On proper refinement of Nash equilibria for bimatrix games

doi:10.1016/j.automatica.2011.07.013

Automatica

Volume 48, Issue 2, February 2012, Pages 297-303

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Abstract

In this paper, we introduce the notion of set of $ϵ$ -proper equilibria for a bimatrix game. We define a 0–1 mixed quadratic program to generate a sequence of $ϵ$ -proper Nash equilibria and show that the optimization results provide reliable indications on strategy profiles that could be used to generate proper equilibria analytically. This approach can be generalized in order to find at least one proper equilibrium for any bimatrix game. Finally, we define another 0–1 mixed quadratic program to identify non-proper extreme Nash equilibria.

Résumé

Dans cet article nous établissons la définition de l’ensemble d’équilibres $ϵ$ -propres pour un jeu bimatriciel. Nous définissons un programme quadratique mixte 0–1 afin de générer une séquence d’équilibres $ϵ$ -propres et de montrer que les résultats de l’optimisation de ce programme permettent d’indiquer les choix stratégiques succeptibles de générer un ou plusieurs équilibres propres analytiquement. Cette approche peut être généralisée afin de trouver au moins un équilibre propre pour tout jeu bimatriciel. Nous définissons aussi un autre programme quadratique mixte 0–1 afin d’identifier les équilibres de Nash non-propres.

Introduction

A bimatrix game is a strategic confrontation of two players, I and II. A bimatrix game $G (A, B)$ is defined by a pair of $n \times m$ payoff matrices $A$ and $B$ . Each player has a finite number of actions to choose from. The deterministic choice of an action is called pure strategy. Player I has to choose between $n$ pure strategies, while player II has to choose between $m$ pure strategies.

Each player attempts to maximize his own payoff by selecting a probability vector over his set of pure strategies. These vectors are combinations of pure strategies, called mixed strategies, and represented by probability vectors $x_{1} \in R^{n}$ and $x_{2} \in R^{m}$ . Hence, player I’s payoff is $x_{1}^{t} A x_{2}$ and player II’s payoff is $x_{1}^{t} B x_{2}$ .

A Nash equilibrium is defined as a profile of strategies such that simultaneously, player I maximizes his payoff given the strategic choice of player II and player II maximizes his payoff given the strategic choice of player I. A number of papers have addressed the problem of enumeration of all Nash extreme equilibria for bimatrix games (see Audet et al., 2006, Audet et al., 2001).

When confronted with a situation where a large number of equilibria can be considered to solve a game, decision makers would have to refine their choices using some other rational concepts in addition to the concept of Nash equilibrium. Perfect and $Proper$ equilibria are two refinements of the concept of Nash equilibrium based on the idea that a reasonable equilibrium should be stable against slight perturbations in the equilibrium strategies. It is also well known that a subgame perfect equilibrium for a two-person extensive game corresponds to a proper equilibrium for its corresponding reduced normal form bimatrix game representation. One can find a short review of these concepts at the end of this paper.

Lack of analytical and numerical tools that can be used to generate such equilibria with robustness properties made these refinements rarely used in practice. This paper tries to answer the following question: How can we automatically detect $proper$ extreme Nash equilibria?

Section 2 recalls the definition of proper refinement concept and introduces the definition of the set of $ϵ$ -proper equilibria. Section 3 proposes a mixed 0–1 quadratic program in order to detect $ϵ$ -proper equilibria. This section details different cases of convergence results and discusses a theoretical procedure to generate proper equilibria and conclude on the non-properness of an equilibrium.

Section snippets

Set of $ϵ$ -proper equilibria

The main idea behind the proper refinement of Nash equilibria is that a reasonable player would try harder to avoid important mistakes than he or she would try to avoid small ones. While any proper equilibrium profile is perfect, a perfect equilibrium profile could be non-proper. Let us note $A_{i}$ and $A_{h}$ respectively as the $i$ th and $h$ th rows of the payoff matrix $A$ . Similarly, we note $B_{j}$ and $B_{l}$ respectively as the $j$ th and $l$ th rows of the payoff matrix $B$ .

Definition 2.1

A bimatrix game profile $(x_{1}, x_{2})$ is said to be

Detection of $ϵ$ -proper equilibria

In order to generate such sequence of positive real numbers, we define a family of parametrized mixed 0–1 quadratic programs such that their solutions define a sequence of $ϵ$ -proper equilibria, when the parameter $σ$ converges to 0.

Proposition 3.1

The perfect equilibrium profile $({\hat{x}}_{1}, {\hat{x}}_{2})$ is a proper equilibrium if and only if the following 0–1 -mixed quadratic program is feasible for all $\bar{σ} > 0$ , and if ${lim}_{σ \to 0^{+}} f (σ) = 0$ . $\begin{matrix} f (σ) & = & min_{(x_{1}, x_{2}) \in Ω_{ϵ}^{σ}, ϵ} & ϵ \\ s.t. & {\hat{x}}_{1 i} - ϵ \leq x_{1 i} \leq {\hat{x}}_{1 i} + ϵ, \forall i \in {1, 2, \dots, n}, \\ {\hat{x}}_{2 j} - ϵ \leq x_{2 j} \leq {\hat{x}}_{2 j} + ϵ, \forall j \in {1, 2, \dots, m}, \\ 0 \leq ϵ \leq 1 . \end{matrix}$

Proof

Let

Conclusion

In this paper we presented a mathematical programming approach for the refinement of Nash equilibria. After complete enumeration of all extreme Nash equilibria, $ϵ$ -proper sequences of equilibria are found using the indications provided by the convergence numerical results of a 0–1 mixed quadratic program. Even in the worst case where no extreme proper equilibrium is found, we have shown that we can always find a pair of extreme perfect equilibria belonging to the same Selten subset in order to

Slim Belhaiza is an Assistant Professor of Mathematics at the King Fahd University of Petroleum and Minerals. His research interests include the development of algorithms for Game theory and Vehicle Routing. He obtained a Ph.D. degree in applied mathematics from the École Polytechnique de Montréal in 2008, and worked for an optimization company in Montréal from 2008 to 2009.

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Charles Audet is a Professor of Mathematics at the École Polytechnique de Montréal.

His research interests include the analysis and development of algorithms for structured global optimization, and blackbox nonsmooth optimization. He obtained a Ph.D. degree in applied mathematics from the École Polytechnique de Montréal in 1998, and worked as a post-doc at the Rice University in Houston, Texas from 1998 to 2000.

Pierre Hansen obtained a Ph.D. degree in Mathematics, from the University of Brussels in 1974. He has taught in Belgium, France, USA, Canada, and for short periods in Italy, Germany, Hong Kong, China and Brazil.

Hansen is currently a Professor and holder of the Data Mining Chair at the HEC Montréal. He is the recipient of several research prizes including the EURO Gold Medal, 1986, the Merit Award of the Canadian Operational Research Society, 1999, and the Pierre Rousseau Prize of ACFAS 2008. He is an author, and most of the time co-author with colleagues and students, of more than 300 papers in refereed journals from various fields. Hansen is a Fellow of the Royal Society of Canada, 1999. He is also a member of the International Academy of Mathematical Chemistry, 2005.

^☆: The material in this paper was partially presented at the 12th Annual Congress of the French National Society of Operations Research and Decision Science (ROADEF 2011), March 2-4, 2011, Saint-Etienne, France. This paper was recommended for publication in revised form under the direction of the Editor, Berç Rüstem.

¹: Tel.: +966 38601054; fax: +966 38602340.

View full text

On proper refinement of Nash equilibria for bimatrix games☆

Abstract

Résumé

Introduction

Section snippets

Set of ϵ-proper equilibria

Detection of ϵ-proper equilibria

Conclusion

Concavity cuts for disjoint bilinear programming

Mathematical Programming

Enumeration of all extreme equilibria in game theory: bimatrix and polymatrix games

Journal of Optimization Theory and Applications

A new sequence form approach for the enumeration and refinement of all extreme Nash equilibria for extensive form games

International Game Theory Review

Enumeration of all extreme equilibrium strategies of bimatrix games

SIAM Journal on Scientific Computing

On the structure of the set of perfect equilibria in bimatrix games

O-R Spektrum

Games with randomly distributed payoffs: a new rationale for mixed-strategy equilibrium points

International Journal of Game Theory

Regularity and stability of equilibrium points of bimatrix games

Mathematics of Operations Research

Regular equilibrium points of bimatrix games

OR Spektrum

Set of $ϵ$ -proper equilibria

Detection of $ϵ$ -proper equilibria