Consider the following modification of the ultimatum game with jealousy preference. There are two players i...

 

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Consider the following modification of the ultimatum game with jealousy preference. There are two players i = 1,2 bargain over how to split 1. Player 1 offers ($1,1-81), 81 € [0, 1]. Then, player 2 either accepts, a, or rejects, r. A " Jealousy" preference is captured by a, 3 € [0,1]: and u₁ (8₁, a) = as₁ + (1 − a)(s₁ - (1—8₁)) = as₁ + (1 -a) (2s₁ - 1) u2 (81, a) = 3(18₁) + (1 - 3)(1 - 81 - 81) = 3(18₁) + (1 - 3)(1-281) As before, rejection yields zero payoff to both players. (1) Describe a pure strategy for each player. (2) Suppose a = Band solve this game by backward induction. (3) Again, solve the game by backward induction for a general a and 3 Consider the following modification of the ultimatum game with jealousy preference. There are two players i = 1,2 bargain over how to split 1. Player 1 offers ($1,1-81), 81 € [0, 1]. Then, player 2 either accepts, a, or rejects, r. A " Jealousy" preference is captured by a, 3 € [0,1]: and u₁ (8₁, a) = as₁ + (1 − a)(s₁ - (1—8₁)) = as₁ + (1 -a) (2s₁ - 1) u2 (81, a) = 3(18₁) + (1 - 3)(1 - 81 - 81) = 3(18₁) + (1 - 3)(1-281) As before, rejection yields zero payoff to both players. (1) Describe a pure strategy for each player. (2) Suppose a = Band solve this game by backward induction. (3) Again, solve the game by backward induction for a general a and 3

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