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By Martin J. Osborne and Ariel Rubinstein

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Thus −snc if s < b/c v(S) = −s[(n − s)c + b] if s ≥ b/c. b. First we argue that since the game is symmetric the core is nonempty if and only if it contains the payoff profile x = (−b, . . , −b). To see this, suppose that x is not in the core. Then for some integer k such that v(S) > −kb for every coalition S with |S| = k. Now let y = x be a feasible payoff profile. Then there exists some coalition T with |T | = k and y(T ) < −kb = v(T ). Thus y is not in the core. Now, if |S| = s ≤ b/c then x(S) = −sb ≥ −snc = v(S) and if |S| = s > b/c then x(S) = −sb ≥ −s[(n − s)c + b] = v(S) and x(N ) = −nb = v(N ) (by the assumption that b ≤ nc).

Let y be the member of Y in which every buyer’s payoff is z b and every seller’s payoff is z . We have yi = z b ≥ zi and yj = z ≥ zj , with at least one strict inequality. Further, yi + yj = z b + z = (1 − z )|L|/|B| + z ≤ 1 = v({i, j}). If we adjust yi and yj slightly to make both of the inequalities yi ≥ zi and yj ≥ zj strict then y {i,j} z. The standard of behavior that this stable set exhibits is “equality among agents of the same type”. Note the difference between this set and a set of the type Yp = {xi = p for all i ∈ L and xj = 1 − p for |L| members of B} for some p, which can be interpreted as the standard of behavior “the price is p”.

A machine that executes si is Qi , qi0 , fi , τi where • Qi = {S1 , . . , Sγ , P1 , . . , Pn }. • qi0 = S1 . • fi (q) = ai (p−j )i • τi (S , a) = if q = S or q = Pi if q = Pj for i = j. Pj S +1 (mod γ) if aj = aj and ai = ai for all i = j otherwise and τi (Pj , a) = Pj for all a ∈ A. 26 Chapter 8. 1 (Example with discounting) We have (v1 , v2 ) = (1, 1), so that the payoff of player 1 in every subgame perfect equilibrium is at least 1. Since player 2’s payoff always exceeds player 1’s payoff we conclude that player 2’s payoff in any subgame perfect equilibria exceeds 1.

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