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 on a compact convex set , we give further conditions which imply that every point belongs to the convex hull of a finite subset of the domain of satisfying .

**[AM]**Robert J. Aumann and Michael B. Maschler,*Repeated games with incomplete information*, MIT Press, Cambridge, MA, 1995. With the collaboration of Richard E. Stearns. MR**1342074****[ES]**Samuel Eilenberg and Norman Steenrod,*Foundations of algebraic topology*, Princeton University Press, Princeton, New Jersey, 1952. MR**0050886****[Ko]**E. Kohlberg,*Optimal strategies in repeated games with incomplete information*, Internat. J. Game Theory**4**(1975), no. 1-2, 7–24. MR**0391993****[MSZ]**MERTENS, J.-F., SORIN, S., and ZAMIR, S. (1994).*Repeated Games*, Core Discussion Papers 9420-22, Université Catholique de Louvain.**[Re]**Jérôme Renault,*On two-player repeated games with lack of information on one side and state-independent signalling*, Math. Oper. Res.**25**(2000), no. 4, 552–572. MR**1855365**, 10.1287/moor.25.4.552.12113**[Si]**SIMON, R. (2002). ``Separation of Joint Plan Equilibrium Payoffs From the Min-Max Functions", to appear in*Games and Economic Behavior*.**[SST]**R. S. Simon, S. Spież, and H. Toruńczyk,*The existence of equilibria in certain games, separation for families of convex functions and a theorem of Borsuk-Ulam type*, Israel J. Math.**92**(1995), no. 1-3, 1–21. MR**1357742**, 10.1007/BF02762067**[So]**S. Sorin,*Some results on the existence of Nash equilibria for non-zero-sum games with incomplete information*, Internat. J. Game Theory**12**(1983), no. 4, 193–205. MR**732248**, 10.1007/BF01769090**[SZ]**Sylvain Sorin and Shmuel Zamir,*A 2-person game with lack of information on 11\over2 sides*, Math. Oper. Res.**10**(1985), no. 1, 17–23. MR**787002**, 10.1287/moor.10.1.17**[Sp]**Edwin H. Spanier,*Algebraic topology*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0210112**

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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:
https://doi.org/10.1090/S0002-9947-02-03098-2

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