Complexity of Pure-Strategy Nash Equilibria in Non-Cooperative Games

2007 ◽  
pp. 45-51
Author(s):  
Juliane Dunke
Keyword(s):  
Cooperative Games ◽  
Nash Equilibria ◽  
Pure Strategy ◽  
Non Cooperative Games

Learning Nash Equilibria in Non-Cooperative Games

2011 ◽  
pp. 1018-1023 ◽  
Author(s):  
Alfredo Garro
Keyword(s):  
Game Theory ◽  
Nash Equilibrium ◽  
Cooperative Games ◽  
Nash Equilibria ◽  
Applied Mathematics ◽  
Solution Concept ◽  
The Other ◽  
Political Sciences ◽  
Definition Of ◽  
Non Cooperative Games

Game Theory (Von Neumann & Morgenstern, 1944) is a branch of applied mathematics and economics that studies situations (games) where self-interested interacting players act for maximizing their returns; therefore, the return of each player depends on his behaviour and on the behaviours of the other players. Game Theory, which plays an important role in the social and political sciences, has recently drawn attention in new academic fields which go from algorithmic mechanism design to cybernetics. However, a fundamental problem to solve for effectively applying Game Theory in real word applications is the definition of well-founded solution concepts of a game and the design of efficient algorithms for their computation. A widely accepted solution concept of a game in which any cooperation among the players must be selfenforcing (non-cooperative game) is represented by the Nash Equilibrium. In particular, a Nash Equilibrium is a set of strategies, one for each player of the game, such that no player can benefit by changing his strategy unilaterally, i.e. while the other players keep their strategies unchanged (Nash, 1951). The problem of computing Nash Equilibria in non-cooperative games is considered one of the most important open problem in Complexity Theory (Papadimitriou, 2001). Daskalakis, Goldbergy, and Papadimitriou (2005), showed that the problem of computing a Nash equilibrium in a game with four or more players is complete for the complexity class PPAD-Polynomial Parity Argument Directed version (Papadimitriou, 1991), moreover, Chen and Deng extended this result for 2-player games (Chen & Deng, 2005). However, even in the two players case, the best algorithm known has an exponential worst-case running time (Savani & von Stengel, 2004); furthermore, if the computation of equilibria with simple additional properties is required, the problem immediately becomes NP-hard (Bonifaci, Di Iorio, & Laura, 2005) (Conitzer & Sandholm, 2003) (Gilboa & Zemel, 1989) (Gottlob, Greco, & Scarcello, 2003). Motivated by these results, recent studies have dealt with the problem of efficiently computing Nash Equilibria by exploiting approaches based on the concepts of learning and evolution (Fudenberg & Levine, 1998) (Maynard Smith, 1982). In these approaches the Nash Equilibria of a game are not statically computed but are the result of the evolution of a system composed by agents playing the game. In particular, each agent after different rounds will learn to play a strategy that, under the hypothesis of agent’s rationality, will be one of the Nash equilibria of the game (Benaim & Hirsch, 1999) (Carmel & Markovitch, 1996). This article presents SALENE, a Multi-Agent System (MAS) for learning Nash Equilibria in noncooperative games, which is based on the above mentioned concepts.

scholarly journals Rational Verification for Probabilistic Systems

2021 ◽  
Author(s):  
Julian Gutierrez ◽  
Lewis Hammond ◽  
Anthony W. Lin ◽  
Muhammad Najib ◽  
Michael Wooldridge
Keyword(s):  
Cooperative Games ◽  
Nash Equilibria ◽  
Deterministic Systems ◽  
Probabilistic Systems ◽  
Verification Problem ◽  
Markov Decision ◽  
Probabilistic Setting ◽  
Multi Agent ◽  
Game Theoretic ◽  
Non Cooperative Games

Rational verification is the problem of determining which temporal logic properties will hold in a multi-agent system, under the assumption that agents in the system act rationally, by choosing strategies that collectively form a game-theoretic equilibrium. Previous work in this area has largely focussed on deterministic systems. In this paper, we develop the theory and algorithms for rational verification in probabilistic systems. We focus on concurrent stochastic games (CSGs), which can be used to model uncertainty and randomness in complex multi-agent environments. We study the rational verification problem for both non-cooperative games and cooperative games in the qualitative probabilistic setting. In the former case, we consider LTL properties satisfied by the Nash equilibria of the game and in the latter case LTL properties satisfied by the core. In both cases, we show that the problem is 2EXPTIME-complete, thus not harder than the much simpler verification problem of model checking LTL properties of systems modelled as Markov decision processes (MDPs).

Computing Nash Equilibria in Non-Cooperative Games

2013 ◽  
Vol 3 (3) ◽  
pp. 29-42
Author(s):  
Alfredo Garro
Keyword(s):  
Game Theory ◽  
Cooperative Games ◽  
Nash Equilibria ◽  
Fundamental Problem ◽  
Solution Concept ◽  
Worst Case ◽  
Interacting Agents ◽  
Definition Of ◽  
Non Cooperative Games ◽  
Computation Of Equilibria

Game Theory has recently drawn attention in new fields which go from algorithmic mechanism design to cybernetics. However, a fundamental problem to solve for effectively applying Game Theory in real word applications is the definition of well-founded solution concepts of a game and the design of efficient algorithms for their computation. A widely accepted solution concept for games in which any cooperation among the players must be self-enforcing (non-cooperative games) is represented by the Nash equilibrium. However, even in the two players case, the best algorithm known for computing Nash equilibria has an exponential worst-case running time; furthermore, if the computation of equilibria with simple additional properties is required, the problem becomes NP-hard. The paper aims to provide a solution for efficiently computing the Nash equilibria of a game as the result of the evolution of a system composed by interacting agents playing the game.

scholarly journals Weak Berge Equilibrium in Finite Three-person Games: Conception and Computation

2020 ◽  
Vol 11 (1) ◽  
pp. 127-134
Author(s):  
Konstantin Kudryavtsev ◽  
Ustav Malkov
Keyword(s):  
Cooperative Games ◽  
Hippocratic Oath ◽  
The Other ◽  
Mixed Strategies ◽  
Finite Games ◽  
Moral Basis ◽  
Other Hand ◽  
Berge Equilibrium ◽  
Do No Harm ◽  
Non Cooperative Games

AbstractThe paper proposes the concept of a weak Berge equilibrium. Unlike the Berge equilibrium, the moral basis of this equilibrium is the Hippocratic Oath “First do no harm”. On the other hand, any Berge equilibrium is a weak Berge equilibrium. But, there are weak Berge equilibria, which are not the Berge equilibria. The properties of the weak Berge equilibrium have been investigated. The existence of the weak Berge equilibrium in mixed strategies has been established for finite games. The weak Berge equilibria for finite three-person non-cooperative games are computed.

scholarly journals Representing pure Nash equilibria in argumentation

2021 ◽  
pp. 1-14
Author(s):  
Bruno Yun ◽  
Srdjan Vesic ◽  
Nir Oren
Keyword(s):  
Normal Form ◽  
Nash Equilibria ◽  
Pure Strategy ◽  
Normal Form Games

In this paper we describe an argumentation-based representation of normal form games, and demonstrate how argumentation can be used to compute pure strategy Nash equilibria. Our approach builds on Modgil’s Extended Argumentation Frameworks. We demonstrate its correctness, showprove several theoretical properties it satisfies, and outline how it can be used to explain why certain strategies are Nash equilibria to a non-expert human user.

A positive approach to non-cooperative games

1986 ◽  
Vol 7 (3) ◽  
pp. 235-251 ◽  
Author(s):  
James W. Friedman ◽  
Robert W. Rosenthal
Keyword(s):  
Cooperative Games ◽  
Positive Approach ◽  
Non Cooperative Games
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