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

By Martin J. Osborne and Ariel Rubinstein

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We proceed in a number of steps. Step 1. m2 ≥ 1 − δM1 . 1. Step 2. M1 ≤ 1 − max{b, δm2 }. Proof. Since Player 2 obtains the payoff b by opting out, we must have M1 ≤ 1 − b. 1. Step 3. m1 ≥ 1 − max{b, δM2 } and M2 ≤ 1 − δm1 . The proof is analogous to those for Steps 1 and 2. Step 4. If δ/(1 + δ) ≥ b then mi ≤ 1/(1 + δ) ≤ Mi for i = 1, 2. Proof. These inequalities follow from the fact that in the subgame perfect equilibrium described in the text player 1 obtains the payoff 1/(1 + δ) in any subgame in which she makes the first offer, and player 2 obtains the same utility in any subgame in which he makes the first offer.

The following proof of uniqueness is taken from Osborne and Rubinstein (1990). Let M1 and M2 be the suprema of player 1’s and player 2’s payoffs over subgame perfect equilibria of the subgames in which players 1 and 2, respectively, make the first offer. Similarly, let m1 and m2 be the infima of these payoffs. Note that (Out, 0) 2 (y ∗ , 1) if and only if b ≤ δ/(1 + δ). We proceed in a number of steps. Step 1. m2 ≥ 1 − δM1 . 1. Step 2. M1 ≤ 1 − max{b, δm2 }. Proof. Since Player 2 obtains the payoff b by opting out, we must have M1 ≤ 1 − b.

Let x be a feasible payoff profile for which xi > 0 for some i ∈ N \ V . Let S be a winning coalition that does not include i. Then x(S) < 1 = v(S), so that x is not in the core. 2 (Zerosum games) If N, v is zerosum and x is in the core of N, v then for any coalition S we have x(S) ≥ v(S) and x(N \ S) ≥ v(N \ S); since x(S) + x(N \ S) = x(N ) = v(N ) = v(S) + v(N \ S) it follows that x(S) = v(S). Thus for all disjoint coalitions S and T we have v(S) + v(T ) = x(S) + x(T ) = x(S ∪ T ) = v(S ∪ T ).

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