A number of experimental studies have addressed the predictions of the perfect equilibrium model in noncooperative game theory. Two games are

Game 6.7

Figure 6.4 Game 6.7: a Centipede Game

particularly important in this connection: the Ultimatum Game and the Centipede Game. On the whole, the experimental results disagree strongly with the predictions of the perfect equilibrium model, if it is considered as an empirical hypothesis. The Ultimatum Game has been discussed in historical context in Chapter 3. Here we will focus on the Centipede Game.

The “Centipede Game” (Rosenthal, 1981; McKelvey and Palfrey, 1992) is illustrated by Figure 6.4. The centipede is a game with two participants and a pot of money payoff dollars. The two participants will be a and b. The game proceeds in stages, and at each stage one or the other of the participants must make a commitment. At the first stage player a can either take or pass a money payment. If he takes it b also gets a smaller payment. If a passes at the first stage, b has an opportunity to take a larger share of the payment, leaving a the smaller share. However, if b passes at the second stage, a in turn gets an opportunity to take the larger share, and the game proceeds in this way. The two players alternate, as shown in Figure 6.4, where the numbers show the payoff to a first and then to b. The game ends after some finite number of steps with each participant getting a specified share of the pot. The size of the total payment to the two players may increase with the number of stages the game continues. This could be a model of “roundabout” production in economics, in that “passing” the pot on an early round allows the resources generated in the first round to be compounded in the later rounds. In some studies the game has subsequent stages, and it may have many stages. If we visualize a game with 100 stages rather than four, the basis of the name “Centipede Game” becomes clear.

A cooperative solution to this game requires a sequence of behavior strategies “pass.” Using backward induction it is clear that the subgame perfect equilibrium in this game is for a to “take the money and run.” Since a knows that b is a rational player, a cannot expect that b will pass on the second round and allow a to grab the larger amount, 15, at the third

Game 6.8, 6.9 in extensive form

Figure 6.5 Game 6.8, 6.9 in extensive form

stage, nor can b expect that if he passes a will allow him the opportunity to grab 20 at the fourth stage. In short, a rational agent cannot “outsmart” another rational agent.

The experimental evidence does not agree, and a variety of outcomes (but only rarely the cooperative outcome of continuing play to the end) are observed. Many of the observed sequences of play are consistent with the possibility that one or both players are trying to “outsmart” one another - with at least one of them failing to do so. Suppose that many experimental subjects commit themselves to inconsistent rationalizable strategies such as “If a passes then I will pass at the second stage and then, if he passes again, I’ll grab at the last stage for 20 rather than 10.” If a conjectures that b has adopted that strategy, a’s best response would be “Pass at the first stage and then, if b passes, grab at the second for 15 rather than 5.” On this interpretation, the evidence suggests that, at least in some circumstances, experimental subjects may adopt inconsistent rationalizable strategies.

In studies of reciprocity, variants on the centipede have given rise to important results. Figure 6.5 will serve as a generic diagram for two extensions of the centipede. These games take place in a maximum of three stages, although it can be cut short by either player.

In Game 6.8, decision node x is player a’s second decision node. (For now, ignore the third payoff number.) It introduces a “punishment” or “threat strategy,” P, that gives a the option of reducing b’s payoff at the cost of some reduction in his own, or of not doing so (arrow N). The basic subgame is the punishment node, and its Nash equilibrium is N. The game is then reduced to a two-step centipede with payoffs of 2, 7 at the second stage and D1 is the subgame perfect behavior strategy for a. The prediction of the subgame perfect equilibrium model is that the punishment node will make no difference and a will grab at the first opportunity, just as in a Centipede Game without the punishment node. However, such punishment is often observed, and cooperative outcomes (with payoffs such as 6,6) are more common in this game than they are when the third stage does not exist in the experiment. Since a decision for P at the punishment node leaves a worse off than he would be otherwise, P would be an instance of negative reciprocity.

Now suppose instead that the third node in Figure 6.5 is not the decision of player a but of a third player, c. The payoffs to c are shown third. This is an example of third-party punishment. As the choice of P makes agent c worse off, it would be an instance of altruistic punishment. It may also be referred to as “third-party reciprocity” (Fehr and Fischbacher, 2004). Once again, noncooperative game theory predicts that such third-party punishment will never occur, and consequently the strategies of a and b will be the same as they would be if there were no third stage; but the experimental evidence does not confirm this prediction. Rather, third-party punishment is observed, and seems consistent with the hypothesis that the third parties place some value on reciprocity between the original two players, and punish deviations from it (Fehr and Fischbacher, 2004).

On the whole, then, experimental evidence does not favor the subgame perfect equilibrium as a general empirical hypothesis. On the other side, we should observe that these experiments have themselves arisen from the subgame perfect equilibrium model. By drawing on that analysis, they have provided more precise evidence on human motivation than earlier experiments were able to supply. Thus, we may best regard the subgame perfect equilibrium as defining one extreme of a spectrum of forms of rationality that we may observe in human action, a point to which we return in Chapter 9.

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