doi: 10.3934/jdg.2022018
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Learning in nonatomic games, part Ⅰ: Finite action spaces and population games

1. 

Tweag I/O, Paris, France

2. 

CNRS (Lamsade, University of Dauphine-PSL), Paris, France, University of Liverpool (Computer Science Department), Liverpool, UK

3. 

Univ. Grenoble Alpes, CNRS, Inria, Grenoble INP, LIG, 38000, Grenoble, France

4. 

Institut de Mathématiques Jussieu-PRG, Sorbonne Université, UPMC, CNRS UMR 7586, Paris, France

*Corresponding author: Panayotis Mertikopoulos

Received  July 2021 Revised  March 2022 Early access August 2022

Fund Project: The authors are grateful to an anonymous referee for their constructive remarks and comments

We examine the long-run behavior of a wide range of dynamics for learning in nonatomic games, in both discrete and continuous time. The class of dynamics under consideration includes fictitious play and its regularized variants, the best reply dynamics (again, possibly regularized), as well as the dynamics of dual averaging / "follow the regularized leader" (which themselves include as special cases the replicator dynamics and Friedman's projection dynamics). Our analysis concerns both the actual trajectory of play and its time-average, and we cover potential and monotone games, as well as games with an evolutionarily stable state (global or otherwise). We focus exclusively on games with finite action spaces; nonatomic games with continuous action spaces are treated in detail in Part Ⅱ.

Citation: Saeed Hadikhanloo, Rida Laraki, Panayotis Mertikopoulos, Sylvain Sorin. Learning in nonatomic games, part Ⅰ: Finite action spaces and population games. Journal of Dynamics and Games, doi: 10.3934/jdg.2022018
References:
[1]

H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd edition, Springer, Cham, 2017. doi: 10.1007/978-3-319-48311-5.

[2]

M. Benaïm, Dynamics of stochastic approximation algorithms, Séminaire de Probabilités XXXIII, 1709 (1999), 1-68.  doi: 10.1007/BFb0096509.

[3]

M. Benaïm and M. Faure, Consistency of vanishingly smooth fictitious play, Math. Oper. Res., 38 (2013), 437-450.  doi: 10.1287/moor.1120.0568.

[4]

M. BenaïmJ. Hofbauer and S. Sorin, Stochastic approximations and differential inclusions, SIAM J. Control Optim., 44 (2005), 328-348.  doi: 10.1137/S0363012904439301.

[5]

M. BenaïmJ. Hofbauer and S. Sorin, Stochastic approximations and differential inclusions, part Ⅱ: Applications, Applications. Math. Oper. Res., 31 (2006), 673-695.  doi: 10.1287/moor.1060.0213.

[6]

C. Berge, Topological Spaces, Dover Publications, Inc., Mineola, NY, 1997.

[7]

M. Bravo and P. Mertikopoulos, On the robustness of learning in games with stochastically perturbed payoff observations, Games Econom. Behav., 103 (2017), 41-66.  doi: 10.1016/j.geb.2016.06.004.

[8]

G. W. Brown, Iterative solutions of games by fictitious play, Activity Analysis of Productions and Allocation, (1951), 374-376. 

[9] N. Cesa-Bianchi and G. Lugosi, Prediction, Learning, and Games, Cambridge University Press, 2006.  doi: 10.1017/CBO9780511546921.
[10]

C. Conley, Isolated Invariant Set and the Morse Index, American Mathematical Society, Providence, RI, 1978.

[11]

S. Dafermos, Traffic equilibrium and variational inequalities, Transportation Sci., 14 (1980), 42-54.  doi: 10.1287/trsc.14.1.42.

[12]

E. van Damme, Stability and Perfection of Nash Equilibria, Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-642-96978-2.

[13]

D. Friedman, Evolutionary games in economics, Econometrica, 59 (1991), 637-666.  doi: 10.2307/2938222.

[14] D. Fudenberg and D. K. Levine, The Theory of Learning in Games, MIT Press, Cambridge, MA, 1998. 
[15]

D. Fudenberg and D. K. Levine, Conditional universal consistency, Games and Economic Behavior, 29 (1999), 104-130.  doi: 10.1006/game.1998.0705.

[16]

I. Gilboa and A. Matsui, Social stability and equilibrium, Econometrica, 59 (1991), 859-867.  doi: 10.2307/2938230.

[17]

S. Hadikhanloo, R. Laraki, P. Mertikopoulos and S. Sorin, Learning in Nonatomic Games, Part Ⅱ: Continuous action spaces and mean-field games. mimeo, 2021.

[18]

A. Héliou, M. Martin, P. Mertikopoulos and T. Rahier, Online non-convex optimization with imperfect feedback, In NeurIPS '20: Proceedings of the 34th International Conference on Neural Information Processing Systems, 2020.

[19]

J. Hofbauer and W. H. Sandholm, On the global convergence of stochastic fictitious play, Econometrica, 70 (2002), 2265-2294. 

[20]

J. Hofbauer and W. H. Sandholm, Stable games and their dynamics, J. Econom. Theory, 144 (2009), 1665-1693.  doi: 10.1016/j.jet.2009.01.007.

[21]

J. HofbauerP. Schuster and K. Sigmund, A note on evolutionarily stable strategies and game dynamics, J. Theoret. Biol., 81 (1979), 609-612.  doi: 10.1016/0022-5193(79)90058-4.

[22] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, UK, 1998.  doi: 10.1017/CBO9781139173179.
[23]

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

[24]

J. HofbauerS. Sorin and Y. Viossat, Time average replicator and best reply dynamics, Math. Oper. Res., 34 (2009), 263-269.  doi: 10.1287/moor.1080.0359.

[25]

A. JuditskyA. S. Nemirovski and C. Tauvel, Solving variational inequalities with stochastic mirror-prox algorithm, Stoch. Syst., 1 (2011), 17-58.  doi: 10.1214/10-SSY011.

[26]

J. Kwon, Stratégies de Descente Miroir pour La Minimisation du Regret et L'approchabilité, PhD thesis, Université Pierre-et-Marie-Curie, 2016.

[27]

J. Kwon and P. Mertikopoulos, A continuous-time approach to online optimization, J. Dyn. Games, 4 (2017), 125-148.  doi: 10.3934/jdg.2017008.

[28]

A. Mas-Colell, On a theorem of Schmeidler, J. Math. Econom., 13 (1984), 201-206.  doi: 10.1016/0304-4068(84)90029-6.

[29]

J. Maynard Smith and G. R. Price, The logic of animal conflict, Nature, 246 (1973), 15-18. 

[30]

D. L. McFadden, Conditional logit analysis of qualitative choice behavior, Frontiers in Econometrics, (1974), 105-142. 

[31]

R. D. McKelvey and T. R. Palfrey, Quantal response equilibria for normal form games, Games Econom. Behav., 10 (1995), 6-38.  doi: 10.1006/game.1995.1023.

[32]

P. Mertikopoulos and A. L. Moustakas, The emergence of rational behavior in the presence of stochastic perturbations, Ann. Appl. Probab., 20 (2010), 1359-1388.  doi: 10.1214/09-AAP651.

[33]

P. Mertikopoulos and W. H. Sandholm, Learning in games via reinforcement and regularization, Math. Oper. Res., 41 (2016), 1297-1324.  doi: 10.1287/moor.2016.0778.

[34]

P. Mertikopoulos and W. H. Sandholm, Riemannian game dynamics, J. Econom. Theory, 177 (2018), 315-364.  doi: 10.1016/j.jet.2018.06.002.

[35]

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.

[36]

P. Mertikopoulos and Z. Zhou, Learning in games with continuous action sets and unknown payoff functions, Math. Program., 173 (2019), 465-507.  doi: 10.1007/s10107-018-1254-8.

[37]

G. J. Minty, On the generalization of a direct method of the calculus of variations, Bull. Amer. Math. Soc., 73 (1967), 315-321.  doi: 10.1090/S0002-9904-1967-11732-4.

[38]

D. Monderer and L. S. Shapley, Potential games, Games Econom. Behav., 14 (1996), 124-143.  doi: 10.1006/game.1996.0044.

[39]

D. Monderer and L. S. Shapley, Fictitious play property for games with identical interests, J. Economic Theory, 68 (1996), 258-265.  doi: 10.1006/jeth.1996.0014.

[40]

A. Nemirovski, Prox-method with rate of convergence ${O}(1/t)$ for variational inequalities with Lipschitz continuous monotone operators and smooth convex-concave saddle point problems, SIAM J. Optim., 15 (2004), 229-251.  doi: 10.1137/S1052623403425629.

[41]

Y. Nesterov, Dual extrapolation and its applications to solving variational inequalities and related problems, Math. Program., 109 (2007), 319-344.  doi: 10.1007/s10107-006-0034-z.

[42]

Y. Nesterov, Primal-dual subgradient methods for convex problems, Math. Program., 120 (2009), 221-259.  doi: 10.1007/s10107-007-0149-x.

[43]

J. Robinson, An iterative method for solving a game, Ann. of Math., 54 (1951), 296-301.  doi: 10.2307/1969530.

[44] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, N. J., 1970. 
[45]

A. Rustichini, Optimal properties of stimulus-response learning models, Games Econom. Behav., 29 (1999), 244-273.  doi: 10.1006/game.1999.0712.

[46]

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

[47] W. H. Sandholm, Population Games and Evolutionary Dynamics, MIT Press, Cambridge, MA, 2010. 
[48]

W. H. Sandholm, Population games and deterministic evolutionary dynamics, Handbook of Game Theory IV, (2015), 703-778. 

[49]

D. Schmeidler, Equilibrium points of nonatomic games, J. Statist. Phys., 7 (1973), 295-300.  doi: 10.1007/BF01014905.

[50]

S. Shalev-Shwartz, Online learning and online convex optimization, Foundations and Trends in Machine Learning, 4 (2011), 107-194. 

[51]

S. Shalev-Shwartz and Y. Singer, Convex repeated games and Fenchel duality, In NIPS' 06: Proceedings of the 19th Annual Conference on Neural Information Processing Systems, (2006), 1265–1272.

[52]

S. Sorin, Exponential weight algorithm in continuous time, Math. Program., 116 (2009), 513-528.  doi: 10.1007/s10107-007-0111-y.

[53]

S. Sorin, Replicator dynamics: Old and new, J. Dyn. Games, 7 (2020), 365-386.  doi: 10.3934/jdg.2020028.

[54]

S. Sorin, No-Regret Algorithms in Online Learning, Games and Convex Optimization, Mimeo, 2021.

[55]

S. Sorin and C. Wan, Finite composite games: Equilibria and dynamics, J. Dyn. Games, 3 (2016), 101-120.  doi: 10.3934/jdg.2016005.

[56]

P. D. Taylor, Evolutionarily stable strategies with two types of player, J. Appl. Probab., 16 (1979), 76-83.  doi: 10.2307/3213376.

[57]

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.

[58] W. Weibull Jörgen, Evolutionary Game Theory, MIT Press, Cambridge, MA, 1995. 

show all references

References:
[1]

H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd edition, Springer, Cham, 2017. doi: 10.1007/978-3-319-48311-5.

[2]

M. Benaïm, Dynamics of stochastic approximation algorithms, Séminaire de Probabilités XXXIII, 1709 (1999), 1-68.  doi: 10.1007/BFb0096509.

[3]

M. Benaïm and M. Faure, Consistency of vanishingly smooth fictitious play, Math. Oper. Res., 38 (2013), 437-450.  doi: 10.1287/moor.1120.0568.

[4]

M. BenaïmJ. Hofbauer and S. Sorin, Stochastic approximations and differential inclusions, SIAM J. Control Optim., 44 (2005), 328-348.  doi: 10.1137/S0363012904439301.

[5]

M. BenaïmJ. Hofbauer and S. Sorin, Stochastic approximations and differential inclusions, part Ⅱ: Applications, Applications. Math. Oper. Res., 31 (2006), 673-695.  doi: 10.1287/moor.1060.0213.

[6]

C. Berge, Topological Spaces, Dover Publications, Inc., Mineola, NY, 1997.

[7]

M. Bravo and P. Mertikopoulos, On the robustness of learning in games with stochastically perturbed payoff observations, Games Econom. Behav., 103 (2017), 41-66.  doi: 10.1016/j.geb.2016.06.004.

[8]

G. W. Brown, Iterative solutions of games by fictitious play, Activity Analysis of Productions and Allocation, (1951), 374-376. 

[9] N. Cesa-Bianchi and G. Lugosi, Prediction, Learning, and Games, Cambridge University Press, 2006.  doi: 10.1017/CBO9780511546921.
[10]

C. Conley, Isolated Invariant Set and the Morse Index, American Mathematical Society, Providence, RI, 1978.

[11]

S. Dafermos, Traffic equilibrium and variational inequalities, Transportation Sci., 14 (1980), 42-54.  doi: 10.1287/trsc.14.1.42.

[12]

E. van Damme, Stability and Perfection of Nash Equilibria, Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-642-96978-2.

[13]

D. Friedman, Evolutionary games in economics, Econometrica, 59 (1991), 637-666.  doi: 10.2307/2938222.

[14] D. Fudenberg and D. K. Levine, The Theory of Learning in Games, MIT Press, Cambridge, MA, 1998. 
[15]

D. Fudenberg and D. K. Levine, Conditional universal consistency, Games and Economic Behavior, 29 (1999), 104-130.  doi: 10.1006/game.1998.0705.

[16]

I. Gilboa and A. Matsui, Social stability and equilibrium, Econometrica, 59 (1991), 859-867.  doi: 10.2307/2938230.

[17]

S. Hadikhanloo, R. Laraki, P. Mertikopoulos and S. Sorin, Learning in Nonatomic Games, Part Ⅱ: Continuous action spaces and mean-field games. mimeo, 2021.

[18]

A. Héliou, M. Martin, P. Mertikopoulos and T. Rahier, Online non-convex optimization with imperfect feedback, In NeurIPS '20: Proceedings of the 34th International Conference on Neural Information Processing Systems, 2020.

[19]

J. Hofbauer and W. H. Sandholm, On the global convergence of stochastic fictitious play, Econometrica, 70 (2002), 2265-2294. 

[20]

J. Hofbauer and W. H. Sandholm, Stable games and their dynamics, J. Econom. Theory, 144 (2009), 1665-1693.  doi: 10.1016/j.jet.2009.01.007.

[21]

J. HofbauerP. Schuster and K. Sigmund, A note on evolutionarily stable strategies and game dynamics, J. Theoret. Biol., 81 (1979), 609-612.  doi: 10.1016/0022-5193(79)90058-4.

[22] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, UK, 1998.  doi: 10.1017/CBO9781139173179.
[23]

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

[24]

J. HofbauerS. Sorin and Y. Viossat, Time average replicator and best reply dynamics, Math. Oper. Res., 34 (2009), 263-269.  doi: 10.1287/moor.1080.0359.

[25]

A. JuditskyA. S. Nemirovski and C. Tauvel, Solving variational inequalities with stochastic mirror-prox algorithm, Stoch. Syst., 1 (2011), 17-58.  doi: 10.1214/10-SSY011.

[26]

J. Kwon, Stratégies de Descente Miroir pour La Minimisation du Regret et L'approchabilité, PhD thesis, Université Pierre-et-Marie-Curie, 2016.

[27]

J. Kwon and P. Mertikopoulos, A continuous-time approach to online optimization, J. Dyn. Games, 4 (2017), 125-148.  doi: 10.3934/jdg.2017008.

[28]

A. Mas-Colell, On a theorem of Schmeidler, J. Math. Econom., 13 (1984), 201-206.  doi: 10.1016/0304-4068(84)90029-6.

[29]

J. Maynard Smith and G. R. Price, The logic of animal conflict, Nature, 246 (1973), 15-18. 

[30]

D. L. McFadden, Conditional logit analysis of qualitative choice behavior, Frontiers in Econometrics, (1974), 105-142. 

[31]

R. D. McKelvey and T. R. Palfrey, Quantal response equilibria for normal form games, Games Econom. Behav., 10 (1995), 6-38.  doi: 10.1006/game.1995.1023.

[32]

P. Mertikopoulos and A. L. Moustakas, The emergence of rational behavior in the presence of stochastic perturbations, Ann. Appl. Probab., 20 (2010), 1359-1388.  doi: 10.1214/09-AAP651.

[33]

P. Mertikopoulos and W. H. Sandholm, Learning in games via reinforcement and regularization, Math. Oper. Res., 41 (2016), 1297-1324.  doi: 10.1287/moor.2016.0778.

[34]

P. Mertikopoulos and W. H. Sandholm, Riemannian game dynamics, J. Econom. Theory, 177 (2018), 315-364.  doi: 10.1016/j.jet.2018.06.002.

[35]

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.

[36]

P. Mertikopoulos and Z. Zhou, Learning in games with continuous action sets and unknown payoff functions, Math. Program., 173 (2019), 465-507.  doi: 10.1007/s10107-018-1254-8.

[37]

G. J. Minty, On the generalization of a direct method of the calculus of variations, Bull. Amer. Math. Soc., 73 (1967), 315-321.  doi: 10.1090/S0002-9904-1967-11732-4.

[38]

D. Monderer and L. S. Shapley, Potential games, Games Econom. Behav., 14 (1996), 124-143.  doi: 10.1006/game.1996.0044.

[39]

D. Monderer and L. S. Shapley, Fictitious play property for games with identical interests, J. Economic Theory, 68 (1996), 258-265.  doi: 10.1006/jeth.1996.0014.

[40]

A. Nemirovski, Prox-method with rate of convergence ${O}(1/t)$ for variational inequalities with Lipschitz continuous monotone operators and smooth convex-concave saddle point problems, SIAM J. Optim., 15 (2004), 229-251.  doi: 10.1137/S1052623403425629.

[41]

Y. Nesterov, Dual extrapolation and its applications to solving variational inequalities and related problems, Math. Program., 109 (2007), 319-344.  doi: 10.1007/s10107-006-0034-z.

[42]

Y. Nesterov, Primal-dual subgradient methods for convex problems, Math. Program., 120 (2009), 221-259.  doi: 10.1007/s10107-007-0149-x.

[43]

J. Robinson, An iterative method for solving a game, Ann. of Math., 54 (1951), 296-301.  doi: 10.2307/1969530.

[44] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, N. J., 1970. 
[45]

A. Rustichini, Optimal properties of stimulus-response learning models, Games Econom. Behav., 29 (1999), 244-273.  doi: 10.1006/game.1999.0712.

[46]

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

[47] W. H. Sandholm, Population Games and Evolutionary Dynamics, MIT Press, Cambridge, MA, 2010. 
[48]

W. H. Sandholm, Population games and deterministic evolutionary dynamics, Handbook of Game Theory IV, (2015), 703-778. 

[49]

D. Schmeidler, Equilibrium points of nonatomic games, J. Statist. Phys., 7 (1973), 295-300.  doi: 10.1007/BF01014905.

[50]

S. Shalev-Shwartz, Online learning and online convex optimization, Foundations and Trends in Machine Learning, 4 (2011), 107-194. 

[51]

S. Shalev-Shwartz and Y. Singer, Convex repeated games and Fenchel duality, In NIPS' 06: Proceedings of the 19th Annual Conference on Neural Information Processing Systems, (2006), 1265–1272.

[52]

S. Sorin, Exponential weight algorithm in continuous time, Math. Program., 116 (2009), 513-528.  doi: 10.1007/s10107-007-0111-y.

[53]

S. Sorin, Replicator dynamics: Old and new, J. Dyn. Games, 7 (2020), 365-386.  doi: 10.3934/jdg.2020028.

[54]

S. Sorin, No-Regret Algorithms in Online Learning, Games and Convex Optimization, Mimeo, 2021.

[55]

S. Sorin and C. Wan, Finite composite games: Equilibria and dynamics, J. Dyn. Games, 3 (2016), 101-120.  doi: 10.3934/jdg.2016005.

[56]

P. D. Taylor, Evolutionarily stable strategies with two types of player, J. Appl. Probab., 16 (1979), 76-83.  doi: 10.2307/3213376.

[57]

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.

[58] W. Weibull Jörgen, Evolutionary Game Theory, MIT Press, Cambridge, MA, 1995. 
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