PLURAL NASH EQUILIBRIA AND THE RATIONALITY POSTULATE
It has been argued (Hargreaves Heap and Varoufakis, 1995; Coleman, 2003) that the multiplicity of Nash equilibria impeaches the rationality postulate basic to game theory (as well as neoclassical economics). When there are plural equilibria, the agents cannot determine their decisions simply through rational procedures. At best, custom and convention come into their decisions, and at worst, the decisions are simply indeterminate. This is said to discredit the rationality postulate. This critique goes too far, however.8
In the case of coordination games, such as Game 4.5, custom can indeed be the decisive determinant of the decisions. But the best-response principle is an explanatory principle without which custom might not determine the decisions in these cases. In Game 4.5, for example, the key point is that following custom is a best response; if people do not predictably follow their best response in Game 4.5 then the custom itself loses its predictive role. The nonrational alternative would be to suppose that people mechanically follow custom regardless of whether it is a best response or not. This in turn would mean that in Game 4.1, the pollution game, a custom of nonpollution would be sufficient to assure the efficient outcome. Game theory predicts the opposite, setting limits to the cases in which custom may be decisive. Indeed, the weakness of custom in limiting the deployment of polluting technologies does seem to contrast with the power of custom in determining that Britons drive on the left side of the road and Americans on the right. In any case, this is an elaboration, not a failure, of the rationality hypothesis.
The case is more difficult in an anticoordination game, such as Game 4.6. In such a game a uniform signal, such as a convention along the lines of “drive on the left,” will not do, so that the decision is all the more likely to be indeterminate. Yet, in fact, a more complex custom may resolve the decision. If the participants in the game are ranked, so that the person of higher rank is given precedence, then the information as to which agent is of the higher rank makes the decision determinate. A difficulty is that the hierarchy of rank must be complete: it must be that every match is between two agents of different rank. Rank in military organizations illustrates this. Combat generally involves coordinated action but differentiated missions and objectives, with great and widely different risks. In battlefield conditions, coordination is crucially important and indeterminate decisions can result in disaster. Thus, it is crucial that some specific person is able to make authoritative decisions, and far less important to make fine calculations about who is best suited to make them. The system of military ranks, with persons of equal rank subordinated by seniority or even age, is well suited as a customary solution to this problem.
On its face, it may seem that the correlated equilibria make the case worse, because they are far more numerous than Nash equilibria in many games. When there are two or more undominated Nash equilibria, there is a continuum of correlated equilibria. However, this is misleading. When the two businessmen agreed to settle their difference by arm-wrestling, the probabilities (whatever they may have been) were probabilities they agreed on, and their agreement specified a single one of the infinitely many possible allocations of probabilities among the two different pure-strategy Nash equilibria. The same is true when the township supervisors of Eastonia and Westoria agree to base their decision on a cost-benefit study, although each is certain that his own town is the best choice, and when two drivers at an intersection happen unpredictably to arrive when the light is red one way and green the other, and of Celtic priests deciding who is to be sacrificed.
It is true that there may not be time to come to agreement on the probabilities or on a mechanism, such as arm-wrestling or drawing the short straw, so that a correlated equilibrium may simply not be available as a means of resolving an indeterminate Nash equilibrium. The meeting of two cars at an intersection has been given as an example - and probably most readers of this book have experienced impasses of this kind. It is also true that, in a game like Game 5.3, Aumann’s game, there is potential of increasing efficiency beyond what an average of the Nash equilibria can support; but this potential can only be realized with private, imperfectly correlated signals, which may be difficult or costly to arrange. But it is not the rationality postulate that underlies these failures - it is the lack of sufficient time or communication.
The widespread observation of equiprobable mechanisms suggest that, in the absence of clear reasons against them, equiprobable correlated equilibria will usually occur when there are multiple undominated Nash equilibria. That equal probabilities are cognitively salient and determine a Schelling focal point, and require little knowledge but are consistent with the principle of insufficient reason, further points in the favor of the prediction of an equiprobable solution. If there is a custom or convention that supplies a solution, then custom will take precedence over an equiprob- able chance mechanism. In the rest of this book, cases of multiple Nash equilibria that cannot be resolved otherwise will be assumed to lead to equiprobable correlated equilibria.