Keywords

1 Introduction

Tendering is a commonly used process where government and public institutions grant bidding opportunities for large projects with a defined bidding target and a defined bidding date. Pharmaceutical tenders represent the process when a number of manufacturers offering the price of similar or comparable products to win the privilege to gain the monopsony in the market. During the confidential bidding process, the winning party is offered the opportunity to sell the bidding product at a pre-defined price for a fixed period of time, which reflects the nature of tendering as “winner takes it all”, as stated by Petrou (2016). From the contract or mechanism design theory perspective, tendering process can be linked to auction theory in view of competition. Thus auction theory is widely used in modeling the price competition during the tendering process. From the social welfare perspective, according to Simoens and Cheung (2019), the tendering process encourages competition and thus cuts the price, which is significantly beneficial to yield short-term savings, especially under extreme budget constraints. In the literature of pharmaceutical tendering, there are two main branches of research: The first stream provides qualitative descriptions on tendering and procurement systems of different countries and their impact, both short-term and long-term, on drug price and market concentration. In addition to overall nationwide impact, some researchers also examine the determinants of bidding behaviors. The second stream focuses on the empirical estimation of the best winning bidding prices in pharmaceutical tenders.

This review focuses on two parts: an overview of tendering systems and elements, and a summary of empirical methods to predict the best bidding price. It will start with an introduction of pharmaceutical bidding in Sect. 1. Then in Sect. 2, the tendering system in four different countries is discussed qualitatively and the impact on local pricing and market landscape is evaluated. Section 3 follows with a definition of the scope of pharmaceutical bidding from auction theory. This includes the common auction elements such as auction time, information asymmetry, and players. Next, in Sect. 4 we summarize the two most commonly used empirical methods for price auction prediction. In this section, we also cover recently involved testing methods for price estimation. Section 5 provides thoughts on different empirical models and proposes best practice to select models based on experimental data.

2 Pharmaceutical Tendering

Pharmaceutical tendering is a common procurement process across countries but could be regulated quite differently depending on the healthcare system of each country. Maniadakis et al. (2018) points out that several factors contribute to these differences, including demographic dynamics, economic growth, and distribution of public bodies. In this section, we summarize the pharmaceutical tendering market from four different countries, both at nation level and at the state or province level. We select the following four countries from four different continents to reflect the diverse nature of pharmaceutical tendering process with different regulatory set up and economic situations. We also provide an overview of the impact of pharmaceutical tendering on drug price and market dynamics.

2.1 Pharmaceutical Tendering Mechanism in Different Countries

In this subsection, we summarize the public pharmaceutical tendering system in different countries. Most countries follow the Public Procurement Act at the country level, while for other large countries such as Brazil and China, procurement for pharmaceutical auctions is organized at state or province level.

2.1.1 Tendering Mechanism in Sao Paulo, Brazil

In Brazil, the Public Procurement Act effective 1993 regulates the procurement process of public goods, including all pharmaceutical procurements, as illustrated by Paulo et al. (2013). The Act requires that all public bodies show a clear description with detailed documentation of the input good, bidding quantity, product quality, origin, and delivery details. According to Paulo et al. (2013), the Brazil legislation specifies two forms for standard purchasing inputs: physical submission and electronic reverse auctions. Paulo et al. (2013) also point out that within the physical submission, there are two types of auctions, open competitive bidding and invited bids. Within electronic auctions, there are three types, first-price sealed bid auction, English auction, and two-stage auction. There are some additional auction requirements, for example, high-value contracts can only be completed through open competitive bidding. Sao Paulo has to follow additional rules due to its large population, unique economic contribution, and complex industrial situation. First, all the procurements in Sao Paulo have to be placed via the common electronic platform, namely, only in the form of an electronic reverse auction. In terms of auction type, there is also a difference where only first-price sealed bid auction or two-stage auction is allowed. The auction type is determined by the value of contracts. Low-value contracts are acquired through first-price sealed bid auctions. The first-price auction is achieved in two steps in Sao Paulo: in the pre-bidding stage, the auction notice is available for five days with detailed description of bidding information, terms, and conditions; in the bidding stage, the bidder submits one single sealed bid before the deadline without knowing the identity of the other bidders. According to Paulo et al. (2013), in Sao Paulo, public hospitals, health agencies, and medical centers obtain prescription drugs in a de-centralized way, in which each entity is responsible to acquire what is needed. These public bodies have to rely on the auction-based tendering process to acquire those generic drugs in Sao Paulo.

2.1.2 Tendering Mechanism in South Africa

The Public Procurement Act in South Africa has been effective since 1982. The scope of procurement by the national government includes the essential drugs in the public healthcare system. According to Wouters et al. (2019), those essential drugs are further divided into 15 different categories and for each tender, the effective period ranges from two to three years. Market authorization is needed to access the South African tendering system. Both local manufacturers and international suppliers are allowed to participate in the bid but in most situations, international suppliers participate through their locally registered subsidiaries and offices. Different from the first-price sealed bid auction mechanism adopted by many countries, South Africa uses a two-stage scoring system to determine the winning supplier. In the first stage, the manufacturer who offers the lowest price among bidders gets 90 points while at the same time remaining suppliers get deductions in proportion to the difference to the winning offer, which is calculated by a publicly announced method. Then, in the second stage, the 10 points left are divided based on an empowerment score, also called “broad-based black economic empowerment score”. This score is decided by the government with a set of criteria, including the diversity of equity owners and managerial roles. When granting the winning bid, price is not the only determinant. Wouters et al. (2019) mention that other ad hoc factors are also considered. For example, under consideration of local economic growth protection, the government may also accept a price mark up to 10 points for national suppliers. In addition to economic factors, industry diversity, labor market balance, and trade situation are also considered. Thus, to keep a proper balance between demand and supply, the government even splits the winning bidding offer between multiple firms if the products are not clearly differentiated. Gray and Suleman (2014) summarize that the bidding process in South Africa is highly regulated and less price originated compared to other countries.

2.1.3 Tendering Mechanism in Cyprus

Petrou (2016) points out that the pharmaceutical market in Cyprus is highly fragmented between the public and private sector, where the public sector has strong regulations on supply and demand. In contrast, regulations only apply to the private market in terms of the price. The wholesale price in the private section can be significantly higher than the official price, where pharmacies add a large mark up. Tendering helps lower the price on both generic and branded products, up to 95 percent of the price for a generic drug and up to 80 percent of the price for a branded drug. Petrou and Talias (2014) state that, however, until 2017, Cyprus was the only country in the EU that did not have a universal coverage national health system (NHS). Petrou and Talias (2015) further state that all the drugs for public healthcare are procured by the Ministry of Healthcare (MoH) by tendering.

2.1.4 Tendering Mechanism in Guangdong, China

Yao and Tanaka (2015) state that Pharmaceutical tendering in China is also implemented at a province level due to the different economic situations across the nation. In Guangdong province, manufacturers can bid for multiple heterogeneous products and complete multiple bids simultaneously. This allows suppliers to combine multiple products together during the bidding process, contrary to the traditional single-product bidding. However, this mechanism is different from package bidding, where bidders can take advantage of the preferred items for higher revenue. In Guangdong, multiple bidding items are allowed but suppliers have to submit the bids separately for each kind of drug. Although the bidding is operated at province level, certain procurement rules still have to be followed at a national level, for example, the rules for drug listings. In China, the central government and provincial municipalities issue multiple layers of drug listings that can be procured, further divided to essential drugs, list A drugs and list B drugs, as introduced by Yao and Tanaka (2015). The above-mentioned categories differ in terms of reimbursement rules and pricing policies. Thus, the price of drugs in China is determined by combining market-driven elements with government regulations. In terms of bidding rules, in Guangdong province, a four-step bidding process is implemented through an online platform. Yao and Tanaka (2015) further explain that: In the first steps, the government screens the access authorization of all interested firms. The Good Manufacturing Practice (GMP) and Good Supply Practice (GSP) certificates are the prerequisites for bidding participation. In the second step, the bidding drugs are classified into one-shot and three-round bidding drugs. Only essential drugs can be procured in the tendering process. One-shot bidding is used for emergency products, low-priced drugs, and drugs for rare diseases or controlled by governments. Compared to three-round drugs, the competition on one-shot drugs is lower. In the third step, all the participating suppliers offer the bidding price in the pre-procurement stage. If the bidding product is a controlled product, then the winner is already secured in this stage. Otherwise, the bidding needs to go through the three rounds of bidding. In the last step, the participants are bidding on the final outcome. Unlike the first-price sealed bid auction where the lowest price bidder wins all, in Guangdong, the process is operated in three rounds in which the highest price bidder is removed from the game after each round. In each round, the supplier is able to set and change the price according to the bidding results from the previous rounds. In addition, in each round a quota is set and short-listed suppliers will stay in the game until the final quota is reached. Thus, the bidding process in Guangdong province China is much more complicated than a normal first-price auction bid.

2.2 Effect of Pharmaceutical Tendering

There has been a lot of discussion on the effectiveness of pharmaceutical tendering system and its impact on individuals and society. Two substantive areas that have been empirically examined are price competition and market concentration.

2.2.1 Pharmaceutical Tendering and Price Competition

Many studies have empirically examined under the pharmaceutical procurement process, and how the price of drugs and volume of bid change in response to increased competition. In the literature, the effect on price is evaluated separately by product attribute, which is divided into generic drugs and brand-name drugs (also called branded drugs). Brand-name products refer to the drug that is originally developed by a pharmaceutical company, approved by the authority for market access, and sold exclusively under brand name for a fixed time period under patent protection. After the expiration of patency, the branded drug becomes a generic drug that contains the same amount of active ingredients as other generic drugs. The key difference between a branded drug and a generic drug is the exclusive patent protection mechanism and this exclusive right significantly changes the manufacturer’s ability to manipulate the price. This is the reason the effect of pharmaceutical tendering is examined separately for these two kinds of drugs.

Petrou (2016) assesses the long-term effect of tendering on price for branded products in the Cyprus market and finds superior price reduction effect through tendering. Using continuous 7-year data from 2006 to 2012 with 36 branded products, Petrou (2016) adopts the repeated measures generalized linear model and empirically proves that tendering retains its capability in reducing the initial drug price [?]. In addition, the downward trend on price exhibits a continuous trajectory. Technically, within the repeated linear model the paper also includes other potentially explanatory variables such as interchangeability, indication, administration status, sales volume as well as triple interaction terms to account for any variances. Finally, Petrou (2016) introduces Greenhouse-Geisser correction to adjust for time-variant elements in price reduction and concludes that tendering shows a significant long-term price reduction effect and the effect is further moderated by product attributes including interchangeability, in-patient status, and indication.

Wouters et al. (2019) also investigate the long-term effect of tendering for medicines and find similar results based on a 14-year period of tendering data for South Africa but focusing on generic drugs. Computing three different types of price index (Laspeyres, Paasche, and Fisher indexes) across multiple medicine categories, the paper finds that the price in most medicine categories drops consistently over time and the medicine procured through public systems is always lower than the ones from private systems (Wouters et al. 2019). Tendering in general is quite effective in securing a lower drug cost.

While most research focuses on the tendering effect either on generic drugs or branded drugs, Paulo et al. (2013) study the interaction of drug entry on drug price between the two types. The authors identify the causal effect of a generic drug’s entry on the bidding participation rate of branded drug manufacturers and the consequences on price paid for the bid. They examine the following three questions: how branded suppliers’ participation decision will change in reaction to the presence of a generic drug entry; how the paid bid price would change after a generic supplier appears in the competitive bidding setting; and whether there is a statistical difference on bids and price bid between generic and branded pharmaceutical manufacturers. By using the Brazil transactional procurement data with 30448 records from 2008 to 2012, Paulo et al. (2013) analyze 3859 different drugs from 425 active ingredients using a 2SLS (Two-Stage Least Squares) approach with Instrumental Variables. In order to establish the causal relationship between generic drug entry and the result of the three questions mentioned above, exogeneity requirements must be satisfied in the generic entry decision. This requirement, widely used in econometrics, mandates that the choice of instrument variable should not correlate with any explanatory variables but only the generic entry decision. Paulo et al. (2013) take advantage of the objective nature of patents expiration and the government setting of auction dates to construct an instrument variable, which is further defined as the difference in days between patent expiration date and the tendering session starting date. By the two-stage regression with the inclusion of fixed effect on product, public body, and time, Paulo et al. (2013) suggest that the bidding price of branded suppliers is lowered in response to the entry of generic supplier and the price paid for pharmaceuticals reduces by seven percent due to the fierce competition by the new entry.

2.2.2 Pharmaceutical Tendering and Market Concentration

In addition to the bidding price change with pharmaceutical tendering, many studies have empirically identified how tenders have changed the market dynamics for pharmaceutical suppliers, including number of participants, market concentration index, and market dynamics.

Paulo et al. (2013) use a 2SLS model to examine the effect of a generic drug entry on the number of branded drug suppliers in the market. It is common knowledge that the fierce competition after the presence of a generic producer will discourage the participation of branded manufacturers. Paulo et al. (2013) validate this empirically with multiple versions of regression including General Least Squares, 2SLS, fixed effects with drug and buyer-specific features, time-specific fixed effects, and the health indicators of the municipalities where the manufacturer is based. The statistically significant negative parameter estimation on the dummy variable representing generic drug entry shows that the presence of a generic drug supplier indicates a reduction in the total number of suppliers for branded drugs in the market by 35 percent. Thus, the increased competition prevents one-third of branded suppliers from staying in the market.

While some researchers focus on quantifying market dynamics using the number of bidders in the market, other researchers such as Wouters et al. (2019) use the Herfindahl-Hirschman (HH Index) index to measure how fierce the market concentration level. Wouters et al. (2019) use the HH index, which not only considers the number of firms/suppliers in the market, but also considers the relative market shares of those firms. It is computed by summing up the squared market share for each supplier in the market. If all firms in the market have equal market shares, the HH index would be minimized. Thus, increasing the HH index indicates uneven distribution of market shares and it is scaled from 0 to 100000 with 0 denoting perfect competition. By calculating the HH indices over a 14-year period, Wouters et al. (2019) find that in general tendering does not change the overall moderate to competitive tendering market in Africa with HH index less than 2500. However, the authors also find that the number of firms actually winning the bid decreased in some drug categories. Overall, the tendering market in South Africa remains moderately competitive in most drug categories.

3 Tendering as Auction: Scope and Concept

Due to the complexity of real-world auction cases, in auction theory there are many different auction types, each of which denotes its own assumptions and requirements. In this section, we summarize the most common auction type for tendering and its assumptions. These assumptions correspond to a priori distribution of models for price estimation. Thus, finding the correct auction type is an essential step before fitting empirical models.

3.1 Tender Versus Auction

As introduced in Sect. 1, a tender is a closed price offer where each bidder keeps their own price, not knowing the other’s price. Auctions are characterized as transactions with a specific set of rules detailing resource allocation according to participants’ bids. They are categorized as games with incomplete information because in the vast majority of auctions, one party will possess information related to the transaction that the other party does not. In a general sense, tender can be viewed as a private form of auction where prices are not transparent to other parties during the bidding price. Thus, auction theory can also be used in the tender process to analyze optimal bidding strategies, find equilibrium bidding price as well as evaluate bidding design.

3.2 Independent Private Value Auction

The basic auction environment consists of the following four elements: the number of players (bidders) n, one object i (drug) to bid on, the actual value of the object \(v_i\), and bidders signal \(s_i\). The signal reflects how much each bidder values the bidding product i, which is not necessarily the same as the actual value of the bidding object. If each bidder has a different view about how much the bidding product means to them, namely each bidder has a different private information (signal) and if knowing the other’s signal may change the perception of their own value, this would be a common value model. In contrast, during the tendering process, each bidder has their own private information about the value of the bidding product. This signal would not change even if information from others is required. This is called the Independent Private Value (IPV) auction. Under the normal tendering setting, tendering is typically an IPV auction. Using the notations, this means, bidder i’s information (signal) is independent of bidder j’s information. Moreover, bidder i’s value is independent of bidder j’s information—so bidder j’s information is private in the sense that it does not affect anyone else’s valuation.

3.3 First Price Sealed Auction

IPV auction focuses on whether the information and value for each bidder is independent. In addition to independency, another very important element in auctions is winning prices. There are multiple forms of auction such as absolute auction, minimum auction, sealed auction, reserve auction, etc., depending on the winning rule and completeness of information on other bidders. In a sealed bid auction, bidders submit their bid \(b_i\), ... \(b_n\) simultaneously, and the bidder who offers the highest bid (in tender case, lowest tendering price) wins the bid and pays the bidding prices offered. “Sealed” means each bidder submits the bid privately so no one else knows the information, while “First Price” means the winning bidder pays the winning price, not the second winning price. It is quite obvious that bidders would not bid on their true values because this brings no profit/benefit. Normally by bidding a little bit lower (in tender cases higher tendering price), bidders can gain margins.

3.4 Number of Bidders

Number of bidders in a single bidding round reflects the scale of a specific bidding process. Number of bidders highly depends on the number of competing manufacturers in the market. For biosimilar bidding, the number of bidders is usually small while for other generic drugs, the number of bidders could be quite large. Yao and Tanaka (2015) empirically examine the determinants of bidding behavior by using the provincial-level data consisting of 2758 bidders, with 757 being the highest number of manufacturers in the same bidding round. So far, this is a truly large-scale study with observational data. For literature incorporating structural models, researchers tend to use a finite number of bidders in the simulation stage and quite a small number of bidders in the empirical estimation. For example, Yao and Tanaka (2015) consider n = 7 bidders in the Monte Carlo Simulation and during the empirical illustration, the paper used California Highway Procurement data with 2–7 bidders. Guerre et al. (2000) used n = 5 bidders in the classical paper proposing non-parametric estimation of first-price auctions. To sum up, for empirical studies using structural models or non-parametric models, the number of bidders normally ranges from 2 to 10. For studies using observational studies, the number of bidders could be much greater.

4 Empirical Methods for Price Auction Estimation

The previous three sections summarized the pharmaceutical tendering background and auction types qualitatively. In this section, we will focus on the methodology to estimate the most important parameter in the tendering price—bidding price. As introduced in Sect. 3, most tendering processes could be modeled theoretically by auction theory. In empirical literature, first-price auctions are the auction type that has been researched most frequently, through structural or reduced form methods. Structural approaches mainly focus on recovering the distribution of observed bids or bidder’s private value in order to extrapolate the unobserved valuation of the bidders. Considering the nature of all structural estimations, they rely heavily on assumptions of the underlying distribution, which is indirectly reflected by the auction type in terms of information asymmetry, information completeness, and the presence of signaling. Reduced form estimations mainly target at solving the selection bias problem, where endogenous factors can bias the estimation results of exogenous variables. By rearranging the equations algebraically until every endogenous variable is on the left side of the equation and all the exogenous variable (also including lagged endogenous ones) is on the right side of the equation so that potential selection bias has been resolved. In this section, we discuss these two methods through the estimation on bidding price.

4.1 Bidding Price Determinants Estimation with Reduced Form Approach

Yao and Tanaka (2015) empirically examine the determinants of suppliers’ bidding behavior in a multiple bidding setting using a provincial pharmaceutical procurement dataset consisting of 19818 biddings from 2758 bidders on 37 groups of drugs from 2007 to 2009. Considering the bidding setting where the winning price is only known to the final winner, bidding price would introduce selection problems when using a non-random subsample to use that price to all participants. Thus a simple OLS or GLS regression with bidding price as the dependent variable is not feasible. In presence of selection bias, the paper used the Heckman Selection Model to correct for endogeneity. Yao and Tanaka (2015) start with the simplest model construction:

$$\begin{aligned} y_{i j t}=\textbf{x}_{i j t} \beta +u_{i j t} \end{aligned}$$
(1)
$$\begin{aligned} s_{i j t}=\textbf{1}\left\{ s_{i j t}^{*}>0\right\} \end{aligned}$$
(2)
$$\begin{aligned} s_{i j t}^{*}=\textbf{z}_{i j t} \gamma +v_{i j t} \end{aligned}$$
(3)

where \(y_{i j t}\) is the bidding price for bidder j on product i in year t, \(x_{i j t}\) are vectors of explanatory variables that would affect bidding prices. The paper includes number of bidders, spatial distance between the bidder and the auctioneer, product specific features including product range and standards, quality specific features including the technological skills, experience, and multiple bidding potential. In Yao and Tanaka (2015)’s definition, \(u_{i j t}\) represents the idiosyncratic error which is not correlated to any of the variables and \(v_{i j t}\) represents the idiosyncratic error during selection. \(s_{i j t}\) is a latent variable and Eqs. (2) and (3) jointly describe the choice made by the auctioneer. Directly estimating the above two equations or using maximum likelihood alternatively both suffer from multicollinearity. The authors used the panel data version of the Heckman selection model with the Mundlak-Chamberlain approach to correct the selection bias as follows:

$$\begin{aligned} y_{i j t}=\textbf{x}_{i j t} \beta +c_{i j}+u_{i j t} \end{aligned}$$
(4)
$$\begin{aligned} \tilde{s}_{i j t}=\textbf{1}\left\{ \eta _{t}+\textbf{z}_{i j t} \gamma _{t}+\textbf{z}_{i j} \varphi _{t}+e_{i j t}>0\right\} \end{aligned}$$
(5)
$$\begin{aligned} e_{i j t}=a_{i j}+v_{i j t} \end{aligned}$$
(6)

where \(c_{i j}\) is the unobservable and \(z_{i j}\) is a vector representing the means of \(x_{i j t}\). Since \(e_{i j t}\) is not depending on \(z_{i j}\), Eq. (1) is converted to the following equation:

$$\begin{aligned} y_{i j t}=\textbf{x}_{i j t} \beta +c_{i j}+\rho E\left( e_{i j t} \mid \textbf{z}_{i j}, \tilde{s}_{i j t}\right) +v_{i j t} \end{aligned}$$
(7)

By first estimating the possibility of selection, pooled OLS is then applied to the selected sample and the paper discovers that more fierce competition and more winning experience encourage suppliers to bid much more aggressively on the bidding prices and consequently make lower bids. In addition, those bidders that are in less competitive groups are less sensitive to the number of participants and their past winning times.

4.2 Structural Estimation of Auction Models

The empirical nature that uses structural econometric modeling approaches to understand firm and consumer behavior has attracted wide interest. The estimation of auction mainly focuses on the private value of the bidders. The earliest literature on structural estimation of auction prices dates back to 1992, where Paarsch (1992) introduce parametric structural models to estimate private and common value first-price sealed auctions. After that, much literature examine the same topic using parametric structural models, including (Donald and Paarsch 1993; Elyakime et al. 1994; Flambard and Perrigne 2022; Campo 2022). All of these papers model the game as a first-price auction game where the Bayes-Nash equilibrium is the end point and the idea to parametrically estimate the distribution of bidder’s private value is achieved by inverting the equilibrium function of private value from observed bids. Guerre et al. (2000) introduce the two-stage optimal estimation of private value in a non-parametric way. This non-parametric method has been widely researched and applied in following studies, with the emergence of another approach, called the quantile-based estimation. In this subsection, we’ll first go over the structural literature of first-price auctions and the experimental evidence of the estimations. Then, we’ll discuss the non-parametric development of structural estimation, including two-stage estimation and quantile-based estimation.

4.2.1 Reasonable Structural Estimation of Price Auction

Bajari and Hortacsu (2003) use experimental data to examine whether four different kinds of structural models give reasonable estimates on bidders’ private value. Many researchers have challenged the strict rationality assumptions imposed by parametric models to be infeasible in reality. The critics focus on mapping the estimation results, bidders’ valuations to the bidders’ true private information. Bajari and Hortacsu (2003) structurally estimated four models under the first-price auction setting: risk neutral Bayes-Nash, risk averse Bayes-Nash, Quantal Response Equilibrium (logit equilibrium model), and an adaptive model of learning.

Assuming there are \(i = 1 \ldots N\) symmetric bidders in the market and \(v_i\) denoting the valuation of one single and indivisible product. Each bidder’s valuation \(v_i\) is iid with cdf F(v) and pdf f(v); \(b_i\) denoting the simultaneously submitted bid by bidder i; \(u_i\) denoting the utility.

Then under the risk neutral Bayes-Nash model, the first-order condition for maximizing expected profit can be expressed by

$$\begin{aligned} v=b+\frac{F(\phi (b))}{f(\phi (b)) \phi ^{\prime }(b)(N-1)} \end{aligned}$$
(8)

Using G(b) and g(b) as the distribution and density of the bids, the above equation can be written as follows:

$$\begin{aligned} v=b+\frac{G(b)}{g(b)(N-1)} \end{aligned}$$
(9)

Next under the risk averse Bayes-Nash model, the first-order condition is

$$\begin{aligned} v_{i}=b_{i}+\frac{\theta _{i}}{Y_{i}\left( b_{i}\right) } \end{aligned}$$
(10)

where 1—\(\theta _i\) represents Bidder i’s risk preference, also known as the coefficient of relative risk aversion.

Thirdly, under the logit equilibrium model, with the extreme value distribution:

$$\begin{aligned} \sigma \left( b_{i} ; v_{i}, \textbf{B}\right) =\frac{\exp \left( \lambda \pi \left( b_{i} ; v_{i}, \textbf{B}\right) \right) }{\sum _{b^{\prime } \in \mathcal {B}} \exp \left( \lambda \pi \left( b^{\prime } ; v_{i}, \textbf{B}\right) \right) } \end{aligned}$$
(11)

where \(\textbf{B}(b \mid v)\) represents a symmetric strategy, which is a measure that gives every bid b a probability based on the condition upon a valuation v. An equilibrium is a bidding function \(\textbf{B}(b \mid v)\) that is a fixed point, that is \(\textbf{B}\left( b \mid v_{i}\right) =\sigma \left( b_{i} ; v_{i}, \textbf{B}\right) \).

Finally, for the simple adaptive model, the first-order condition for maximizing expected profit is

$$\begin{aligned} \hat{v}_{i t}=b_{i t}+\frac{\hat{G}\left( b_{i t} \mid h_{i t}\right) }{\hat{g}\left( b_{i t} \mid h_{i t}\right) (N-1)} \end{aligned}$$
(12)

By using the structural estimation to measure the closeness of the estimated valuation distribution of the above four models with the true distribution of bidders’ private value, Bajari and Hortacsu (2003) calculate the Kolmogorov-Smirnov distance between the distribution and true values. The paper notices that given the number of bidders is large enough, the three models except QRE are able to uncover the deep parameter. When the number of bidder is small, the estimation result is not stable enough and is more sensitive to the choice of models. Rational models give better estimated results than the behavioral models.

4.2.2 Two-Stage Non-parametric Structural Estimation of Price Auction

Guerre et al. (2000) point out that the structural econometrics approach on price auction has to rely exclusively on the parametric requirement of the distribution of bidders’ private value. However, the strong assumption may not be met completely in both observational and experimental data. Guerre et al. (2000) also mention the computational limitation for structural models due to its dependency on complex numerical computations and simulations to find out the Bayesian Nash equilibrium. Instead, Guerre et al. (2000) propose a two-stage indirect process to estimate the distribution of bidder’s valuation from observed bids without computing Bayesian Equilibrium or imposing any parametric assumptions on the observed bids.

The main idea of Guerre et al. (2000)s two-stage approach is that constructing a function with the distribution of observed bids, corresponding bids, and corresponding density function to represent the private value of each bid. This idea is implemented in the first step by making a set of pseudo-private values according to the observed bids’ kernel distribution and density function. Then, the density of the bidder’s private value can be estimated non-parametrically using the pseudo samples. Guerre et al. (2000) prove the uniform consistency property of the estimator by showing it has the best uniform convergence rate in estimating the latent density.

For a more detailed model specification, let i denote bidder \(i = 1\ldots I\) and \(v_i\) as the private value for the bidding product, and \(p_0\) as the reservation (lowest possible) price. Let \(F(v_i)\) denote the common distribution of private values and \(f(v_i)\) denote the density. Under the Bayesian Nash Equilibrium of symmetric bidders, the equilibrium bid \(b_i\) for bidder i is:

$$\begin{aligned} 1=\left( v_{i}-s\left( v_{i}\right) \right) (I-1) \frac{f\left( v_{i}\right) }{F\left( v_{i}\right) } \frac{1}{s^{\prime }\left( v_{i}\right) } \end{aligned}$$
(13)

With non-parametric identification, the previous equation can be rewritten that now expresses the individual private value vi as a function of the individual’s equilibrium bid \(b_i\), its distribution \(G(\cdot )\), its density \(g(\cdot )\) as follows:

$$\begin{aligned} v_{i}=\xi \left( b_{i}, G, I\right) \equiv b_{i}+\frac{1}{I-1} \frac{G\left( b_{i}\right) }{g\left( b_{i}\right) } \end{aligned}$$
(14)

For estimation, considering L auctions, \(G(\cdot )\) and \(g(\cdot )\) can be estimated by

$$\begin{aligned} \tilde{G}(b)=\frac{1}{I L} \sum _{l=1}^{L} \sum _{p=1}^{I} {1}\left( B_{p l} \le b\right) \end{aligned}$$
(15)
$$\begin{aligned} \tilde{g}(b)=\frac{1}{I L h_{g}} \sum _{l=1}^{L} \sum _{p=1}^{I} K_{g}\left( \frac{b-B_{p l}}{h_{g}}\right) \end{aligned}$$
(16)

where \(h_{\rho }\) is a bandwidth and \(K_{g}(\cdot )\) is a kernel with a compact support. In this paper, the author uses a tri-weight kernel and uses 1.06 as the bandwidth. We’ll discuss more about the choice of kernel and bandwidth in the last section of this review with real data.

Then as an illustration, the paper conducts a Monte Carlo Simulation with 200 auctions and 5 bidders over 1000 observed bids to empirically verify that the latent density of bidder’s private values can be estimated from available bids.

4.3 Quantile-Based Non-parametric Estimation of Private Value

After Guerre et al. (2000)s paper on the two-stage optimal non-parametric structural estimation approach, numerous studies have focused on releasing the assumption on the distribution of observed bids. Marmer and Shneyerov (2012) identify the idea to estimate a bidder’s valuation distribution in another flavor, estimating it based on the quantile representation of the first-order condition. Under the risk neutral Bayes-Nash model, the Guerre et al. (2000) paper transform the first-order condition for optimal bids and expresses a bidder’s value as an explicit function of the submitted bid, the Probability Density Function (PDF) and Cumulative Distribution Function (CDF) of bids, as shown in Eq. (10). Marmer and Shneyerov (2012) further propose to estimate the valuation distribution based on the quantile representation of the first-order condition. This indicated, when there is the strict monotone condition on the equilibrium bidding strategy, the valuation quantile function \(Q_{v}(\cdot )\) can be expressed as

$$\begin{aligned} Q_{v}(\alpha )=Q_{b}(\alpha )+\frac{1}{I-1} \frac{\alpha }{g\left( Q_{b}(\alpha )\right) ^{\prime }} \quad 0 \le \alpha \le 1 \end{aligned}$$
(17)

where \(Q_{b}(\cdot )\) is the bid quantile function. In Eq. (17), Marmer and Shneyerov (2012) propose to first estimate \(Q_{v}(\cdot )\) using plug-in estimators for \(g(\cdot )\) and \(Q_{b}(\cdot )\) and subsequently estimate the valuation density using the relationship \(f(v)=1 / Q_{v}^{\prime }\left( Q_{v}^{-1}(v)\right) \). They prove that the quantile-based estimator is asymptotically normal and has the optimal rate of Guerre et al. (2000)s definition of Private Value (GPV).

Marmer et al. (2010) also use the quantile-based non-parametric approach to infer the PDF of private values on first-price auctions under the Independent Private Value (IPV) construction. They disclose a fully kernel-based estimator of the quantiles and PDF over observed bids and estimate it non-parametrically. They are also able to achieve the optimal rate with Guerre’s paper and under proper choice of bandwidth, the estimator is also asymptotically normal. While Marmer et al. (2010)s paper have already released one of the steps in Guerre et al. (2000)s paper on constructing pseudo values, Luo and Wan (2017) go one step further and propose a fully tuning-parameter-free estimator for the valuation quantile function. This means to estimate the quantile function of the valuation requires neither the choice of a kernel nor a bandwidth. They provide a trimming-free smoothing estimator and this estimator is also asymptotically normal and is the same as the optimal rate of Marmer et al. (2010)s paper. Liu and Luo (2017) investigate the comparison of valuations in first-price auctions using non-parametric tests. Building on the fact that two distributions of private valuations would be the same if and only if the integrated quantile functions are identical, the paper proposed a test statistic that measures the square distance between the sample analogues of the linear functional for bid samples.

5 Non-parametric Estimation on Observational Data

In this section, we briefly introduce the process of applying non-parametric structural estimation on real-world observational tendering data. This includes a description of a sample dataset coming from one of the affiliates of a large pharmaceutical company, the initial findings from basic data visualization, the process to formulate the empirical model, the choice of parameters during the non-parametric estimation, and the conclusions from the estimation.

5.1 Dataset

The dataset comes from one of the affiliates of a large multi-national pharmaceutical company, containing around 1500 tendering records over a three-year period from 2018 to 2020. Tender records cover three main generic products in oncology with biosimilar competition. Tendering process in the concerned country follows the typical first-price sealed bid auction where competitors place the tender offer simultaneously without knowing the other’s price as signals. In addition, through market research and discussion, for the three products involved in the tendering process, we can assume there are three main participants in the market (including the manufacturer). Figure 1 gives the description of the dataset:

Fig. 1
figure 1

Three dimensions of tender dataset

Full size image

5.2 Visualization and Descriptive Analysis

With real-world tendering result datasets, based on our experience, we first start by visualizing and cleaning the dataset with a few plots and descriptive statistics. We will explain in more detail which variables and information have been used to perform exploratory data analysis and what information could be relevant to down-streaming modeling tasks. The first dimension to explore is the tender-specific information and we plot basic descriptive statistics like bar plot density plot on categorical variables such as Tender Type, Hospital (Customer), Hospital Type, Tender Result, Bidding Product, Discount Rate, and kernel density plot on numerical variables such as Manufacturing Price, Winning Price, and Wholesale Price Margin. This information gives us an overview of the distribution of tender-related attributes. Similar to our assumptions to real-world bidding setting, most of the categorical are skew-distributed with highly imbalanced classes. We account for that in our estimation by using re-sampling methods to create more balanced classes by imposing different class weights.

In addition to visualizing variables in single dimensions, we also perform pivot analysis across multiple dimensions, such as bar plot of tender results by product type, bar plot of tender type by product, bar plot of tender result by tender type, and bar plot of tender result by tender quantity (convert numerical quantity to categorical intervals). For confidentiality reason, we only describe the methodologies in visualization, rather than the actual results. These plots provide us with the interactions between variables and help us to verify some of the business assumptions, which will be further explained in more detail in the next section. To summarize, exploratory data analysis and descriptive statistics are a critical step to perform on a real-world dataset as the starting point before modeling and estimation. It provides initial insights and helps validate some of the key assumptions to choose the most accurate auction forms.

5.3 Assumptions Validation

As described in Sect. 3, Tendering Scope and Auction Forms are defined by a set of conditions and assumptions. Choosing the most accurate auction form is the most critical step for modeling pharmaceutical tendering in a real-world setting. However, in most real-world cases, the datasets do not completely follow one exact type of auction form and some of the assumptions need to be verified for modeling. In this part, we’ll illustrate how to perform validity checks on real-world datasets for the four most important assumptions to determine the corresponding auction form.

First, from an auction theory perspective, the starting point is to examine the number of players in the game, which maps the number of manufacturers (bidders) in a pharmaceutical tendering setting. This information is examined by a simple frequency count of the Manufacturer (bidder) in the dataset. One important check is to confirm that the products from manufacturers are perceived as “Biosimilars” to each other with the same standards of quality, safety, and efficacy since first-sealed auction form requires each bidder to bid on non-differentiated products. For non-competing biosimilar manufacturers, which offer low quality products or products with different biologic compositions are not eligible to be counted as players (bidders) in the modeling process. Thus the key point to check is the count of manufacturers in the dataset and keep those who offer biosimilar products.

Next, the validity check should focus on the bidding rule. This includes both qualitative and quantitative checks. Qualitative check deals with the following questions relevant to tender managers or tender specialists who manage and perform the actual bidding process: What is the tender submission process? Does the participating entity have any information about the other competitors before the bidding process? Are bidding prices submitted simultaneously in each round? Or sequentially where some competitors could signal the others’ behavior? How many rounds of bidding are contained in a full bidding process? Is there any entry threshold for each round and how is it defined? These questions require qualitative checks with business stakeholders or tender managers who actually participate in the tendering process. In addition, the number of rounds in a tender process can be quantitatively verified in the dataset by grouping records by unique tender id (if it exists) and count the number of records under each tender id.

Then, the most important validity check is on the winning rule since first-price sealed bid auction requires “lowest price winning rule” which means the winner is solely determined by the raised bidding price, with no other factor taken into account such as brand perception and loyalty, quality, and packaging. This assumption check is performed quantitatively on the dataset by first grouping all the bidding records by the unique identifier of a tender, for example, tender id (if it exists), and extracting the lowest bidding price under the same id; next, extract the final winning price of each tender and comparing it with the lowest bidding price to check whether the two figures are equivalent. Theoretically, with the “lowest price winning rule” the lowest bidding price in a tender should be the same as the final winning price of the bid. However, in real-world bidding datasets, sometimes the two figures can be different. The underlying reason could be that the manufacturer who offered the lowest bidding price could not fulfill the bidding quantity completely due to logistics blocker or insufficient remaining quota. Under this situation, those tender results should be excluded from the modeling to avoid adding noise to the models and estimation. Thus it is very crucial to perform a quantitative check to validate the “lowest price winning rule”.

Finally, based on our analysis on the dataset, verifying the winning quota is the last step for the validity check. This includes would the winner take all the quota as submitted in the bid (Winners take all)? Or the ending quota could be smaller/larger than the submitted amount? In some cases, the winner cannot get exactly the same amount as submitted during the bidding price. This happens most frequently when there is a mix of biosimilar manufacturers and non-competing biosimilar manufacturers in the bidding round. And in the above case, there is a very high probability that the previous “lowest price winning rule” does not hold. Winners may only get a subset or part of the submitted bidding quantity and the remaining small portion may go to the non-competing biosimilars due to their extremely low price compared to biosimilars. This assumption on quota is generally examined in the dataset by comparing the column indicating submitted quota and the column indicating winning quota. If these two columns are not the same, normally we should dive deeper to check which one is larger and whether the pattern is consistent. At the same time, this assumption is always examined together with the previous one “winning rule”. If the winning quota diverges with the submitted quota, we must be very cautious on choosing the auction form since the first-price sealed bid auction does not apply anymore.

5.4 Modeling and Estimation

After checking the assumptions on the real-world tendering dataset, we confirm that for the country concerned, the bidding form can be well modeled as a first-price sealed bid auction. After data cleaning and processing, we follow the methodology proposed by Guerre et al. (2000) with Two-Stage Non-Parametric Structural Estimation to compute the Cumulative Density Function of the private value for bidders.

Using the Two-Stage Non-Parametric approach by Guerre et al. (2000), we extract the Wholesaler Price after Discount (MTS price) in the dataset to obtain the observed bids and use I = 3 to represent the three main suppliers in the market. Following the two-stage estimation process, first according to Eq. (14), we are able to construct a sample of pseudo-private values using non-parametric estimates of the distribution and density functions of observed bids (MTS Price). Then for the second step, using Eqs. (15) and (16), we are able to get the distribution of the bidders’ valuation with the choice of bandwidth and kernel. Similar to Guerre et al. (2000)s method, we also try the Triweight Kernel and the Epanechnikov Kernel. For bandwidth, we use a loop to select the most reasonable bandwidth ranging from 0.35 to 2.6 (the above range is decided by observing the pattern of the distribution plot). After obtaining the distribution of valuation, we use KS test (Kolmogorov-Smirnov test) and standardization to map back to the bidding price of actual bids in order to get the Probability Density Function and Cumulative Distribution Function for the probability to win. After obtaining the Cumulative Distribution Function of bidder’s private values, given a constant bidding price x, with the definition of Cumulative Distribution Function, we can easily get the probability that the X will take a value less than or equal to x. According to the bidding setting, if the bidding price of other competitors is less than the bidder’s offering price, the bidder will lose this bid. This means the area under the Cumulative Distribution Function curve and to the left of the raised price x denotes the probability that the competitor price will be lower than the raised bid. Thus, the winning probability is obtained by subtracting the probability from 1.

5.5 Adjustments and Lessons Learned

In the previous section, we discussed how to perform validity checks on the real-world tendering data to determine the most appropriate auction forms. This is the most important step since in real-world settings, it is hard to meet all the assumptions of a specific auction form due to the complexity of bidding system design and actual bidding process implementation. At the same time, policy and regulatory requirements also impose some constraints on meeting all the requirements and assumptions of an auction type. This directly leads to the fact that during the modeling and estimation process, we often need to use “rule of thumb” to determine some of the parameters that best fit and depict the dataset, instead of pre-defined parameters from previous assumptions. One example in our estimation case is the choice of Kernels and their bandwidth using the two-stage methodology in the Guerre et al. (2000) paper.

All the estimation process in the real-world tendering data is implemented in Python 3.6. When selecting the kernel, there is no clear evidence that the tri-weight kernel should outperform other kernels. Also for the bandwidth choice, it is more of a rule of thumb instead of mathematically proved constant. As explained in the previous paragraph, finding the best combination of kernel and bandwidth with a real-world tendering dataset requires iterative computation and it is a more heuristic process. Thus, we tried all the possible combinations of bandwidth and kernel to get the most meaningful valuation distribution.

In terms of take-aways and lessons learned from fitting observational data, we find that a well-established structural model is the theoretical foundation. In addition to that, adjustments should also be made to understand the data and give intuitive explanations to the data. Different choice of parameters results in different distributions of valuations, where the probability to win should not have multiple spikes in real-world bidding practices. Thus, fitting non-parametric structural models should always consider the underlying business rationale.