Abstract
Allowing for games with a continuous action space, we investigate how evolutionary stability, the existence of a uniform invasion barrier, local superiority and asymptotic stability relate to each other. This is done without restricting the populations of which we want to investigate the stability to monomorphic population states or to strategies with finite support.
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References
Bickel, P.J., Klaassen, C.A., Ritov, Y., Wellner, J.A.: Efficient and Adaptive Estimation for Semiparametric Models. Johns Hopkins University Press, Baltimore (1993)
Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968)
Bomze, I.M.: Dynamical aspects of evolutionary stability. Monatshefte Math 110, 189–206 (1990)
Bomze, I.M.: Cross entropy minimization in uninvadable states of complex populations. J Math Biol 30, 73–87 (1991)
Bomze, I.M., Pötscher, B.M.: Game Theoretical Foundations of Evolutionary Stability. Springer, Berlin (1989)
Cressman, R.: Stability of the replicator equation with continuous strategy space. Math Soc Sci 50, 127–147 (2005)
Cressman, R., Hofbauer, J.: Measure dynamics on a one-dimensional continuous trait space: theoretical foundations for adaptive dynamics. Theor Pop Biol 67, 47–59 (2005)
Cressman, R., Hofbauer, J., Riedel, F.: Stability of the replicator equation for a single-species with a multi-dimensional trait space. J Theor Biol 239, 273–288 (2006)
Csiszár, I.: Informationstheoretische Konvergenzbegriffe im Raum der Wahrscheinlichkeitsverteilungen. Magyar Tudomanyos Akademia Matematikai Kutato Intezetenek Közlemenyei 7, 137–158 (1962)
Csiszár, I.: Über topologische und metrische Eigenschaften der relativen Information der ordnung α. In: Transactions of the third Prague Conference on Information Theory, Statistical Decision Functions and Random Processes, Prague, pp. 63–73 (1964)
Csiszár, I.: On topological properties of f-divergences. Stud Sci Math Hungarica 2, 329–339 (1967)
Csiszár, I.: I-divergence geometry of probability distributions and minimization problems. Ann Probab 3, 46–158 (1975)
Hofbauer, J., Schuster, P., Sigmund, K.: A note on evolutionary stable strategies and game dynamics. J Theor Biol 81, 609–612 (1979)
Oechssler, J., Riedel, F.: On the dynamic foundation of evolutionary stability in continuous models. J Econ Theory 107, 223–252 (2002)
Oechssler, J., Riedel, F.: Evolutionary dynamics on infinite strategy spaces. Econ Theory 17, 141–162 (2001)
Pervin, W.J.: Foundations of General Topology. Textbooks in Mathematics. Academic Press, New York (1964)
Reiss, R-D.: Approximate Distributions of Order Statistics. Springer, Berlin (1998)
Seymour, R.: Dynamics for infinite dimensional games. Working Paper, University College London (2000)
Sigmund, K.L.: Game dynamics, mixed strategies and gradient systems. Theor Pop Biol 32, 114–126 (1987)
Sierpinski, W.: General Topology. Toronto: University of Toronto Press (1952, 1966)
Van Veelen, M.: Evolution in games with a continuous action space. Tinbergen discussion paper 2001-068/1 (2001)
Weibull, J.W.: Evolutionary Game Theory. MIT Press, Cambridge MA. (1996)
Williams, D.: Probability with Martingales. Cambridge University Press, Cambridge (1991)
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This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
The authors have benefitted from careful and precise comments by Gerard van der Laan, Jan van Mill, the editor and an anonymous referee. Matthijs van Veelen gratefully acknowledges financial support by the Netherlands’ Organization for Scientific Research (NWO).
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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van Veelen, M., Spreij, P. Evolution in games with a continuous action space. Econ Theory 39, 355–376 (2009). https://doi.org/10.1007/s00199-008-0338-8
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DOI: https://doi.org/10.1007/s00199-008-0338-8
Keywords
- Evolutionary stability
- Uniform invasion barrier
- Local superiority
- Asymptotic stability
- Continuous action space
- Topology