research-article

Game dynamics as the meaning of a game

Authors:

Christos Papadimitriou,

Georgios PiliourasAuthors Info & Claims

ACM SIGecom Exchanges, Volume 16, Issue 2

Pages 53 - 63

https://doi.org/10.1145/3331041.3331048

Published: 07 May 2019 Publication History

Abstract

Learning dynamics have traditionally taken a secondary role to Nash equilibria in game theory. We propose a new approach that places the understanding of game dynamics over mixed strategy profiles as the central object of inquiry. We focus on the stable recurrent points of the dynamics, i.e. states which are likely to be revisited infinitely often; obviously, pure Nash equilibria are a special case of such behavior. We propose a new solution concept, the Markov-Conley Chain (MCC), which has several favorable properties: It is a simple randomized generalization of the pure Nash equilibrium, just like the mixed Nash equilibrium; every game has at least one MCC; an MCC is invariant under additive constants and positive multipliers of the players' utilities; there is a polynomial number of MCCs in any game, and they can be all computed in polynomial time; the MCCs can be shown to be, in a well defined sense, surrogates or traces of an important but elusive topological object called the sink chain component of the dynamics; finally, it can be shown that a natural game dynamics surely ends up at one of the MCCs of the game.

References

[1]

Arora, S., Hazan, E., and Kale, S. 2012. The multiplicative weights update method: a meta-algorithm and applications. Theory of Computing 8, 1, 121--164.

[2]

Basu, K. and Weibull, J. W. 1991. Strategy subsets closed under rational behavior. Economics Letters 36, 2, 141--146.

[3]

Benaim, M. 1996. A dynamical system approach to stochastic approximations. SIAM Journal on Control and Optimization 34, 2, 437--472.

Digital Library

[4]

Benaïm, M. and Faure, M. 2012. Stochastic approximation, cooperative dynamics and super-modular games. The Annals of Applied Probability 22, 5, 2133--2164.

[5]

Benaim, M. and Hirsch, M. W. 1999. Mixed equilibria and dynamical systems arising from fictitious play in perturbed games. Games and Economic Behavior 29, 1--2, 36--72.

[6]

Benaïm, M., Hofbauer, J., and Sorin, S. 2012. Perturbations of set-valued dynamical systems, with applications to game theory. Dynamic Games and Applications 2, 2, 195--205.

[7]

Candogan, O., Menache, I., Ozdaglar, A., and Parrilo, P. A. 2011. Flows and decompositions of games: Harmonic and potential games. Mathematics of Operations Research 36, 3, 474--503.

Digital Library

[8]

Conley, C. 1978. Isolated invariant sets and the morse index. cbms regional conference series in mathematics, 38. American Mathematical Society, Providence, RI 16.

[9]

Goemans, M., Mirrokni, V., and Vetta, A. 2005. Sink equilibria and convergence. In Foundations of Computer Science, 2005. FOCS 2005. 46th Annual IEEE Symposium on. IEEE, 142--151.

Digital Library

[10]

Harsanyi, J. C. and Selten, R. 1988. A general theory of equilibrium selection in games. MIT Press Books 1.

[11]

Hart, S. and Mas-Colell, A. 2003. Uncoupled dynamics do not lead to nash equilibrium. American Economic Review 93, 5, 1830--1836.

[12]

Hart, S. and Schmeidler, D. 1989. Existence of correlated equilibria. Mathematics of Operations Research 14, 1, 18--25.

[13]

Hirsch, M. W., Smale, S., and Devaney, R. L. 2012. Differential equations, dynamical systems, and an introduction to chaos. Academic press.

[14]

Hofbauer, J. and Sandholm, W. H. 2002. On the global convergence of stochastic fictitious play. Econometrica 70, 6, 2265--2294.

[15]

Hofbauer, J. and sigmund, K. 1998. Evolutionary Games and Population Dynamics. Cambridge University Press, Cambridge.

[16]

Littlestone, N. and Warmuth, M. K. 1994. The weighted majority algorithm. Inf. Comput. 108, 2 (Feb.), 212--261.

Digital Library

[17]

Mertikopoulos, P., Papadimitriou, C., and piliouras, G. 2018. Cycles in adversarial regularized learning. In ACM-SIAM Symposium on Discrete Algorithms (SODA).

Digital Library

[18]

Myerson, R. B. 1997. Dual reduction and elementary games. Games and Economic Behavior 21, 1--2, 183--202.

[19]

Myerson, R. B. 1999. Nash equilibrium and the history of economic theory. Journal of Economic Literature 37, 3, 1067--1082.

[20]

Nash, J. F. et al. 1950. Equilibrium points in n-person games. Proceedings of the national academy of sciences 36, 1, 48--49.

[21]

Panageas, I. and Piliouras, G. 2016. Average case performance of replicator dynamics in potential games via computing regions of attraction. In Proceedings of the 2016 ACM Conference on Economics and Computation. ACM, 703--720.

Digital Library

[22]

Papadimitriou, C. and piliouras, G. 2018. Games are algorithms: Game dynamics, chain components and markov-conley chains. Arxiv Preprints.

[23]

Piliouras, G. and Shamma, J. S. 2014. Optimization despite chaos: Convex relaxations to complex limit sets via poincaré recurrence. In Proceedings of the twenty-fifth annual ACM-SIAM symposium on Discrete algorithms. SIAM, 861--873.

Digital Library

[24]

Sandholm, W. H. 2010. Population Games and Evolutionary Dynamics. MIT Press.

[25]

Sato, Y., Akiyama, E., and Farmer, J. D. 2002. Chaos in learning a simple two-person game. Proceedings of the National Academy of Sciences 99, 7, 4748--4751.

[26]

Schuster, P. and Sigmund, K. 1983. Replicator dynamics. Journal of Theoretical Biology 100, 3, 533 -- 538.

[27]

Smith, J. M. 1982. Evolution and the Theory of Games. Cambridge university press.

[28]

Taylor, P. D. and Jonker, L. B. 1978. Evolutionary stable strategies and game dynamics. Mathematical Biosciences 40, 12, 145 -- 156.

[29]

Weibull, J. W. 1995. Evolutionary Game Theory. MIT Press; Cambridge, MA: Cambridge University Press.

[30]

Young, H. P. 1993. The evolution of conventions. Econometrica: Journal of the Econometric Society, 57--84.

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Das PNortmann BRatliff LGupta VMylvaganam T(2024)Learning in Stochastic Stackelberg Games2024 American Control Conference (ACC)10.23919/ACC60939.2024.10644265(3557-3562)Online publication date: 10-Jul-2024
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Information

Published In

cover image ACM SIGecom Exchanges

ACM SIGecom Exchanges Volume 16, Issue 2

June 2018

63 pages

EISSN:1551-9031

DOI:10.1145/3331041

Editor:
Hu Fu
University of British Columbia

Issue’s Table of Contents

Copyright © 2019 Authors.

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 07 May 2019

Published in SIGECOM Volume 16, Issue 2

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Cited By

Das PNortmann BRatliff LGupta VMylvaganam T(2024)Learning in Stochastic Stackelberg Games2024 American Control Conference (ACC)10.23919/ACC60939.2024.10644265(3557-3562)Online publication date: 10-Jul-2024
https://doi.org/10.23919/ACC60939.2024.10644265
Konda RChandan RGrimsman DMarden J(2024)Optimal Utility Design of Greedy Algorithms in Resource Allocation GamesIEEE Transactions on Automatic Control10.1109/TAC.2024.337525269:10(6592-6604)Online publication date: Oct-2024
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https://dl.acm.org/doi/10.1007/978-3-031-71033-9_12
Konda RChandan RMarden J(2023)Quality of Non-Convergent Best Response Processes in Multi-Agent Systems through Sink Equilibria2023 62nd IEEE Conference on Decision and Control (CDC)10.1109/CDC49753.2023.10383372(6996-7001)Online publication date: 13-Dec-2023
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Engel MPiliouras G(2023)A stochastic variant of replicator dynamics in zero-sum games and its invariant measuresPhysica D: Nonlinear Phenomena10.1016/j.physd.2023.133940456(133940)Online publication date: Dec-2023
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