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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: 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 $.


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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