On the existence of Nash equilibrium in discontinuous games

This paper offers an equilibrium existence theorem in discontinuous games. We introduce a new notion of very weak continuity, called quasi-weak transfer continuity that guarantees the existence of pure strategy Nash equilibrium in compact and quasiconcave games. We also consider possible extensions and improvements of the main result. We present applications to show that our conditions allow for economically meaningful payoff discontinuities.

[1]  Paul R. Milgrom,et al.  Rationalizability, Learning, and Equilibrium in Games with Strategic Complementarities , 1990 .

[2]  Kazuo Nishimura,et al.  Existence of Nash Equilibrium in n Person Games without Quasi-Concavity , 1981 .

[3]  P. Reny.  Further Results on the Existence of Nash Equilibria in Discontinuous Games∗ , 2009 .

[4]  D. M. Topkis.  Equilibrium Points in Nonzero-Sum n-Person Submodular Games , 1979 .

[5]  X. Vives.  Nash equilibrium with strategic complementarities , 1990 .

[6]  Leo K. Simon,et al.  Games with Discontinuous Payoffs , 1987 .

[7]  M. McManus.  EQUILIBRIUM, NUMBERS AND SIZE IN COURNOT OLIGOPOLY1 , 1964 .

[8]  Jianxin Zhou,et al.  Transfer continuities, generalizations of the Weierstrass and maximum theorems: A full characterization , 1995 .

[9]  Gerard Debreu,et al.  A Social Equilibrium Existence Theorem* , 1952, Proceedings of the National Academy of Sciences.

[10]  J. Nash,et al.  NON-COOPERATIVE GAMES , 1951, Classics in Game Theory.

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

[12]  Guoqiang Tian,et al.  Characterizations of the Existence of Equilibria in Games with Discontinuous and Non-quasiconcave Payoffs , 1993 .

[13]  Guoqiang Tian,et al.  Existence of Equilibria in Discontinuous Games , 2010 .

[14]  Guoqiang Tian,et al.  The Existence of Equilibria in Games with Arbitrary Strategy Spaces and Payoffs: A Full Characterization , 2009 .

[15]  W. Zame,et al.  Discontinuous Games and Endogenous Sharing Rules , 1987 .

[16]  H. Hotelling.  Stability in Competition , 1929 .

[17]  Guilherme Carmona,et al.  An Existence Result for Discontinuous Games , 2008, J. Econ. Theory.

[18]  Guoqiang Tian.  Existence of equilibrium in abstract economies with discontinuous payoffs and non-compact choice spaces , 1992 .

[19]  Robert J. Weber,et al.  Distributional Strategies for Games with Incomplete Information , 1985, Math. Oper. Res..

[20]  J. Goodman.  Note on Existence and Uniqueness of Equilibrium Points for Concave N-Person Games , 1965 .

[21]  Nicholas C. Yannelis,et al.  Existence of Maximal Elements and Equilibria in Linear Topological Spaces , 1983 .

[22]  Alejandro Jofré,et al.  RECIPROCAL UPPER SEMICONTINUITY AND BETTER REPLY SECURE GAMES: A COMMENT , 2006 .

[23]  E. Maskin,et al.  The Existence of Equilibrium in Discontinuous Economic Games, I: Theory , 1986 .

[24]  Necessary and Sufficient Conditions for Maximization of a Class of Preference Relations , 1993 .

[25]  M. Shubik,et al.  On 'On the Foundations of the Theory of Monopolistic Competition' , 1978 .

[26]  Discontinuous Payoffs, Shared Resources, and Games of Fiscal Competition: Existence of Pure Strategy Nash Equilibrium , 2006 .

[27]  Guoqiang Tian.  Generalizations of the FKKM theorem and the Ky Fan minimax inequality, with applications to maximal elements, price equilibrium, and complementarity , 1992 .

[28]  Jacqueline Morgan,et al.  Pseudocontinuous functions and existence of Nash equilibria , 2007 .

[29]  Guoqiang Tian.  On the existence of equilibria in generalized games , 1992 .

[30]  The maximum Theorem and the existence of Nash equilibrium of (generalized) games without lower semicontinuities , 1992 .

[31]  P. Reny.  On the Existence of Pure and Mixed Strategy Nash Equilibria in Discontinuous Games , 1999 .

[32]  Matthew O. Jackson,et al.  Non-existence of equilibrium in Vickrey, second-price, and English auctions , 2009 .

[33]  Jen-Chih Yao.  Nash equilibria in N-person games without convexity , 1992 .