What Does It Take

What Does It Take To Eliminate The Use of A Strategy
Strictly Dominated by a Mixture?

John Van Huyck, Frederick Rankin, and Raymond Battalio

June 1998

Abstract: This paper reports an experiment to determine whether subjects will learn to stop using a strictly dominated strategy that can be an above average reply. It is difficult to find an experimental design that eliminates the play of the strictly dominated strategy completely. The least effective treatment used money to motivate behavior directly. The most effective treatment uses a binary-lottery to induce preferences, but even this treatment requires giving subjects plenty of experience. Doing so reduced the play of the strictly dominated strategy to around 10 percent by the end of a session. There is no evidence for the explosive cycling needed to make the strictly dominated strategy an above average reply. The most surprising result was observing cycles in the frequency with which the strictly dominated action was played or spite cycles. Spite cycles appear using both evolutionary and fixed pairs protocols.

Introduction

A strictly dominated strategy does worse than the strategy that dominates it in all possible outcomes. It is never in the set of best replies to any feasible belief. Since it can never be a best reply, rational choice requires that a player avoid the use of strictly dominated strategies. Advising players to avoid using strictly dominated strategies does not require any theory of mutually consistent behavior. Eliminating strictly dominated strategies ought to be intuitively appealing to anyone who understands the definition.

The advice to avoid using strictly dominated strategies is sometimes thought to be unsound, because it can conflict with efficiency. Binmore (1994, chapter 3) provides a critical survey of the "circle-squarers" who think it is bad advice. Much of this confusion comes from a failure to distinguish strict dominance arguments from iterated dominance arguments, which require a model of how others behave.

The large experimental literature on repeated dominance solvable games, like the prisoners' dilemma and many oligopoly and public goods experiments, seems particularly prone to this confusion. The use of evolutionary protocols to implement strategic form games, which allows subjects to learn while mitigating repeated game effects, has proven to be an effective way to observe behavior that is much closer to the predictions of strict dominance. For example, Cooper et al. (1995) report that using an evolutionary protocol reduces the play of a strictly dominated strategy in the prisoners' dilemma game to 12 percent in a fairly short experiment. Of course, observing the use of strictly dominated strategies undermines our confidence in the usefulness of rational choice models for prediction.

The prisoners' dilemma has been widely studied because it poses a conflict between rationality and efficiency. In this paper, we report an experiment that does not involve a conflict between rationality and efficiency. Rationality and efficiency will both require the elimination of the strictly dominated strategy.

This paper examines the use of a strategy that is not strictly dominated by any pure strategy but is strictly dominated by a mixture of pure strategies. Some players may find the strictly dominated strategy attractive because it is a second best reply to all of the pure strategies in the support of the dominating mixed strategy. It is, however, the worst reply to itself.

General theories of adaptive behavior, like Milgrom and Roberts (1991), predict that learning will eliminate the use of a strategy strictly dominated by a mixture under fairly weak conditions. After eliminating the dominated strategy in the game studied in the experiment what remains is the rock-scissors-paper game. It takes much stronger assumptions to predict that learning will converge to a mixed strategy equilibrium. In the rock-scissors-paper game, some dynamics, especially deterministic dynamics, can get caught in cycles and never converge to the unique equilibrium. For the game studied in the experiment, Dekel and Scotchmer (1992) have proved that for almost all initial conditions, the strictly dominated strategy is not eliminated by a discrete time replicator dynamic. The explosive cycle in the rock-scissors-paper component of the game makes the dominated strategy an above average reply sufficiently often that it never goes extinct.[1]

This paper reports an experiment to determine whether subjects will learn to stop using a strictly dominated strategy that can be an above average reply. It is difficult to find an experimental design that eliminates the play of the strictly dominated strategy completely. The least effective treatments used money to motivate behavior directly. The most effective treatment uses a binary-lottery to induce preferences, but even this treatment requires giving subjects plenty of experience. Doing so reduced the play of the strictly dominated strategy to around 10 percent by the end of a session.

There is no evidence for the explosive cycling needed to make the strictly dominated strategy an above average reply. The most surprising result was observing cycles in the frequency with which the strictly dominated action was played or spite cycles. Spite cycles appear using both evolutionary and fixed pairs protocols.

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Conclusion

In all treatments we observe persistent play of a strictly dominated strategy, D. This phenomena, while predicted by the discrete replicator dynamics, can not be explained by it, because the explosive cycles through R, S, P, which make D an above average reply, never occur. D is almost never an above average reply.

In other work, we have found the replicator dynamic to be a useful selection dynamic. In this paper, it doesn't make accurate predictions. A lesson from the experiment is that one should discount models that predict deterministic cycles. Humans just don't seem to get caught in them. It follows then that when a simple deterministic dynamic predicts deterministic cycles the situation ought to be re-examined using a probabilistic choice learning dynamic.

Experience reduced the frequency of D in all treatments except the pairs treatment. Somewhat paradoxically, the ability to "teach" the other participant to stop using D did not reduce the frequency with which D was played over time.

Using a binary lottery significantly reduced the frequency of D. The emphasis put on the use of binary lotteries when testing mixed strategy equilibria in the literature seems well founded. The RSPD game should be added to the examples in Prasnikar (1993) and Reitz (1993) who have previously compared the performance of induced value techniques and found it important to control for unobserved risk preferences using a binary lottery, see Roth (1995, 81-83).[2]

We observed cycles over time in the frequency of D, which we called spite cycles. Pairs seemed particularly prone to spite cycles, but the other treatments also exhibit some cycling. The least cycling occurs in the binary lottery treatment. A bad outcome can be attributed to luck rather than necessarily due to the other participants behavior in the binary lottery treatment. A possibility worth considering in future work is that the binary lottery may be effective not because it controls risk preference, but rather because the binary lottery reduces the likelihood of spite cycles.

It is possible to find a noise parameter for the logit equilibrium such that our non-parametric tests fail to reject the hypothesis that the data was generated by the fitted logit equilibrium. We were surprised by the small size of the noise parameter, it was only one third as large as the smallest parameter we had previously estimated. Nevertheless, we think the probabilistic choice framework provides a promising way to organize experimental data.

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Footnotes

1. This striking result turns out to depend on the particular way Dekel and Scotchmer discretize the replicator dynamic, see Cabrales and Sobel (1992).

2. John Kagel in personal communication tells us that he has used the binary lottery technique in signaling games and found an increase in the play of strictly dominated strategies.