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By Martin J. Osborne and Ariel Rubinstein

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Given that δM2 ≤ b we have m1 ≥ 1 − b by Step 3, so that m1 = 1 − b by Step 6. Further, M2 ≤ 1 − δm1 = 1 − δ(1 − b) by Step 3, so that M2 = 1 − δ(1 − b) by Step 6. Thus in each case the subgame perfect equilibrium outcome is unique. 1. 2 (Risk of breakdown) The argument that the strategy pair is a subgame perfect equilibrium is straightforward. 1, with 1 − α playing the role of δi for i = 1, 2. 1 (Three-player bargaining) First we argue that in any subgame perfect equilibrium the offer made by each player is immediately accepted.

Now, by repeatedly applying inflation–deflation we obtain a game of perfect information. Repeated applications of the principle of coalescing of moves yields a game with a single non-terminal history. 1 (Example of mixed and behavioral strategies) At the initial history choose A and B each with probability 21 ; at the second information set choose . 1 (Mixed and behavioral strategies and imperfect recall ) If player 1 uses the mixed strategy that assigns probability 12 to and probability 12 to rr then she obtains the payoff of 12 regardless of player 2’s strategy.

An interpretation is that the fact that 3, 4, and 5 always act as a block dulls the competition between 1 and 2, who cannot now compete with each other by forming (efficient) coalitions consisting of only two of the three members of 3, 4, and 5. 2 (Existence of competitive equilibrium in market) First note that the two sets are nonempty and convex and their interiors are disjoint, so that indeed they can be separated. Thus there exists (α, β) ∈ R × R, not equal to 0, such that zi∗ + β α · z + βy ≤ α · i∈N fi (zi∗ ) for all (z, y) ∈ X.

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A Course in Game Theory. SOLUTIONS by Martin J. Osborne and Ariel Rubinstein

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