New Maximal Numbers of Equilibria in Bimatrix Games

Stengel, Bernhard von

doi:10.1007/pl00009438

Cited by 41 publications

(33 citation statements)

References 28 publications

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“…This holds if the game is nondegenerate, defined by the property that the number of pure best responses to any mixed strategy never exceeds the size of its support (see von Stengel (2002) for a detailed discussion). The LH algorithm can be extended to degenerate games by standard lexicographic perturbation techniques.…”

Section: Games Polytopes and The Lemke-howson Algorithmmentioning

confidence: 99%

“…The equilibrium payoff u, normalized to 1 in Cz ≤ 1, is the scaling factor. The converse mapping from x to z defines a projective transformation of a polyhedron that represents the "upper envelope" of expected payoffs to the polytope S (see von Stengel (2002)). …”

Section: Qedmentioning

confidence: 99%

“…This geometric view gives a good insight into the structure of Nash equilibria of two-player games (see von Stengel (2002)). The set of each player's mixed strategies together with the best response payoff to the other player is described by a polyhedron.…”

Section: Introductionmentioning

confidence: 99%

“…These are polyhedra for which the vertex-defining inequalities are known in arbitrary dimension (Gale (1963)). This purely combinatorial information can also be used to construct bimatrix games with many equilibria (von Stengel (1999)). …”

Section: Introductionmentioning

confidence: 99%

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Hard-to-Solve Bimatrix Games

2006

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The Lemke-Howson algorithm is the classical method for finding one Nash equilibrium of a bimatrix game. This paper presents a class of square bimatrix games for which this algorithm takes, even in the best case, an exponential number of steps in the dimension d of the game. Using polytope theory, the games are constructed using pairs of dual cyclic polytopes with 2d suitably labeled facets in d-space. The construction is extended to nonsquare games where, in addition to exponentially long Lemke-Howson computations, finding an equilibrium by support enumeration takes on average exponential time.

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Section: Games Polytopes and The Lemke-howson Algorithmmentioning

confidence: 99%

Section: Qedmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hard-to-Solve Bimatrix Games

2006

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“…For example, McKelvey and McLennan (1997) show that games with ten players with two strategies each, can have u p t o 1.3 million competely mixed (regular) equilibria. (See also Keiding (1997), McLennan (1997), and von Stengel (1999) for more literature on the maximal numbers of Nash equilibria of normal form games.) Although it needs to be checked exactly to what extent s u c h n umbers are related to say the number of Nash equivalence classes, it suggests that they could be large.…”

Section: Resultsmentioning

confidence: 99%

On some geometry and equivalence classes of normal form games

Germano

2006

Int J Game Theory

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Equivalence classes of normal form games are de ned using the geometry of correspondences of standard equilibrium concepts like correlated, Nash, and robust equilibrium, or risk dominance and rationalizability. Resulting equivalence classes are fully characterized and compared across di erent equilibrium concepts for 2 2 games. It is argued that the procedure can lead to broad and gametheoretically meaningful distinctions of games as well as to alternative w ays of viewing and testing equilibrium concepts. Larger games are also brie y considered. Keywords Non-cooperative games, classi cation and equivalence classes, experimental games, geometry of games. JEL Classi cation C 7 0 , C 7 2 , C 7 8 , C 9 0 , C 9 1 .

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Nash Equilibrium (Pure and Mixed)

Jones

2011

Wiley Encyclopedia of Operations Research and Management Science

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This article defines and examines the Nash equilibrium solution concept for noncooperative strategic form games. Examples of canonical (2 × 2) bimatrix form games are used to demonstrate that a noncooperative strategic form game may have a unique pure or mixed strategy Nash equilibrium (Prisoners' Dilemma and Matching Pennies, respectively), may have both pure and mixed strategy Nash equilibria (Battle of the Sexes), or an infinite number of Nash equilibria. Nash's application of the Kakutani fixed point theorem to prove the existence of equilibria is discussed and applied to specific games. Motivated by the possible multiplicity of Nash equilibria in noncooperative strategic form games, the number of equilibria, the construction of games with a unique, prespecified equilibrium, and the calculation of equilibria are considered. The article concludes with a short description of the attempts to refine the Nash equilibrium concept, including perfect and proper equilibria, as well as evolutionary stability.

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New Maximal Numbers of Equilibria in Bimatrix Games

Abstract: Abstract. This paper presents a new lower bound of 2.

Cited by 41 publications

References 28 publications

Hard-to-Solve Bimatrix Games

Hard-to-Solve Bimatrix Games

On some geometry and equivalence classes of normal form games

Nash Equilibrium (Pure and Mixed)

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