By Martin J. Osborne and Ariel Rubinstein

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**Example text**

Thus M2 (G2 ) < c2 . 2) implies that m1 (G1 ) ≥ 1, so that m1 (G1 ) = 1 and hence M1 (G1 ) = 1. 2) we have m2 (G2 ) ≥ c1 , so that M2 (G2 ) = m2 (G2 ) = c1 . 1. b. 1 is a subgame perfect equilibrium. Refer to this equilibrium as E(x∗ ). Now suppose that c < 31 . An example of an equilibrium in which agreement is reached with delay is the following. Player 1 begins by proposing (1, 0). Player 2 rejects this proposal, and play continues as in the equilibrium E( 31 , 23 ). Player 2 rejects also any proposal x in which x1 > c and accepts all other proposals; in each of these cases play continues as in the equilibrium E(c, 1 − c).

Bm ) for some h ∈ Ii , where the sequence of actions of player i in the sequence (a1 , . . , ak ) is the same as the sequence of actions of player i in the sequence (b1 , . . , bm ) Now suppose that (β, µ) is a consistent assessment, let βi be a strategy of player i, ˆ a , a ) be a let β = (β−i , βi ), let Ii and Ii be information sets of player i, and let h = (h, ˆ ˆ terminal history, where a and a are sequences of actions, h ∈ Ii , and (h, a ) ∈ Ii . We begin by showing that O(β , µ|Ii )(h) = O(β , µ|Ii )(h) · Pr(β , µ|Ii )(Ii ).

The first equilibrium has the following undesirable feature. Player 2’s strategy d is optimal only if he believes that each of the two histories in his information set occurs with probability 12 . If he derives such a belief from beliefs about the behavior of players 1 and 3 then he must believe that player 1 chooses c with positive probability and player 3 chooses e with positive probability. But then it is no longer optimal for him to choose d: and r both yield him 2, while d yields less than 2.

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

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