October  2020, 7(4): 365-386. doi: 10.3934/jdg.2020028

Replicator dynamics: Old and new

Institut de Mathématiques Jussieu-PRG, Sorbonne Université, Campus P. & M. Curie, CNRS UMR 7586, 4 Place Jussieu, 75005 Paris, France

Received  October 2019 Published  October 2020 Early access  September 2020

Fund Project: Part of this work was presented at "Journées Franco-Chiliennes d'Optimisation", Toulouse, July 2017, and dedicated to the memory of Felipe Alvarez. This research was partially supported by a PGMO grant COGLED. The author thanks Josef Hofbauer for many constructive comments and a referee for an extremely precise and helpful report

We introduce the unilateral version associated to the replicator dynamics and describe its connection to on-line learning procedures, in particular to the multiplicative weight algorithm. We show the interest of handling simultaneously discrete and continuous time analysis.

We then survey recent results on extensions of this dynamics as maximization of the cumulative outcome with alternative regularization functions and variable weights. This includes no regret algorithms, time average version and link to best reply dynamics in two person games, application to equilibria and variational inequalities, convergence properties in potential and dissipative games.

Citation: Sylvain Sorin. Replicator dynamics: Old and new. Journal of Dynamics and Games, 2020, 7 (4) : 365-386. doi: 10.3934/jdg.2020028
References:
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E. Akin, The Geometry of Population Genetics, Springer, 1979.

[2]

E. Akin and J. Hofbauer, Recurrence of the unfit, Mathematical Biosciences, 61 (1982), 51-62.  doi: 10.1016/0025-5564(82)90095-5.

[3]

F. AlvarezJ. Bolte and O. Brahic, Hessian Riemannian gradient flows in convex programming, SIAM Journal on Control and Optimization, 43 (2004), 477-501.  doi: 10.1137/S0363012902419977.

[4]

S. AroraE. Hazan and S. Kale, The multiplicative weights update method: A meta algorithm and applications, Theory of Computing, 8 (2012), 121-164.  doi: 10.4086/toc.2012.v008a006.

[5]

P. Auer, N. Cesa–Bianchi, Y. Freund and R. E. Shapire, The nonstochastic multiarmed bandit problem, SIAM J. Comput., 32 (2002), 48–77. doi: 10.1137/S0097539701398375.

[6]

R. J. Aumann, Subjectivity and correlation in randomized strategies, Journal of Mathematical Economics, 1 (1974), 67-96.  doi: 10.1016/0304–4068(74)90037–8.

[7]

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

[8]

M. Benaim and M. Faure, Consistency of vanishingly smooth fictitious play, Mathematics of Operations Research, 38 (2013), 437–450. doi: 10.1287/moor.1120.0568.

[9]

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

[10]

M. Benaim, J. Hofbauer and S. Sorin, Stochastic approximations and differential inclusions. Part Ⅱ: Applications, Mathematics of Operations Research, 31 (2006), 673–695. doi: 10.1287/moor.1060.0213.

[11]

M. Benaim, J. Hofbauer and S. Sorin, Perturbations of set–valued dynamical systems, with applications to game theory, Dynamic Games and Applications, 2 (2012), 195–205. doi: 10.1007/s13235–012–0040–0.

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D. Blackwell, An analog of the minimax theorem for vector payoffs, Pacific Journal of Mathematics, 6 (1956), 1–8. doi: 10.2140/pjm.1956.6.1.

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A. Blum and Y. Mansour, From external to internal regret, Journal of Machine Learning Reserach, 8 (2007), 1307–1324. doi: 10.1007/11503415_42.

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S. Hart and A. Mas–Colell, Regret–based continuous time dynamics, Games and Economic Behavior, 45 (2003), 375–394. doi: 10.1016/S0899–8256(03)00178–7.

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J. Hofbauer and S. Sorin, Best response dynamics for continuous zero–sum games, Discrete and Continuous Dynamical Systems–series B, 6 (2006), 215–224. doi: 10.3934/dcdsb.2006.6.215.

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J. Hofbauer, S. Sorin and Y. Viossat, Time average replicator and best reply dynamics, Mathematics of Operations Research, 34 (2009), 263–269. doi: 10.1287/moor.1080.0359.

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A. Kalai and S. Vempala, Efficient algorithms for online decision problems, Journal of Computer and System Sciences, 71 (2005), 291-307.  doi: 10.1016/j.jcss.2004.10.016.

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

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show all references

References:
[1]

E. Akin, The Geometry of Population Genetics, Springer, 1979.

[2]

E. Akin and J. Hofbauer, Recurrence of the unfit, Mathematical Biosciences, 61 (1982), 51-62.  doi: 10.1016/0025-5564(82)90095-5.

[3]

F. AlvarezJ. Bolte and O. Brahic, Hessian Riemannian gradient flows in convex programming, SIAM Journal on Control and Optimization, 43 (2004), 477-501.  doi: 10.1137/S0363012902419977.

[4]

S. AroraE. Hazan and S. Kale, The multiplicative weights update method: A meta algorithm and applications, Theory of Computing, 8 (2012), 121-164.  doi: 10.4086/toc.2012.v008a006.

[5]

P. Auer, N. Cesa–Bianchi, Y. Freund and R. E. Shapire, The nonstochastic multiarmed bandit problem, SIAM J. Comput., 32 (2002), 48–77. doi: 10.1137/S0097539701398375.

[6]

R. J. Aumann, Subjectivity and correlation in randomized strategies, Journal of Mathematical Economics, 1 (1974), 67-96.  doi: 10.1016/0304–4068(74)90037–8.

[7]

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

[8]

M. Benaim and M. Faure, Consistency of vanishingly smooth fictitious play, Mathematics of Operations Research, 38 (2013), 437–450. doi: 10.1287/moor.1120.0568.

[9]

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

[10]

M. Benaim, J. Hofbauer and S. Sorin, Stochastic approximations and differential inclusions. Part Ⅱ: Applications, Mathematics of Operations Research, 31 (2006), 673–695. doi: 10.1287/moor.1060.0213.

[11]

M. Benaim, J. Hofbauer and S. Sorin, Perturbations of set–valued dynamical systems, with applications to game theory, Dynamic Games and Applications, 2 (2012), 195–205. doi: 10.1007/s13235–012–0040–0.

[12]

D. Blackwell, An analog of the minimax theorem for vector payoffs, Pacific Journal of Mathematics, 6 (1956), 1–8. doi: 10.2140/pjm.1956.6.1.

[13]

A. Blum and Y. Mansour, From external to internal regret, Journal of Machine Learning Reserach, 8 (2007), 1307–1324. doi: 10.1007/11503415_42.

[14]

I. M. Bomze, Dynamic aspects of evolutionary stability, Monatshefte Math., 110 (1990), 189–206. doi: 10.1007/BF01301675.

[15]

G. W. Brown, Some notes on computation of games solutions, Report P–78, The Rand Corporation, 1949.

[16]

G. W. Brown, Iterative solution of games by fictitious play, in Koopmans T.C. (ed.), Activity Analysis of Production and Allocation, Wiley, (1951), 374–376.

[17]

S. Bubeck, Convex optimization: Algorithms and complexity, Fondations and Trends in Machine Learning, 8 (2015), 231-357. 

[18]

N. Cesa–Bianchi and G. Lugosi, Potential–based algorithms in on–line prediction and game theory, Computational Learning Theory (Amsterdam, 2001), 48–64, Lecture Notes in Comput. Sci., 2111, Lecture Notes in Artificial Intelligence, Springer, Berlin, 2001. doi: 10.1007/3–540–44581–1_4.

[19] N. Cesa–Bianchi and G. Lugosi, Prediction, Learning and Games, Cambridge University Press, 2006.  doi: 10.1017/CBO9780511546921.
[20]

M.–W. Cheung, Imitative dynamics for games with continuous strategy space, Games and Economic Behavior, 99 (2016), 206-223.  doi: 10.1016/j.geb.2016.08.003.

[21]

P. CoucheneyB. Gaujal and P. Mertikopoulos, Penalty–regulated dynamics and robust learning procedures in games, Mathematics of Operations Research, 40 (2015), 611-633.  doi: 10.1287/moor.2014.0687.

[22]

T. Cover, Universal portfolios, Math. Finance, 1 (1991), 1-29.  doi: 10.1111/j.1467–9965.1991.tb00002.x.

[23]

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

[24]

P. Dupuis and A. Nagurney, Dynamical systems and variational inequalities, Ann. Oper. Res., 44 (1993), 9-42.  doi: 10.1007/BF02073589.

[25]

M. FaureP. GaillardB. Gaujal and V. Perchet, Online learning and game theory. A quick overview with recent results and applications, ESAIM: Proceedings and Surveys, 51 (2015), 246-271.  doi: 10.1051/proc/201551014.

[26]

D. Foster and R. Vohra, A randomization rule for selecting forecasts, Operations Research, 41 (1993), 704-707. 

[27]

D. Foster and R. Vohra, Regret in the on–line decision problem, Games and Economic Behavior, 29 (1999), 7–35. doi: 10.1006/game.1999.0740.

[28]

Y. Freund and R. E. Schapire, Adaptive game playing using multiplicative weights, Games and Economic Behavior, 29 (1999), 79–103. doi: 10.1006/game.1999.0738.

[29]

D. Fudenberg and D. K. Levine, Consistency and cautious fictitious play, Journal of Economic Dynamics and Control, 19 (1995), 1065–1089. doi: 10.1016/0165–1889(94)00819–4.

[30] D. Fudenberg and D. K. Levine, The Theory of Learning in Games, MIT Press, 1998. 
[31]

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

[32]

A. Gaunersdorfer and J. Hofbauer, Fictitious play, Shapley polygons and the replicator equation, Games and Economic Behavior, 11 (1995), 279-303.  doi: 10.1006/game.1995.1052.

[33]

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

[34]

J. Hannan, Approximation to Bayes risk in repeated plays, Contributions to the Theory of Games, III, Drescher M., A.W. Tucker and P. Wolfe eds., Princeton University Press, (1957), 97–139.

[35]

C. Harris, On the rate of convergence of continuous time fictitious play, Games and Economic Behavior, 22 (1998), 238–259. doi: 10.1006/game.1997.0582.

[36]

S. Hart, Adaptive heuristics, Econometrica, 73 (2005), 1401–1430. doi: 10.1111/j.1468–0262.2005.00625.x.

[37]

S. Hart and A. Mas–Colell, A simple adaptive procedure leading to correlated equilibria, Econometrica, 68 (2000), 1127–1150. doi: 10.1111/1468–0262.00153.

[38]

S. Hart and A. Mas–Colell, A general class of adaptive strategies, Journal of Economic Theory, 98 (2001), 26–54. doi: 10.1006/jeth.2000.2746.

[39]

S. Hart and A. Mas–Colell, Regret–based continuous time dynamics, Games and Economic Behavior, 45 (2003), 375–394. doi: 10.1016/S0899–8256(03)00178–7.

[40]

S. Hart and A. Mas–Colell, Uncoupled dynamics do not lead to Nash equilibria, American Economic Review, 93 (2003), 1830–1836.

[41]

S. Hart and A. Mas Colell, Simple Adaptive Strategies: From Regret–Matching to Uncoupled Dynamics, World Scientific Publishing, 2013. doi: 10.1142/8408.

[42]

E. Hazan, The convex optimization approach to regret minimization, Optimization for machine learning, S. Sra, S. Nowozin, S. Wright eds, MIT Press, (2011), 287–303.

[43]

E. Hazan, Optimization for Machine Learning, https://arxiv.org/pdf/1909.03550.pdf, 2019.

[44]

J. Hofbauer, From Nash and Brown to Maynard Smith: Equilibria, dynamics and ESS, Selection, 1(2000), 81–88.

[45]

J. Hofbauer, Deterministic evolutionary game dynamics, in Evolutionary Game Dynamics, K. Sigmund ed., Proceedings of Symposia in Applied Mathematics, A.M.S., 69 (2011), 61–79. doi: 10.1090/psapm/069/2882634.

[46]

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

[47]

J. Hofbauer and W. H. Sandholm, Stable games and their dynamics, Journal of Economic Theory, 144 (2009), 1665–1693, 1693.e4. doi: 10.1016/j.jet.2009.01.007.

[48]

J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge U.P., 1998. doi: 10.1017/CBO9781139173179.

[49]

J. Hofbauer and K. Sigmund, Evolutionary game dynamics, Bulletin of the A.M.S., 40 (2003), 479–519. doi: 10.1090/S0273–0979–03–00988–1.

[50]

J. Hofbauer and S. Sorin, Best response dynamics for continuous zero–sum games, Discrete and Continuous Dynamical Systems–series B, 6 (2006), 215–224. doi: 10.3934/dcdsb.2006.6.215.

[51]

J. Hofbauer, S. Sorin and Y. Viossat, Time average replicator and best reply dynamics, Mathematics of Operations Research, 34 (2009), 263–269. doi: 10.1287/moor.1080.0359.

[52]

A. Kalai and S. Vempala, Efficient algorithms for online decision problems, Journal of Computer and System Sciences, 71 (2005), 291-307.  doi: 10.1016/j.jcss.2004.10.016.

[53]

J. Kwon and P. Mertikopoulos, A continuous time approach to on–line optimization, Journal of Dynamics and Games, 4 (2017), 125-148.  doi: 10.3934/jdg.2017008.

[54]

N. Littlestone and M. K. Warmuth, The weighted majority algorithm, Information and Computation, 108 (1994), 212–261. doi: 10.1006/inco.1994.1009.

[55]

J. Maynard Smith, Evolution and the Theory of Games, Cambridge U.P., 1982.

[56]

P. Mertikopoulos and W. H. Sandholm, Learning in games via reinforcement and regularization, Mathematics of Operations Research, 41 (2016), 1297-1324.  doi: 10.1287/moor.2016.0778.

[57]

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

[58]

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Table1 
0 1 -1
-1 0 1
1 -1 0
0 1 -1
-1 0 1
1 -1 0
Table2 
0 0
0 0
0 0
0 0
0 0
0 0
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