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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Equilibrium existence and topology in some repeated games with incomplete information


Authors: Robert S. Simon, Stanislaw Spiez and Henryk Torunczyk
Journal: Trans. Amer. Math. Soc. 354 (2002), 5005-5026
MSC (2000): Primary 55M20, 91A20; Secondary 54C60, 52A20, 91A05, 91A10
Published electronically: August 1, 2002
MathSciNet review: 1926846
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Abstract | References | Similar Articles | Additional Information

Abstract: This article proves the existence of an equilibrium in any infinitely repeated, un-discounted two-person game of incomplete information on one side where the uninformed player must base his behavior strategy on state-dependent information generated stochastically by the moves of the players and the informed player is capable of sending nonrevealing signals.

This extends our earlier result stating that an equilibrium exists if additionally the information is standard. The proof depends on applying new topological properties of set-valued mappings. Given a set-valued mapping $F$ on a compact convex set $P\subset \mathbb R^n$, we give further conditions which imply that every point $p_0\in P $ belongs to the convex hull of a finite subset $P _0$ of the domain of $F$satisfying $\bigcap_{x\in P_0} F(x)\ne \emptyset $.


References [Enhancements On Off] (What's this?)

  • [AM] AUMANN, R. and MASCHLER, M. (1995). Repeated Games with Incomplete Information. With the collaboration of R. Stearns. M.I.T. Press, Cambridge, MA. MR 96k:90001
  • [ES] EILENBERG, S. and STEENROD, N. (1952). Foundations of Algebraic Topology. Princeton University Press, Princeton, N.J. MR 14:398b
  • [Ko] KOHLBERG, E. (1975). ``Optimal Strategies in Repeated Games with Incomplete Information." International Journal of Game Theory, 4, 7-24. MR 52:12811
  • [MSZ] MERTENS, J.-F., SORIN, S., and ZAMIR, S. (1994). Repeated Games, Core Discussion Papers 9420-22, Université Catholique de Louvain.
  • [Re] RENAULT, J. (2000). ``On Two-player Repeated Games with Lack of Information on One Side and State-independent Signalling", Mathematics of Operations Research, 25, No. 4, pp. 552-572. MR 2002g:91033
  • [Si] SIMON, R. (2002). ``Separation of Joint Plan Equilibrium Payoffs From the Min-Max Functions", to appear in Games and Economic Behavior.
  • [SST] SIMON, R., SPIEZ, S. and TORUNCZYK, H. (1995). ``The Existence of Equilibria in Certain Games, Separation for Families of Convex Functions and a Theorem of Borsuk-Ulam Type," Israel Journal of Mathematics 92, 1-21. MR 97f:90132
  • [So] SORIN, S. (1983). ``Some Results on the Existence of Nash Equilibria for Non-zero-sum Games with Incomplete Information,'' International Journal of Game Theory 12, 193-205. MR 85c:90104
  • [SZ] SORIN, S. and ZAMIR, S. (1985). ``A Two-Person Game with Lack of Information on One and One-Half Sides," Mathematics of Operations Research 10, 17-23. MR 86c:90133
  • [Sp] SPANIER, E. H. (1966). ``Algebraic Topology", McGraw-Hill, New York. MR 35:1007

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Additional Information

Robert S. Simon
Affiliation: Universität Göttingen, Institut für Mathematische Stochastik, Lotze strasse 13, 37083 Göttingen, Germany
Email: simon@math.uni-goettingen.de

Stanislaw Spiez
Affiliation: Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, P.O.B. 137, 00-950 Warszawa, Poland
Email: S.Spiez@impan.gov.pl

Henryk Torunczyk
Affiliation: Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warszawa, Poland
Email: H.Torunczyk@impan.gov.pl

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03098-2
PII: S 0002-9947(02)03098-2
Received by editor(s): August 14, 2000
Received by editor(s) in revised form: May 10, 2002
Published electronically: August 1, 2002
Additional Notes: The first named author wishes to thank Christof Wehrsig of the Sociology Department of the University of Bielefeld for introducing him to game theory. The research of this author was supported by the German Science Foundation (Deutsche Forschungsgemeinschaft), the Institute of Mathematical Economics (Bielefeld), the Institute of Mathematical Stochastics (Goettingen), the Center for Rationality and Interactive Decision Theory (Jerusalem), the Department of Mathematics of the Hebrew University (Jerusalem), and the Edmund Landau Center for Research in Mathematical Analysis (Jerusalem), sponsored by the Minerva Foundation (Germany). The Stefan Banach International Center at the Institute of Mathematics of the Polish Academy of Sciences enabled for meetings of the three authors while the paper was in preparation.
Article copyright: © Copyright 2002 American Mathematical Society