Vincent P. Crawford

An "evolutionary" interpretation of Van Huyck, Battalio, and Beil's experimental results on coordination

This paper proposes an adaptive interpretation of the results of some recent experiments with repeated tacit coordination games. These experiments revealed several behavioral regularities, including a systematic discrimination between strict Nash equilibria in certain games, that appear to be driven by strategic uncertainty, and are not explained by traditional equilibrium refinements. The observed patterns of discrimination correspond closely to predictions based on Maynard Smiths notion of evolutionary stability. An adaptive model, in the spirit of the evolutionary dynamics but recognizing the important differences between learning in human populations and evolution, promises to yield a unified explanation of the results. Journal of Economic Literature Classification Numbers: 020, 026.

Social Security and the Retirement Decision

The effect of Social Security and private pensions on individual retirement decisions is modeled, relaxing in turn three commonly maintained assumptions—perfect capital markets, actuarial fairness, and certain lifetimes—which together imply that there is no effect. In each case, raising the contribution level can cause systematic changes (of either sign in general) in individual retirement decisions. For Social Security, the effects associated with forced saving and deviations from actuarial fairness probably tend to advance retirement. But those effects that arise solely from the insurance aspect of Social Security and private pensions are ambiguous in sign, owing to the presence of a substitution effect that tends to delay retirement because the insurance benefits can be fully realized only by working longer.

Equilibrium without independence

Abstract Because players whose preferences violate the von neumann-Morgenstern independence axiom may be unwilling to randomize as mixed-strategy Nash equilibrium would require, a Nash equilibrium may not exist without independence. This paper generalizes Nashs definition of equilibrium, retaining its rational-expectations spirit but relaxing its requirement that a player must bear as much uncertainty about his own strategy choice as other players do. The resulting notion, “equilibrium in beliefs”, is equivalent to Nash equilibrium when independence is satisfied, but exists without independence. This makes it possible to study the robustness of equilibrium comparative statics results to violations of independence.

Comparative statics in matching markets

Abstract This paper studies the comparative statics of adding agents to matching markets that generalize the marriage and college-admissions markets of D. Gale and L. Shapley (Amer. Math. Monthly 69, 1962, 9–15). It is shown, for the wide class of matching markets studied by A. Roth (Econometrica 52, 1984, 47–57), that adding an agent to one side of the market weakens the competitive positions of the other agents on that side and strengthens the competitive positions of the agents on the other side.

COMPARING MODELS OF STRATEGIC THINKING IN VAN HUYCK, BATTALIO, AND BEIL'S COORDINATION GAMES

This paper compares the leading models of strategic thinking with subjects’ initial responses to Van Huyck, Battalio, and Beil (1990, 1991) coordination games. Among the refined “equilibrium plus noise” models we compare, payoff-dominant equilibrium performs better than risk-dominant or maximin equilibrium. Among the individualistic models we compare, level-k and cognitive hierarchy models usually fit better than logit quantal response equilibrium or noisy introspection models. In Van Huyck, Battalio, and Beil games, payoff-dominant equilibrium usually fits better than level-k or cognitive hierarchy. The data favor versions of the models in which people model others as if they were perfectly correlated over the standard, independent versions. (JEL: C51, C72, C92)

Distortion of preferences and the Nash theory of bargaining

Abstract It is shown that in Nash bargaining over division of a single good, when agents are allowed to distort their von Neumann-Morgenstern utility functions into any (weakly) concave form, reporting linear utility functions constitutes a unique dominant-strategy Nash equilibrium.

Learning behavior and mixed-strategy Nash equilibria

Abstract The paper considers whether an adaptive justification, like those commonly available for non-interactive optimization models, can be found for the mixed-strategy Nash equilibrium. Although it is known that such a justification is frequently available for pure-strategy equilibria, it is shown that all members of a wide class of behaviorally plausible learning mechanisms must fail to converge in ‘almost all’ games in which the equilibrium involves mixed strategies. An exact formal analogy is developed, which allows these learning mechanisms to be viewed as discrete taˆtonnement processes in properly chosen competitive exchange economies. In the analogy, the instability derives from the independence of excess demand functions (except at the boundaries) from ‘own’ prices. This independence arises because of the linearity in probabilities of von Neumann-Morgenstern risk preferences, and therefore does not extend to pure-strategy equilibria in general. Thus, assuming that agents will play equilibrium strategies implicitly assumes more sophistication when the equilibrium involves mixed strategies than when it involves only pure strategies.

BEHAVIORALLY OPTIMAL AUCTION DESIGN: EXAMPLES AND OBSERVATIONS

This paper begins to explore behavioral mechanism design, replacing equilibrium by a model based on “level-k” thinking, which has strong support in experiments. In representative examples, we consider optimal sealed-bid auctions with two symmetric bidders who have independent private values, assuming that the designer knows the distribution of level-k bidders. We show that in a first-price auction, level-k bidding changes the optimal reserve price and often yields expected revenue that exceeds Myerson’s (1981) bound; and that an exotic auction that exploits bidders’ non-equilibrium beliefs can far exceed the revenue bound. We close with some general observations about level-k auction design. (JEL: C72, C92)

Players of Matching Pennies automatically imitate opponents’ gestures against strong incentives

There is a large body of evidence of apparently spontaneous mimicry in humans. This phenomenon has been described as “automatic imitation” and attributed to a mirror neuron system, but there is little direct evidence that it is involuntary rather than intentional. Cook et al. supplied the first such evidence in a unique strategic game design that gave all subjects a pecuniary incentive to avoid imitation [Cook R, Bird G, Lünser G, Huck S, Heyes C (2012) Proc Biol Sci 279(1729):780–786]. Subjects played Rock-Paper-Scissors repeatedly in matches between fixed pairs, sometimes with one and sometimes with both subjects blindfolded. The frequency of draws in the blind-blind condition was at chance, but in the blind-sighted condition it was significantly higher, suggesting automatic imitation had occurred. Automatic imitation would raise novel issues concerning how strategic interactions are modeled in game theory and social science; however, inferring automatic imitation requires significant incentives to avoid it, and subjects’ incentives were less than 3 US cents per 60-game match. We replaced Cook et al.’s Rock-Paper-Scissors with a Matching Pennies game, which allows far stronger incentives to avoid imitation for some subjects, with equally strong incentives to imitate for others. Our results are important in providing evidence of automatic imitation against significant incentives. That some of our subjects had incentives to imitate also enables us clearly to distinguish intentional responding from automatic imitation, and we find evidence that both occur. Thus, our results strongly confirm the occurrence of automatic imitation, and illuminate the way that automatic and intentional processes interact in a strategic context.

Introduction to Experimental Game Theory

The title Experimental Game Theory refers to experiments whose goal is to learn about general principles of strategic behavior, as opposed to the performance of specific institutions. Twenty-five years ago it would have been startling to see ‘‘experimental’’ modifying ‘‘game theory’’ in this way or to find an entire issue of an economics journal devoted to game experiments. If this now seems natural, it is a tribute to the increasingly