On the Strategic Advantages of a Lack of Common Knowledge

I illustrate the surprising strategic effects a lack of common knowledge can have in a simple example. I analyze a two-person bargaining game in which the seller makes a single take-it-or-leave-it offer. The seller knows everything except possibly what information the buyer has. If it is common knowledge that the buyer knows the value of the object or if it is common knowledge that the buyer does not know the value, then the buyer's expected payoffis zero. However, if the seller does not know what the buyer knows, then the buyer may be able to obtain some of the gains from trade. In the exogenous information case, where there is a fixed probability q that the buyer is informed, if q is far enough from zero and one, the buyer's expected payoff is strictly positive. However, if the seller is almost sure of what the buyer knows, the buyer's expected payoff is zero. Surprisingly, the endogenous information case, where, after seeing the seller's offer, the buyer decides whether or not to become informed, is quite different. If the cost of obtaining information is zero, we get the same outcome as it is common knowledge that the buyer is informed. However, if the cost of obtaining the information is small but strictly positive, the buyer's payoff is strictly positive. As the cost goes to zero, the limit of the buyer's payoff is strictly positive and may even be virtually all of the gains from trade. 2. I N T R O D U C T I O N Much research in economics over the past decade has demonstrated the fact that the distribution of knowledge is of crucial importance for predicting outcomes. Changing the specification of "who knows what and when they learn it" can greatly affect the equilibrium outcome. 1 What is less well understood is how knowledge about the knowledge of others affects equilibrium outcomes. (The small amount of literature on this subject is discussed below.) Typically, we assume that "who knows what and when they learn it" is common knowledge. Informally, a statement p is said to be common knowledge among the players if all statements of the form "everyone knows that everyone knows that . . . knows that p" are true. In this paper, I show by example that relaxing the assumption that "who knows what" is common knowledge can have very surprising implications. 1 For classic examples , see Kreps and Wilson [1982a] and Milgrom and Rober t s [1082].