scholarly journals Nash Equilibria in Large Games

Game Theory ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Dionysius Glycopantis
Keyword(s):  
Nash Equilibrium ◽  
Nash Equilibria ◽  
Pure Strategy ◽  
General Setting ◽  
Large Games ◽  
Finite Games ◽  
Infinite Game ◽  
Anonymous Game

This paper adds to the discussion, in a general setting, that given a Nash-Schmeidler (nonanonymous) game it is not always possible to define a Mas-Colell (anonymous) game. In the two games, the players have different strategic behaviours and the formulations of the two problems are different. Also, we offer a novel explanation for the lack of a Nash equilibrium in an infinite game. We consider this game as the limit of a sequence of approximate, finite games for which an equilibrium exists. However, the limiting pure strategy function is not measurable.

scholarly journals Approximation and characterization of Nash equilibria of large games

2020 ◽  
Author(s):  
Guilherme Carmona ◽  
Konrad Podczeck
Keyword(s):  
Nash Equilibrium ◽  
Nash Equilibria ◽  
Large Games ◽  
Finite Games ◽  
Continuum Of Players ◽  
The Relationship

Abstract We characterize Nash equilibria of games with a continuum of players in terms of approximate equilibria of large finite games. This characterization precisely describes the relationship between the equilibrium sets of the two classes of games. In particular, it yields several approximation results for Nash equilibria of games with a continuum of players, which roughly state that all finite-player games that are sufficiently close to a given game with a continuum of players have approximate equilibria that are close to a given Nash equilibrium of the non-atomic game.

Pure-strategy Nash equilibria in large games: characterization and existence

2015 ◽  
Vol 45 (3) ◽  
pp. 685-697 ◽  
Author(s):  
Haifeng Fu ◽  
Ying Xu ◽  
Luyi Zhang
Keyword(s):  
Nash Equilibria ◽  
Pure Strategy ◽  
Large Games

scholarly journals Refinement of acyclic-and-asymmetric payoff aggregates of pure strategy efficient Nash equilibria in finite noncooperative games by maximultimin and superoptimality

2021 ◽  
Vol 4 (2) ◽  
pp. 178-199
Author(s):  
Vadim Romanuke ◽  
Keyword(s):  
Nash Equilibrium ◽  
Nash Equilibria ◽  
Pure Strategy ◽  
Noncooperative Games ◽  
Partial Reduction

A theory of refining pure strategy efficient Nash equilibria in finite noncooperative games under uncertainty is outlined. The theory is based on guaranteeing the corresponding payoffs for the players by using maximultimin, which is an expanded version of maximin. If a product of the players’ maximultimin subsets contains more than one efficient Nash equilibrium, a superoptimality rule is attached wherein minimization is substituted with summation. The superoptimality rule stands like a backup plan, and it is involved if maximultimin fails to produce just a single refined efficient equilibrium (a metaequilibrium). The number of the refinement possible outcomes is 10. There are 3 single-metaequilibrium cases, 3 partial reduction cases, and 4 fail cases. Despite successfulness of refinement drops as the game gets bigger, pessimistic estimation of its part is above 54 % for games with no more than four players.

scholarly journals Learning Deviation Payoffs in Simulation-Based Games

2019 ◽  
Vol 33 ◽  
pp. 2173-2180 ◽  
Author(s):  
Samuel Sokota ◽  
Caleb Ho ◽  
Bryce Wiedenbeck
Keyword(s):  
Neural Networks ◽  
Nash Equilibrium ◽  
Nash Equilibria ◽  
Mixed Strategy ◽  
State Of The Art ◽  
Pure Strategy ◽  
Normal Form Games ◽  
Novel Approach ◽  
Simulation Based ◽  
Expected Values

We present a novel approach for identifying approximate role-symmetric Nash equilibria in large simulation-based games. Our method uses neural networks to learn a mapping from mixed-strategy profiles to deviation payoffs—the expected values of playing pure-strategy deviations from those profiles. This learning can generalize from data about a tiny fraction of a game’s outcomes, permitting tractable analysis of exponentially large normal-form games. We give a procedure for iteratively refining the learned model with new data produced by sampling in the neighborhood of each candidate Nash equilibrium. Relative to the existing state of the art, deviation payoff learning dramatically simplifies the task of computing equilibria and more effectively addresses player asymmetries. We demonstrate empirically that deviation payoff learning identifies better approximate equilibria than previous methods and can handle more difficult settings, including games with many more players, strategies, and roles.

scholarly journals The Frequency of Convergent Games under Best-Response Dynamics

2021 ◽  
Author(s):  
Samuel C. Wiese ◽  
Torsten Heinrich
Keyword(s):  
Nash Equilibrium ◽  
Nash Equilibria ◽  
Pure Strategy ◽  
Response Dynamics ◽  
Novel Approach ◽  
Best Response ◽  
Random Games ◽  
Payoff Matrices ◽  
Best Response Dynamic ◽  
Strategy Nash Equilibrium

AbstractWe calculate the frequency of games with a unique pure strategy Nash equilibrium in the ensemble of n-player, m-strategy normal-form games. To obtain the ensemble, we generate payoff matrices at random. Games with a unique pure strategy Nash equilibrium converge to the Nash equilibrium. We then consider a wider class of games that converge under a best-response dynamic, in which each player chooses their optimal pure strategy successively. We show that the frequency of convergent games with a given number of pure Nash equilibria goes to zero as the number of players or the number of strategies goes to infinity. In the 2-player case, we show that for large games with at least 10 strategies, convergent games with multiple pure strategy Nash equilibria are more likely than games with a unique Nash equilibrium. Our novel approach uses an n-partite graph to describe games.

Quasi-Transfer Continuity and Nash Equilibrium

2019 ◽  
Vol 21 (04) ◽  
pp. 1950004
Author(s):  
Rabia Nessah ◽  
Tarik Tazdait
Keyword(s):  
Nash Equilibrium ◽  
Nash Equilibria ◽  
Existence Result ◽  
Mixed Strategy ◽  
Existence Theorems ◽  
Pure Strategy ◽  
Econ Theory ◽  
Discontinuous Games ◽  
Normal Form Games ◽  
Existence Of Equilibria

We introduce a new notion of continuity, called quasi-transfer continuity, and show that it is enough to guarantee the existence of Nash equilibria in compact, quasiconcave normal form games. This holds true in a large class of discontinuous games. We show that our result strictly generalizes the pure strategy existence theorem of Carmona [Carmona, G. [2009] An existence result for discontinuous games, J. Econ. Theory 144, 1333–1340]. We also show that our result is neither implied by nor does it imply the existence theorems of Reny [Reny, J. P. [1999] On the existence of pure and mixed strategy Nash equilibria in discontinuous games, Econometrica 67, 1029–1056] and Baye et al. [Baye, M. R., Tian, G. and Zhou, J. [1993] Characterizations of the existence of equilibria in games with discontinuous and nonquasiconcave payoffs, Rev. Econ. Studies 60, 935–948].

scholarly journals A Mixed 0-1 Linear Programming Approach to the Computation of All Pure-Strategy Nash Equilibria of a Finiten-Person Game in Normal Form

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Zhengtian Wu ◽  
Chuangyin Dang ◽  
Hamid Reza Karimi ◽  
Changan Zhu ◽  
Qing Gao
Keyword(s):  
Linear Programming ◽  
Nash Equilibrium ◽  
Normal Form ◽  
Nash Equilibria ◽  
Selection Process ◽  
Pure Strategy ◽  
Programming Approach ◽  
Main Concern ◽  
Person Game ◽  
Strategy Nash Equilibrium

A main concern in applications of game theory is how to effectively select a Nash equilibrium, especially a pure-strategy Nash equilibrium for a finiten-person game in normal form. This selection process often requires the computation of all Nash equilibria. It is well known that determining whether a finite game has a pure-strategy Nash equilibrium is an NP-hard problem and it is difficult to solve by naive enumeration algorithms. By exploiting the properties of pure strategy and multilinear terms in the payoff functions, this paper formulates a new mixed 0-1 linear program for computing all pure-strategy Nash equilibria. To our knowledge, it is the first method to formulate a mixed 0-1 linear programming for pure-strategy Nash equilibria and it may work well for similar problems. Numerical results show that the approach is effective and this method can be easily distributed in a distributed way.

scholarly journals A new evolutionary approach for computing Nash equilibria in bimatrix games with known support

2012 ◽  
Vol 2 (2) ◽  
Author(s):  
Urszula Boryczka ◽  
Przemyslaw Juszczuk
Keyword(s):  
Nash Equilibrium ◽  
Nash Equilibria ◽  
Probability Distributions ◽  
Pure Strategy ◽  
Global Optimum ◽  
Adaptive Mutation ◽  
De Algorithm ◽  
Approximate Nash Equilibria ◽  
Zero Sum ◽  
Continuous Problem

AbstractIn this paper, we present the application of the Differential Evolution (DE) algorithm to the problem of finding approximate Nash equilibria in matrix, non-zero sum games for two players with finite number of strategies. Nash equilibrium is one of the main concepts in game theory. It may be classified as continuous problem, where two probability distributions over the set of strategies of both players should be found. Every deviation from the global optimum is interpreted as Nash approximation and called ε-Nash equilibrium. The main advantage of the proposed algorithm is self-adaptive mutation operator, which direct the search process. The approach used in this article is based on the probability of chosing single pure strategy. In optimal mixed strategy, every strategy has some probability of being chosen. Our goal is to determine this probability and maximize payoff for a single player.

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