Partial exposure in large games

doi:10.1016/j.geb.2009.09.006

Games and Economic Behavior

Volume 68, Issue 2, March 2010, Pages 602-613

https://doi.org/10.1016/j.geb.2009.09.006 Get rights and content

Abstract

In this work we introduce the notion of partial exposure, in which the players of a simultaneous-move Bayesian game are exposed to the realized types and chosen actions of a subset of the other players. We show that in any large simultaneous-move game, each player has very little regret even after being partially exposed to other players. If players are given the opportunity to be exposed to others at the expense of a small decrease in utility, players will decline this opportunity, and the original Nash equilibria of the game will survive.

References (17)

Y. Azrieli
Categorizing others in a large game
Games Econ. Behav.
(2009)
N.I. Al-Najjar et al.
Pivotal players and the characterization of influence
J. Econ. Theory
(2000)
M. Blonski
Characterization of pure strategy equilibria in finite anonymous games
J. Math. Econ.
(2000)
G. Carmona et al.
On the existence of pure strategy Nash equilibria in large games
J. Econ. Theory
(2009)
G. Carmona
Purification of Bayesian–Nash equilibria in large games with compact type and action spaces
J. Math. Econ.
(2008)
D. Fudenberg et al.
When are non-anonymous players negligible?
J. Econ. Theory
(1998)
O. Haggstrom et al.
A law of large numbers for weighted majority
Adv. Appl. Math.
(2006)
J. Halpern
A computer scientist looks at game theory
Games Econ. Behav.
(2003)

There are more references available in the full text version of this article.

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For a similar conclusion while allowing correlated signals, Cartwright and Wooders (2009) worked with a finite signal space and a countable action space. More generalizations concerning robustness were achieved, for instance, by Gradwohl and Reingold (2010), Carmona and Podczeck (2012), and Deb and Kalai (2015). There have been works dealing with both an infinite number of players and incomplete information.
We investigate a nonatomic game (NG) involving a random state of the world. Every player receives only a signal of the state’s realization. Not knowing the true state, the player would strive for a higher average payoff conditioned on her received signal. We demonstrate that equilibria exist for the game under reasonable conditions. Not only are they in existence, but these equilibrium points are also useful. When the state space is finite, such an equilibrium would help induce asymptotically equilibrium behaviors as $n$ tends to $+ \infty$ in $n$ -player Bayesian games whose player profiles are randomly generated from the original NG’s player distribution. More important, pure $ϵ$ -equilibria would likely emerge even though the NG equilibrium might well be mixed to start with. Pure versions of the latter would exist when the NG is anonymous.
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We consider Nash equilibria of large anonymous games (i.e., each player's payoff depends on his choice and the distribution of the choices made by others). We show that pure strategy Nash equilibria exist in all sufficiently large finite-player games with finite action spaces and for generic distributions of players' payoff functions. We also show that equilibrium distributions of non-atomic games are asymptotically implementable in terms of Nash equilibria of large finite-player games. Extensions of these results to games with general compact metric action spaces are provided.
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In particular, the Nash equilibria of such games are ex post Nash: A player's strategy remains a best response even after he sees the chosen actions of all the other players. In an attempt to generalize the work of Kalai, we (Gradwohl and Reingold, 2010) introduced the notion of partial exposure. In that paper we show that in any large game, players' Nash strategies remain nearly optimal even after they are exposed to the chosen actions of some of the other players.
A Nash equilibrium is an optimal strategy for each player under the assumption that others play according to their respective Nash strategies, but it provides no guarantees in the presence of irrational players or coalitions of colluding players. In fact, no such guarantees exist in general. However, in this paper we show that large games are innately fault tolerant. We quantify the ways in which two subclasses of large games – λ-continuous games and anonymous games – are resilient against Byzantine faults (i.e. irrational behavior), coalitions, and asynchronous play. We also show that general large games have some non-trivial resilience against faults.
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^☆: We would like to thank Ehud Kalai and Yaron Azrieli for interesting and useful conversations. We are grateful to Ariel Yadin and Amir Yehudayoff for helpful discussions throughout the course of this research. We also thank seminar participants at Northwestern University and at the Technion. Finally, we are grateful to the anonymous referees for valuable input.

¹: Work done while the author was a student at the Department of Computer Science and Applied Mathematics, The Weizmann Institute of Science.

²: Research supported by US–Israel Binational Science Foundation Grants 2002246 and 2006060.

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Partial exposure in large games☆

Abstract

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