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Games of fixed rank: A hierarchy of bimatrix games [article]

Ravi Kannan, Thorsten Theobald
2005 arXiv   pre-print
Specifically, we investigate a hierarchy of bimatrix games (A,B) which results from restricting the rank of the matrix A+B to be of fixed rank at most k.  ...  For every fixed k, this class strictly generalizes the class of zero-sum games, but is a very special case of general bimatrix games.  ...  Although the problem of finding a Nash equilibrium in a game of fixed rank is a very special case of the problem of finding a Nash equilibrium in an arbitrary bimatrix game, we do not know if there exists  ... 
arXiv:cs/0511021v1 fatcat:6bzfl6h6e5cynlodqflzeax2za

Games of fixed rank: a hierarchy of bimatrix games

Ravi Kannan, Thorsten Theobald
2009 Economic Theory  
We propose and investigate a hierarchy of bimatrix games (A, B), whose (entry-wise) sum of the pay-off matrices of the two players is of rank k, where k is a constant.  ...  We will say the rank of such a game is k. For every fixed k, the class of rank kgames strictly generalizes the class of zero-sum games, but is a very special case of general bimatrix games.  ...  From the viewpoint of game theory, it provides a flexible hierarchy between zero-sum games and general bimatrix games.  ... 
doi:10.1007/s00199-009-0436-2 fatcat:srt5zgxqwbfcnppfvvwvg6qsi4

Enumerating the Nash equilibria of rank 1-games [article]

Thorsten Theobald
2007 arXiv   pre-print
A bimatrix game (A,B) is called a game of rank k if the rank of the matrix A+B is at most k. We consider the problem of enumerating the Nash equilibria in (non-degenerate) games of rank 1.  ...  game of rank 1.  ...  ENUMERATING THE NASH EQUILIBRIA OF RANK 1-GAMES  ... 
arXiv:0709.1263v1 fatcat:wktlp4xtardbjepgs6bocza7ru

Enumerating the Nash equilibria of rank-1 games [chapter]

Thorsten Theobald
2009 CRM Proceedings and Lecture notes AMS  
A bimatrix game (A, B) is called a game of rank k if the rank of the matrix A + B is at most k. We consider the problem of enumerating the Nash equilibria in (non-degenerate) games of rank 1.  ...  game of rank 1.  ...  Recently, Kannan and Theobald [9] have introduced a hierarchy of bimatrix games in which the matrix A + B is restricted to be of rank at most k, for some fixed constant k.  ... 
doi:10.1090/crmp/048/06 fatcat:ep4dn2oki5hx3jbhqhrgcgwasi

Bilinear Games: Polynomial Time Algorithms for Rank Based Subclasses [article]

Jugal Garg and Albert Xin Jiang and Ruta Mehta
2011 arXiv   pre-print
We consider a rank based hierarchy of bilinear games, where rank of a game (A,B) is defined as rank(A+B).  ...  (ET'09), for bimatrix games with low rank matrices.  ...  Next, we extend Kannan and Theobald's [17] rank-based hierarchy for bimatrix games to bilinear games, by defining the rank of a bilinear game with payoff matrices (A, B) as the rank of (A + B).  ... 
arXiv:1109.6182v1 fatcat:cuairizybbhvzjmvb3d7dgalgy

Fast Algorithms for Rank-1 Bimatrix Games [article]

Bharat Adsul, Jugal Garg, Ruta Mehta, Milind Sohoni, Bernhard von Stengel
2019 arXiv   pre-print
The rank of a bimatrix game is the matrix rank of the sum of the two payoff matrices.  ...  In contrast, such a path-following method finds only one equilibrium of a bimatrix game. (4) The number of equilibria of a rank-1 game may be exponential. (5) There is a homeomorphism between the space  ...  We thank two anonymous referees of Operations Research for their detailed comments which helped improve the manuscript. Heinrich Nax suggested the "trade game" (70) in Section 10.  ... 
arXiv:1812.04611v4 fatcat:3yepxkylvjacreouuxdcoztuam

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

Bharat Adsul, Jugal Garg, Ruta Mehta, Milind Sohoni
2010 arXiv   pre-print
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.  ...  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  ...  Kannan and Theobald [8] defined a hierarchy of bimatrix games using the rank of (A + B) and gave a polynomial time algorithm to compute an approximate Nash equilibrium for games of a fixed rank k.  ... 
arXiv:1010.3083v2 fatcat:dnmsy2qeabhblogfnnybwl7z3a

Exploiting Concavity in Bimatrix Games: New Polynomially Tractable Subclasses [chapter]

Spyros Kontogiannis, Paul Spirakis
2010 Lecture Notes in Computer Science  
It is though incomparable to the subclass of games with fixed rank [16]: Even rank−1 games may not be mutually concave (eg, Prisoner's dilemma), but on the other hand, there exist mutually concave games  ...  This subclass entirely contains the most popular subclass of polynomial-time solvable bimatrix games, namely, all the constantsum games (rank−0 games).  ...  [16] introduced a hierarchy of the bimatrix games, according to the rank of the matrix R + C of the game R, C , which was called the rank of the game.  ... 
doi:10.1007/978-3-642-15369-3_24 fatcat:ukvdyvu6qffhrhkm3tfyeuvrtm

On mutual concavity and strategically-zero-sum bimatrix games

Spyros Kontogiannis, Paul Spirakis
2012 Theoretical Computer Science  
We conclude our discussion with a comparison of MC-games (or, SZS-games) to k-rank games, which are known to admit for 2NASH a FPTAS when k is fixed [18] , and a polynomialtime algorithm for k = 1 [2].  ...  between the spaces of CE and NE points in a bimatrix game (e.g., [15, 26, 33] ).  ...  We also observed that the class of MC-games, which entirely contains all constant-sum games, is incomparable to the class of bimatrix games with fixed rank: even rank-1 games may be non-MC-games, and even  ... 
doi:10.1016/j.tcs.2012.01.016 fatcat:ihoq65e2nzdlpfv3nyhpujdnfa

Semidefinite Programming and Nash Equilibria in Bimatrix Games [article]

Amir Ali Ahmadi, Jeffrey Zhang
2019 arXiv   pre-print
We explore the power of semidefinite programming (SDP) for finding additive epsilon-approximate Nash equilibria in bimatrix games.  ...  Furthermore, we prove that if a rank-2 solution to our SDP is found, then a 5/11-NE can be recovered for any game, or a 1/3-NE for a symmetric game.  ...  equilibria of a bimatrix game.  ... 
arXiv:1706.08550v3 fatcat:5zwr6lcukbhu3m3fvv54zba2ri

A Semidefinite Hierarchy for Disjointly Constrained Multilinear Programming [article]

Kai Kellner
2016 arXiv   pre-print
For nondegenerate bimatrix games, a Nash equilibrium can be computed by the sum of squares approach in finitely many steps.  ...  Based on a reformulation of the problem in terms of sum-of-squares polynomials, we study a hierarchy of semidefinite relaxations to the problem.  ...  Originally, bilinear programming has been considered as a generalization of bimatrix games in game theory.  ... 
arXiv:1603.03634v1 fatcat:happcknkmzacfbb4q2c774zxky

A Polynomial-Time Algorithm for 1/2-Well-Supported Nash Equilibria in Bimatrix Games [article]

Argyrios Deligkas, Michail Fasoulakis, Evangelos Markakis
2022 arXiv   pre-print
Since the seminal PPAD-completeness result for computing a Nash equilibrium even in two-player games, an important line of research has focused on relaxations achievable in polynomial time.  ...  Our algorithm is based on linear programming and in particular on exploiting suitably defined zero-sum games that arise from the payoff matrices of the two players.  ...  Observe that for any fixed value of k, the size of the support of a k-uniform strategy is at most k. This implies that in a n × n bimatrix game, there are at most O(n k ) kuniform strategy profiles.  ... 
arXiv:2207.07007v1 fatcat:hukr5lrbnzg4dlreeilyagyvqi

Exchangeable Equilibria, Part I: Symmetric Bimatrix Games [article]

Noah D. Stein, Asuman Ozdaglar, Pablo A. Parrilo
2014 arXiv   pre-print
We introduce the notion of exchangeable equilibria of a symmetric bimatrix game, defined as those correlated equilibria in which players' strategy choices are conditionally independently and identically  ...  We give several game-theoretic interpretations and a version of the "revelation principle".  ...  This research was funded by the National Science Foundation under Award 1027922 and the Air Force Office of Scientific Research under grant FA9550-11-1-0305.  ... 
arXiv:1307.3586v3 fatcat:46yq7yo7dvcvfpkb66t5afy6lm

Similarity Suppresses Cyclicity: Why Similar Competitors Form Hierarchies [article]

Christopher Cebra, Alexander Strang
2022 arXiv   pre-print
We consider a series of canonical bimatrix games and an ensemble of random performance functions that demonstrate the generality of our mechanism, even when faced with highly cyclic games.  ...  To test that theory, we run a series of evolution experiments designed to mimic genetic training algorithms.  ...  These include a set of illustrative bimatrix games, and a sequence of randomly generated games with tuneable structure designed to demonstrate generality.  ... 
arXiv:2205.08015v1 fatcat:jwjzw3o4gzdkbo5f2jylzajy3q

Complexity Aspects of Fundamental Questions in Polynomial Optimization [article]

Jeffrey Zhang
2020 arXiv   pre-print
We show that for a symmetric game, a 1/3-Nash equilibrium can be efficiently recovered from any rank-2 solution to this relaxation.  ...  We also give a new characterization of coercive polynomials that lends itself to a hierarchy of SDPs.  ...  quadratic objective function over the set of Nash equilibria of a bimatrix game.  ... 
arXiv:2008.12170v1 fatcat:jdlr2ydcorcg3gmxhwhhcjngmq
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