Geanakoplos and Sebenius model with noise

Abstract

Assume that two risk neutral agents with asymmetric information simultaneously expect a gain from zero-sum betting. Geanakoplos and Sebenius (1983) (henceforth GS) consider the case where the agents may re-evaluate the profitability of betting successively before the payments are realized. They prove that one of the players must reject the proposed bet within some finite number (N0) of re-evaluation rounds. This paper extends the GS model to the case where there exists a small probability ɛ that players accept the bet when they should reject it. We claim that, generically, the GS results are not effected by small noise. That is, when ɛ is low, the players will reject the bet withinn0 iterations with probability close to one. Surprisingly, we find a non-generic example where — for every positive ɛ — the agents keep expecting a gain from betting forever. Our main Theorem, however, says that even when the noise ɛ is large and even in such non generic examples one of the following two alternatives must hold: (A) Some agent expects a loss from betting (and thus rejects the bet with high probability) after some finite number of reevaluation rounds, or (B) The expected gain from betting goes to zero for both agents. Moreover, if there is a small cost to entertaining the bet, then some player must expect a loss from betting eventually.