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October  2017, 4(4): 319-333. doi: 10.3934/jdg.2017017

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

Received  March 2017 Revised  June 2017 Published  September 2017

Fund Project: This research was partially supported by CONACYT Grant 221291.

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.

Keywords: Evolutionary games, population games, replicator dynamics, space of finite signed measures.
Mathematics Subject Classification: Primary: 91A22, 91A10; Secondary: 34A34, 34G20.
Citation: Saul Mendoza-Palacios, Onésimo Hernández-Lerma. Stability of the replicator dynamics for games in metric spaces. Journal of Dynamics & Games, 2017, 4 (4) : 319-333. doi: 10.3934/jdg.2017017
References:
[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.  Google Scholar

[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.  Google Scholar

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A. Bobrowski, Functional Analysis for Probability and Stochastic Processes: An Introduction, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511614583.  Google Scholar

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I. M. Bomze, Dynamical aspects of evolutionary stability, Monatshefte für Mathematik, 110 (1990), 189-206.  doi: 10.1007/BF01301675.  Google Scholar

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I. M. Bomze, Cross entropy minimization in uninvadable states of complex populations, Journal of Mathematical Biology, 30 (1991), 73-87.  doi: 10.1007/BF00168008.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9781139173179.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

A. Mas-Colell, M. D. Whinston and J. R. Green, Microeconomic Theory, Oxford university Press, 1995. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

J. R. Munkres, Topology, Second edition, Prentice Hall, 2000. Google Scholar

[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.  Google Scholar

[23]

J. Oechssler and F. Riedel, Evolutionary dynamics on infinite strategy spaces, Economic Theory, 17 (2001), 141-162.  doi: 10.1007/PL00004092.  Google Scholar

[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.  Google Scholar

[25]

G. K. Pedersen, Analysis Now, Springer, New York, 1989. doi: 10.1007/978-1-4612-1007-8.  Google Scholar

[26]

R. -D. Reiss, Approximate Distributions of Order Statistics, Springer, New York, 1989. doi: 10.1007/978-1-4613-9620-8.  Google Scholar

[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.  Google Scholar

[28]

H. L. Royden, Real Analysis, Third edition, Macmillan, New York, 1988.  Google Scholar

[29]

W. H. Sandholm, Population Games and Evolutionary Dynamics, MIT press, 2010.  Google Scholar

[30]

A. N. Shiryaev, Probability, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4757-2539-1.  Google Scholar

[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.  Google Scholar

[32]

C. Villani, Optimal Transport: Old and New, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.  Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[3]

A. Bobrowski, Functional Analysis for Probability and Stochastic Processes: An Introduction, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511614583.  Google Scholar

[4]

I. M. Bomze, Dynamical aspects of evolutionary stability, Monatshefte für Mathematik, 110 (1990), 189-206.  doi: 10.1007/BF01301675.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9781139173179.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

A. Mas-Colell, M. D. Whinston and J. R. Green, Microeconomic Theory, Oxford university Press, 1995. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

J. R. Munkres, Topology, Second edition, Prentice Hall, 2000. Google Scholar

[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.  Google Scholar

[23]

J. Oechssler and F. Riedel, Evolutionary dynamics on infinite strategy spaces, Economic Theory, 17 (2001), 141-162.  doi: 10.1007/PL00004092.  Google Scholar

[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.  Google Scholar

[25]

G. K. Pedersen, Analysis Now, Springer, New York, 1989. doi: 10.1007/978-1-4612-1007-8.  Google Scholar

[26]

R. -D. Reiss, Approximate Distributions of Order Statistics, Springer, New York, 1989. doi: 10.1007/978-1-4613-9620-8.  Google Scholar

[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.  Google Scholar

[28]

H. L. Royden, Real Analysis, Third edition, Macmillan, New York, 1988.  Google Scholar

[29]

W. H. Sandholm, Population Games and Evolutionary Dynamics, MIT press, 2010.  Google Scholar

[30]

A. N. Shiryaev, Probability, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4757-2539-1.  Google Scholar

[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.  Google Scholar

[32]

C. Villani, Optimal Transport: Old and New, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.  Google Scholar

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