doi: 10.3934/jdg.2021021
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Large deviations and Stochastic stability in Population Games

1. 

Maastricht University, Department of Data Science and Knowledge Engineering, Paul-Henri-Spaaklaan 1, 6229 EN Maastricht, The Netherlands

2. 

Technion-Israel Institute of Technology, Faculty of Industrial Engineering and Management, Haifa, 32000003, Israel

3. 

University of Wisconsin, Department of Economics, 1180 Observatory Drive, Madison, WI 53706, USA

* Corresponding author: Mathias Staudigl

Received  October 2020 Early access July 2021

Fund Project: M. Staudigl is supported by the COST Action CA-16228.Srinivas Arigapudi is supported in part at the Technion by a Fine Fellowship

In this article we review a model of stochastic evolution under general noisy best-response protocols, allowing the probabilities of suboptimal choices to depend on their payoff consequences. We survey the methods developed by the authors which allow for a quantitative analysis of these stochastic evolutionary game dynamics. We start with a compact survey of techniques designed to study the long run behavior in the small noise double limit (SNDL). In this regime we let the noise level in agents' decision rules to approach zero, and then the population size is formally taken to infinity. This iterated limit strategy yields a family of deterministic optimal control problems which admit an explicit analysis in many instances. We then move in by describing the main steps to analyze stochastic evolutionary game dynamics in the large population double limit (LPDL). This regime refers to the iterated limit in which first the population size is taken to infinity and then the noise level in agents' decisions vanishes. The mathematical analysis of LPDL relies on a sample-path large deviations principle for a family of Markov chains on compact polyhedra. In this setting we formulate a set of conjectures and open problems which give a clear direction for future research activities.

Citation: Mathias Staudigl, Srinivas Arigapudi, William H. Sandholm. Large deviations and Stochastic stability in Population Games. Journal of Dynamics & Games, doi: 10.3934/jdg.2021021
References:
[1]

S. Arigapudi, Exit from equilibrium in coordination games under probit choice, Games Econom. Behav., 122 (2020), 168-202.  doi: 10.1016/j.geb.2020.04.003.  Google Scholar

[2]

S. Arigapudi, Transitions between equilibria in bilingual games under logit choice, J. Math. Econom., 86 (2020), 24-34.  doi: 10.1016/j.jmateco.2019.10.004.  Google Scholar

[3] M. BeckmannC. B. McGuire and C. B. Winsten, Studies in the Economics of Transportation, Yale University Press, New Haven, CT, 1956.   Google Scholar
[4]

M. Benaïm, Recursive algorithms, urn processes, and chaining number of chain recurrent sets, Ergodic Theory Dynam. Systems, 18 (1998), 53-87.  doi: 10.1017/S0143385798097557.  Google Scholar

[5]

M. Benaïm and J. W. Weibull, Deterministic approximation of stochastic evolution in games, Econometrica, 71 (2003), 873-903.  doi: 10.1111/1468-0262.00429.  Google Scholar

[6]

M. Benaïm and J. W. Weibull, Mean-Field Approximation of Stochastic Population Processes in Games, Université de Neuchâtel and Stockholm School of Economics, 2009. Available from: https://hal.archives-ouvertes.fr/hal-00435515/document. Google Scholar

[7]

L. E. Blume, How noise matters, Games Econom. Behav., 44 (2003), 251-271.  doi: 10.1016/S0899-8256(02)00554-7.  Google Scholar

[8]

L. E. Blume, The statistical mechanics of strategic interaction, Games Econom. Behav., 5 (1993), 387-424.  doi: 10.1006/game.1993.1023.  Google Scholar

[9]

I. M. Bomze, Regularity versus degeneracy in dynamics, games, and optimization: A unified approach to different aspects, SIAM Rev., 44 (2002), 394-414.  doi: 10.1137/S00361445003756.  Google Scholar

[10]

A. Bovier and F. den Hollander, Metastability. A Potential-Theoretic Approach, Grundlehren der Mathematischen Wissenschaften, 351, Springer, Cham, 2015. doi: 10.1007/978-3-319-24777-9.  Google Scholar

[11]

O. Catoni, Simulated annealing algorithms and {M}arkov chains with rare transitions, in Séminaire de Probabilités XXXIII, Lecture Notes in Math., 1709, Springer, Berlin, 1999, 69–119. doi: 10.1007/BFb0096510.  Google Scholar

[12]

G. C. Chasparis, Stochastic stability of perturbed learning automata in positive-utility games, IEEE Trans. Automat. Control, 64 (2019), 4454-4469.  doi: 10.1109/TAC.2019.2895300.  Google Scholar

[13]

I.-K. ChoN. Williams and T. J. Sargent, Escaping Nash inflation, Rev. Econom. Stud., 69 (2002), 1-40.  doi: 10.1111/1467-937X.00196.  Google Scholar

[14]

E. Dokumacı and W. H. Sandholm, Large deviations and multinomial probit choice, J. Econom. Theory, 146 (2011), 2151-2158.  doi: 10.1016/j.jet.2011.06.013.  Google Scholar

[15]

P. Dupuis and R. S. Ellis, A Weak Convergence Approach to the Theory of Large Deviations, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1997. doi: 10.1002/9781118165904.  Google Scholar

[16]

G. Ellison, Basins of attraction, long-run stochastic stability, and the speed of step-by-step evolution, Rev. Econom. Stud., 67 (2000), 17-45.  doi: 10.1111/1467-937X.00119.  Google Scholar

[17]

D. Foster and P. Young, Stochastic evolutionary game dynamics, Theoret. Population Biol., 38 (1990), 219-232.  doi: 10.1016/0040-5809(90)90011-J.  Google Scholar

[18]

M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Grundlehren der Mathematischen Wissenschaften, 260, Springer, New York, 1998. doi: 10.1007/978-1-4612-0611-8.  Google Scholar

[19] J. C. Harsanyi and R. Selten, A General Theory of Equilibrium Selection in Games, MIT Press, Cambridge, MA, 1988.   Google Scholar
[20]

S. Hart and A. Mas-Colell, Uncoupled dynamics do not lead to Nash equilibrium, Amer. Econom. Rev., 93 (2003), 1830-1836.  doi: 10.1257/000282803322655581.  Google Scholar

[21]

J. Hofbauer, Deterministic evolutionary game dynamics, in Evolutionary Game Dynamics, Proc. Sympos. Appl. Math., 69, AMS Short Course Lecture Notes, Amer. Math. Soc., Providence, RI, 2011, 61–79. doi: 10.1090/psapm/069/2882634.  Google Scholar

[22]

J. Hofbauer, Stability for the Best Response Dynamics, unpublished manuscript, University of Vienna, 1995. Google Scholar

[23]

J. HofbauerJ. Oechssler and F. Riedel, Brown-von Neumann-Nash dynamics: The continuous strategy case, Games Econom. Behav., 65 (2009), 406-429.  doi: 10.1016/j.geb.2008.03.006.  Google Scholar

[24]

J. Hofbauer and W. H. Sandholm, Evolution in games with randomly disturbed payoffs, J. Econom. Theory, 132 (2007), 47-69.  doi: 10.1016/j.jet.2005.05.011.  Google Scholar

[25]

J. Hofbauer and W. H. Sandholm, Survival of dominated strategies under evolutionary dynamics, Theor. Econ., 6 (2011), 341-377.  doi: 10.3982/TE771.  Google Scholar

[26]

J. Hofbauer and K. Sigmund, Evolutionary game dynamics, Bull. Amer. Math. Soc. (N.S.), 40 (2003), 479-519.  doi: 10.1090/S0273-0979-03-00988-1.  Google Scholar

[27]

J. Hofbauer and S. Sorin, Best response dynamics for continuous zero-sum games, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 215-224.  doi: 10.3934/dcdsb.2006.6.215.  Google Scholar

[28]

Y. Kifer, A discrete-time version of the Wentzell-Freidlin theory, Ann. Probab, 18 (1990), 1676-1692.  doi: 10.1214/aop/1176990641.  Google Scholar

[29]

Y. Kifer, Random Perturbations of Dynamical Systems, Progress in Probability and Statistics, 16, Birkhäuser Boston, Inc., Boston, MA, 1988. doi: 10.1007/978-1-4615-8181-9.  Google Scholar

[30]

H. J. Kushner and G. G. Yin, Stochastic Approximation Algorithms and Applications, Applications of Mathematics (New York), 35, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4899-2696-8.  Google Scholar

[31]

J. R. Marden and J. S. Shamma, Game-Theoretic Learning in Distributed Control, Handbook of Dynamic Game Theory, Springer, Cham, 2018,511–546. doi: 10.1007/978-3-319-44374-4_9.  Google Scholar

[32]

J. R. MardenH. P. Young and L. Y. Pao, Achieving pareto optimality through distributed learning, SIAM J. Control Optim., 52 (2014), 2753-2770.  doi: 10.1137/110850694.  Google Scholar

[33]

J. Maynard Smith and G. R. Price, The logic of animal conflict, Nature, 246 (1973), 15-18.  doi: 10.1038/246015a0.  Google Scholar

[34]

P. Mertikopoulos and M. Staudigl, On the convergence of gradient-like flows with noisy gradient input, SIAM J. Optim., 28 (2018), 163-197.  doi: 10.1137/16M1105682.  Google Scholar

[35]

K. Michihiro and R. Rob, Bandwagon effects and long run technology choice, Games Econom. Behav., 22 (1998), 30-60.  doi: 10.1006/game.1997.0563.  Google Scholar

[36]

J. F. Nash Jr., Non-Cooperative Games, Ph.D thesis, Princeton University, 1950.  Google Scholar

[37]

D. Oyama and S. Takahashi, Contagion and uninvadability in local interaction games: The bilingual game and general supermodular games, J. Econom. Theory, 157 (2015), 100-127.  doi: 10.1016/j.jet.2014.12.012.  Google Scholar

[38]

N. QuijanoC. Ocampo-MartinezJ. Barreiro-GomezG. ObandoA. Pantoja and E. Mojica-Nava, The role of population games and evolutionary dynamics in distributed control systems: The advantages of evolutionary game theory, IEEE Control Syst., 37 (2017), 70-97.  doi: 10.1109/MCS.2016.2621479.  Google Scholar

[39] R. T. Rockafellar, Convex Analysis, Princeton Mathematical Series, 28, Princeton University Press, Princeton, NJ, 1970.   Google Scholar
[40]

R. W. Rosenthal, A class of games possessing pure-strategy Nash equilibria, Internat. J. Game Theory, 2 (1973), 65-67.  doi: 10.1007/BF01737559.  Google Scholar

[41]

W. H Sandholm, Evolution and equilibrium under inexact information, Games Econom. Behav., 44 (2003), 343-378.  doi: 10.1016/S0899-8256(03)00026-5.  Google Scholar

[42]

W. H. Sandholm, Orders of limits for stationary distributions, stochastic dominance, and stochastic stability, Theor. Econ., 5 (2010), 1-26.  doi: 10.3982/TE554.  Google Scholar

[43] W. H. Sandholm, Population Games and Evolutionary Dynamics, Economic Learning and Social Evolution, MIT Press, Cambridge, MA, 2010.   Google Scholar
[44]

W. H. Sandholm, Potential games with continuous player sets, J. Econom. Theory, 97 (2001), 81-108.  doi: 10.1006/jeth.2000.2696.  Google Scholar

[45]

W. H. Sandholm, Stochastic evolutionary game dynamics: Foundations, deterministic approximation, and equilibrium selection, in Evolutionary Game Dynamics, Proc. Sympos. Appl. Math., 69, AMS Short Course Lecture Notes, Amer. Math. Soc., Providence, RI, 2011,111–141. doi: 10.1090/psapm/069/2882636.  Google Scholar

[46]

W. H. Sandholm and M. Staudigl, Large deviations and stochastic stability in the small noise double limit, Theor. Econ., 11 (2016), 279-355.  doi: 10.3982/TE1905.  Google Scholar

[47]

W. H. Sandholm and M. Staudigl, Sample path large deviations for stochastic evolutionary game dynamics, Math. Oper. Res., 43 (2018), 1348-1377.  doi: 10.1287/moor.2017.0908.  Google Scholar

[48]

W. H. Sandholm, H. V. Tran and S. Arigapudi, Hamilton-Jacobi equations with semilinear costs and state constraints, with applications to large deviations in games, Math. Oper. Res.. doi: 10.1287/moor.2020.1114.  Google Scholar

[49]

K. SatsukawaK. Wada and T. Iryo, Stochastic stability of dynamic user equilibrium in unidirectional networks: Weakly acyclic game approach, Trans. Res. Part B: Methodological, 125 (2019), 229-247.  doi: 10.1016/j.trb.2019.05.015.  Google Scholar

[50]

D. Shah and J. Shin, Dynamics in congestion games, in Proceedings of the ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Systems, SIGMETRICS '10, New York, 2010,107–118. doi: 10.1145/1811039.1811052.  Google Scholar

[51]

M. Staudigl, Co-evolutionary dynamics and Bayesian interaction games, Internat. J. Game Theory, 42 (2013), 179-210.  doi: 10.1007/s00182-012-0331-0.  Google Scholar

[52]

M. Staudigl, Potential games in volatile environments, Games Econom. Behav., 72 (2011), 271-287.  doi: 10.1016/j.geb.2010.08.004.  Google Scholar

[53]

M. Staudigl, Stochastic stability in asymmetric binary choice coordination games, Games Econom. Behav., 75 (2012), 372-401.  doi: 10.1016/j.geb.2011.11.003.  Google Scholar

[54]

C. TaylorD. FudenbergA. Sasaki and M. A. Nowak, Evolutionary game dynamics in finite populations, Bull. Math. Biol., 66 (2004), 1621-1644.  doi: 10.1016/j.bulm.2004.03.004.  Google Scholar

[55]

P. D. Taylor and L. B. Jonker, Evolutionary stable strategies and game dynamics, Math. Biosci., 40 (1978), 145-156.  doi: 10.1016/0025-5564(78)90077-9.  Google Scholar

[56]

A. Traulsen and C. Hauert, Stochastic evolutionary game dynamics, in Reviews of Nonlinear Dynamics and Complexity. Vol. 2, Wiley-VCH Verlag, Weinheim, 2009, 25–61.  Google Scholar

[57]

N. Vielle, Small perturbations and stochastic games, Israel J. Math., 119 (2000), 127-142.  doi: 10.1007/BF02810665.  Google Scholar

[58]

J. W. Weibull, The mass action interpretation, J. Econom. Theory, 69 (1996), 165-171.   Google Scholar

[59]

H. P. Young, The evolution of conventions, Econometrica, 61 (1993), 57-84.  doi: 10.2307/2951778.  Google Scholar

[60]

D. Zhou and H. Qian, Fixation, transient landscape, and diffusion dilemma in stochastic evolutionary game dynamics, Phys. Rev. E, 84 (2011). doi: 10.1103/PhysRevE.84.031907.  Google Scholar

show all references

References:
[1]

S. Arigapudi, Exit from equilibrium in coordination games under probit choice, Games Econom. Behav., 122 (2020), 168-202.  doi: 10.1016/j.geb.2020.04.003.  Google Scholar

[2]

S. Arigapudi, Transitions between equilibria in bilingual games under logit choice, J. Math. Econom., 86 (2020), 24-34.  doi: 10.1016/j.jmateco.2019.10.004.  Google Scholar

[3] M. BeckmannC. B. McGuire and C. B. Winsten, Studies in the Economics of Transportation, Yale University Press, New Haven, CT, 1956.   Google Scholar
[4]

M. Benaïm, Recursive algorithms, urn processes, and chaining number of chain recurrent sets, Ergodic Theory Dynam. Systems, 18 (1998), 53-87.  doi: 10.1017/S0143385798097557.  Google Scholar

[5]

M. Benaïm and J. W. Weibull, Deterministic approximation of stochastic evolution in games, Econometrica, 71 (2003), 873-903.  doi: 10.1111/1468-0262.00429.  Google Scholar

[6]

M. Benaïm and J. W. Weibull, Mean-Field Approximation of Stochastic Population Processes in Games, Université de Neuchâtel and Stockholm School of Economics, 2009. Available from: https://hal.archives-ouvertes.fr/hal-00435515/document. Google Scholar

[7]

L. E. Blume, How noise matters, Games Econom. Behav., 44 (2003), 251-271.  doi: 10.1016/S0899-8256(02)00554-7.  Google Scholar

[8]

L. E. Blume, The statistical mechanics of strategic interaction, Games Econom. Behav., 5 (1993), 387-424.  doi: 10.1006/game.1993.1023.  Google Scholar

[9]

I. M. Bomze, Regularity versus degeneracy in dynamics, games, and optimization: A unified approach to different aspects, SIAM Rev., 44 (2002), 394-414.  doi: 10.1137/S00361445003756.  Google Scholar

[10]

A. Bovier and F. den Hollander, Metastability. A Potential-Theoretic Approach, Grundlehren der Mathematischen Wissenschaften, 351, Springer, Cham, 2015. doi: 10.1007/978-3-319-24777-9.  Google Scholar

[11]

O. Catoni, Simulated annealing algorithms and {M}arkov chains with rare transitions, in Séminaire de Probabilités XXXIII, Lecture Notes in Math., 1709, Springer, Berlin, 1999, 69–119. doi: 10.1007/BFb0096510.  Google Scholar

[12]

G. C. Chasparis, Stochastic stability of perturbed learning automata in positive-utility games, IEEE Trans. Automat. Control, 64 (2019), 4454-4469.  doi: 10.1109/TAC.2019.2895300.  Google Scholar

[13]

I.-K. ChoN. Williams and T. J. Sargent, Escaping Nash inflation, Rev. Econom. Stud., 69 (2002), 1-40.  doi: 10.1111/1467-937X.00196.  Google Scholar

[14]

E. Dokumacı and W. H. Sandholm, Large deviations and multinomial probit choice, J. Econom. Theory, 146 (2011), 2151-2158.  doi: 10.1016/j.jet.2011.06.013.  Google Scholar

[15]

P. Dupuis and R. S. Ellis, A Weak Convergence Approach to the Theory of Large Deviations, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1997. doi: 10.1002/9781118165904.  Google Scholar

[16]

G. Ellison, Basins of attraction, long-run stochastic stability, and the speed of step-by-step evolution, Rev. Econom. Stud., 67 (2000), 17-45.  doi: 10.1111/1467-937X.00119.  Google Scholar

[17]

D. Foster and P. Young, Stochastic evolutionary game dynamics, Theoret. Population Biol., 38 (1990), 219-232.  doi: 10.1016/0040-5809(90)90011-J.  Google Scholar

[18]

M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Grundlehren der Mathematischen Wissenschaften, 260, Springer, New York, 1998. doi: 10.1007/978-1-4612-0611-8.  Google Scholar

[19] J. C. Harsanyi and R. Selten, A General Theory of Equilibrium Selection in Games, MIT Press, Cambridge, MA, 1988.   Google Scholar
[20]

S. Hart and A. Mas-Colell, Uncoupled dynamics do not lead to Nash equilibrium, Amer. Econom. Rev., 93 (2003), 1830-1836.  doi: 10.1257/000282803322655581.  Google Scholar

[21]

J. Hofbauer, Deterministic evolutionary game dynamics, in Evolutionary Game Dynamics, Proc. Sympos. Appl. Math., 69, AMS Short Course Lecture Notes, Amer. Math. Soc., Providence, RI, 2011, 61–79. doi: 10.1090/psapm/069/2882634.  Google Scholar

[22]

J. Hofbauer, Stability for the Best Response Dynamics, unpublished manuscript, University of Vienna, 1995. Google Scholar

[23]

J. HofbauerJ. Oechssler and F. Riedel, Brown-von Neumann-Nash dynamics: The continuous strategy case, Games Econom. Behav., 65 (2009), 406-429.  doi: 10.1016/j.geb.2008.03.006.  Google Scholar

[24]

J. Hofbauer and W. H. Sandholm, Evolution in games with randomly disturbed payoffs, J. Econom. Theory, 132 (2007), 47-69.  doi: 10.1016/j.jet.2005.05.011.  Google Scholar

[25]

J. Hofbauer and W. H. Sandholm, Survival of dominated strategies under evolutionary dynamics, Theor. Econ., 6 (2011), 341-377.  doi: 10.3982/TE771.  Google Scholar

[26]

J. Hofbauer and K. Sigmund, Evolutionary game dynamics, Bull. Amer. Math. Soc. (N.S.), 40 (2003), 479-519.  doi: 10.1090/S0273-0979-03-00988-1.  Google Scholar

[27]

J. Hofbauer and S. Sorin, Best response dynamics for continuous zero-sum games, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 215-224.  doi: 10.3934/dcdsb.2006.6.215.  Google Scholar

[28]

Y. Kifer, A discrete-time version of the Wentzell-Freidlin theory, Ann. Probab, 18 (1990), 1676-1692.  doi: 10.1214/aop/1176990641.  Google Scholar

[29]

Y. Kifer, Random Perturbations of Dynamical Systems, Progress in Probability and Statistics, 16, Birkhäuser Boston, Inc., Boston, MA, 1988. doi: 10.1007/978-1-4615-8181-9.  Google Scholar

[30]

H. J. Kushner and G. G. Yin, Stochastic Approximation Algorithms and Applications, Applications of Mathematics (New York), 35, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4899-2696-8.  Google Scholar

[31]

J. R. Marden and J. S. Shamma, Game-Theoretic Learning in Distributed Control, Handbook of Dynamic Game Theory, Springer, Cham, 2018,511–546. doi: 10.1007/978-3-319-44374-4_9.  Google Scholar

[32]

J. R. MardenH. P. Young and L. Y. Pao, Achieving pareto optimality through distributed learning, SIAM J. Control Optim., 52 (2014), 2753-2770.  doi: 10.1137/110850694.  Google Scholar

[33]

J. Maynard Smith and G. R. Price, The logic of animal conflict, Nature, 246 (1973), 15-18.  doi: 10.1038/246015a0.  Google Scholar

[34]

P. Mertikopoulos and M. Staudigl, On the convergence of gradient-like flows with noisy gradient input, SIAM J. Optim., 28 (2018), 163-197.  doi: 10.1137/16M1105682.  Google Scholar

[35]

K. Michihiro and R. Rob, Bandwagon effects and long run technology choice, Games Econom. Behav., 22 (1998), 30-60.  doi: 10.1006/game.1997.0563.  Google Scholar

[36]

J. F. Nash Jr., Non-Cooperative Games, Ph.D thesis, Princeton University, 1950.  Google Scholar

[37]

D. Oyama and S. Takahashi, Contagion and uninvadability in local interaction games: The bilingual game and general supermodular games, J. Econom. Theory, 157 (2015), 100-127.  doi: 10.1016/j.jet.2014.12.012.  Google Scholar

[38]

N. QuijanoC. Ocampo-MartinezJ. Barreiro-GomezG. ObandoA. Pantoja and E. Mojica-Nava, The role of population games and evolutionary dynamics in distributed control systems: The advantages of evolutionary game theory, IEEE Control Syst., 37 (2017), 70-97.  doi: 10.1109/MCS.2016.2621479.  Google Scholar

[39] R. T. Rockafellar, Convex Analysis, Princeton Mathematical Series, 28, Princeton University Press, Princeton, NJ, 1970.   Google Scholar
[40]

R. W. Rosenthal, A class of games possessing pure-strategy Nash equilibria, Internat. J. Game Theory, 2 (1973), 65-67.  doi: 10.1007/BF01737559.  Google Scholar

[41]

W. H Sandholm, Evolution and equilibrium under inexact information, Games Econom. Behav., 44 (2003), 343-378.  doi: 10.1016/S0899-8256(03)00026-5.  Google Scholar

[42]

W. H. Sandholm, Orders of limits for stationary distributions, stochastic dominance, and stochastic stability, Theor. Econ., 5 (2010), 1-26.  doi: 10.3982/TE554.  Google Scholar

[43] W. H. Sandholm, Population Games and Evolutionary Dynamics, Economic Learning and Social Evolution, MIT Press, Cambridge, MA, 2010.   Google Scholar
[44]

W. H. Sandholm, Potential games with continuous player sets, J. Econom. Theory, 97 (2001), 81-108.  doi: 10.1006/jeth.2000.2696.  Google Scholar

[45]

W. H. Sandholm, Stochastic evolutionary game dynamics: Foundations, deterministic approximation, and equilibrium selection, in Evolutionary Game Dynamics, Proc. Sympos. Appl. Math., 69, AMS Short Course Lecture Notes, Amer. Math. Soc., Providence, RI, 2011,111–141. doi: 10.1090/psapm/069/2882636.  Google Scholar

[46]

W. H. Sandholm and M. Staudigl, Large deviations and stochastic stability in the small noise double limit, Theor. Econ., 11 (2016), 279-355.  doi: 10.3982/TE1905.  Google Scholar

[47]

W. H. Sandholm and M. Staudigl, Sample path large deviations for stochastic evolutionary game dynamics, Math. Oper. Res., 43 (2018), 1348-1377.  doi: 10.1287/moor.2017.0908.  Google Scholar

[48]

W. H. Sandholm, H. V. Tran and S. Arigapudi, Hamilton-Jacobi equations with semilinear costs and state constraints, with applications to large deviations in games, Math. Oper. Res.. doi: 10.1287/moor.2020.1114.  Google Scholar

[49]

K. SatsukawaK. Wada and T. Iryo, Stochastic stability of dynamic user equilibrium in unidirectional networks: Weakly acyclic game approach, Trans. Res. Part B: Methodological, 125 (2019), 229-247.  doi: 10.1016/j.trb.2019.05.015.  Google Scholar

[50]

D. Shah and J. Shin, Dynamics in congestion games, in Proceedings of the ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Systems, SIGMETRICS '10, New York, 2010,107–118. doi: 10.1145/1811039.1811052.  Google Scholar

[51]

M. Staudigl, Co-evolutionary dynamics and Bayesian interaction games, Internat. J. Game Theory, 42 (2013), 179-210.  doi: 10.1007/s00182-012-0331-0.  Google Scholar

[52]

M. Staudigl, Potential games in volatile environments, Games Econom. Behav., 72 (2011), 271-287.  doi: 10.1016/j.geb.2010.08.004.  Google Scholar

[53]

M. Staudigl, Stochastic stability in asymmetric binary choice coordination games, Games Econom. Behav., 75 (2012), 372-401.  doi: 10.1016/j.geb.2011.11.003.  Google Scholar

[54]

C. TaylorD. FudenbergA. Sasaki and M. A. Nowak, Evolutionary game dynamics in finite populations, Bull. Math. Biol., 66 (2004), 1621-1644.  doi: 10.1016/j.bulm.2004.03.004.  Google Scholar

[55]

P. D. Taylor and L. B. Jonker, Evolutionary stable strategies and game dynamics, Math. Biosci., 40 (1978), 145-156.  doi: 10.1016/0025-5564(78)90077-9.  Google Scholar

[56]

A. Traulsen and C. Hauert, Stochastic evolutionary game dynamics, in Reviews of Nonlinear Dynamics and Complexity. Vol. 2, Wiley-VCH Verlag, Weinheim, 2009, 25–61.  Google Scholar

[57]

N. Vielle, Small perturbations and stochastic games, Israel J. Math., 119 (2000), 127-142.  doi: 10.1007/BF02810665.  Google Scholar

[58]

J. W. Weibull, The mass action interpretation, J. Econom. Theory, 69 (1996), 165-171.   Google Scholar

[59]

H. P. Young, The evolution of conventions, Econometrica, 61 (1993), 57-84.  doi: 10.2307/2951778.  Google Scholar

[60]

D. Zhou and H. Qian, Fixation, transient landscape, and diffusion dilemma in stochastic evolutionary game dynamics, Phys. Rev. E, 84 (2011). doi: 10.1103/PhysRevE.84.031907.  Google Scholar

Figure 1.  Optimal exit paths with non-binding state constraints
Figure 2.  Optimal exit paths with indirect exit and binding state constraints
Figure 3.  Optimal transition paths in Bilingual games
Figure 4.  Phase portrait of the logit dynamics $ V^{0.1} $ for example (46)
Table 1.  Coordination Game
Table 2.  Bilingual Game
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