Optimization incentives and coordination failure in laboratory stag hunt games

Optimization Incentives And Coordination Failure In Laboratory Stag Hunt Games

Abstract: This paper reports an experiment comparing three stag hunt games that have the same best-response correspondence and the same expected payoff from the mixed equilibrium, but differ in the incentive to play a best response rather than an inferior response. In each game, risk dominance conflicts with payoff dominance and selects an inefficient pure strategy equilibrium. We find statistically and economically significant evidence that the differences in the incentive to optimize help explain observed behavior.

© 1997-1999 by the authors. All rights reserved. The first draft of this paper was called "Risk Dominance, Payoff Dominance, and Probabilistic Choice Learning," which was drafted while Van Huyck was on faculty development leave at the University of Pittsburgh.

The abstraction assumptions specifying feasible strategies, preferences over consequences, and substantively rational players, which define a strategic form game, construct a powerful framework within which to analyze strategic behavior. These abstraction assumptions in turn can be summarized by the best response correspondence. One need only know the best response correspondence of a strategic form game to identify its Nash equilibria.

This paper reports an investigation of three stag hunt games using the experimental method. The three games have the same best response correspondence as well as similar payoff magnitudes, but produce different behavior. Games 2R, R, and 0.6R, shown in Figures 1, 2, and 3, were used in the experiment.

Under the assumption that players maximize the expected value of their earnings, games 2R, R, and 0.6R have identical best response correspondences. Strategy X is a strict best response to any mixture that attaches a probability greater than k* to X, where k* = 0.8, while Y is a strict best response to any mixture attaching a lower probability to X. Each game has two pure-strategy equilibria, where (X,X) is payoff dominant and (Y,Y) is risk dominant,[1] as well as a mixed equilibrium in which X is played with probability k*. To the extent possible, the games also involve payoffs of similar magnitudes. In particular, the expected payoff from the mixed equilibrium is 36 for all three games.

	X	Y		X	Y		X	Y
X	45,45	0,35	X	45,45	0,40	X	45,45	0,40
Y	35,0	40,40	Y	40,0	20,20	Y	42,40	12,12

The classical approach to games typically either exploits only the information contained in a game’s best response correspondence, or augments this information with risk-dominance and payoff-dominance considerations in order to choose between strict Nash equilibria. In either case, games 2R, R and 0.6R are treated identically.[2]

Our analysis of games 2R, R, and 0.6R is motivated by the observation that the pecuniary incentive to select a best response to an opponent’s strategy is twice as large in game 2R as it is in game R and six tenths as large in game 0.6R as it is in game R. We call this incentive the optimization premium: the difference between the payoff of the best response to an opponent’s strategy and the inferior response. The optimization premium may be irrelevant to substantively rational agents, but we expect people to more readily learn to play a best response when the optimization premium is large, and expect the differing optimization premia of games 2R, R, and 0.6R to induce systematically different play in laboratory experiments.

It is well known that increasing the stakes for which subjects play games can affect their behavior, see Smith (1976) or Slonim and Roth (1998). We maintain approximately the same stakes in all of our games by fixing the payoff to the mixed equilibrium at 36 cents. One can think of the optimization premium as describing the steepness, rather than the level, of the payoff function near an equilibrium. A larger optimization premium implies not that equilibrium payoffs are higher, but rather that the penalty to suboptimal play is larger.

Our experimental results provide evidence that changing the optimization premium between X and Y influences behavior. The sensitivity of individual subjects to the history of opponents’ play is greater in games with a larger optimization premium. Behavior converges more quickly in game 2R than in R, and more quickly in game R than in game 0.6R. The payoff dominant equilibrium is more likely to emerge the smaller is the optimization premium.

The following section describes the experiment. Section 3 discusses how play may be expected to differ across the three games. Section 4 presents the experimental results. The penultimate section relates our findings to the literature and the final section contains concluding comments.

Our results provide evidence that more than the best response correspondence matters when predicting human behavior in laboratory experiments. We have focused on the optimization premium, that is, the expected earnings difference between the two actions, in three stag hunt games that have the same best response correspondence, the same mixed strategy equilibrium, and the same expected payoff at this mixed strategy equilibrium, but have different pecuniary incentives to play a best response. We find statistically and economically significant evidence that the optimization premium helps explain observed behavior. The sensitivity of individual subjects to the history of opponents’ play is greater in games with a larger optimization premium. Behavior converges more quickly the larger the optimization premium. The risk dominant equilibrium is more likely to emerge the larger is the optimization premium.

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[2] Hillas (1990) introduces a reformulation of Kohlberg and Mertens’ (1986) strategic stability that makes the exclusive reliance on the best response correspondence particularly obvious. Among theories that make an equilibrium selection in the stag hunt game, Carlsson and van Damme (1993) and Harsanyi (1995) choose the risk dominant equilibrium, while Anderlini (1990) and Harsanyi and Selten (1988) choose the payoff dominant equilibrium.