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Game theoretical modelling of a dynamically evolving network Ⅰ: General target sequences
Stability of the replicator dynamics for games in metric spaces
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 |
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.
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. |
[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. |
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. |
[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. |
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