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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.  ...  In particular, we show that even for games of rank 1 not all equilibria can be reached by a Lemke-Howson path and present a parametric simplex-type algorithm for enumerating all Nash equilibria of a non-degenerate  ...  Thanks to the reviewers for very helpful comments and corrections.  ... 
doi:10.1090/crmp/048/06 fatcat:ep4dn2oki5hx3jbhqhrgcgwasi

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.  ...  In particular, we show that even for games of rank 1 not all equilibria can be reached by a Lemke-Howson path and present a parametric simplex-type algorithm for enumerating all Nash equilibria of a non-degenerate  ...  Thanks to the reviewers for very helpful comments and corrections. ENUMERATING THE NASH EQUILIBRIA OF RANK 1-GAMES  ... 
arXiv:0709.1263v1 fatcat:wktlp4xtardbjepgs6bocza7ru

A note on bimatrix game maximal Selten subsets

Slim Belhaiza, Charles Audet, Pierre Hansen
2014 Arabian Journal of Mathematics  
We present the Eχ -MIPerfect and the EEE-Perfect algorithms which enumerate all extreme perfect Nash equilibria.  ...  In this paper, we implement automatic procedures to enumerate all Nash maximal subsets of a bimatrix game and compute their dimensions.  ...  Enumeration of extreme perfect Nash equilibria To detect all extreme perfect Nash equilibria of bimatrix games directly, the two state of the art algorithms Eχ -MIP [2, 3] and the EEE [1] were modified  ... 
doi:10.1007/s40065-014-0101-x fatcat:lnpurtxaz5hz7gw53eavtzvgxe

Computing Bayes-Nash Equilibria through Support Enumeration Methods in Bayesian Two-Player Strategic-Form Games

Sofia Ceppi, Nicola Gatti, Nicola Basilico
2009 2009 IEEE/WIC/ACM International Joint Conference on Web Intelligence and Intelligent Agent Technology  
The computation of equilibria in games is a challenging task.  ...  The literature studies the problem of finding Nash equilibria with complete-information games in depth, but not enough attention is paid to searching for equilibria in Bayesian games.  ...  ACKNOWLEDGEMENTS The authors are glad to thank Giorgio Patrini and Marco Rocco for their contribution in the implementation and evaluation of the algorithms.  ... 
doi:10.1109/wi-iat.2009.209 dblp:conf/iat/CeppiGB09 fatcat:h6ortuemrrelzemrzn7nntmqt4

Fast Algorithms for Rank-1 Bimatrix Games [article]

Bharat Adsul, Jugal Garg, Ruta Mehta, Milind Sohoni, Bernhard von Stengel
2019 arXiv   pre-print
This paper comprehensively analyzes games of rank one, and shows the following: (1) For a game of rank r, the set of its Nash equilibria is the intersection of a generically one-dimensional set of equilibria  ...  of parameterized games of rank r-1 with a hyperplane. (2) One equilibrium of a rank-1 game can be found in polynomial time. (3) All equilibria of a rank-1 game can be found by following a piecewise linear  ...  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

An Empirical Study of Finding Approximate Equilibria in Bimatrix Games [article]

John Fearnley, Tobenna Peter Igwe, Rahul Savani
2015 arXiv   pre-print
While there have been a number of studies about the efficacy of methods to find exact Nash equilibria in bimatrix games, there has been little empirical work on finding approximate Nash equilibria.  ...  We extend the breadth and depth of our study by including new interesting families of bimatrix games, and studying bimatrix games upto size 2000 × 2000.  ...  We denote their games as SGC. In these game the only equilibrium in a (2k − 1) × (k − 1) game has support sizes k for both players, which makes these games hard for support enumeration.  ... 
arXiv:1502.04980v3 fatcat:7w35hwbyxrcctaifydney52v3y

An Empirical Study of Finding Approximate Equilibria in Bimatrix Games [chapter]

John Fearnley, Tobenna Peter Igwe, Rahul Savani
2015 Lecture Notes in Computer Science  
While there have been a number of studies about the efficacy of methods to find exact Nash equilibria in bimatrix games, there has been little empirical work on finding approximate Nash equilibria.  ...  We extend the breadth and depth of our study by including new interesting families of bimatrix games, and studying bimatrix games upto size 2000×2000.  ...  We denote their games as SGC. In these game the only equilibrium in a (2k − 1) × (k − 1) game has support sizes k for both players, which makes these games hard for support enumeration.  ... 
doi:10.1007/978-3-319-20086-6_26 fatcat:rgjkbsd545gtpbhrux36seyvmu

Roemer on the Rationality of Cooperation

Peter Vallentyne
2020 Erasmus Journal for Philosophy and Economics  
Kantian optimization among people who trust each other to cooperate solves two major problems that confront Nash optimization: (1) the inefficiency of Nash equilibria in the presence of negative externalities  ...  The point is rather that rationality, I claim, limits the role of cooperative reasoning, if any, to the selection of Nash equilibria. To illustrate this, consider, for example, the game in Table 3 .  ... 
doi:10.23941/ejpe.v13i2.502 fatcat:mfkb43qgrneebmzejurpiy5yhe

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
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.  ...  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  ...  The second algorithm (Enumeration) enumerates all the Nash equilibria of a rank-1 game.  ... 
arXiv:1010.3083v2 fatcat:dnmsy2qeabhblogfnnybwl7z3a

Structure of extreme correlated equilibria: a zero-sum example and its implications

Noah D. Stein, Asuman Ozdaglar, Pablo A. Parrilo
2011 International Journal of Game Theory  
In finite games the ratio of extreme correlated to extreme Nash equilibria can be greater than exponential in the size of the strategy spaces.  ...  the set of Nash equilibria which is always expressible in terms of finitely many moments.  ...  Silva for many discussions about ergodic theory, and in particular for the simple proof of Corollary 4.11 using Birkhoff's ergodic theorem.  ... 
doi:10.1007/s00182-010-0267-1 fatcat:gdyy5wehc5dkfdwsudiwcvj4b4

A note on approximate Nash equilibria

Constantinos Daskalakis, Aranyak Mehta, Christos Papadimitriou
2009 Theoretical Computer Science  
be reduced to the case of win-lose games (games with all utilities 0 or 1), and that an approximation of 5 6 is possible, contingent upon a graph-theoretic conjecture.  ...  Subsequent work extends the 1 4 impossibility result of Ingo Althöfer's paper, as mentioned above, to 1 2 [Tomás Feder, Hamid Nazerzadeh, Amin Saberi, Approximating nash equilibria using small-support  ...  Acknowledgments The first and third authors were supported by NSF grant CCF-0515259. The second author's work was done while at the IBM Almaden Research Center.  ... 
doi:10.1016/j.tcs.2008.12.031 fatcat:m4bm47yox5b4bab7apukx4nuna

Page 5372 of Mathematical Reviews Vol. , Issue 2002G [page]

2002 Mathematical Reviews  
, QC); Savard, Gilles (3-MTRLP-GI; Montreal, QC) Enumeration of all extreme equilibria of bimatrix games.  ...  Comput. 23 (2001), no. 1, 323-338 (electronic). The paper under review suggests an algorithm for the computa- tion of the set of Nash equilibrium points in a bimatrix game.  ... 

Parameterized Two-Player Nash Equilibrium [article]

Danny Hermelin and Chien-Chung Huang and Stefan Kratsch and Magnus Wahlstrom
2010 arXiv   pre-print
We study the computation of Nash equilibria in a two-player normal form game from the perspective of parameterized complexity.  ...  These cases occur in the previously studied settings of sparse games and unbalanced games as well as in the newly considered case of locally bounded treewidth games that generalizes both these two cases  ...  Since the result of Chen and Deng [6] , the focus on computing Nash equilibria in bimatrix games was directed either towards finding approximate Nash equilibria [3, 7-9, 12, 13, 23] , or towards finding  ... 
arXiv:1006.2063v1 fatcat:dbtxjagizrgrfcsyb5wdcksqkq

Exclusion Method for Finding Nash Equilibrium in Multiplayer Games

Kimmo Berg, Tuomas Sandholm
2017 PROCEEDINGS OF THE THIRTIETH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE AND THE TWENTY-EIGHTH INNOVATIVE APPLICATIONS OF ARTIFICIAL INTELLIGENCE CONFERENCE  
We present a complete algorithm for finding an epsilon-Nash equilibrium, for arbitrarily small epsilon, in games with more than two players.  ...  The run time grows rapidly with the game size; this reflects the dimensionality of this difficult problem.  ...  Acknowledgments This material is based on work supported by the NSF under grants IIS-1617590, IIS-1320620, and IIS-1546752, the ARO under award W911NF-16-1-0061.  ... 
doi:10.1609/aaai.v31i1.10581 fatcat:hldnqfxgefgcnjm6hnk5wmceji

Parameterized Two-Player Nash Equilibrium

Danny Hermelin, Chien-Chung Huang, Stefan Kratsch, Magnus Wahlström
2012 Algorithmica  
We study the problem of computing Nash equilibria in a twoplayer normal form (bimatrix) game from the perspective of parameterized complexity.  ...  Our results are based on a graph-theoretic representation of a bimatrix game, and on applying graph-theoretic tools on this representation.  ...  Since the result of Chen and Deng, the focus on computing Nash equilibria in bimatrix games was directed either towards finding approximate Nash equilibria [4, 7-9, 12, 13, 25-27, 30] , or towards finding  ... 
doi:10.1007/s00453-011-9609-z fatcat:nqnnvasqtzfflmutpeaqe5tymm
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