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# Nash Equilibria of Games with a Continuum of Players

## Author

Listed:
• Guilherme Carmona

## Abstract

We characterize Nash equilibria of games with a continuum of players (Mas-Colell (1984)) in terms of approximate equilibria of large finite games. For the concept of $(\epsilon,\epsilon)$ - equilibrium --- in which the fraction of players not $\epsilon$ - optimizing is less than $\epsilon$ --- we show that a strategy is a Nash equilibrium in a game with a continuum of players if and only if there exists a sequence of finite games such that its restriction is an $(\epsilon_n,\epsilon_n)$ - equilibria, with $\epsilon_n$ converging to zero. The same holds for $\epsilon$ - equilibrium --- in which almost all players are $\epsilon$ - optimizing --- provided that either players' payoff functions are equicontinuous or players' action space is finite. Furthermore, we give conditions under which the above results hold for all approximating sequences of games. In our characterizations, a sequence of finite games approaches the continuum game in the sense that the number of players converges to infinity and the distribution of characteristics and actions in the finite games converges to that of the continuum game. These results render approximate equilibria of large finite economies as an alternative way of obtaining strategic insignificance.

## Suggested Citation

• Guilherme Carmona, 2004. "Nash Equilibria of Games with a Continuum of Players," Game Theory and Information 0412009, University Library of Munich, Germany.
• Handle: RePEc:wpa:wuwpga:0412009
Note: Type of Document - pdf; pages: 31. This is a revised version of part of my paper ''Nash and Limit Equilibria of Games with a Continuum of Players''
as

File URL: https://econwpa.ub.uni-muenchen.de/econ-wp/game/papers/0412/0412009.pdf

## References listed on IDEAS

as
1. Drew Fudenberg & David Levine, 2008. "Limit Games and Limit Equilibria," World Scientific Book Chapters,in: A Long-Run Collaboration On Long-Run Games, chapter 2, pages 21-39 World Scientific Publishing Co. Pte. Ltd..
2. Khan, M. Ali & Rath, Kali P. & Sun, Yeneng, 1997. "On the Existence of Pure Strategy Equilibria in Games with a Continuum of Players," Journal of Economic Theory, Elsevier, vol. 76(1), pages 13-46, September.
3. Barlo, Mehmet & Carmona, Guilherme, 2015. "Strategic behavior in non-atomic games," Journal of Mathematical Economics, Elsevier, vol. 60(C), pages 134-144.
4. Dubey, Pradeep & Mas-Colell, Andreau & Shubik, Martin, 1980. "Efficiency properties of strategies market games: An axiomatic approach," Journal of Economic Theory, Elsevier, vol. 22(2), pages 339-362, April.
5. Hildenbrand, W & Mertens, J F, 1972. "Upper Hemi-Continuity of the Equilibrium-Set Correspondence for Pure Exchange Economies," Econometrica, Econometric Society, vol. 40(1), pages 99-108, January.
6. Guilherme Carmona, 2003. "Symmetric Approximate Equilibrium Distributions with Finite Support," Game Theory and Information 0311006, EconWPA.
7. Guilherme Carmona, 2004. "On the Existence of Pure Strategy Nash Equilibria in Large Games," Game Theory and Information 0412008, EconWPA.
8. Novshek, William & Sonnenschein, Hugo, 1983. "Walrasian equilibria as limits of noncooperative equilibria. Part II: Pure strategies," Journal of Economic Theory, Elsevier, vol. 30(1), pages 171-187, June.
9. Rashid, Salim, 1983. "Equilibrium points of non-atomic games : Asymptotic results," Economics Letters, Elsevier, vol. 12(1), pages 7-10.
10. Green, Edward J, 1984. "Continuum and Finite-Player Noncooperative Models of Competition," Econometrica, Econometric Society, vol. 52(4), pages 975-993, July.
11. Wooders, M. & Selten, R. & Cartwright, E., 2001. "Some First Results for Noncooperative Pregames : Social Conformity and Equilibrium in Pure Strategies," The Warwick Economics Research Paper Series (TWERPS) 589, University of Warwick, Department of Economics.
12. Mas-Colell, Andreu, 1983. "Walrasian equilibria as limits of noncooperative equilibria. Part I: Mixed strategies," Journal of Economic Theory, Elsevier, vol. 30(1), pages 153-170, June.
Full references (including those not matched with items on IDEAS)

## Citations

Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
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Cited by:

1. Yang, Jian, 2011. "Asymptotic interpretations for equilibria of nonatomic games," Journal of Mathematical Economics, Elsevier, vol. 47(4-5), pages 491-499.
2. Guilherme Carmona, 2004. "On the Existence of Pure Strategy Nash Equilibria in Large Games," Game Theory and Information 0412008, EconWPA.
3. Carmona, Guilherme & Podczeck, Konrad, 2009. "On the existence of pure-strategy equilibria in large games," Journal of Economic Theory, Elsevier, vol. 144(3), pages 1300-1319, May.
4. Robin Nicole & Peter Sollich, 2017. "Dynamical selection of Nash equilibria using Experience Weighted Attraction Learning: emergence of heterogeneous mixed equilibria," Papers 1706.09763, arXiv.org.
5. Bodoh-Creed, Aaron, 2013. "Efficiency and information aggregation in large uniform-price auctions," Journal of Economic Theory, Elsevier, vol. 148(6), pages 2436-2466.
6. Aaron Bodoh-Creed & Brent Hickman, 2016. "College Assignment as a Large Contest," Working Papers 2016-27, Becker Friedman Institute for Research In Economics.
7. repec:eee:jetheo:v:175:y:2018:i:c:p:88-126 is not listed on IDEAS
8. Guilherme Carmona, 2009. "Intermediate Preferences and Behavioral Conformity in Large Games," Journal of Public Economic Theory, Association for Public Economic Theory, vol. 11(1), pages 9-25, February.

### Keywords

Nash equilibrium; Games with a continuum of players; Games with finitely many players; approximate equilibria.;

### JEL classification:

• C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

### NEP fields

This paper has been announced in the following NEP Reports:

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