Computation of Equilibria in Noncooperative Games

1. I N T R O D U C T I O N 1.1. T h e E q u i l i b r i u m P r o b l e m in C l a s s i c a l G a m e T h e o r y In recent years, t he re has b e e n a p ro l i fe ra t ion of app l ica t ions of n o n c o o p e r a t i v e g a m e t h e o r y t o economics~ pol i t ica l science, evo lu t i ona ry biology, and o the r disciplines. A typ ica l research *Supported by DARPA/ARO Contract DAAL03-88-K-0195, DARPA/ISTO Contract N00014-88-K-0458, Office of Naval Research under contract N00014-87-K-0310, and Air Force contract AFSO1~87-0386. Also Supported by Duke University's James B. Duke Fellowship for Advanced Studies. tSupported by NSF Grants MCS-8120790 and SES-8420114. ~Supported by DARPA/ARO Contract DAAL03-88-K-0195, DARPA/ISTO Contract N00014-88-K-0458, OffÉce of Naval Research under contract N00014-87-K-0310, and Air Force contract AFSOR-87-0386. Also, subsequently supported by National Science Foundation Grants CCF-0523555, CCF-0432038, CCF-0432047, ITR-03261573 EIA-0218376, EIA-0218359, and EIA-0086015. 0898-1221/05/$ see front matter (~) 2005 Elsevier Ltd. All rights reserved. Typeset by ,4.~¢b-q-TEX doi:10.1016/j.camwa.2005.02.015

[1]  John F. Canny,et al.  A new algebraic method for robot motion planning and real geometry , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

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

[3]  J. Neumann,et al.  Theory of games and economic behavior , 1945, 100 Years of Math Milestones.

[4]  J. Reif,et al.  Decision algorithms for multiplayer noncooperative games of incomplete information , 2002 .

[5]  J. Renegar,et al.  On the Computational Complexity and Geometry of the First-Order Theory of the Reals, Part I , 1989 .

[6]  C. A. R. Hoare,et al.  An axiomatic basis for computer programming , 1969, CACM.

[7]  Maurice Mignotte,et al.  On Mechanical Quantifier Elimination for Elementary Algebra and Geometry , 1988, J. Symb. Comput..

[8]  L. Peter Deutsch An interactive program verifier , 1973 .

[9]  D. Evelyn,et al.  Proving Programs Correct , 1979 .

[10]  SIDNEY L. HANTLER,et al.  An Introduction to Proving the Correctness of Programs , 1976, CSUR.

[11]  L. Csanky,et al.  Fast parallel matrix inversion algorithms , 1975, 16th Annual Symposium on Foundations of Computer Science (sfcs 1975).

[12]  Robert S. Boyer,et al.  The Correctness Problem in Computer Science , 1982 .

[13]  Manuel Blum,et al.  Designing programs that check their work , 1989, STOC '89.

[14]  In-Koo Cho,et al.  A Refinement of Sequential Equilibrium , 1987 .

[15]  FrancezNissim,et al.  A Proof System for Communicating Sequential Processes , 1980 .

[16]  John H. Reif,et al.  The complexity of elementary algebra and geometry , 1984, STOC '84.

[17]  A. McLennan Consistent conditional systems in noncooperative game theory , 1989 .

[18]  J. Reif,et al.  Lower bounds for multiplayer noncooperative games of incomplete information , 2001 .

[19]  J. Mertens,et al.  ON THE STRATEGIC STABILITY OF EQUILIBRIA , 1986 .

[20]  David Gries,et al.  A proof technique for communicating sequential processes , 1981, Acta Informatica.

[21]  James Renegar,et al.  On the Computational Complexity of Approximating Solutions for Real Algebraic Formulae , 1992, SIAM J. Comput..

[22]  James Renegar,et al.  On the Computational Complexity and Geometry of the First-Order Theory of the Reals, Part III: Quantifier Elimination , 1992, J. Symb. Comput..

[23]  David Pearce Rationalizable Strategic Behavior and the Problem of Perfection , 1984 .

[24]  Christos H. Papadimitriou,et al.  Games against nature , 1985, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[25]  Ephraim Feig,et al.  A fast parallel algorithm for determining all roots of a polynomial with real roots , 1986, STOC '86.

[26]  Chee-Keng Yap,et al.  Algebraic cell decomposition in NC , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[27]  S. Smale,et al.  On a theory of computation and complexity over the real numbers; np-completeness , 1989 .

[28]  A. K. Chandra,et al.  Alternation , 1976, 17th Annual Symposium on Foundations of Computer Science (sfcs 1976).

[29]  John H. Reif,et al.  The Complexity of Two-Player Games of Incomplete Information , 1984, J. Comput. Syst. Sci..

[30]  A. McLennan Justifiable Beliefs in Sequential Equilibrium , 1985 .

[31]  Willem P. de Roever,et al.  A Proof System for Communicating Sequential Processes , 1980, ACM Trans. Program. Lang. Syst..