Empirical Application: Experimental Evidence on Coordination-Failure Games

Coordination-failure models have more than one Nash equilibrium. Traditional game theory predicts that such economies will arrive at one of their equilibria, but does not predict which one. Various theories of equilibrium refinements make predictions about which equilibrium will be reached. For example, a common view is that Pareto-superior equilibria are focal, and that economies where there is the potential for coordination failure therefore attain the best possible equilibrium. There are other possibilities as well. For example, it may be that each agent is unsure about what rule others are using to choose among the possible outcomes, and that as a result such economies do not reach any of their equilibria.

One approach to testing theories that has been pursued extensively in recent years, especially in game theory, is the use of experiments. Experiments have the advantage that they allow researchers to control the economic

Models with multiple, Pareto-ranked equilibria are known as coordination-failure models. The possibility of coordination failure implies that the economy can get stuck in an underemployment equilibrium. That is, output can be inefficiently low just because everyone believes that it will be. In such a situation, there is no force tending to restore output to normal. As a result, there may be scope for government policies that coordinate expectations on a high-output equilibrium; for example, a temporary stimulus might permanently move the economy to a better equilibrium.

There is an important link between multiple equilibria and our earlier discussion of real rigidity. Recall that there is a high degree of real rigidity- when, in response to an increase in the price level and the consequent decline in aggregate output, the representative firm wants to reduce its relative price only slightly. In terms of output, this corresponds to a reaction function with a slope slightly less than 1: when aggregate output falls, the representative firm wants its sales to decline almost as much as others. The existence of multiple equilibria requires that over some range, declines in aggregate output cause the representative firm to want to raise its price and thus reduce its sales relative to others; that is, what is needed is that the reaction function have a slope greater than 1 over some range. In short, coordination failure requires that real rigidity be very strong over some range.

One imphcation of this observation is that, since there are many potential sources of real rigidity, there are many potential sources of coordination failure. Thus there are many possible models that fit Cooper and Johns general framework. Examples include Bryant (1983); Heller (1986); Kiyotaki 11988); Shleifer (1986); Murphy, Shleifer, and Vishny (1989b, 19890; Durlauf 11993); and Lamont (1994).

In addition, they add a constant of S0.60 to the payoff function so that no one car. lose money.

environment precisely. They have the disadvantages, however, that they are often not feasible and that behavior may be different in the laboratory than in similar situations in practice.

\an Huyck, Battaho, and Bell (1990,1991) and Cooper, DeJong, Forsythe. and Ross (1990,1992) test coordination-failure theories experimentally. Van Huyck, Battalio, and Bell (1990) consider the coordination-failure game proposed by Bryant (1983). In Bryants game, each of N agents chooses an effort level over the range [0,¸]. The payoff to agent i is

U, = amm[ei,e2,...,eN]~ pe,, a > p > 0. (6.97»

The best equilibrium is for every agent to choose the maximum effort lev el. ¸; this gives each agent a payoff of (a - )¸. But any common effort level in [0, eJ is also a Nash equilibrium: if every agent other than agent z sets his or her effort to some level e, i also wants to choose effort of e. Since each agents payoff is increasing in the common effort level, Bryants game is a coordination-failure model with a continuum of equilibria.

\an Huyck, Battaho, and Bell consider a version of Bryants game with effort restricted to the integers 1 through 7, a = $0.20,/3 = $0.10, and .V between 14 and 16. They report several main results. The first concerns the first time a group plays the game; since Bryants model is not one of repeated play, this situation may correspond most closely to the model Van Huyck, Battaho, and Beil find that in the first play, the players do not reach any of the equilibria. The most common levels of effort are 5 and 7, but there is a great deal of dispersion. Thus, no deterministic theory of equilibrium selection successfully describes behavior.

Second, repeated play of the game results in rapid movement toward lo\\ effort. Among five of the seven experimental groups, the minimum effort m the first period is more than 1. But in all seven groups, by the fourth play the minimum level of effort reaches 1 and remains there in every subsequent round. Thus there is strong coordination failure.

Third, the game fails to converge to any equilibrium. Each group pla\ ec the game 10 times, for a total of 70 trials. Yet in none of the 70 trials do al. of the players choose the same effort. Even in the last several trials, which are preceded in every group by a string of trials where the minimum effor is 1, over a quarter of players choose effort greater than 1.

Finally, even modifying the payoff function to induce "coordination successes" does not prevent reversion to inefficient outcomes. After the initial 10 trials, each group played 5 trials with the parameter p in (6.97) set tc zero. With jS = 0, there is no cost to higher effort; as a result, most (though not aU) of the groups converge to the Pareto-efficient outcome of e, == 7 for all players. But when p is changed back to $0.10, there is rapid reversion tc the situation where most players choose the minimum effort.

(6.98)

j>(x) =

l~{ayf/dy) dx

(6.99)

FIGURE 6.9 A reaction function that implies a unique but fragile equilibrium

Van Huyck, Battalio, and Bells results suggest that predictions from deductive theories of behavior should be treated with caution: even though Bryants game is fairly simple, actual behavior does not correspond well with the predictions of any standard theory. The results also suggest that coordination-failure models can give rise to complicated behavior and dynamics.

Real Non-Walrasian Theories

Substantial real rigidity, even if it is not strong enough to cause multiple equilibria, can make the equilibrium highly sensitive to disturbances. Consider the case where the reaction function is upward-sloping with a slope slightly less than 1. As shown in Figure 6.9, this leads to a unique equilibrium. Now let X be some variable that shifts the reaction function; thus we now write the reaction function as y, = y*(y,x). The equilibrium level of for a given x, denoted y(x), is defined by the condition y*(>>(x),x) = >>(x). Differentiating this condition with respect to x yields