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.

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 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.

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.

Bayesian Rationality and Social Epistemology

2014 ◽  
Author(s):  
Herbert Gintis
Keyword(s):  
Game Theory ◽  
Nash Equilibrium ◽  
Social Norms ◽  
Social Epistemology ◽  
Correlated Equilibrium ◽  
Epistemic Game Theory ◽  
Famous Theorem ◽  
Rational Agents ◽  
Natural Equilibrium ◽  
Bayesian Rationality

This chapter uses epistemic game theory to expand on the notion of social norms as choreographer of a correlated equilibrium, and to elucidate the socio-psychological prerequisites for the notion that social norms implement correlated equilibria. The correlated equilibrium is a much more natural equilibrium criterion than the Nash equilibrium, because of a famous theorem of Aumann (1987), who showed that Bayesian rational agents in an epistemic game G with a common subjective prior play a correlated equilibrium of G. Thus, while rationality and common priors do not imply Nash equilibrium, these assumptions do imply correlated equilibrium and social norms act not only as choreographer, but also supply the epistemic conditions for common priors.

On the Foundations of Nash Equilibrium

1996 ◽  
Vol 12 (1) ◽  
pp. 67-88 ◽  
Author(s):  
Hans Jørgen Jacobsen
Keyword(s):  
Game Theory ◽  
Nash Equilibrium ◽  
Cooperative Game Theory ◽  
Mixed Strategy ◽  
Pure Strategy ◽  
Mixed Strategies ◽  
Positive Theory ◽  
Equilibrium Concept ◽  
Strategy Equilibrium ◽  
Pure Strategies

The most important analytical tool in non-cooperative game theory is the concept of a Nash equilibrium, which is a collection of possibly mixed strategies, one for each player, with the property that each player's strategy is a best reply to the strategies of the other players. If we do not go into normative game theory, which concerns itself with the recommendation of strategies, and focus instead entirely on the positive theory of prediction, two alternative interpretations of the Nash equilibrium concept are predominantly available.In the more traditional one, a Nash equilibrium is a prediction of actual play. A game may not have a Nash equilibrium in pure strategies, and a mixed strategy equilibrium may be difficult to incorporate into this interpretation if it involves the idea of actual randomization over equally good pure strategies. In another interpretation originating from Harsanyi (1973a), see also Rubinstein (1991), and Aumann and Brandenburger (1991), a Nash equilibrium is a ‘consistent’ collection of probabilistic expectations, conjectures, on the players. It is consistent in the sense that for each player each pure strategy, which has positive probability according to the conjecture about that player, is indeed a best reply to the conjectures about others.

Flexibility mechanisms and credit discounting

2003 ◽  
Vol 8 (2) ◽  
pp. 247-260 ◽  
Author(s):  
Sungwhee Shin ◽  
Sang-Chul Suh
Keyword(s):  
Climate Change ◽  
Game Theory ◽  
Nash Equilibrium ◽  
Kyoto Protocol ◽  
Cooperative Game ◽  
Cooperative Game Theory ◽  
International Treaties ◽  
Equilibrium Outcome

We consider the international treaties on climate change as self-enforcing agreements. Applying non-cooperative game theory, we interpret the UN framework Convention on Climate Change as a Nash equilibrium outcome of an international abatement game where only domestic abatement is allowed. The Kyoto Protocol is a Nash equilibrium outcome of the game where flexibility mechanisms are introduced. We also show that credit discounting on foreign abatement can Pareto improve the welfare of all countries.

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.

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