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

Borm, Peter; Jansen, M.J.M.; Potters, J.A.M.; Tijs, S.H.

doi:10.1007/bf01783413

Cited by 13 publications

(15 citation statements)

References 11 publications

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“…(x 1 , x 2 ) ∈ T and (y 1 , y 2 ) ∈ T imply that (x 1 , y 2 ) ∈ T and (y 1 , x 2 ) ∈ T . A Nash subset T is called maximal if it is not properly contained in another Nash subset (Jansen [6]). Any extreme equilibrium is an extreme point of one of these maximal Nash subsets.…”

Section: Enumeration Of All Maximal Nash Subsetsmentioning

confidence: 99%

“…We then use exact arithmetics as in [3] to determine the rank of the corresponding matrices A(T ) and B(T ). Example 2.2 For the bimatrix game taken from Borm et al [6], we have determined the dimensions of the six maximal Nash subsets found (Table 2).…”

Section: Maximal Nash Subset Geometry and Dimensionmentioning

confidence: 99%

“…When decision-makers face a large set of Nash equilibria to choose from, additional rational concepts can be used to refine the set. As described by Borm et al [6], a maximal Selten subset is a set of interchangeable perfect equilibria. Each maximal Selten subset is a subset of a maximal Nash subset and each extreme point of a maximal Selten subset corresponds to an extreme perfect equilibrium.…”

Section: Extreme Perfect Equilibriamentioning

confidence: 99%

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A note on bimatrix game maximal Selten subsets

2014

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In this paper, we implement automatic procedures to enumerate all Nash maximal subsets of a bimatrix game and compute their dimensions. We propose a linear programming approach to identify extreme perfect Nash equilibria, enumerate all Selten maximal subsets and compute their dimensions. We present the Eχ-MIPerfect and the EEE-Perfect algorithms which enumerate all extreme perfect Nash equilibria. We finally report and comment computational experiments on randomly generated bimatrix games with different size and density.

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Section: Enumeration Of All Maximal Nash Subsetsmentioning

confidence: 99%

Section: Maximal Nash Subset Geometry and Dimensionmentioning

confidence: 99%

Section: Extreme Perfect Equilibriamentioning

confidence: 99%

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A note on bimatrix game maximal Selten subsets

2014

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“…Her constructive proof generates a correspondence whose fixed points are precisely the perfect equilibria of a given finite game. For bimatrix games, Borm et al [22] described a maximal Selten subset as a set of interchangeable perfect equilibria. Each maximal Selten subset is a subset of a maximal Nash subset and each extreme point of a maximal Selten subset corresponds to an extreme perfect equilibrium.…”

Section: Motivation Decision Makers Confronted To Multiplementioning

confidence: 99%

“…A Nash subset is called maximal if it is not properly contained in another Nash subset [22]. Enumeration of all maximal Nash subsets can be achieved using an algorithm for the enumeration of all maximal cliques of a graph [26].…”

Section: (5)mentioning

confidence: 99%

On Perfect Nash Equilibria of Polymatrix Games

Belhaiza

2014

Game Theory

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When confronted with multiple Nash equilibria, decision makers have to refine their choices. Among all known Nash equilibrium refinements, theperfectnessconcept is probably the most famous one. It is known that weakly dominated strategies of two-player games cannot be part of a perfect equilibrium. In general, thisundominanceproperty however does not extend ton-player games (E. E. C. van Damme, 1983). In this paper we show that polymatrix games, which form a particular class ofn-player games, verify the undominance property. Consequently, we prove that every perfect equilibrium of a polymatrix game is undominated and that every undominated equilibrium of a polymatrix game is perfect. This result is used to set a new characterization of perfect Nash equilibria for polymatrix games. We also prove that the set of perfect Nash equilibria of a polymatrix game is a finite union of convex polytopes. In addition, we introduce a linear programming formulation to identify perfect equilibria for polymatrix games. These results are illustrated on two small game applications. Computational experiments on randomly generated polymatrix games with different size and density are provided.

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On the set of proper equilibria of a bimatrix game

Jansen

1993

Int J Game Theory

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In this paper it is proved that the set of proper equilibria of a bimatrix game is the finite union of polytopes. To that purpose we split up the strategy space of each player into a finite number of equivalence classes and consider for a given e > 0 the set of all e-proper pairs within the cartesian product of two equivalence classes. If this set is non-empty, its closure is a polytope. By considering this polytope as e goes to zero, we obtain a (Myerson) set of proper equilibria. A Myerson set appears to be a polytope.

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

Abstract: On the structure of the set of perfect equilibria in bimatrix games Borm, Peter; Jansen, M.J.M.; Potters, J.A.M.; Tijs, S.H.

Cited by 13 publications

References 11 publications

A note on bimatrix game maximal Selten subsets

A note on bimatrix game maximal Selten subsets

On Perfect Nash Equilibria of Polymatrix Games

On the set of proper equilibria of a bimatrix game

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