Adaptive behavior and coordination failure

Adaptive Behavior And Coordination Failure

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

TAMU Economic Research Laboratory
Department of Economics
Texas A&M University
College Station, TX 77843-4228

Abstract: We use the experimental method to study people's adaptive behavior in a generic game with multiple Pareto ranked equilibria. The experiment was designed to discover if behavior diverged at the separatrix predicted by the fictitious play dynamic. The equilibrium selected was sensitive to small differences in initial conditions as predicted. The experiment provides some striking examples of coordination failure growing from small historical accidents.

Key Words: adaptive learning, fictitious play, path dependence, coordination failure, predictive success.

JEL classification: c720, c920.

Acknowledgements: Artie Powell made valuable comments on the experimental design and helped run the experiments reported in this paper. The National Science Foundation provided financial support.

Introduction

The power of the equilibrium method derives from the ability to abstract from the dynamic process that produces mutually consistent behavior and to abstract from the historical accident that initiated the process. This ability depends on an appeal to the long run: a time when adaptive behavior will have converged to a unique stable equilibrium, see Lucas (1987). In this paper, we consider two related problems with this traditional defense of the equilibrium method: non-convergence and non-uniqueness. [1]

First, models of adaptive behavior do not guarantee convergence to Nash equilibria in general. For example, Cournot's (1960) myopic best response dynamic and Brown's (1951) fictitious play dynamic can lead to cycles or chaos. Hence, an open question is whether we should expect strategic behavior to converge to an outcome that satisfies a mutual consistency condition, like Nash equilibrium, or to an outcome that does not, like rationalizability.

Second, multiple equilibria undermine the usefulness of an analysis that abstracts from historical accident and dynamic process. Multiple equilibria arise in many economic contexts. For example, multiple Pareto ranked equilibria arise in both macroeconomic models with production, search, or trading externalities and microeconomic models of monopolistic competition, technology adoption and diffusion, and manufacturing with non-convexities. These superficially dissimilar market and non-market models share the common property that a decision maker's best "level of effort" depends positively upon other decision makers' "level of effort." This property is called strategic complementarity in the coordination failure literature. [2]

While it is tempting to assume that behavior will converge to an efficient equilibrium in situations with multiple Pareto ranked equilibria, doing so ignores the role of historical accident and dynamic process in producing mutually consistent behavior. Models of adaptive behavior often predict barriers that separate the space of outcomes into regions in which behavior does and regions in which behavior does not converge to an efficient equilibrium. The selected equilibrium is path dependent, that is, the equilibrium predicted to emerge depends on the historical accident of the initial condition, rather than on deductive concepts of efficiency.

The experiment reported in this paper was designed to discover if the predicted separation is observed. Our search was conducted in a generic game with strategic complementarities. While a deductive equilibrium analysis predicts multiple Pareto ranked equilibria, the observed behavior was systematic. We did observe the separation predicted by the fictitious play dynamic. Moreover, the equilibrium selected was sensitive to small differences in initial conditions as predicted. The experiment provides some striking examples of coordination failure growing from small historical accidents.

Finally, the paper introduces a measure of the origin of mutually consistent behavior. Our measure searches the space of -equilibria to find the value (t)* that maximizes Selten's (1991) measure of predictive success. Our measure reveals that, while initially naive subjects behave more like decision theorists than game theorists, their behavior becomes mutually consistent in the sense that (t)* is decreasing with time. While adaptive behavior leads to mutually consistent behavior, the mutually consistent behavior that emerges is path dependent.

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Conclusion

This paper has reported on subjects' adaptive behavior in a generic game with strategic complementarities. Both the myopic best response dynamic and the fictitious play dynamic predict a separatrix between a median of 7 and 8 and both predict that behavior will move away from this barrier. Given the high precision of these models of adaptive behavior, they make remarkably accurate predictions. Specifically, all ten sessions started with an initial median contained in the set {7,8,9,10,11}; after period six none of the observed medians were contained in the set {7,8,9,10,11}; moreover, the median never crossed the separatrix, that is, subjects trapped in the low equilibrium's basin of attraction never escaped.

Initially, subjects behave more like decision theorists than game theorists in the sense that a concern for security rather than mutual consistency appears to influence their choices. For example, 44 percent of the subjects chose their secure action or their best response to a diffuse prior over the median in the first period of the ten sessions. We expected this and designed payoff table G to exploit subjects' systematic response to security in order to generate initial conditions that are close to the separatrix without having to resort to crude methods, like forced trials, etc.

Over time subjects' behavior becomes more consistent. Using the concept of a mutual -best response outcome to measure the degree of mutually consistent behavior, we found that equal to 14 cents after 15 periods and to 2 cents after 40 periods maximized Selten's measure of predictive success. This convergence to mutually consistent behavior did not necessarily lead to more efficient outcomes.

The average subject in the first fifteen periods of the five low sessions earned $9.71. The average subject in the first fifteen periods of the five high sessions earned $15.57. The small differences in the distribution of subjects' period one choices result in large differences in average earnings. Specifically, the average subject in a high session earns about $6 (or 60 percent) more than the average subject in a low session.

The myopic best response dynamic had a slightly higher hit rate than the fictitious play dynamic. However, game G(T) is not well suited for discriminating between the myopic best response dynamic and the fictitious play dynamic, since both predict the same attractors and the same basins of attraction: see Van Huyck, Cook, and Battalio (1994) for an environment in which fictitious play easily does better than myopic best response dynamics. Whether it is possible to create an accurate and precise model of adaptive behavior in repeated games remains an open question. Our results suggest that at least for some strategic situations it should be possible to construct accurate models of adaptive behavior that are much more precise than conventional wisdom suggests.

The following analogy may help explain our discovery. Mark Twain (1962, p.86) describes a remarkable spring at the summit of a Rocky Mountain pass that "spent its water through two outlets and set it in opposite directions." One of the streams starts a journey westward to the Gulf of California and the Pacific Ocean. The other starts a journey eastward to the Gulf of Mexico and the Atlantic Ocean. Our search was for a spring that straddles a barrier dividing a continent of human behavior. Payoff table G is such a spring and either the myopic best response dynamic or the fictitious play dynamic predict such a barrier. Unlike the Rocky Mountains, this barrier is hard to see without a vision informed by models of adaptive behavior.

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Footnotes

[1] See Kreps (1990) for discussion, examples, and references.

[2] See Cooper and John (1988) and Milgrom and Roberts (1990) for examples and references.