Cheating in a Duopoly Supergame

Abstract

THIS PAPER examines an approach to oligopoly theory, first proposed in a relatively neglected paper by GeorgeJ. Stigler [ I 964]. Suppose that the firms in an oligopoly have achieved, tacitly or explicitly, a collusive agreement, and that open defection, in a sense to be made more precise below, does not pay. Given that the other firms observe the agreenient, it pays any one of them to cheat, that is, to make secret price cuts to selected buyers in order to secure sales it may not otherwise have made. Stigler then sets out to define the likely extent of the gain a cheat can make without detection and to relate this to observable market parameters. This would allow prediction of the probable incidence of cheating in a market and hence of the effectiveness of the collusive agreement. Stigler first argues that in a purely deterministic world, there could be no scope for cheating. Each firm's sales could be fully anticipated and any deviation from the expected level would be immediate evidence of cheating, which becomes, in effect, open defection. Implicit in this, of course, is the assumption that firms can observe each others' outputs at negligible cost-a reasonable assumption which will be maintained throughout the rest of this paper. If the world is not in fact deterministic, then scope for cheating may arise. Suppose that in each period any one buyer buys from any one seller with a probability between zero and one. Then, each seller's sales in each period are a random variable and it is possible that a seller may attribute to chance a well above average sales level which in fact was due to cheating. In deciding whether to accept this, the other sellers are involved in hypothesis testing and the critical level of this test will determine how far a cheat dare go. Stigler's development of this idea in a formal model is open to a number of specific criticisms. McKinnon [ I 966] criticized the excessively simplistic decision-theoretic basis of Stigler's model. In deciding whether a rival has gained above average sales by cheating or simply by chance, a firm is assumed to use a standard significance test with conventional critical levels, the choice of which is entirely unexplained in the model. There is also an unnecessary separation of tests according to the precise source of the buyers won by cheating-previous customers of the cheat, other firms' customers or newly-