scholarly journals From Nash Equilibria to Chain Recurrent Sets: An Algorithmic Solution Concept for Game Theory

Entropy ◽  
2018 ◽  
Vol 20 (10) ◽  
pp. 782 ◽  
Author(s):  
Christos Papadimitriou ◽  
Georgios Piliouras
Keyword(s):  
Game Theory ◽  
Nash Equilibrium ◽  
Dynamical Systems ◽  
Nash Equilibria ◽  
Equilibrium Solution ◽  
Solution Concept ◽  
Constructive Proof ◽  
Specific Information ◽  
Potential Game ◽  
Chain Recurrence

In 1950, Nash proposed a natural equilibrium solution concept for games hence called Nash equilibrium, and proved that all finite games have at least one. The proof is through a simple yet ingenious application of Brouwer’s (or, in another version Kakutani’s) fixed point theorem, the most sophisticated result in his era’s topology—in fact, recent algorithmic work has established that Nash equilibria are computationally equivalent to fixed points. In this paper, we propose a new class of universal non-equilibrium solution concepts arising from an important theorem in the topology of dynamical systems that was unavailable to Nash. This approach starts with both a game and a learning dynamics, defined over mixed strategies. The Nash equilibria are fixpoints of the dynamics, but the system behavior is captured by an object far more general than the Nash equilibrium that is known in dynamical systems theory as chain recurrent set. Informally, once we focus on this solution concept—this notion of “the outcome of the game”—every game behaves like a potential game with the dynamics converging to these states. In other words, unlike Nash equilibria, this solution concept is algorithmic in the sense that it has a constructive proof of existence. We characterize this solution for simple benchmark games under replicator dynamics, arguably the best known evolutionary dynamics in game theory. For (weighted) potential games, the new concept coincides with the fixpoints/equilibria of the dynamics. However, in (variants of) zero-sum games with fully mixed (i.e., interior) Nash equilibria, it covers the whole state space, as the dynamics satisfy specific information theoretic constants of motion. We discuss numerous novel computational, as well as structural, combinatorial questions raised by this chain recurrence conception of 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 PERIODIC STRATEGIES II: GENERALIZATIONS AND EXTENSIONS

2020 ◽  
Vol 23 (02) ◽  
pp. 2050005
Author(s):  
V. K. OIKONOMOU ◽  
J. JOST
Keyword(s):  
Game Theory ◽  
Nash Equilibrium ◽  
Nash Equilibria ◽  
Cooperative Game Theory ◽  
Solution Concept ◽  
Epistemic Game Theory ◽  
Bayesian Games ◽  
Mathematical Properties ◽  
Games With Incomplete Information ◽  
Theory Framework

At a mixed Nash equilibrium, the payoff of a player does not depend on her own action, as long as her opponent sticks to his. In a periodic strategy, a concept developed in a previous paper [V. K. Oikonomou and J. Jost, Periodic strategies: A new solution concept and an algorithm for nontrivial strategic form games, Adv. Compl. Syst. 20(5) (2017) 1750009], in contrast, the own payoff does not depend on the opponent’s action. Here, we generalize this to multi-player simultaneous perfect information strategic form games. We show that also in this class of games, there always exists at least one periodic strategy, and we investigate the mathematical properties of such periodic strategies. In addition, we demonstrate that periodic strategies may exist in games with incomplete information; we shall focus on Bayesian games. Moreover, we discuss the differences between the periodic strategies formalism and cooperative game theory. In fact, the periodic strategies are obtained in a purely non-cooperative way, and periodic strategies are as cooperative as the Nash equilibria are. Finally, we incorporate the periodic strategies in an epistemic game theory framework, and discuss several features of this approach.

scholarly journals Holding a group together: non-game-theory vs. game-theory

2021 ◽  
Author(s):  
Michael Richter ◽  
Ariel Rubinstein
Keyword(s):  
Game Theory ◽  
Nash Equilibrium ◽  
Competitive Equilibrium ◽  
Solution Concept ◽  
Equilibrium Concept

Abstract Each member of a group chooses a position and has preferences regarding his chosen position. The group’s harmony depends on the profile of chosen positions meeting a specific condition. We analyse a solution concept (Richter and Rubinstein, 2020) based on a permissible set of individual positions, which plays a role analogous to that of prices in competitive equilibrium. Given the permissible set, members choose their most preferred position. The set is tightened if the chosen positions are inharmonious and relaxed if the restrictions are unnecessary. This new equilibrium concept yields more attractive outcomes than does Nash equilibrium in the corresponding game.

scholarly journals Smoothing Method for Approximate Extensive-Form Perfect Equilibrium

2017 ◽  
Author(s):  
Christian Kroer ◽  
Gabriele Farina ◽  
Tuomas Sandholm
Keyword(s):  
Nash Equilibrium ◽  
Imperfect Information ◽  
Large Scale ◽  
Nash Equilibria ◽  
Convex Polytope ◽  
Solution Concept ◽  
Extensive Form ◽  
Game Tree ◽  
Equilibrium Refinements ◽  
Perfect Equilibria

Nash equilibrium is a popular solution concept for solving imperfect-information games in practice. However, it has a major drawback: it does not preclude suboptimal play in branches of the game tree that are not reached in equilibrium. Equilibrium refinements can mend this issue, but have experienced little practical adoption. This is largely due to a lack of scalable algorithms.Sparse iterative methods, in particular first-order methods, are known to be among the most effective algorithms for computing Nash equilibria in large-scale two-player zero-sum extensive-form games. In this paper, we provide, to our knowledge, the first extension of these methods to equilibrium refinements. We develop a smoothing approach for behavioral perturbations of the convex polytope that encompasses the strategy spaces of players in an extensive-form game. This enables one to compute an approximate variant of extensive-form perfect equilibria. Experiments show that our smoothing approach leads to solutions with dramatically stronger strategies at information sets that are reached with low probability in approximate Nash equilibria, while retaining the overall convergence rate associated with fast algorithms for Nash equilibrium. This has benefits both in approximate equilibrium finding (such approximation is necessary in practice in large games) where some probabilities are low while possibly heading toward zero in the limit, and exact equilibrium computation where the low probabilities are actually zero.

scholarly journals A Cooperative Dual to the Nash Equilibrium for Two-Person Prescriptive Games

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
H. W. Corley ◽  
Phantipa Kwain
Keyword(s):  
Game Theory ◽  
Nash Equilibrium ◽  
Normal Form ◽  
Fundamental Solution ◽  
Solution Concept ◽  
Alternative Solution ◽  
Noncooperative Game ◽  
Noncooperative Game Theory ◽  
Best Interest

An alternative to the Nash equilibrium (NE) is presented for two-person, one-shot prescriptive games in normal form, where the outcome is determined by an arbiter. The NE is the fundamental solution concept in noncooperative game theory. It is based on the assumption that players are completely selfish. However, NEs are often not played in practice, so we present a cooperative dual as an alternative solution concept by which an arbiter can assign the players' actions. In this dual equilibrium (DE), each player acts in the other's best interest. We formally define prescriptive games and the DE, then summarize the duality relationships between the NE and DE for two players. We also apply the DE to some prescriptive games and compare it to other outcomes.

Study of potential games using Ising interaction

2021 ◽  
Author(s):  
U. Tejasvi ◽  
R. D. Eithiraj ◽  
S. Balakrishnan
Keyword(s):  
Game Theory ◽  
Nash Equilibrium ◽  
Large Population ◽  
Real Life ◽  
Potential Games ◽  
Potential Game ◽  
Countable Number ◽  
Symmetric Games ◽  
Close Observation ◽  
Hamiltonian Analysis

Problems can be handled properly in game theory as long as a countable number of players are considered, whereas, in real life, we have a large number of players. Hence, games at the thermodynamic limit are analyzed in general. There is a one-to-one correspondence between classical games and the modeled Hamiltonian at a particular equilibrium condition, usually the Nash equilibrium. Such a correspondence is arrived for symmetric games, namely the Prisoner’s Dilemma using the Ising Hamiltonian. In this work, we have shown that another class of games known as potential games can be analyzed with the Ising Hamiltonian. Analysis of this work brings out very close observation with real-world scenarios. In other words, the model of a potential game studied using Ising Hamiltonian predicts behavioral aspects of a large population precisely.

A Countermeasure Model about Food Safety Regulatory

2013 ◽  
Vol 781-784 ◽  
pp. 1546-1549 ◽  
Author(s):  
Li Xia Cao ◽  
Wei Wen Chai
Keyword(s):  
Game Theory ◽  
Nash Equilibrium ◽  
Food Safety ◽  
Equilibrium Solution ◽  
Mixed Strategy ◽  
Game Model ◽  
Mixed Strategy Nash Equilibrium ◽  
Regulatory Strategy ◽  
The Status ◽  
Strategy Nash Equilibrium

By studying the status of the food safety regulatory, this paper points out the drawbacks of existing researches, that is, lack of operability; Relying on the equilibrium theory of game theory, this paper establishes a practical and effective regulatory game model, gives the models mixed strategy Nash equilibrium solution and a food safety regulatory strategy. Finally, to illustrate the effectiveness and feasibility of the model, an illustrative example is shown.

scholarly journals Constructive Proof of the Existence of Nash Equilibrium in a Finite Strategic Game with Sequentially Locally Nonconstant Payoff Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Yasuhito Tanaka
Keyword(s):  
Nash Equilibrium ◽  
Nash Equilibria ◽  
Constructive Proof ◽  
Constructive Mathematics ◽  
Strategic Game ◽  
Sperner’S Lemma ◽  
Approximate Nash Equilibria ◽  
Payoff Functions

We will constructively prove the existence of a Nash equilibrium in a finite strategic game with sequentially locally nonconstant payoff functions. The proof is based on the existence of approximate Nash equilibria which is proved by Sperner's lemma. We follow the Bishop-style constructive mathematics.

Lane Change Decision Analysis Based on Drivers' Perception-Judgment and Game Theory

2013 ◽  
Vol 361-363 ◽  
pp. 1875-1879 ◽  
Author(s):  
Jin Shuan Peng ◽  
Ying Shi Guo ◽  
Yi Ming Shao
Keyword(s):  
Game Theory ◽  
Decision Making ◽  
Nash Equilibrium ◽  
Equilibrium Solution ◽  
Information Source ◽  
Driving Safety ◽  
External Information ◽  
Lane Change ◽  
Safety Level ◽  
Lane Changing

To clearly understand the mechanism of drivers lane-changing decision, based on drivers perception of external information, integrated cognitive judgment and game theory, the decision-making model was established, then the structure and operating mechanism of the model were detailedly analyzed. By introducing game theory-related knowledge, the non-cooperative mixed strategy game between the object vehicle and the following vehicle in the target lane was further discussed. Then, the benefits and Nash equilibrium solution of the participants in the game were deeply researched. Analysis shows that lane-changing decision is composed of information perception and three judgment-decision processes, the factors which would affect decision-making level include information source characteristics, the ability of drivers perception and comprehensive cognitive judgment, driving behavior characteristics and so on. The Nash equilibrium solution of the lane change game is determined by driving safety level, journey time and importance degree of the revenues.

scholarly journals Game-theory models of binary collective behavior

2020 ◽  
Vol 12 (2) ◽  
pp. 3-19
Author(s):  
Владимир Валетинович Бреер ◽  
Vladimir Breer
Keyword(s):  
Game Theory ◽  
Nash Equilibrium ◽  
Collective Behavior ◽  
General Model ◽  
Nash Equilibria ◽  
Dynamic Models ◽  
Utility Functions ◽  
Point Of View ◽  
Algebraic Equations ◽  
Game Theoretic

Game-theoretic models were investigated not from the point of view of the maxima of the players' utility functions, as is usually done, but by solving algebraic equations that characterize the Nash equilibrium. This characterization is obtained for models of binary collective behavior, in which players choose one of two possible strategies. Based on the results for the general model, game-theoretic models of conformal threshold Binary Collective Behavior (BCB) are studied, provided the collective is divided into L groups. The conditions for the existence of Nash equilibria is proved. For each Nash equilibrium, its structure is defined. The results obtained are illustrated by two examples of conformal threshold BCB when the group coincides with the whole team and when the latter is divided into two groups. It is shown that the Nash equilibria in the first and second examples are analogues of the equilibria in the dynamic models of M. Granovetter and T. Schelling, respectively.

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