1. AN APPLICATION OF LOVASZ LOCAL LEMMA: THERE EXISTS AN INFINITE 01-SEQUENCE CONTAINING NO NEAR IDENTICAL INTERVALS
2. Proof of Step 1. Suppose to the contrary that player 1 accepts the demand for sure. Player 2's payoff will be (1?q)H +c 1 . We shall argue that (1?q)H +c 1 is strictly dominated and cannot be an equilibrium demand. Consider another demand qH ? c 2 . If player 1 is type G, then he will reject it and player 2's payoff will be (1 ? q)H ? c 2 ; if player 1 is type B, then whether or not he rejects qH ? c 2 , player 2 will earn qH ? c 2 in expectation. Therefore, player 2's expected payoff is p(1 ? q)H + (1 ? p)qH ? c 2;Then, type B must reject this demand with a positive probability, and hence the equilibrium posterior belief after a rejection but before the expert verdict does not exceed p
3. Step 2. Fix any p < p * * . One of the following holds: (a) V (p) = V ; or (b) player 1 weakly prefers to reject any equilibrium demand and the equilibrium posterior immediately after the rejection (before the expert verdict) does not exceed p. Proof of Step 2. There are two cases to consider. Case 1 : (1 ? q)H + c 1 is demanded with a positive probability in equilibrium. Then, by Step 1, (b) holds. Case 2 : (1 ? q)H + c 1 is demanded with probability 0 in equilibrium. In this case only qH ? c 2 can be possibly accepted by Proposition 4. -If type B's equilibrium strategy prescribes that qH ? c 2 be rejected for sure, then the belief will not change after rejection; hence, (b) holds. -If it prescribes that qH ? c 2 be accepted with a positive probability, then all demands greater than (1 ? q)H + c 1 but less than qH ? c 2 is going to be accepted for sure;Since (1 ? q)H + c 1 is rejected with positive probability, all higher demands are rejected for sure by Lemma 1. It follows that in this case rejection reduces the posterior belief