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

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

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]**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:
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