1 π A (a) = P[a | A is behavioral] and π B (b) = P[b | B is behavioral]. We will denote this incomplete information game by Γ(r, α, z A , z B ), where r = (r A , r B 1 , r B 2 ) and α = (α 1 , α 2 ). The parameters A, B, π A and π B are held fixed throughout. Letā = max A andb = max B. We assume that [min A] +b > 1,ā + [min B] > 1, and that π A (a) > 0 and π B (b) > 0 for all a ∈ A and b ∈ B. Hereafter, we find it convenient to call A the rational player A and B k the rational player B with

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... ount rate r B k , k = 1, 2. We will require that A's initial demand belongs to A, and similarly that B k 's initial demand belongs to B. There is no restriction on either player's subsequent demands. 4 In equilibrium, A chooses an initial posture a ∈ A with probability φ A (a), and after observing a, B k chooses an initial posture b ∈ B with probability φ B k (b|a). A pair of choices (a, b) ∈ A × B with a + b > 1 leads to the subgame Γ(r,α 1 (a, b),α 2 (a, b),ẑ A (a),ẑ B (a, b), a, b), wherê are the posterior probabilities that player A is behavioral, and that player B is behavioral or B k , respectively. For simplicity, we will often omit the arguments (a, b) and simply write, for example,ẑ A andẑ B instead ofẑ A (a) andẑ B (a, b). 4 This is without loss of generality. The choice ofā ∈ A for A weakly dominates any a / ∈ A, and when A demands a ∈ A, a counteroffer b / ∈ B yields a unique equilibrium in which B k concedes to a right away with probability 1, k = 1, 2. This is another expression of Coasean dynamics (see Section 8.8 of Myerson (1991) , Proposition 4 of Abreu and Gul (2000) and Lemma 1 of Abreu and Pearce (2007) ).