Assume that two risk neutral agents with asymmetric information simultaneously expect a gain from zero-sum betting. Geanakoplos and Sebenius (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 ϵ that players accept the bet when they should reject it. We claim that, generically, the GS results are not affected by small noise. That is, when ϵ is low, the players will reject the bet within N0 interations with probability close to one. Surprisingly, we find a non-generic example where – for every positive ϵ – the agents keep expecting a gain from betting forever. Our main Theorem, however, says that even when the noise ϵ 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 re-evaluation 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.
