite subset of the states that could be reached by the beginning of phase i such that the following condition holds: Under any pair of strategies, for any $s \in S(t_{i-1})$, conditioned under the event that s has been visited at some time $t \le t_{i-1}$, the probability of being inside $S_i$ at time $t_{i-1}+1$ is $\ge 1-(\varepsilon _{t_{i-1}}/2)$. Under condition (A), we can simply take $S_i \overset{{\textrm{def}}}{=}R(t_{i-1}+1)$, since that is finite. Under condition (B), since $S(t_{i-1})$ is finite and the players have finite action sets, we obtain a suitable $S_i$ by cutting suitably small tails off distributions. Note that $S_i$ does not depend on the pair of strategies. The lengths of the phases $\Phi _1, \Phi _2, \ldots$ are determined as follows.

• (LO) Each odd phase $\Phi _i$ is long enough (i.e., $t_i$ is chosen large enough) so that we have $\inf _\pi {{\mathcal {P}}}_{\bar{{\mathcal {G}}},s,\bar{\sigma },\pi }(\texttt{Reach}_{\Phi _i}(\{t\})) \ge \frac{\texttt{val}(s)}{2}$ for all $s \in S_i \cap S_\textit{opt}$, where we write $\texttt{Reach}_{\Phi _i}(\{t\})$ for the event that t is reached within $t_i - t_{i-1}$ steps, i.e., the length of phase $\Phi _i$. That is, if $\Phi _i$ begins at a state $s \in S_i \cap S_\textit{opt}$ and if Minimizer does not give a gift during $\Phi _i$, the Maximizer strategy $\bar{\sigma }$ realizes at least half of the value of s already within $\Phi _i$. The length of $\Phi _i$ can be chosen finite, because $S_i$ is finite and Minimizer has finite action sets.

• (LE) For each even phase $\Phi _i$, by Lemma 5, there is an MD Maximizer strategy $\sigma _i$ so that $\Phi _i$ can be made long enough so that we have $\inf _\pi {{\mathcal {P}}}_{{\mathcal {G}},s,\sigma _i,\pi }(\texttt{Reach}_{\Phi _i}(\{t\})) \ge \texttt{val}(s) - (\varepsilon _{t_{i-1}}/2)$ for all $s \in S_i$. Again the length of $\Phi _i$ can be chosen finite, because $S_i$ is finite and Minimizer has finite action sets. If Minimizer has given a gift from a state $s \in S(t_{i-1})$ in the previous phase, then this gift will be $\ge \varepsilon _{t_{i-1}}$. Moreover, by the definition of $S_i$, we will then be in a state in $S_i$ at the beginning of phase $\Phi _i$ with very high conditional probability $\ge 1-(\varepsilon _{t_{i-1}}/2)$ (or even surely under condition (A)). Thus, in the phase $\Phi _i$, the Maximizer strategy $\sigma _i$ can undercut Minimizer’s gift and realizes most of the value of s already within $\Phi _i$.

We now define a deterministic Maximizer strategy $\sigma$ from $s_0$ that uses a step counter and 1 bit of public memory. Later we show that $\sigma$ is optimal from $s_0$. Strategy $\sigma$ uses two memory modes, $\textsf{m}_0$ and $\textsf{m}_1$, where $\textsf{m}_0$ is the initial mode. Strategy $\sigma$ updates the mode as follows.

• (U1) While in $\textsf{m}_0$ and in an odd phase $\Phi _i$: if Minimizer gives a gift from a state $s \in S(t_i)$ switch to $\textsf{m}_1$. I.e., Maximizer uses the bit to remember that Minimizer has given a gift and will undercut it in the next even phase. The size of the gift is lower-bounded by $\varepsilon _{t_i} >0$. Note that Maximizer ignores all Minimizer gifts from states outside $S(t_i)$ (which can only happen under condition (B)).

• (U2) While in $\textsf{m}_0$ and upon entering an odd phase: if the new state does not admit an optimal strategy, switch to $\textsf{m}_1$. This can only happen if Minimizer has given a gift in some previous even phase (and not at all under condition (B)). If Minimizer had given a gift in some previous odd phase then the memory mode would already be $\textsf{m}_1$. If Minimizer has never given a gift then the current state would still admit an optimal strategy, since Maximizer never voluntarily leaves $S_\textit{opt}$.

Note that once the mode has been switched to $\textsf{m}_1$ it is never switched back to $\textsf{m}_0$. Strategy $\sigma$ plays as follows.

• (P1) While in $\textsf{m}_0$ and in $S_\textit{opt}$: play $\bar{\sigma }$. This keeps the game in $S_\textit{opt}$, at least until possibly Minimizer gives a gift. (Under condition (A), the play might leave $S_\textit{opt}$ after a Minimizer gift. Under condition (B), all plays stay inside $S_\textit{opt}$.)

• (P2) While in $\textsf{m}_0$ and in a state $s \in S_\Box \setminus S_\textit{opt}$: choose a value-preserving transition, i.e., $s {\longrightarrow } s^{\prime }$ with $\texttt{val}(s) = \texttt{val}(s^{\prime })$. Such a transition must exist, due to the finite Maximizer branching in ${\mathcal {G}}$.

• (P3) While in $\textsf{m}_1$ during an odd phase: choose a value-preserving transition, i.e., $s {\longrightarrow } s^{\prime }$ with $\texttt{val}(s) = \texttt{val}(s^{\prime })$. Such a transition must exist, due to the finite Maximizer branching in ${\mathcal {G}}$. Intuitively, Maximizer has recorded the fact that Minimizer has given a non-ignored gift, but waits until the next even phase to capitalize on it.

• (P4) While in $\textsf{m}_1$ during an even phase $\Phi _i$: play the MD strategy $\sigma _i$ from the definition (LE) of the even phase $\Phi _i$. It follows from (U1) and (U2) that $\sigma _i$ has been played from the beginning of $\Phi _i$. (Here Maximizer undercuts Minimizer’s previous gift.)

Note that not all possible gifts by Minimizer are detected, i.e., result in a switch to memory mode $\textsf{m}_1$. First, gifts in phase $\Phi _i$ from states outside $S(t_i)$ are ignored. Moreover, Minimizer could give a gift in an even phase while staying in $S_\textit{opt}$, or the game might just temporarily leave $S_\textit{opt}$ but move back to $S_\textit{opt}$ before the next odd phase, thus avoiding rule (U2). However, this is not a problem for Maximizer: Since the game returns to $S_\textit{opt}$ before the next odd phase, Maximizer is fine to just continue playing $\bar{\sigma }$ by (P1), because he will realize at least half of the value (at least of most states; those in $S_i$) during the next odd phase.

To show that $\sigma$ is optimal from $s_0$, fix an arbitrary Minimizer strategy $\pi$ for the rest of this proof, and assume that the target t is a sink. Let us write ${{\mathcal {P}}}$ for ${{\mathcal {P}}}_{{\mathcal {G}},s_0,\sigma ,\pi }$ and ${\mathcal {E}}$ for the associated expectation. We need to show that $\texttt{val}(s_0) \le {{\mathcal {P}}}(\texttt{Reach}(\{t\}))$.

For a play $s_0 s_1 \cdots \in \{s_0\} S^\omega$, define a random variable $\tau _1$, taking values in $\mathbb {N}\cup \{\infty \}$, such that $\tau _1 = \infty$ if Minimizer never gives a gift or all Minimizer gifts are ignored, and $\tau _1 = j < \infty$ if $s_j {\longrightarrow } s_{j+1}$ is the first non-ignored Minimizer gift. Also define a random variable $\tau _2$, taking values in $\mathbb {N}\cup \{\infty \}$, such that $\tau _2 = \infty$ if no mode switch from $\textsf{m}_0$ to $\textsf{m}_1$ ever occurs, and $\tau _2 = k < \infty$ if k is the beginning of the even phase following a mode switch from $\textsf{m}_0$ to $\textsf{m}_1$. The random variables $\tau _1+1, \tau _2$ are both stopping times. As long as Minimizer does not give any non-ignored gift, Maximizer plays $\bar{\sigma }$ and, by (P1), keeps the game in $S_\textit{opt}$, and thus, by (U1) and (U2), the mode remains $\textsf{m}_0$. Hence, $\tau _1 < \tau _2 \le \infty$ or $\tau _1 = \infty = \tau _2$.

Further, define random variables $V_0, V_1, \ldots$ and $W_0, W_1, \ldots$, taking values in [0, 1], by $V_i \overset{{\textrm{def}}}{=}\texttt{val}(s_i)$ for all $i \le \tau _1$, and $V_i = V_{\tau _1}$ for all $i \ge \tau _1$, and $W_i \overset{{\textrm{def}}}{=}\texttt{val}(s_i)$ for all $i \le \tau _2$, and $W_i = W_{\tau _2}$ for all $i \ge \tau _2$. By (P1), (P2) and (P3), Maximizer preserves the value in each of his transitions, at least until $\tau _2$. Thus, $W_0, W_1, \ldots$ is a submartingale. Minimizer cannot decrease the value, but might increase it when giving a gift. Under condition (B), Minimizer might give ignored gifts before $\tau _1$. Thus, $V_0, V_1, \ldots$ is a submartingale. Under condition (A), Minimizer gifts are never ignored, and thus $V_0, V_1, \ldots$ is even a martingale. By Theorem 23, $V_0, V_1, \ldots$ and $W_0, W_1, \ldots$ converge almost surely to random variables, which we may call, without risk of confusion, $V_{\tau _1}$ (which equals $V_{\tau _1+1}$) and $W_{\tau _2}$, respectively. Again by Theorem 23, we have

$$\begin{aligned} \texttt{val}(s_0) \ = \ {\mathcal {E}}V_0 \ {\le } \ {\mathcal {E}}V_{\tau _1+1} \ = \ {\mathcal {E}}V_{\tau _1}\,. \end{aligned}$$

(27)

Now consider the event $\tau _2=\infty$. By (U1) and (U2), Minimizer does not give any non-ignored gift in any odd phase and the state at the beginning of every odd phase admits an optimal strategy for Maximizer. This property is ensured by each of the conditions (A) and (B). Under condition (A), Minimizer gifts are never ignored. Condition (B) ensures that the game is always in a state that admits an optimal strategy for Maximizer, and thus in particular at the beginning of every odd phase.

Under condition (A), Minimizer will not give any gift at all in any odd phase, and thus by (P1) Maximizer plays the strategy $\bar{\sigma }$ undisturbed in every odd phase. Since (A) implies that $S_i \overset{{\textrm{def}}}{=}R(t_{i-1}+1)$, definition (LO) ensures that in every odd phase Maximizer realizes at least half of the value of the state at the beginning of the phase.

Under condition (B), Minimizer might still give ignored gifts in odd phases, which could disrupt the attainment of Maximizer’s strategy $\bar{\sigma }$. However, by Claim 25, except in a nullset of plays, there are only finitely many ignored Minimizer gifts in a play. I.e., almost every play is eventually undisturbed by ignored Minimizer gifts in odd phases. Moreover, no part of the value is lost, since $V_0, V_1, \ldots$ is a submartingale, and every state admits an optimal strategy for Maximizer. Hence, except in a nullset, by (LO), Maximizer eventually realizes at least half of the value of each state $s \in S_i$ that it is in at the beginning of every odd phase $\Phi _i$. Finally, since $S(t_{i-1}+1) \subseteq S_i$, the probability of being in a state $s \in S_i$ at the beginning of an odd phase $\Phi _i$ converges to 1 as $i \rightarrow \infty$.

Therefore, under either condition (A) or (B), we have that

$$\begin{aligned} {{\mathcal {P}}}(\{\tau _2 = \infty \} \cap \{W_\infty > 0\} \setminus \texttt{Reach}(\{t\})) \ =\ 0\,. \end{aligned}$$

(28)

Thus,

$$\begin{aligned} \begin{aligned} {\mathcal {E}}(V_\infty \mid \tau _1 = \infty ) \ &\le \ {{\mathcal {P}}}(V_\infty> 0 \mid \tau _1 = \infty ) \\&= \ {{\mathcal {P}}}(W_\infty > 0 \mid \tau _1 = \infty ) \ \le \ {{\mathcal {P}}}(\texttt{Reach}(\{t\}) \mid \tau _1 = \infty )\,. \end{aligned} \end{aligned}$$

Hence, continuing (27),

$$\begin{aligned} \begin{aligned} \texttt{val}(s_0) \ &{\le } \ {{\mathcal {P}}}(\tau _1 = \infty ) \cdot {\mathcal {E}}(V_\infty \mid \tau _1 = \infty ) + \sum _{0 \le j< \infty } {{\mathcal {P}}}(\tau _1 = j) \cdot {\mathcal {E}}(V_j \mid \tau _1 = j) \\&\le \ {{\mathcal {P}}}(\texttt{Reach}(\{t\}), \tau _1 = \infty ) + \sum _{0 \le j < \infty } {{\mathcal {P}}}(\tau _1 = j) \cdot {\mathcal {E}}(V_j \mid \tau _1 = j)\,, \end{aligned} \end{aligned}$$

(29)

where here and henceforth, to avoid clutter, we may write “,” for the intersection of events. Let $j \in \mathbb {N}$. It follows from the definitions of $\tau _1$ and $\varepsilon _j$ that on $\tau _1=j$ we have $W_{j+1} \ge W_j + \varepsilon _j$. Thus,

$$\begin{aligned} \begin{aligned}&{\mathcal {E}}(V_j \mid \tau _1=j) \\&=\ {\mathcal {E}}(W_j \mid \tau _1=j) & \text {by def. of }V_j, W_j \\&\le \ -\varepsilon _j + {\mathcal {E}}(W_{j+1} \mid \tau _1 = j) & \text {as explained above}\\&\le \ -\varepsilon _j + {\mathcal {E}}(W_{\tau _2} \mid \tau _1 = j) & \text {Theorem}~23 \\&=\ -\varepsilon _j + {{\mathcal {P}}}(\tau _2=\infty \mid \tau _1=j) \cdot {\mathcal {E}}(W_\infty \mid \tau _1=j, \tau _2=\infty ) + \\&\quad + \! \sum _{j+1 \le k < \infty } {{\mathcal {P}}}(\tau _2=k \mid \tau _1=j) \cdot {\mathcal {E}}(W_k \mid \tau _1=j, \tau _2=k)\,. \end{aligned} \end{aligned}$$

(30)

Concerning the first expectation, we have

$$\begin{aligned} \begin{aligned} {\mathcal {E}}(W_\infty \mid \tau _1=j, \tau _2=\infty ) \ &\le \ {{\mathcal {P}}}(W_\infty >0 \mid \tau _1=j, \tau _2=\infty ) \\ \ &\le \ {{\mathcal {P}}}(\texttt{Reach}(\{t\}) \mid \tau _1=j, \tau _2=\infty ) & \text {by }(28). \end{aligned} \end{aligned}$$

(31)

Concerning expectations under the sum, let $k > j$, and denote by H(j, k) the set of histories $s_0 \cdots s_k \in \{s_0\} S^k$ such that for some (hence, all) extension(s) $r = s_0 \cdots s_k s_{k+1} \cdots$ we have $\tau _1(r) = j$ and $\tau _2(r) = k$. Then we have

$$\begin{aligned}&{{\mathcal {P}}}(\tau _1=j, \tau _2=k) \cdot {\mathcal {E}}(W_k \mid \tau _1=j, \tau _2=k) \\&\quad =\ \sum _{h = s_0 \cdots s_k \in H(j,k)} {{\mathcal {P}}}(\{h\} S^\omega ) \cdot \texttt{val}(s_k) & \text {by the defs.} \\&\quad \le \ \sum _{h \in H(j,k)} {{\mathcal {P}}}(\{h\} S^\omega ) \cdot \big ({{\mathcal {P}}}(\texttt{Reach}(\{t\}) \mid \{h\}S^\omega )+ {\varepsilon _{k-1}/2 + \varepsilon _{k-1}/2}\big ) & \text {by (P4), (LE)} \\&\text {Minimizer's gift at time }j\text { happens in some odd phase }\Phi _i\text { and is }\ge \varepsilon _{t_i}.\\&\text {The next even phase begins at time }k=t_i +1.\\&\text {Each of the two errors in this even phase are }\le \varepsilon _{t_i}/2=\varepsilon _{k-1}/2.\\&\quad \le \ \sum _{h \in H(j,k)} {{\mathcal {P}}}(\{h\} S^\omega ) \cdot \big ({{\mathcal {P}}}(\texttt{Reach}(\{t\}) \mid \{h\} S^\omega ) + \varepsilon _j\big ) & \varepsilon _{k-1} \le \varepsilon _j \\&\quad =\ {{\mathcal {P}}}(\tau _1=j, \tau _2=k) \cdot \big ( {{\mathcal {P}}}(\texttt{Reach}(\{t\}) \mid \tau _1=j, \tau _2=k) + \varepsilon _j \big )\,. \end{aligned}$$

Thus,

$$\begin{aligned} \begin{aligned} \sum _{j+1 \le k< \infty }&{{\mathcal {P}}}(\tau _2=k \mid \tau _1=j) \cdot {\mathcal {E}}(W_k \mid \tau _1=j, \tau _2=k) \\&\le \ \sum _{j+1 \le k< \infty } {{\mathcal {P}}}(\tau _2=k \mid \tau _1=j) \cdot \big ( {{\mathcal {P}}}(\texttt{Reach}(\{t\}) \mid \tau _1=j, \tau _2=k) + \varepsilon _j \big ) \\&\le \ {{\mathcal {P}}}(\texttt{Reach}(\{t\})\mid \tau _2 < \infty ,\tau _1=j) + \varepsilon _j\,. \end{aligned} \end{aligned}$$

Combined with Eqs. (30) and (31) this gives

$$\begin{aligned} \begin{aligned} {\mathcal {E}}(V_j \mid \tau _1=j)&~\le ~{{\mathcal {P}}}(\texttt{Reach}(\{t\}) \mid \tau _2=\infty , \tau _1=j) + {{\mathcal {P}}}(\texttt{Reach}(\{t\})\mid \tau _2 < \infty , \tau _1=j) \\&\le \ {{\mathcal {P}}}(\texttt{Reach}(\{t\}) \mid \tau _1=j)\,. \end{aligned} \end{aligned}$$

Combined with (29), we obtain

$$\begin{aligned} \texttt{val}(s_0) \ \le \ {{\mathcal {P}}}(\texttt{Reach}(\{t\}), \tau _1=\infty ) + {{\mathcal {P}}}(\texttt{Reach}(\{t\}), \tau _1<\infty ) \ = \ {{\mathcal {P}}}(\texttt{Reach}(\{t\}))\,, \end{aligned}$$

as required. $\square$

The following example shows a corresponding lower bound to Theorem 22, i.e., even if both conditions (A) and (B) hold, just a step counter does not suffice for optimal Maximizer strategies.

Fig. 10

Finitely branching turn-based reachability game ${\mathcal {G}}$, where optimal Maximizer strategies cannot be Markov. For clarity, we have drawn several copies of the target state t. The number $y_i$ is defined to be $\frac{1}{2} - \frac{1}{2^{i+1}}$; see Proposition 26

Proposition 26

There exists a finitely branching turn-based reachability game ${\mathcal {G}}$ with initial state $u_1$ and objective $\texttt{Reach}(\{t\})$, as shown in Fig. 10, such that

• 1. From every state in ${\mathcal {G}}$, Maximizer has an optimal strategy.

• 2. Every randomized Maximizer strategy from $u_1$ that uses only a step counter and no memory is not optimal.

Proof

Consider first a version of ${\mathcal {G}}$, say ${\mathcal {G}}^{\prime }$, in which the only outgoing transition from the states $u_i$ is the horizontal one, shown in red in Fig. 10. I.e., in ${\mathcal {G}}^{\prime }$ Minimizer does not have any choice and thus ${\mathcal {G}}^{\prime }$ can be regarded as a maximizing MDP. Let $\sigma$ be the MD Maximizer strategy that chooses at all states $s_i$ the horizontal outgoing transition, shown in red in Fig. 10. Then we have

$$\begin{aligned} {{\mathcal {P}}}_{{\mathcal {G}}^{\prime },u_i,\sigma }(\texttt{Reach}(\{t\})) \ = \ \sum _{j=0}^\infty \frac{1}{2^j} \cdot \left( \frac{1}{4} - \frac{1}{2^{i+j+1}}\right) \ = \ \frac{1}{2} - \frac{1}{3} \cdot \frac{1}{2^{i-1}}\,. \end{aligned}$$

In ${\mathcal {G}}^{\prime }$, strategy $\sigma$ is optimal for Maximizer everywhere. Indeed the only alternative is to take the vertical outgoing transition at some state $s_i$, which is suboptimal by the following. Consider a strategy $\sigma ^{\prime }$ that chooses at state $s_i$ the vertical outgoing transition. Then we have

$$\begin{aligned} \begin{aligned} {{\mathcal {P}}}_{{\mathcal {G}}^{\prime },s_i,\sigma ^{\prime }}(\texttt{Reach}(\{t\}))&= \frac{1}{2} - \frac{1}{2^{i+1}}\\&< \frac{1}{2} - \frac{1}{3} \cdot \frac{1}{2^i} \\&= {{\mathcal {P}}}_{{\mathcal {G}}^{\prime },u_{i+1},\sigma }(\texttt{Reach}(\{t\}))\\&= {{\mathcal {P}}}_{{\mathcal {G}}^{\prime },s_{i},\sigma }(\texttt{Reach}(\{t\}))\\&\le {\texttt{val}_{{\mathcal {G}}^{\prime }}(s_i)} \,. \end{aligned} \end{aligned}$$

(32)

Consider now the original game ${\mathcal {G}}$ as shown in Fig. 10. Since Minimizer has additional options, the value at each state is not larger than at the corresponding state in ${\mathcal {G}}^{\prime }$.

However, we show that, in ${\mathcal {G}}$, Maximizer still has an optimal strategy ${\hat{\sigma }}$ from every state $s$. It suffices to show this property for states $s= u_k$ for any $k \ge 1$. At states $s= s_k$, the optimal move is always to go right to $u_{k+1}$, because the vertical transition is suboptimal by (32), and at random states no decision can be made until the next step (or ever).

We show that, starting from $u_k$, strategy ${\hat{\sigma }}$ attains the same value ($\frac{1}{2} - \frac{1}{3} \cdot \frac{1}{2^{k-1}}$) in ${\mathcal {G}}$ as in ${\mathcal {G}}^{\prime }$. Namely, define ${\hat{\sigma }}$ so that as long as Minimizer chooses the horizontal (red) outgoing transitions at $u_i$, Maximizer chooses the horizontal (red) outgoing transition at $s_i$; once Minimizer deviates and chooses the non-horizontal outgoing transition at, say, $u_i$, then Maximizer responds by choosing the vertical outgoing transition at $s_i$. (The strategy ${\hat{\sigma }}$ is a deterministic public 1-bit strategy, but we do not need that here.)

Intuitively, for Minimizer a “deviation”, i.e., choosing a non-horizontal outgoing transition, is value-increasing and thus suboptimal. But she may try to lay a trap for Maximizer and trick him into visiting all states $u_i, s_i$. To stop this from happening, Maximizer, using ${\hat{\sigma }}$, responds to a Minimizer deviation by also deviating, i.e., by choosing a vertical outgoing transition. Such a deviation is suboptimal for him, but the game is constructed so that a Maximizer deviation decreases the value less than Minimizer has previously increased it by her deviation. In effect, with ${\hat{\sigma }}$, Maximizer attains as much as in ${\mathcal {G}}^{\prime }$ if Minimizer never deviates; if Minimizer deviates, Maximizer attains slightly more than in ${\mathcal {G}}^{\prime }$. Thus, ${\hat{\sigma }}$ is optimal.

Formally, let $\pi$ be any Minimizer strategy. Denote by $D_i$ the event that Minimizer deviates at $u_i$ (for some $i \ge k$), i.e., chooses the non-horizontal outgoing transition at $u_i$. Since the Maximizer strategy ${\hat{\sigma }}$ responds by choosing the vertical outgoing transition at $s_i$, we have

$$\begin{aligned} {{\mathcal {P}}}_{{\mathcal {G}},{u_k},{\hat{\sigma }},\pi }(\texttt{Reach}(\{t\}) \mid D_i) \ = \ \frac{1}{2} - \frac{1}{2^{i+1}} \ > \ \frac{1}{2} - \frac{1}{3} \cdot \frac{1}{2^{i-1}} \ = \ {{\mathcal {P}}}_{{\mathcal {G}}^{\prime },u_i,\sigma }(\texttt{Reach}(\{t\})) \,, \end{aligned}$$

i.e., by deviating at $u_i$, Minimizer increases the probability of reaching t compared to her not deviating at $u_i$ or thereafter (which corresponds to playing in ${\mathcal {G}}^{\prime }$). We have already argued that ${{\mathcal {P}}}_{{\mathcal {G}}^{\prime },u_i,\sigma }(\texttt{Reach}(\{t\})) = {\texttt{val}_{{\mathcal {G}}^{\prime }}(u_i)} \ge {\texttt{val}_{{\mathcal {G}}}(u_i)}$. It follows that ${\hat{\sigma }}$ is optimal, which concludes the proof of Item 1.

Towards Item 2, note that in ${\mathcal {G}}$ the step counter from $u_1$ is implicit in the current state. In particular, starting from $u_1$, if a state $s_i$ is visited then it is visited as the 3i-th state. It follows that a step counter is not useful for Maximizer strategies. Thus, it suffices to show that no memoryless strategy for Maximizer is optimal. Let $\sigma$ be any memoryless Maximizer strategy. If $\sigma$ chooses at every $s_i$ the horizontal outgoing transition, the probability of reaching t is zero if Minimizer never chooses the horizontal outgoing transition at any $u_i$; thus, $\sigma$ is not optimal. Hence, we can assume that there is a state $s_i$ at which $\sigma$ chooses with a positive probability the vertical outgoing transition. Denote by $E_i$ the event that Maximizer chooses the vertical outgoing transition at $s_i$. Let $\pi$ be the Minimizer strategy that at all $u_j$ chooses the horizontal outgoing transition. Recall that $\pi$ is optimal for Minimizer everywhere. Similarly to (32) above, we have

$$\begin{aligned} {{\mathcal {P}}}_{{\mathcal {G}},u_1,\sigma ,\pi }(\texttt{Reach}(\{t\}) \mid E_i) \ = \ \frac{1}{2} - \frac{1}{2^{i+1}} \ < \ \frac{1}{2} - \frac{1}{3} \cdot \frac{1}{2^i} \ = \ {\texttt{val}_{{\mathcal {G}}}(u_{i+1})} \ = \ {\texttt{val}_{{\mathcal {G}}}(s_i)}\,. \end{aligned}$$

Thus, $\sigma$ is not optimal. As $\sigma$ was chosen arbitrarily, Maximizer does not have an optimal memoryless strategy. This proves Item 2. $\square$

In the example in Fig. 10, subgame-perfect Maximizer strategies cannot guarantee any positive probability of reaching the target state, because they would always choose the step $s_i \rightarrow u_{i+1}$ for all $i \in \mathbb {N}$. Thus an optimal Maximizer strategy may need to take steps that are locally sub-optimal in subgames.

However, in those turn-based reachability games with finite Minimizer action sets where optimal subgame-perfect Maximizer strategies do exist, there also exist such strategies that are memoryless and deterministic by [30, Theorem 5].

9.2 Concurrent Games

The lower bounds for turn-based games from Sect. 9.1 immediately carry over to concurrent games. It is an open question whether the upper bounds carry over. We conjecture that a suitably adapted version of Theorem 22 might hold for concurrent games (e.g., condition (A) might be generalized by requiring that all probability distributions have finite support). However, such a generalization faces several obstacles. In concurrent games, it is more difficult to define what it means for Minimizer to “give a gift”, and how to define a restricted version of the game where such gift-giving is forbidden. Also one would need a suitably generalized version of Lemma 24.

A special case of optimal Maximizer strategies are those that win almost surely. Here no memory is needed at all, and these strategies can even be made uniform. The following upper bound for concurrent games trivially carries over to turn-based games (with finite action sets).

Theorem 27

Given a concurrent game with finite action sets and a reachability objective, there exists some randomized memoryless Maximizer strategy that is almost surely winning from every state that admits an almost surely winning strategy (i.e., the same strategy works from all these states).

Proof

Let ${\mathcal {G}}$ be a concurrent game with state space $S$, and let $\texttt{Reach}(T)$ be a reachability objective.

Without restriction, we can assume that all states in $S$ admit an almost surely winning strategy. Otherwise, we consider the subgame ${\mathcal {G}}^{\prime }$ obtained by restricting ${\mathcal {G}}$ to $S^{\prime }$, ie the game on the subgraph induced by $S^{\prime }$, where $S^{\prime } \subseteq S$ is the subset of states that admit an almost surely winning strategy in ${\mathcal {G}}$. Then all states in $S^{\prime }$ admit an almost surely winning strategy in ${\mathcal {G}}^{\prime }$. (Note that this construction of ${\mathcal {G}}^{\prime }$ would not work if we replaced the “almost surely winning” condition by the weaker condition of “having value 1”.)

In order to construct a memoryless Maximizer strategy $\hat{\sigma }$ that wins almost surely from every state, we inductively define a sequence of modified games ${\mathcal {G}}_i$ in which the strategy of Maximizer is already fixed on a finite subset of the state space, and where all states in ${\mathcal {G}}_i$ still admit an almost surely winning strategy. Fix an enumeration $s_1, s_2, \ldots$ of $S$ in which very state $s$ appears infinitely often.

For the base case we have ${\mathcal {G}}_0 \overset{{\textrm{def}}}{=}{\mathcal {G}}$ and the property holds by our assumption on ${\mathcal {G}}$.

Given ${\mathcal {G}}_i$, we construct ${\mathcal {G}}_{i+1}$ as follows. We use Lemma 5 to get a memoryless strategy $\sigma _i$ and a finite subset of states $R_i$ s.t. $\inf _\pi {{\mathcal {P}}}_{{\mathcal {G}}_i,s_i,\sigma _i,\pi }(\texttt{Reach}_{R_i}(T)) \ge {\texttt{val}_{{\mathcal {G}}_i}(s_i)} - 2^{-i} = 1 - 2^{-i}$.

Let ${\mathcal {G}}_i^{\prime }$ be the subgame of ${\mathcal {G}}_i$ that is restricted to $R_i$ and further let

$$\begin{aligned} R_i^{\prime } \overset{{\textrm{def}}}{=}\{s\in R_i \mid \inf _\pi {{\mathcal {P}}}_{{\mathcal {G}}_i^{\prime },s,\sigma _i,\pi }(\texttt{Reach}_{R_i}(T)) >0\} \end{aligned}$$

be the subset of states in $R_i$ where $\sigma _i$ has strictly positive attainment in ${\mathcal {G}}_i^{\prime }$. In particular, we have $s_i \in R_i^{\prime }$ for all $i \ge 1$. Since $R_i^{\prime }$ is finite, we have

$$\begin{aligned} \lambda _i \overset{{\textrm{def}}}{=}\min _{s\in R_i^{\prime }} \inf _\pi {{\mathcal {P}}}_{{\mathcal {G}}_i^{\prime },s,\sigma _i,\pi }(\texttt{Reach}_{R_i}(T))> 0. \end{aligned}$$

We now construct ${\mathcal {G}}_{i+1}$ by modifying ${\mathcal {G}}_i$ as follows. For every state $s\in R_i^{\prime }$ we fix Maximizer’s (randomized) action according to $\sigma _i$. Then $\inf _\pi {{\mathcal {P}}}_{{\mathcal {G}}_{i+1},s_i,\sigma ,\pi }(\texttt{Reach}(T)) \ge 1 - 2^{-i}$ and $\inf _\pi {{\mathcal {P}}}_{{\mathcal {G}}_{i+1},s,\sigma ,\pi }(\texttt{Reach}_{R_i^{\prime }}(T)) \ge \lambda _i$ for all $s\in R_i^{\prime }$ and all $\sigma \in \Sigma _{{\mathcal {G}}_{i+1}}$ (and thus in particular for the strategy $\hat{\sigma }$ that we will construct).

Now we show that in ${\mathcal {G}}_{i+1}$ all states $s$ still have an almost surely winning strategy.

Let $\sigma$ be an a.s. winning Maximizer strategy from $s$ in ${\mathcal {G}}_i$, which exists by the induction hypothesis. We now define an a.s. winning Maximizer strategy $\sigma ^{\prime }$ from $s$ in ${\mathcal {G}}_{i+1}$.

If the game does not enter $R_i^{\prime }$ then $\sigma ^{\prime }$ plays exactly as $\sigma$ (which is possible since outside $R_i^{\prime }$ no Maximizer actions have been fixed). If the game enters $R_i^{\prime }$ then it will reach the target within $R_i^{\prime }$ (i.e., before exiting $R_i^{\prime }$, if ever) with probability $\ge \lambda _i >0$. Plays that do not stay inside $R_i^{\prime }$ then exit $R_i^{\prime }$ at some state $s^{\prime } \notin R_i^{\prime }$. Then, from $s^{\prime }$, $\sigma ^{\prime }$ plays an a.s. winning strategy w.r.t. ${\mathcal {G}}_i$ (which exists by the induction hypothesis).

Now we show that $\sigma ^{\prime }$ wins almost surely from $s$ in ${\mathcal {G}}_{i+1}$. The plays from $s$ can be partitioned into the following three subsets. The first set of plays visit $R_i^{\prime }$ only finitely often and eventually forever follow an a.s. winning strategy outside of $R_i^{\prime }$ and thus (except for a nullset) eventually reach the target. The second set of plays enter $R_i^{\prime }$ infinitely often and the third set of plays eventually forever remain in $R_i^{\prime }$. For plays in both the second and third sets, the probability of reaching the target from the current state does not converge to zero, since $\lambda _i >0$. Hence, by Lévy’s 0-1 law, the probability of reaching the target must converge to 1, and thus (except for a nullset) the plays in the second and third set also reach the target. Therefore $\sigma ^{\prime }$ almost surely wins from $s$ in ${\mathcal {G}}_{i+1}$.

Finally, we can construct the memoryless Maximizer strategy $\hat{\sigma }$. Since our enumeration of the states $s_1, s_2, \dots$ contains every state $s\in S$ infinitely often, in particular it contains every state in $S$. Moreover, $s_i \in R_i^{\prime }$ for every $i \ge 1$. Thus, in the limit of the games ${\mathcal {G}}_\infty$, all Maximizer choices are fixed. The memoryless Maximizer strategy $\hat{\sigma }$ plays according to these fixed choices, i.e., it plays like $\sigma _i$ at state $s_i$ for all $i \in \mathbb {N}$. Note that if $s_i \in R_i^{\prime }$ then, for all $j > i$, the mixed action of $\sigma _j$ at $s_i$ coincides with the mixed action of $\sigma _i$ at $s_i$, because $\sigma _j$ is defined in a game where Maximizer’s mixed action in $s_i$ is already fixed.

Since $\hat{\sigma }$ plays like $\sigma _i$ inside $R_i^{\prime }$, we obtain $\inf _\pi {{\mathcal {P}}}_{{\mathcal {G}},s_i,\hat{\sigma },\pi }(\texttt{Reach}(T)) \ge \inf _\pi {{\mathcal {P}}}_{{\mathcal {G}}_i,s_i,\sigma _i,\pi } (\texttt{Reach}_{R_i}(T)) \ge 1 - 2^{-i}$ for all $i \in \mathbb {N}$. Let $s \in S$. Since our enumeration of the states contains every state infinitely often, $s=s_i$ holds for infinitely many i, and thus we obtain $\inf _\pi {{\mathcal {P}}}_{{\mathcal {G}},s,\hat{\sigma },\pi }(\texttt{Reach}(T)) = 1$ as required. $\square$

10 Minimizer Strategies

In the previous sections we have considered the strategy complexity of Maximizer’s strategies. In this section we complete the picture of the strategy complexity of Minimizer. In reachability games, Minimizer strategies are generally simpler than Maximizer strategies, because they do not need to make progress towards the target. By [44, Thm. 1], we already know that Minimizer always has optimal (and thus $\varepsilon$-optimal) MR strategies in concurrent reachability games with finite action sets. In [10, Thm. 3.1], this result is strengthened in the context of finitely branching turn-based games, where it is shown that Minimizer always has MD such strategies. In the sequel, as depicted in Table 2, we close the remaining gaps in the theory by studying the strategy complexity of Minimizer in infinitely branching turn-based reachability games. We prove that $\varepsilon$-optimal Minimizer strategies in infinitely branching turn-based reachability games can be chosen as deterministic and Markov (Theorem 29). In contrast, optimal Minimizer strategies need not always exist in infinitely branching turn-based reachability games. However, even if optimal Minimizer strategies do exist, a step counter plus finite private memory is not sufficient in general (Proposition 30).

We begin by considering games on acyclic graphs. Memoryless strategies in acyclic games yield Markov strategies in general games, since an encoded step counter makes the graph acyclic. In fact, the following result about acyclic games is slightly more general, since not all acyclic graphs yield an implicit step counter, i.e., the same state might be reached via paths of different lengths.

Lemma 28

For every acyclic turn-based reachability game ${\mathcal {G}}=(S,(S_\Box ,S_\Diamond ,S_\ocircle ),{\longrightarrow },P)$, reachability target $T\subseteq S$ and every $0<\varepsilon <1$ there exists an MD Minimizer strategy $\pi$ which satisfies, for every state $s_0\in S$ and every Maximizer strategy $\sigma$, that ${{\mathcal {P}}}_{{\mathcal {G}},s_0,\sigma ,\pi }(\texttt{Reach}(T)) \le {\texttt{val}_{{\mathcal {G}},\texttt{Reach}(T)}(s_0)} (1+ \varepsilon )$. Hence, acyclic turn-based reachability games admit uniformly $\varepsilon$-optimal MD strategies for Minimizer.

Proof

Let us shortly write $\texttt{val}(s)={\texttt{val}_{{\mathcal {G}},\texttt{Reach}(T)}(s)}$ for the value of a state $s$ and let $\iota :S\rightarrow \mathbb {N}\setminus \{0\}$ be an enumeration of the state space starting at 1. Define $\pi$ as the MD Minimizer strategy that, at any state $s\in S_\Diamond$, picks a successor $s^{\prime }$ such that

$$\begin{aligned} \texttt{val}(s^{\prime })\quad \le \quad \texttt{val}(s) (1+ \ln (1+\varepsilon ) 2^{-\iota (s)}). \end{aligned}$$

To show that this strategy $\pi$ satisfies the claim we (over)estimate the error by $L(s) \overset{{\textrm{def}}}{=}\prod _{s^{\prime }\in \textit{Post}^*(s)}(1+\ln (1+\varepsilon )2^{-\iota (s^{\prime })})$ where $\textit{Post}^*(s)\subseteq S$ is the set of states reachable from state $s\in S$ (under any pair of strategies). Notice that this guarantees that

$$\begin{aligned} \nonumber 1 < L(s)&\le \prod _{i>0}\left( 1+\ln (1+\varepsilon )2^{-i}\right) \\&\le \exp \left( \sum _{i>0}\ln (1+\varepsilon )2^{-i}\right) \\&\le \exp (\ln (1+\varepsilon )) = 1+\varepsilon \nonumber \end{aligned}$$

(33)

where the third inequality uses that $1+x \le \exp (x)$.

Let $\sigma$ be an arbitrary Maximizer strategy. For this pair $\sigma ,\pi$ of strategies let’s consider plays $(X_{i})_{i\ge 0}$ that start in $s_0\in S$ and proceed according to $\sigma ,\pi$, and let ${\mathcal {E}}_{{\mathcal {G}},s_0,\sigma ,\pi }$ be the expectation with respect to ${{\mathcal {P}}}_{{\mathcal {G}},s_0,\sigma ,\pi }$.

An induction on n using our choice of strategy gives, for every initial state $s_0\in S$, that

$$\begin{aligned} {\mathcal {E}}_{{\mathcal {G}},s_0,\sigma ,\pi }(\texttt{val}(X_{n})) \le \texttt{val}(s_0)L(s_0). \end{aligned}$$

(34)

Indeed, this trivially holds for $n=0$ as ${\mathcal {E}}_{{\mathcal {G}},s_0,\sigma ,\pi }(\texttt{val}(X_{0}))=\texttt{val}(s_0)$ and $L(s_0) > 1$. For the induction step there are three cases.

Case 1: $s_0\in S_\Diamond$ and $\pi (s_0)=s$. Let $\sigma {[s_0\rightarrow s]}$ denote the Maximizer strategy from s that behaves just like $\sigma$ does after observing the first step, i.e., satisfies $\sigma {[s_0\rightarrow s]}(sh) = \sigma (s_0sh)$ for all suffix histories $h\in S^*$. Then

$$\begin{aligned} {\mathcal {E}}_{{\mathcal {G}},s_0,\sigma ,\pi }(\texttt{val}(X_{n+1}))&= {\mathcal {E}}_{{\mathcal {G}},s,\sigma {[s_0\rightarrow s],\pi }} (\texttt{val}(X_{n}))\\&\le \texttt{val}(s)L(s)&\text {ind. hyp.}\\&\le \texttt{val}(s_0) \left( 1+\ln (1+\varepsilon )2^{-\iota (s_0)}\right) L(s)&\text {def. of }\pi \\&\le \texttt{val}(s_0)L(s_0)&\text {acyclicity; def. of } L(s_0). \end{aligned}$$

Case 2: $s_0\in S_\Box$. Again, for any state s let $\sigma {[s_0\rightarrow s]}$ denote the suffix strategy consistent with $\sigma$ after the first step. Then

$$\begin{aligned} {\mathcal {E}}_{{\mathcal {G}},s_0,\sigma ,\pi }(\texttt{val}(X_{n+1}))&= \sum _{s \in S}\sigma (s_0)(s) \cdot {\mathcal {E}}_{{\mathcal {G}},s,\sigma {[s_0\rightarrow s]},\pi }(\texttt{val}(X_{n})) \\&\le \sum _{s \in S}\sigma (s_0)(s) \cdot \texttt{val}(s)L(s) \\&\le \sum _{s \in S}\sigma (s_0)(s) \cdot \texttt{val}(s) \left( 1+\ln (1+\varepsilon )2^{-\iota (s_0)}\right) L(s) \\&\le \sum _{s\in S}\sigma (s_0)(s) \cdot \texttt{val}(s) L(s_0) \\&\le \texttt{val}(s_0)L(s_0), \end{aligned}$$

where the first inequality holds by induction hypothesis, the second holds because $1<{(1+\ln (1+\varepsilon )2^{-\iota (s_0)})}$, and the third is acyclicity and the definition of $L(s_0)$.

Case 3: $s_0\in S_\ocircle$ is analogous to case 2, with the only difference that the initial successor distribution is $P(s_0)$, the one fixed by the game, instead of $\sigma (s_0)$ and the last inequality becomes an equality.

Together with the observation (Eq. 33) that $L(s_0)\le (1+\varepsilon )$ for every $s_0$, we derive that

$$\begin{aligned} \liminf _{n\rightarrow \infty } {\mathcal {E}}_{{\mathcal {G}},s_0,\sigma ,\pi }(\texttt{val}(X_{n)}) \le \texttt{val}(s_0)(1+\varepsilon ). \end{aligned}$$

(35)

Finally, to show the claim, let $[X_{n} \in T] : S^\omega \rightarrow \{0,1\}$ be the random variable that indicates that the nth state is in $T$. Note that $[X_{n} \in T] \le \texttt{val}(X_{n)}$ because target states have value 1. Recall that $\texttt{Reach}_{n}(T)$ denotes the objective of visiting $T$ within at most n rounds of the game. We conclude that

$$\begin{aligned} {{\mathcal {P}}}_{{\mathcal {G}},s_0,\sigma , \pi }(\texttt{Reach}(T))&=\quad {{\mathcal {P}}}_{{\mathcal {G}},s_0,\sigma , \pi }\left( \bigcup _{i=0}^\infty {\texttt{Reach}_{i}(T)}\right) & \text {semantics of}~\texttt{Reach}(T) \\&=\quad \mathop {\lim }\limits _{n\rightarrow \infty } {{\mathcal {P}}}_{{\mathcal {G}},s_0,\sigma , \pi }\left( \bigcup _{i=0}^n\texttt{Reach}_{i}(T) \right) & \text {continuity of measures} \\&=\quad \mathop {\lim }\limits _{n\rightarrow \infty } {{\mathcal {P}}}_{{\mathcal {G}},s_0,\sigma , \pi }\left( \texttt{Reach}_{n}(T)\right) & T\text { is a sink} \\&=\quad \mathop {\lim }\limits _{n\rightarrow \infty } {\mathcal {E}}_{{\mathcal {G}},s_0,\sigma , \pi }([X_{n} \in T]) & \text {definition of }[X_{n} \in T] \\&\le \quad \liminf _{n\rightarrow \infty } {\mathcal {E}}_{{\mathcal {G}},s_0,\sigma , \pi }(\texttt{val}(X_{n)}) & \text {as }[X_{n} \in T] \le \texttt{val}(X_{n)}\\&\le \quad \texttt{val}(s_0)(1+\varepsilon ) & \text {by Eq.}~(35). \ \end{aligned}$$

$\square$

Theorem 29

Turn-based games, even infinitely branching ones, with reachability objective admit uniformly $\varepsilon$-optimal strategies for Minimizer that are deterministic and Markov.

Proof

For a given game ${\mathcal {G}}=(S,(S_\Box ,S_\Diamond ,S_\ocircle ),{\longrightarrow },P)$ and reachability target $T\subseteq S$, one can construct the acyclic game that encodes the stage (clock value) into the states: $\mathcal {G}^{\prime }=(S,(S_\Box ^{\prime },S_\Diamond ^{\prime },S_\ocircle ^{\prime }),{\longrightarrow }^{\prime },P^{\prime })$ where $S^{\prime }=S\times \mathbb {N}$, $S_\Box ^{\prime }=S_\Box \times \mathbb {N}$, $S_\Diamond ^{\prime }=S_\Diamond \times \mathbb {N}$ $S_\ocircle ^{\prime }=S_\ocircle \times \mathbb {N}$, and for all $i\in \mathbb {N}$, $(s,i){\longrightarrow }^{\prime }(t,i+1) \iff s{\longrightarrow }t$ and $P((s,i))((t,i+1)) = P(s)(t)$.

Every Markov strategy in $\mathcal {G}$ uniquely gives rise to a memoryless strategy in $\mathcal {G}^{\prime }$ and vice versa. The claim now follows from Lemma 28. $\square$

In infinitely branching turn-based reachability games, optimal Minimizer strategies need not exist [31]. When they do exist, they may need infinite memory. We slightly improve the result of [36, Proposition 5.6.(a)] by showing that even a step counter does not help Minimizer.

Fig. 11

The game ${\mathcal {G}}$ from Definition 14

Definition 14

We define an infinitely branching turn-based reachability game ${\mathcal {G}}$ with initial state $s_0$ and target state t. See Fig. 11 for a depiction. Let $s_0$ be Maximizer-controlled. We have transitions $s_0 \rightarrow s_i^{\prime }$ for all $i \ge 1$. All states $s_i^{\prime }$ are random states with $P(s_i^{\prime })(t) = 1/2 - 2^{-i}$ and $P(s_i^{\prime })(u) = 1/2 + 2^{-i}$. The state u is Minimizer-controlled with transitions $u \rightarrow s_i^{\prime \prime }$ for all $i \ge 1$. All states $s_i^{\prime \prime }$ are random states with $P(s_i^{\prime \prime } )(t) = 2^{-i}$ and $P(s_i^{\prime \prime } )(b) = 1-2^{-i}$ for a losing sink state b.

Proposition 30

There exists an infinitely branching turn-based reachability game ${\mathcal {G}}$ with initial state $s_0$ and objective $\texttt{Reach}(\{t\})$ as in Definition 14, such that

• 1. Minimizer has an optimal strategy from $s_0$.

• 2. Every randomized Minimizer strategy from $s_0$ that uses only a step counter and finite private memory is not optimal.

Proof

Towards Item 1, we note that ${\texttt{val}_{{\mathcal {G}}}(u)}=0$ and thus ${\texttt{val}_{{\mathcal {G}}}(s_0)}=1/2$. Minimizer’s optimal strategy $\pi$ from $s_0$ is defined as follows. In plays where the state u is not reached, Minimizer does not make any decisions. If state u is reached, Minimizer considers the history of this play: If Maximizer made the step $s_0 \rightarrow s_i^{\prime }$ for some $i\ge 1$, then Minimizer plays $u \rightarrow s_i^{\prime \prime }$ for the same i. Now we show that $\pi$ is optimal for Minimizer from $s_0$. Let $\sigma$ be an arbitrary Maximizer strategy from $s_0$ and let $x_i$ be the probability that $\sigma$ chooses the step $s_0 \rightarrow s_i^{\prime }$. This must be a distribution, i.e., $\sum _{i\ge 1} x_i = 1$. Then we have

$$\begin{aligned} \begin{aligned} {{\mathcal {P}}}_{{\mathcal {G}},s_0,\sigma ,\pi }(\texttt{Reach}(\{t\}))&= \sum _{i\ge 1} x_i((1/2 - 2^{-i}) + (1/2 + 2^{-i})2^{-i})\\&\le \sum _{i\ge 1} x_i(1/2) = 1/2 = {\texttt{val}_{{\mathcal {G}}}(s_0)} \end{aligned} \end{aligned}$$

(36)

as required.

Towards Item 2, we note that the step counter from $s_0$ is implicit in the states of ${\mathcal {G}}$, and thus superfluous for Minimizer strategies. Hence it suffices to prove the property for Minimizer strategies with finite memory. Let $\pi$ be an FR Minimizer strategy with finitely many memory modes $\{1,\dots ,k\}$. In state u this strategy $\pi$ can base its decision only on the current memory mode $\textsf{m}\in \{1,\dots ,k\}$. Let $X(\textsf{m}) \overset{{\textrm{def}}}{=}{{\mathcal {P}}}_{{\mathcal {G}},u,\sigma ,\pi [\textsf{m}]}(\texttt{Reach}(\{t\}))$ be the probability of reaching the target if $\pi$ is in mode $\textsf{m}$ at state u. (From state u only Minimizer plays, thus Maximizer has no influence.) Since $X(\textsf{m}) >0$ and the memory is finite, we have $Y \overset{{\textrm{def}}}{=}\min _{\textsf{m}\in \{1,\dots ,k\}} X(\textsf{m}) > 0$. There exists a number i sufficiently large such that $2^{-i} < Y/2$. Let $\sigma$ be a Maximizer strategy from $s_0$ that chooses the transition $s_0 \rightarrow s_i^{\prime }$. Then we have

$$\begin{aligned} {{\mathcal {P}}}_{{\mathcal {G}},s_0,\sigma ,\pi }(\texttt{Reach}(\{t\})) \ge (1/2 - 2^{-i}) + (1/2 + 2^{-i})Y > 1/2 = {\texttt{val}_{{\mathcal {G}}}(s_0)} \end{aligned}$$

and thus $\pi$ is not optimal. $\square$

11 Conclusion and Outlook

Our results closed many gaps about the strategy complexity of reachability games; cf. Table 1 and Table 2. To summarize our main contributions, we return to the open questions raised in Sect. 1, which are now answered.

Q1.:

The negative result of [44] can be strengthened. There are no uniformly $\varepsilon$-optimal memoryless Maximizer strategies in countably infinite reachability games, not even if the game is turn-based and finitely branching; cf. Theorem 7. This highlights the difference between (turn-based) 2-player stochastic games and MDPs. In the latter, there do exist uniformly $\varepsilon$-optimal memoryless strategies for reachability [46].

Q2.:

In concurrent reachability games with finite action sets, uniformly $\varepsilon$-optimal Maximizer strategies exist and they require only 1 bit of public memory. In turn-based games, these strategies can even be chosen as deterministic. See Theorem 6.

Q3.:

If Minimizer is allowed infinite action sets then reachability games are much more difficult for Maximizer. Even in turn-based reachability games with infinitely branching Minimizer states, Maximizer strategies based on a step counter plus arbitrary finite private memory are insufficient. In general, they cannot guarantee any positive attainment against all Minimizer strategies, even if the start state has value 1. In fact, the counterexample in Theorem 15 satisfies the even stronger property that all states in it admit an almost surely winning Maximizer strategy.

Open questions for further work concern the strategy complexity of optimal Maximizer strategies, where they exist. In general, a step counter plus finite private memory is not sufficient for optimal Maximizer strategies, even in turn-based reachability games, by Proposition 21. However, under certain mild conditions, a step counter plus 1 bit of public memory suffices for optimal Maximizer strategies in turn-based reachability games, by Theorem 22. A similar theorem might hold for concurrent reachability games with finite action sets under suitably adapted conditions.

Technical Lemmas

The following inequality is due to Weierstrass.

Proposition 31

([12, p. 104–105]) Given an infinite sequence of real numbers $a_n$ with $0 \le a_n \le 1$, the following holds for all $n\in \mathbb {N}$.

$$\begin{aligned} \prod _{k=1}^n (1-a_k) \le \frac{1}{1+ \sum _{k=1}^n a_k} \end{aligned}$$

Proof

By induction on n. In case $n=1$ we have $(1-a_1)(1+a_1)={(1-a_1^2)}\le 1$ as required. For the induction hypothesis we assume

$$\begin{aligned} \prod _{k=1}^n (1-a_k)\left( 1+\sum _{k=1}^n a_k\right) \le 1 \end{aligned}$$

For the induction step we have

$$\begin{aligned} \prod _{k=1}^{n+1} (1-a_k)\left( 1+\sum _{k=1}^{n+1} a_k\right)&= (1-a_{n+1})\prod _{k=1}^n (1-a_k)\left( \left( 1+\sum _{k=1}^n a_k\right) +a_{n+1}\right) \\&\le (1-a_{n+1})\left( 1 + a_{n+1}\prod _{k=1}^n (1-a_k)\right) \\&\le (1-a_{n+1})(1 + a_{n+1})\\&= (1-a_{n+1}^2) \le 1 \ \end{aligned}$$

$\square$

Proposition 32

Given an infinite sequence of real numbers $a_n$ with $0 \le a_n < 1$, we have

$$\begin{aligned} \prod _{n=1}^\infty (1-a_n) > 0 \quad \Leftrightarrow \quad \sum _{n=1}^\infty a_n < \infty . \end{aligned}$$

and the “$\Rightarrow$” implication holds even for the weaker assumption $0 \le a_n \le 1$.

Proof

If $a_n=1$ for any n then the “$\Rightarrow$” implication is vacuously true, but the “$\Leftarrow$” implication does not hold in general. In the following we assume $0 \le a_n < 1$.

In the case where $a_n$ does not converge to zero, the property is trivial. In the case where $a_n \rightarrow 0$, it is shown by taking the logarithm of the product and using the limit comparison test as follows.

Taking the logarithm of the product gives the series

$$\begin{aligned} \sum _{n=1}^{\infty } \ln (1 - a_n) \end{aligned}$$

whose convergence (to a finite number $\le 0$) is equivalent to the positivity of the product. It is also equivalent to the convergence (to a number $\ge 0$) of its negation $\sum _{n=1}^{\infty } -\ln (1 - a_n)$. But observe that (by L’Hôpital’s rule)

$$\begin{aligned} \lim _{x \rightarrow 0} \frac{-\ln (1-x)}{x} = 1. \end{aligned}$$

Since $a_n \rightarrow 0$ we have

$$\begin{aligned} \lim _{n \rightarrow \infty } \frac{-\ln (1-a_n)}{a_n} = 1. \end{aligned}$$

By the limit comparison test, the series $\sum _{n=1}^{\infty } -\ln (1 - a_n)$ converges if and only if the series $\sum _{n=1}^\infty a_n$ converges. $\square$

Proposition 33

Given an infinite sequence of real numbers $a_n$ with $0 \le a_n \le 1$,

$$\begin{aligned} \prod _{n=1}^\infty a_n> 0 \quad \Rightarrow \quad \forall \varepsilon >0\,\exists N.\, \prod _{n=N}^\infty a_n \ge (1-\varepsilon ). \end{aligned}$$

Proof

If there is n with $a_n=0$ or if $\varepsilon \ge 1$ then the property is vacuously true. In the following we assume $a_n >0$ and $\varepsilon < 1$. Since $\prod _{n=1}^\infty a_n > 0$, by taking the logarithm we obtain $\sum _{n=1}^\infty \ln (a_n) > -\infty$. Thus for every $\delta >0$ there exists an N s.t. $\sum _{n=N}^\infty \ln (a_n) \ge -\delta$. By exponentiation we obtain $\prod _{n=N}^\infty a_n \ge \exp (-\delta )$. By picking $\delta = -\ln (1-\varepsilon )$ the result follows. $\square$