From Imitation Games to Kakutani

We give a full proof of the Kakutani (1941) fixed point theorem that is brief, elementary, and based on game theoretic concepts. This proof points to a new family of algorithms for computing approximate fixed points that have advantages over simplicial subdivision methods. An imitation game is a finite two person normal form game in which the strategy spaces for the two agents are the same and the goal of the second player is to choose the same strategy as the first player. These appear in our proof, but are also interesting from other points of view.

[1]  E. M. Hartwell.  Boston , 1906 .

[2]  L. Brouwer.  Über Abbildung von Mannigfaltigkeiten , 1911 .

[3]  S. Kakutani.  A generalization of Brouwer’s fixed point theorem , 1941 .

[4]  D. Montgomery,et al.  Fixed Point Theorems for Multi-Valued Transformations , 1946 .

[5]  H. F. Bohnenblust,et al.  On a Theorem of Ville , 1949 .

[6]  J. Nash.  Equilibrium Points in N-Person Games. , 1950, Proceedings of the National Academy of Sciences of the United States of America.

[7]  G. Debreu,et al.  Nonnegative Square Matrices , 1953 .

[8]  G. Dantzig,et al.  Notes on Linear Programming: Part 1. The Generalized Simplex Method for Minimizing a Linear Form under Linear Inequality Restraints , 1954 .

[9]  J. Milnor.  Topology from the differentiable viewpoint , 1965 .

[10]  C. E. Lemke,et al.  Bimatrix Equilibrium Points and Mathematical Programming , 1965 .

[11]  H. Scarf.  The Core of an N Person Game , 1967 .

[12]  Herbert E. Scarf,et al.  The Approximation of Fixed Points of a Continuous Mapping , 1967 .

[13]  H. Scarf,et al.  On The Applications of a Recent Combinatorial Algorithm , 1969 .

[14]  B. Eaves.  Computing Kakutani Fixed Points , 1971 .

[15]  K. Arrow,et al.  General Competitive Analysis , 1971 .

[16]  L. Shapley.  On Balanced Games Without Side Payments , 1973 .

[17]  Francesco Mallegni,et al.  The Computation of Economic Equilibria , 1973 .

[18]  Herbert E. Scarf,et al.  The Computation of Economic Equilibria , 1974 .

[19]  L. Shapley.  A note on the Lemke-Howson algorithm , 1974 .

[20]  M. Todd.  The Computation of Fixed Points and Applications , 1976 .

[21]  John Milnor,et al.  Analytic Proofs of the “Hairy Ball Theorem” and the Brouwer Fixed Point Theorem , 1978 .

[22]  Editors , 1986, Brain Research Bulletin.

[23]  Christos H. Papadimitriou,et al.  Exponential lower bounds for finding Brouwer fixed points , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[24]  Katta G. Murty,et al.  Linear complementarity, linear and nonlinear programming , 1988 .

[25]  T. M. Doup,et al.  Simplicial Algorithms on the Simplotope , 1988 .

[26]  李幼升,et al.  Ph , 1989 .

[27]  Eitan Zemel,et al.  Nash and correlated equilibria: Some complexity considerations , 1989 .

[28]  Lloyd S. Shapley,et al.  On Kakutani's fixed point theorem, the K-K-M-S theorem and the core of a balanced game , 1991 .

[29]  Richard W. Cottle,et al.  Linear Complementarity Problem. , 1992 .

[30]  B. M. Fulk.  MATH , 1992 .

[31]  P. Gács,et al.  Algorithms , 1992 .

[32]  Christos H. Papadimitriou,et al.  On the Complexity of the Parity Argument and Other Inefficient Proofs of Existence , 1994, J. Comput. Syst. Sci..

[33]  Walter D. Morris,et al.  Lemke Paths on Simple Polytopes , 1994, Math. Oper. Res..

[34]  B. Stengel,et al.  COMPUTING EQUILIBRIA FOR TWO-PERSON GAMES , 1996 .

[35]  H. Kuk.  On equilibrium points in bimatrix games , 1996 .

[36]  P. Jean-Jacques Herings,et al.  An extremely simple proof of the K-K-M-S Theorem , 1997 .

[37]  Ilse C. F. Ipsen,et al.  THE IDEA BEHIND KRYLOV METHODS , 1998 .

[38]  Eric van Damme,et al.  Non-Cooperative Games , 2000 .

[39]  Christos H. Papadimitriou,et al.  Algorithms, Games, and the Internet , 2001, ICALP.

[40]  Janet Rosenbaum.  The Computational Complexity of Nash Equilibria , 2002 .

[41]  Oper , 2002 .

[42]  Vincent Conitzer,et al.  Complexity Results about Nash Equilibria , 2002, IJCAI.

[43]  Bernhard von Stengel,et al.  Exponentially many steps for finding a Nash equilibrium in a bimatrix game , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[44]  Bruno Codenotti,et al.  On the computational complexity of Nash equilibria for (0, 1) bimatrix games , 2005, Inf. Process. Lett..

[45]  Xi Chen,et al.  3-NASH is PPAD-Complete , 2005, Electron. Colloquium Comput. Complex..

[46]  Vincenzo Bonifaci,et al.  On the Complexity of Uniformly Mixed Nash Equilibria and Related Regular Subgraph Problems , 2005, FCT.

[47]  J. M. Bilbao,et al.  Contributions to the Theory of Games , 2005 .

[48]  Christos H. Papadimitriou,et al.  Three-Player Games Are Hard , 2005, Electron. Colloquium Comput. Complex..

[49]  Xiaotie Deng,et al.  Settling the Complexity of Two-Player Nash Equilibrium , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[50]  Paul W. Goldberg,et al.  The complexity of computing a Nash equilibrium , 2006, STOC '06.

[51]  Paul W. Goldberg,et al.  Reducibility among equilibrium problems , 2006, STOC '06.

[52]  Rahul Savani,et al.  Hard‐to‐Solve Bimatrix Games , 2006 .

[53]  Amin Saberi,et al.  Leontief economies encode nonzero sum two-player games , 2006, SODA '06.

[54]  Jörg Bewersdorff,et al.  Symmetric Games , 2022, Luck, Logic, and White Lies.