Selection dynamics, asymptotic stability, and adaptive behavior

Selection Dynamics, Asymptotic Stability, and Adaptive Behavior

John B. Van Huyck, Joseph P. Cook, and Raymond C. Battalio

[ Introduction | Conclusion | References | Footnotes | John's Web ]

Abstract: Selection dynamics are often used to distinguish stable and unstable equilibria. This is particularly useful when multiple equilibria prevent a priori comparative static analysis. This paper reports an experiment designed to test the accuracy of the myopic best response dynamic and an inertial selection dynamic. The inertial selection dynamic makes both more precise and more accurate predictions.

Key Words: equilibrium selection, relaxation algorithms, stability, adaptive learning, human behavior.

JEL classification: c720, c920, d830.

Acknowledgments: The National Science Foundation provided financial support. We thank Tom Sargent for encouraging remarks on a previous draft of this paper titled "Selection Dynamics and Adaptive Behavior," and Eric Battalio for implementing the experimental design on the TAMU economic laboratory network.

Introduction

Stability arguments are often used to select amongst multiple equilibria. This approach to the equilibrium selection problem is based on the interpretation of an equilibrium point as a potential convention that might arise amongst players interacting repeatedly. Mutually consistent behavior is not deduced from the description of the situation, but rather is the outcome of some evolutive libration.

An equilibrium point is unstable if it does not correspond to an asymptotically stable fixed point of some explicit selection dynamic. An unstable equilibrium point is unlikely to emerge as the result of an evolutive process and is an unlikely convention. Hence, the analyst should select from the set of stable equilibria. When there is a unique stable equilibrium this approach may appear to preserve the analyst's ability to abstract from the evolutive process itself with its undesirable dependence on historical accident.[1]

However, the asymptotic stability of an equilibrium point depends on the assumed selection dynamic, compare Lucas (1987) and Woodford (1990) for example and see Guesnerie and Woodford (1993) for a survey of alternative stability concepts. Moreover, selection dynamics need not converge to any fixed point corresponding to any equilibrium of the model. Even in simple settings it is possible to construct examples of selection dynamics that predict cyclical or chaotic behavior.

A venerable selection dynamic is the myopic best response dynamic, which dates back at least to Cournot's (1838,[1960]) duopoly analysis of firms that best respond to the other firm's last action. Alternative selection dynamics are often used when the myopic best response dynamic fails to converge to a fixed point. Selection dynamics based on a best response to slowly changing beliefs, inertial beliefs, will often converge when the myopic best response dynamic does not, see for example Bray (1982), Marcet and Sargent (1989), and Thorlund-Peterson (1990). The class of selection dynamics we consider here are "relaxation algorithms," which include the myopic best response dynamic, the partial adjustment dynamic, and least squares learning, see Sargent (1993).

Lucas (1987, p.241) discusses stability theory based on adaptive behavior and concludes that to be useful: "stability theory must be more than simply a fancy way of saying that one does not want to think about certain equilibria. I prefer to view it as an experimentally testable hypothesis, as a special instance of the adaptive laws that we believe govern all human behavior."

This paper examines human behavior in a generic game with multiple equilibria in which the myopic best response dynamic and inertial selection dynamics make different predictions about stability. The inertial selection dynamics are more precise and make more accurate predictions than the myopic best response dynamic in our experiment.

Conclusion

The inertial selection dynamic, L map, accurately predicts the behavior observed in our experiment. The myopic best response dynamic does not. Given our results, an accurate selection theory must characterize the interior equilibrium of ( ) as a stable fixed point of the selection dynamic. Observed behavior is inertial and an accurate selection dynamic must reflect this inertia. Whether this inertia is increasing with the reciprocal of time, as in the L map, is a question for future research.

It is also an open question whether the L map is an accurate model of adaptive behavior. Milgrom and Roberts (1991) general model of adaptive learning is not more accurate and is much less precise than the L map. A point theory rather than an area theory or a distribution theory can accurately explain the observed behavior in our experiment. Hence, one is tempted to use the L map as a model of adaptive behavior, see also Boylan and El-Gamal (1993, p.212) who use a Bayesian analysis to conclude that an inertial selection dynamic, specifically fictitious play, is "infinitely more likely" than the myopic best response dynamic. Marimon and Sunder's (1990) mixed results in favor of least squares learning make us cautious about giving in to this temptation.

Finally, this experiment does not contradict the traditional view of stability analysis. Behavior always converged to the unique stable fixed point of the inertial selection dynamic. It does so remarkably quickly. In game ( ), it seems reasonable to abstract from the evolutive process and to conduct comparative static exercises. One can be confident that the transition to equilibrium will be brief.

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Footnotes

[1] For those who think path dependence is a fact of life, see Van Huyck, Cook, and Battalio (1993), Van Huyck, Battalio, and Beil (1991), and Van Huyck, Battalio, Mathur, Ortmann, and Van Huyck (1992).