A general structure theorem for the Nash equilibrium correspondence

Predtetchinski, Arkadi

doi:10.1016/j.geb.2008.10.005

Cited by 7 publications

(7 citation statements)

References 6 publications

(11 reference statements)

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“…Moreover, the homeomorphism result also validates the homotopy methods devised to compute a Nash equilibrium [6,7]. The structural result has been extended for more general game spaces [16], however, to the best of our knowledge, no such result is known for special subspaces of the bimatrix game space. Such a result may pave a way to device better algorithms for the Nash equilibrium computation or to prove the hardness of computing a Nash equilibrium for the games in the subspace.…”

Section: Introductionsupporting

confidence: 61%

Rank-1 bimatrix games

Adsul

Garg

Mehta

et al. 2011

Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing

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Given a rank-1 bimatrix game (A, B), i.e., where rank(A + B) = 1, we construct a suitable linear subspace of the rank-1 game space and show that this subspace is homeomorphic to its Nash equilibrium correspondence. Using this homeomorphism, we give the first polynomial time algorithm for computing an exact Nash equilibrium of a rank-1 bimatrix game. This settles an open question posed in [8,19]. In addition, we give a novel algorithm to enumerate all the Nash equilibria of a rank-1 game and show that a similar technique may also be applied for finding a Nash equilibrium of any bimatrix game. Our approach also provides new proofs of important classical results such as the existence and oddness of Nash equilibria, and the index theorem for bimatrix games. Further, we extend the rank-1 homeomorphism result to a fixed rank game space, and give a fixed point formulation on [0, 1] k for solving a rank-k game. The homeomorphism and the fixed point formulation are piece-wise linear and considerably simpler than the classical constructions.

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Section: Introductionsupporting

confidence: 61%

Rank-1 bimatrix games

Adsul

Garg

Mehta

et al. 2011

Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing

View full text Add to dashboard Cite

show abstract

“…Moreover, the homeomorphism result also validates the homotopy methods devised to compute a Nash equilibrium [5,7]. The structural result has been extended for more general game spaces [17], however, to the best of our knowledge, no such result is known for special subspaces of the bimatrix game space. Such a result may pave a way to device a better algorithm for the Nash equilibrium computation or to prove the hardness of computing a Nash equilibrium, for the games in the subspace.…”

Section: Introductionsupporting

confidence: 61%

Rank-1 Bi-matrix Games: A Homeomorphism and a Polynomial Time Algorithm

Adsul¹,

Garg²,

Mehta³

et al. 2010

Preprint

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Given a rank-1 bimatrix game (A, B), i.e., where rank(A+ B) = 1, we construct a suitable linear subspace of the rank-1 game space and show that this subspace is homeomorphic to its Nash equilibrium correspondence. Using this homeomorphism, we give the first polynomial time algorithm for computing an exact Nash equilibrium of a rank-1 bimatrix game. This settles an open question posed in [8,21]. In addition, we give a novel algorithm to enumerate all the Nash equilibria of a rank-1 game and show that a similar technique may also be applied for finding a Nash equilibrium of any bimatrix game. This technique also proves the existence, oddness and the index theorem of Nash equilibria in a bimatrix game. Further, we extend the rank-1 homeomorphism result to a fixed rank game space, and give a fixed point formulation on [0, 1] k for solving a rank-k game. The homeomorphism and the fixed point formulation are piece-wise linear and considerably simpler than the classical constructions.

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“…For every u 2 U , the strategy profile x is a Nash equilibrium of (u) (see [17,18]), and since i (u) is equal to u i up to an element of A i , we obtain that (u) 2 U (by assumption, as…”

Section: Proof Of Corollary 21mentioning

confidence: 95%

“…The proof can be found in [17,18], which is an extension of Kohlberg-Mertens' structure theorem ( [12]). For completness, we recall that the homeomorphism ⌘ : N !…”

mentioning

confidence: 98%

“…Concavity (or at least quasiconcavity) of payo↵ functions u i (x i , x i ) of each player i with respect to x i is a standard assumption in the literature when considering conditions for the existence of a Nash equilibrium. this requires to be able to prove some extension of ) applied to each U k , which can be done by an adequate choice of these sets (applying Tarski-Seidenberg's theorem), 4 and by some application of a recent result of Predtetchinsky ( [17,18]). 5 Our paper is organized as follows.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Oddness of the number of Nash equilibria: The case of polynomial payoff functions

Bich,

Fixary

2024

Games and Economic Behavior

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A general structure theorem for the Nash equilibrium correspondence

Cited by 7 publications

References 6 publications

Rank-1 bimatrix games

Rank-1 bimatrix games

Rank-1 Bi-matrix Games: A Homeomorphism and a Polynomial Time Algorithm

Oddness of the number of Nash equilibria: The case of polynomial payoff functions

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