Stability of the replicator dynamics for games in metric spaces

Saul Mendoza-Palacios; Onésimo Hernández-Lerma

doi:10.3934/jdg.2017017

Article Contents

2017, Volume 4, Issue 4: 319-333. Doi: 10.3934/jdg.2017017

This issue Previous Article Game theoretical modelling of a dynamically evolving network Ⅰ: General target sequences Next Article Nonlinear dynamics from discrete time two-player status-seeking games

Stability of the replicator dynamics for games in metric spaces

Saul Mendoza-Palacios^1, , and
Onésimo Hernández-Lerma^2,

1.
Centro de Estudios Económicos, El Colegio de México, Entronque Picacho-ajusco 20, Fuentes del Pedregal, 10740 México City, México
2.
Mathematics Department, CINVESTAV-IPN, A. Postal 14-740, México City 07000, México

* Corresponding author: Saul Mendoza-Palacios
* Corresponding author: Saul Mendoza-Palacios

Received: February 28, 2017

Revised: May 31, 2017

Published: November 2017

This research was partially supported by CONACYT Grant 221291.

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Abstract

In this paper we study the stability of the replicator dynamics for symmetric games when the strategy set is a separable metric space. In this case the replicator dynamics evolves in a space of measures. We study stability criteria with respect to different topologies and metrics on the space of probability measures. This allows us to establish relations among Nash equilibria (of a certain normal form game) and the stability of the replicator dynamics in different metrics. Some examples illustrate our results.

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Mathematics Subject Classification: Primary: 91A22, 91A10; Secondary: 34A34, 34G20.

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[1]	K. Bagwell and A. Wolinsky, Game theory and industrial organization, Handbook of Game Theory with Economic Applications, 3 (2002), 1851-1895. doi: 10.2139/ssrn.239431.
[2]	D. Bishop and C. Cannings, A generalized war of attrition, Journal of Theoretical Biology, 70 (1978), 85-124. doi: 10.1016/0022-5193(78)90304-1.
[3]	A. Bobrowski, Functional Analysis for Probability and Stochastic Processes: An Introduction, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511614583.
[4]	I. M. Bomze, Dynamical aspects of evolutionary stability, Monatshefte für Mathematik, 110 (1990), 189-206. doi: 10.1007/BF01301675.
[5]	I. M. Bomze, Cross entropy minimization in uninvadable states of complex populations, Journal of Mathematical Biology, 30 (1991), 73-87. doi: 10.1007/BF00168008.
[6]	R. Cressman, Local stability of smooth selection dynamics for normal form games, Mathematical Social Sciences, 34 (1997), 1-19. doi: 10.1016/S0165-4896(97)00009-7.
[7]	R. Cressman, Stability of the replicator equation with continuous strategy space, Mathematical Social Sciences, 50 (2005), 127-147. doi: 10.1016/j.mathsocsci.2005.03.001.
[8]	R. Cressman and J. Hofbauer, Measure dynamics on a one-dimensional continuous trait space: Theoretical foundations for adaptive dynamics, Theoretical Population Biology, 67 (2005), 47-59. doi: 10.1016/j.tpb.2004.08.001.
[9]	R. Cressman, J. Hofbauer and F. Riedel, Stability of the replicator equation for a single species with a multi-dimensional continuous trait space, Journal of Theoretical Biology, 239 (2006), 273-288. doi: 10.1016/j.jtbi.2005.07.022.
[10]	I. Eshel and E. Sansone, Evolutionary and dynamic stability in continuous population games, Journal of Mathematical Biology, 46 (2003), 445-459. doi: 10.1007/s00285-002-0194-2.
[11]	C. R. Givens and R. M. Shortt, A class of Wasserstein metrics for probability distributions, The Michigan Mathematical Journal, 31 (1984), 231-240. doi: 10.1307/mmj/1029003026.
[12]	J. C. Harsanyi, Oddness of the number of equilibrium points: A new proof, International Journal of Game Theory, 2 (1973), 235-250. doi: 10.1007/BF01737572.
[13]	J. Hofbauer, J. Oechssler and F. Riedel, Brown--von Neumann--Nash dynamics: The continuous strategy case, Games and Economic Behavior, 65 (2009), 406-429. doi: 10.1016/j.geb.2008.03.006.
[14]	J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9781139173179.
[15]	J. Hofbauer and K. Sigmund, Evolutionary game dynamics, Bulletin of the American Mathematical Society, 40 (2003), 479-519. doi: 10.1090/S0273-0979-03-00988-1.
[16]	J. Hofbauer and J. W. Weibull, Evolutionary selection against dominated strategies, Journal of Economic Theory, 71 (1996), 558-573. doi: 10.1006/jeth.1996.0133.
[17]	R. Lahkar and F. Riedel, The logit dynamic for games with continuous strategy sets, Games and Economic Behavior, 91 (2015), 268-282. doi: 10.1016/j.geb.2015.03.009.
[18]	A. Mas-Colell, M. D. Whinston and J. R. Green, Microeconomic Theory, Oxford university Press, 1995.
[19]	J. Maynard Smith and G. A. Parker, The logic of asymmetric contests, Animal Behaviour, 24 (1976), 159-175. doi: 10.1016/S0003-3472(76)80110-8.
[20]	S. Mendoza-Palacios and O. Hernández-Lerma, Evolutionary dynamics on measurable strategy spaces: Asymmetric games, Journal of Differential Equations, 259 (2015), 5709-5733. doi: 10.1016/j.jde.2015.07.005.
[21]	J. R. Munkres, Topology, Second edition, Prentice Hall, 2000.
[22]	T. W. Norman, Dynamically stable sets in infinite strategy spaces, Games and Economic Behavior, 62 (2008), 610-627. doi: 10.1016/j.geb.2007.05.005.
[23]	J. Oechssler and F. Riedel, Evolutionary dynamics on infinite strategy spaces, Economic Theory, 17 (2001), 141-162. doi: 10.1007/PL00004092.
[24]	J. Oechssler and F. Riedel, On the dynamic foundation of evolutionary stability in continuous models, Journal of Economic Theory, 107 (2002), 223-252. doi: 10.1006/jeth.2001.2950.
[25]	G. K. Pedersen, Analysis Now, Springer, New York, 1989. doi: 10.1007/978-1-4612-1007-8.
[26]	R. -D. Reiss, Approximate Distributions of Order Statistics, Springer, New York, 1989. doi: 10.1007/978-1-4613-9620-8.
[27]	K. Ritzberger, The theory of normal form games from the differentiable viewpoint, International Journal of Game Theory, 23 (1994), 207-236. doi: 10.1007/BF01247316.
[28]	H. L. Royden, Real Analysis, Third edition, Macmillan, New York, 1988.
[29]	W. H. Sandholm, Population Games and Evolutionary Dynamics, MIT press, 2010.
[30]	A. N. Shiryaev, Probability, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4757-2539-1.
[31]	M. Van Veelen and P. Spreij, Evolution in games with a continuous action space, Economic Theory, 39 (2009), 355-376. doi: 10.1007/s00199-008-0338-8.
[32]	C. Villani, Optimal Transport: Old and New, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.