COORDINATION AND ANTICOORDINATION GAMES
Game 5.1, with its two Nash equilibria, has something in common with Games 4.5 and 4.6, the coordination and anticoordination games. Indeed, when the equilibrium of a coordination game is determined by custom, the custom might be considered as a signal that supports a correlated equilibrium. In some superspace of different possible histories, perhaps, driving on the left and driving on the right are equally probable - as witness the opposite customs in different countries. But, as we recall, anticoordination games present more difficult problems.
Stoplights as a Paradigm
The example of an anticoordination game is the intersection game - two cars approaching an intersection. Which will go through, and which will pause? Two Nash equilibria exist, where each car takes one of these roles. If they had time, the two drivers could get out and flip a coin to decide - but that would defeat the purpose of the exercise, which is to get through the intersection quickly. If the intersection is controlled by a stoplight, though, the car with the green light will go ahead and the car with the red light will stop. This is a correlated strategy equilibrium. The probability of getting a green light may be equal for the two, depending on when they arrive, or it may be unequal, if one of the streets is a major artery and so has longer green lights to accommodate heavier traffic; but, as we have seen, equal probability is not a requirement.
The traffic light is a fascinating twentieth-century innovation in the practice of interdependent decisions!2 It also illustrates the difference that makes anticoordination games more difficult than coordination games. For a coordination game, such as Game 4.5, the two decision makers receive the same signal, such as a customary practice, and that is sufficient so that they can coordinate their decisions. However, in the anticoordination game, different, correlated signals are required.
In a two-person game, given time, there is little difficulty in contriving this. However, the intersection game is a bit artificial, in that highway traffic is not really a two-person game. Rather it is a many-person game with drivers randomly matched to interact at intersections. In the individual matches, there will not be time enough for the drivers to get out and arm-wrestle. What is needed is a correlated set of signals for the entire population. The stoplights provide that correlated set of signals. But notice that the provision of common signals for this large population is a public good. It is no accident that stoplights are provided by government, although, in the early stages of motoring, some traffic direction was provided by private initiative.3
This defines a function of the public authority that has not been explicitly recognized, although it is implicit in some existing public activities such as the provision of stoplights and signage. As we will see it is also implicit in some aspects of economic policy. Perhaps explicit consideration of this role of the public authority will lead to innovations that can improve the results of private decisions in new ways.