Aldo Rustichini

David K Levine

English

Research Papers in Economics

Title: Introduction to the Dynamic Games Special Issue
Authors: David K. Levine,
  Department of Economics
  UCLA
  Los Angeles, CA 90095
  phone/fax 310-825-3810
  email dlevine@ucla.edu
                 Aldo Rustichini
  Department of Economics
  Boston University
  Boston, MA, 02215
  phone 617 353 6323
  fax 617 353 4001
  email raldo@bu.edu
Proposed Running Head: IntroductionIntroduction to the Dynamic Games Special Issue
February 12, 2000
This special issue of the Review of Economic Dynamics is devoted to the subject
of dynamic games. There are two classes of problems that have excited the interest of
game theory researchers over the last several years. The first is the topic of repeated
games and the folk theorem; the second is the way in which learning in games may lead
to the selection of particular equilibria. This special issue contains some of the most
recent research on both of these topics.
The most significant result on repeated games is the folk-theorem, which says
roughly that if players are sufficiently patient then any socially feasible individually
rational payoff vector can be supported as a subgame perfect equilibrium. Recent years
have seen a strong effort to extend the theorem to environments with imperfect
information. Harrison Cheng’s “Folk Theorem with One-Sided Information” is one such
effort. By focusing on the special case of one-sided information, Cheng is able to
simplify and extend the folk-theorem of Fudenberg, Levine and Maskin [1994]. The
simplification comes about by introducing an easy to verify criterion called admissability.
The extension comes about by showing that in the case of one-sided moral hazard, the
Nash threats folk theorem can be improved on to give a full minimax folk theorem.
The other recent topic in the folk-theorem literature has been to try to expand the
scope of the theorem beyond the setting of repeated games, to incorporate storage and
other state variables. Ligon, Thomas and Worral in “Mutual Insurance, Individual savings
and Limited Commitment” examine a setting with a linear storage technology dynamic
that allows self-insurance as a substitute for mutual insurance. The striking fact is that the
welfare effects of the storage technology are ambiguous. Higher levels of utility are
possible because of self-insurance: however, this reduces the equilibrium level of mutual
insurance, and hence the welfare level. In numerical examples, they find that storage and
the gain from insurance are correlated: more storage gives more utility from autarky, so
to sustain insurance these households must have more to gain from it.
By way of contrast David Levine’s “Castle on the Hill” shows just how badly the
folk theorem can fail when the repeated game assumption is dropped. Dutta [1991]
showed that the folk theorem continues to hold with state variables when players cannot
prevent states from being reached. Levine studies a simple game with irreversibility:
in one state one player is a Lord living in a castle and the other is a serf;
while in the other state the roles are reversed. However, the Lord may, by a
costly defence of his castle prevent the roles from being reversed. In repeated
games, the set of possible equilibria generally expands with the discount
factor, and for discount factors near one there are many equilibria. In Castle
on the Hill, for a broad range of discount factors, including those close to one,
equilibrium is unique. Moreover, the equilibrium for large discount factors is
Pareto dominated by the equilibrium for low discount factors. A unique cyclic
equilibrium is also possible for intermediate ranges of discount factors.
The remaining papers in our special issue examine learning. Dimitri’s
“Correlation Learning and the Robustness of Cooperation” studies learning in a repeatedgame framework. In particular he examines a repeated Prisoner’s Dilemma game, where
players learn using fictitious play. The novelty of the model is that players may think that
the probabilities of the choices are correlated rather than independent. Remarkably, in
this framework cooperation can survive with positive probability: If cooperative players
meet sufficiently often the estimated conditional probability of being matched with a
cooperative player will remain sufficiently high.
Perez’s “Private Experience in Adaptive Learning Models” turns to the
connection between learning theory and the experimental laboratory. The experimental
motivation is the discrepancy between the Marimon-Sunder [1993] results and the
learning theory of Marcet and Sargeant [1989a,b]. Perez finds that if we allow different
agents to participate and acquire information at different levels the experimental
anomalies disappear.
Finally, Cabrales and Ponti in “Implementation, Elimination of Weakly
Dominated Strategies and Evolutionary Dynamics” study the role of learning dynamics in
mechanism design. Specifically, they examine the dynamic implementation of the
Siostrom mechanism. In this mechanism players simultaneous announce their
preferences, and the preferences of their immediate neighbors. The truth telling
equilibrium is the only one that survives the first round of elimination of weakly
dominated strategies. However, as is often the case, this mechanism has many equilibria.
The authors study dynamic implementation: an outcome is dynamically implemented if it
is the limit point of some dynamic process. They focus on monotone dynamics and best
response dynamics. Monotone dynamics is not a strong enough assumption to pin down
an equilibrium: every component of the Nash equilibrium set contains a limit point of the
monotonic dynamics. The best-response dynamic leads to a much sharper conclusion:
any interior solution of the best reply dynamics converges to the equilibrium whose
outcome is desired by the social planner.
References
Cheng, H. (2000), “The Folk Theorem with One-Sided Information,” Review of
Economic Dynamics, this issue.
Dimitri, N. (2000), “Correlation, Learning and the Robustness of Cooperation,” Review
of Economic Dynamics, this issue.
Dutta, P. K. (1991), “Folk Theorem for Stochastic Games,” University of
Rochester, Rochester Center for Economic Research, Working Paper
No. 293.
Fudenberg, D., D. K. Levine and E. Maskin (1994), “The Folk Theorem with
Imperfect Public Information,” Econometrica, 62: 997-1039.
Levine, D. K. (2000), “The Castle on the Hill,” Review of Economic Dynamics, this
issue.
Ligon, Thomas, and Worrall (2000), “Mutual Insurance Individual Savings and
Limited Commitment,” Review of Economic Dynamics, this issue.
Marcet, A. and T. J. Sargent (1989a), “Convergence of Least Squares Learning
Mechanisms in Self Referential Linear Stochastic Models,” Journal of Economic
Theory, 48, 337-368.Marcet, A. and T. J. Sargent (1989b): “Convergence of Least Squares Learning in
Environments with Hidden State Variables and Private Information,” Journal of
Political Economy, 97, 1306-1322.
Marimon, Ramon, and Shyam Sunder (1993), “Indeterminacy of Equilibria in a
Hyperinflationary World: Experimental Evidence,” Econometrica, 61, 1073-1107.
Perez, F. (2000), “Private Experience in Adaptive Learning Models,” Review of
Economic Dynamics, this issue.
Ponti and Cabrales (2000), “Implementation, Elimination of Weakly Dominated
Strategies and Evolutionary Dynamics,” Review of Economic Dynamics, this
issue.

Correlation, Learning and the Robustness of Cooperation,” Review of Economic Dynamics, this issue.

Implementation, Elimination of Weakly Dominated Strategies and Evolutionary Dynamics,” Review of Economic Dynamics, this issue.

Indeterminacy of Equilibria in a Hyperinflationary World:

Private Experience in Adaptive Learning Models,” Review of Economic Dynamics, this issue.

Sargent (1989a), “Convergence of Least Squares Learning Mechanisms in Self Referential Linear Stochastic Models,”

The Castle on the Hill,” Review of Economic Dynamics, this issue.

The Folk Theorem with One-Sided Information,” Review of Economic Dynamics, this issue.

Introduction: The Dynamic Games Special Issue

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Introduction: The Dynamic Games Special Issue

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