Fiscal Decentralization in Mexico: The Bailout Problem

Abstract In this paper we review known minimax theorems with applications in game theory and show that these theorems can be proved using the first minimax theorem for a two-person zero-sum game with finite strategy sets published by von Neumann in 1928. Among these results are the well known minimax theorems of Wald, Ville and Kneser and their generalizations due to Kakutani, Ky Fan, Konig, Neumann and Gwinner–Oettli. Actually, it is shown that these results form an equivalent chain and this chain includes the strong separation result in finite dimensional spaces between two disjoint closed convex sets of which one is compact. To show the implications the authors only use simple properties of compact sets and the well-known Weierstrass–Lebesgue lemma.

[1]  A. Wald.  Generalization of a Theorem By v. Neumann Concerning Zero Sum Two Person Games , 1945 .

[2]  W. Rudin.  Principles of mathematical analysis , 1964 .

[3]  Jonathan M. Borwein,et al.  On Fan's minimax theorem , 1986, Math. Program..

[4]  Vaithilingam Jeyakumar,et al.  A generalization of a minimax theorem of Fan via a theorem of the alternative , 1986 .

[5]  M. Sion.  On general minimax theorems , 1958 .

[6]  Jay F. Nunamaker,et al.  What does it mean for an organisation to be intelligent? Measuring intellectual bandwidth for value creation , 2001, Proceedings of the 35th Annual Hawaii International Conference on System Sciences.

[7]  J. Aubin.  Optima and Equilibria: An Introduction to Nonlinear Analysis , 1993 .

[8]  Lyn C. Thomas,et al.  Game Theory-Mathematical Models of Conflict , 1980 .

[9]  Joachim Gwinner,et al.  Theorems of the Alternative and Duality for INF-SUP Problems , 1994, Math. Oper. Res..

[10]  Wen Song,et al.  On the Equivalence of Minimax Theorems , 1999 .

[11]  Erwin van der Laan,et al.  The Discovery of New Export Products in Ecuador , 2010 .

[12]  E. Rowland.  Theory of Games and Economic Behavior , 1946, Nature.

[13]  M. Setnes,et al.  Fuzzy Modeling of Client Preference in Data-Rich Marketing Environments , 2000 .

[14]  T. H. Kjeldsen.  John von Neumann's Conception of the Minimax Theorem: A Journey Through Different Mathematical Contexts , 2001 .

[15]  J. Neumann.  Zur Theorie der Gesellschaftsspiele , 1928 .

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

[17]  K Fan,et al.  Minimax Theorems. , 1953, Proceedings of the National Academy of Sciences of the United States of America.

[18]  R. A. Silverman,et al.  Introductory Real Analysis , 1972 .

[19]  J. Hiriart-Urruty,et al.  Convex analysis and minimization algorithms , 1993 .

[20]  I. Joó,et al.  On some convexities , 1989 .

[21]  Biagio Ricceri,et al.  Minimax theory and applications , 1998 .

[22]  J. Szép,et al.  Introduction to the theory of games , 1985 .

[23]  B. Fuchssteiner,et al.  New Versions of the Hahn-Banach Theorem , 1980 .

[24]  S. Simons.  Minimax and monotonicity , 1998 .

[25]  J. B. G. Frenk,et al.  The level set method of Joó and its use in minimax theory , 2006, Math. Program..

[26]  J. B. G. Frenk,et al.  Minimax Results and Finite-Dimensional Separation , 2002 .

[27]  Michael M. Neumann.  Bemerkungen zum von Neumannschen Minimaxtheorem , 1977 .

[28]  T. Illés,et al.  Theorems of the Alternative and Optimality Conditions for Convexlike and General Convexlike Programming , 1999 .

[29]  J. E. L. Peck,et al.  Games On a Compact Set , 1957, Canadian Journal of Mathematics.

[30]  Abraham Wald,et al.  Statistical Decision Functions , 1951 .

[31]  T. Illés,et al.  Farkas type theorems for generalized convexities , 1994 .

[32]  Bor-Luh Lin,et al.  Nonlinear Two Functions Minimax Theorems , 1998 .

[33]  Vaithilingam Jeyakumar,et al.  Equivalence of a Ky Fan type minimax theorem and a Gordan type alternative theorem , 1986 .

[34]  József Kolumbán,et al.  On a generalized sup-inf problem , 1996 .

[35]  R. Teunter,et al.  Average Costs versus Net Present Value , 2000 .

[36]  Paul Hudson.  Games Theory — Lectures for Economists and Systems Scientists , 1977 .

[37]  H. König.  Über das von Neumannsche Minimax-Theorem , 1968 .

[38]  I. Joó,et al.  Note on my paper “a simple proof for von Neumann's minimax theorem” , 1984 .

[39]  P. Pardalos,et al.  Minimax and applications , 1995 .

[40]  Stephen Simons,et al.  Minimax Theorems and Their Proofs , 1995 .

[41]  H. Nikaidô.  On von Neumann’s minimax theorem , 1954 .

[42]  Gábor Kassay.  A simple proof for König's minimax theorem , 1994 .