July 21, 2006.
Consider a two player game that is to be played once. The players receive information that they use to help them predict the choices made by each other. A decision rule for each player captures how each player uses the information received in making their choices. Priors in this context are probability distributions over the information that may be received and over the decision rules that their opponents may use. I investigate the existence of prior beliefs for each player that satisfy the following properties: (R) they do not rule out their opponent using a rational decision rule, (K) they do not rule out the existence of information that would reveal the choice made by the opponent, and (SU) they do not rule out strategic uncertainty - a belief diversity condition. In this paper I show that for a large class of games there are no prior beliefs that satisfy properties (R), (K), and (SU). In the paper I discuss the implications of this result, in particular regarding whether one should expect a Nash equilibrium to arise in a game that is to be played once.
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