Computing Supergame Equilibria
Kenneth L. Judd
Hoover Institution
Stanford, CA 94305
[email protected]
Sevin Yeltekin
KGSMMEDS
Northwestern University
2001 Sheridan Rd.
Evanston, IL 60208
[email protected]
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 threestep 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 SES9012128, SES9309613, and SES9708991. 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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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
ﬃ
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 &selfgenerating 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
&
nitehorizon games with state variables.
The recursive theory of supergames introduces the concept of selfgenerating 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 starshaped, and does not equal
the closure of its interior. One suspects that equilibrium values sets are often di
ﬃ
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
ﬃ
cient ways
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 Numerical Analysis, Supergame Equilibria

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