juddworkingpapersgame - Computing Supergame Equilibria...

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Computing Supergame Equilibria Kenneth L. Judd Hoover Institution Stanford, CA 94305 judd@hoover.stanford.edu Sevin Yeltekin KGSM-MEDS Northwestern University 2001 Sheridan Rd. Evanston, IL 60208 s-yeltekin@kellogg.nwu.edu James Conklin Lehman Brothers New York, New York This revision: September, 2000 Abstract. We present a general method for computing the set of supergame equilibria in in & nitely repeated games with perfect monitoring and public random- ization. We present a three-step algorithm which constructs a convex set containing the set of equilibrium values, constructs another convex set contained in the set of equilibrium values, and produces strategies which support equilibrium values. We explore the properties of this algorithm by applying it to familiar games. We & nd that the algorithm produces high quality approximations at moderate computational cost. Keywords: Computational methods, repeated games, Nash equilibrium This work was supported by NSF grant SES-9012128, SES-9309613, and SES-9708991. This paper is an extension and & nal version of Conklin and Judd (1993). We gratefully acknowledge comments of seminar participants at Northwestern University and SITE. 1
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Computing Supergame Equilibria 2 1. Introduction The nature of repeated interaction has been extensively studied in the repeated game literature. The theory of repeated games has produced important qualitative results, but it is di cult to apply the theory quantitatively. This paper presents methods for computing the set of subgame perfect equilibria in in & nitely repeated games with perfect monitoring. Our approach is recursive, following the &self-generating set± constructions of Abreu (1988), Abreu, Pearce and Stacchetti (1986, 1990), and Cronshaw and Luenberger (1994). Our algorithm can quantitatively investigate many of the properties of the set of all Nash equilibrium payo f sinin & nitely repeated games. The method presented in this paper is general, being applicable to a large variety of games studied in industrial organization, contract theory, and dynamic policy analysis. Furthermore, the method can be extended in many directions such as in & nite-horizon games with state variables. The recursive theory of supergames introduces the concept of self-generating sets of payo f s to characterize the set of equilibria. The set of equilibrium values may not be well- behaved. For example, Sorin (1986) studies an in & nitely repeated Prisoner²s Dilemma game and shows that its equilibrium value set is a square plus two line segments. In particular, the equilibrium value set is neither convex nor star-shaped, and does not equal the closure of its interior. One suspects that equilibrium values sets are often di cult to approximate in a & nitistic, computable fashion in any reliable way. To make the problem tractable, we allow public randomization, as in Cronshaw and Luenberger (1994). This is advantageous since the resulting set of equilibrium values is convex and convex sets can be approximated in several ways. The numerical methods presented use e cient ways
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juddworkingpapersgame - Computing Supergame Equilibria...

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