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Three players bargain over the division of 1 dollar. There are at most three rounds of bargain- ing. In the first round, player 1 proposes a division = (#₁, #2, #3), with x₁ + x2 + x3 = 1. After observing the proposed x, first player 2 chooses whether to accept or reject it, then player 3 does. If both accept the proposal, it is implemented and each player i receives payoff ₁. If either rejects the proposal, then we move to the second round of bargaining, in which player 2 proposes a division y = (y1, 92, 93) with y₁ + y2 + y3 = 1, which player 3 first decides whether to accept or reject, followed by player 1. If both 3 and 1 accept, the proposal is implemented, and each player i receives payoff dyi, where & € (0,1) is a common discount factor. If either rejects the proposal, we move to the third round, in which player 3 makes a proposal z = (21, 22, 23) with z1+z2+z3 = 1 that player 1 accepts or rejects, followed by player 2. If both 1 and 2 accept, each player i receives payoff 6²%, whereas if either rejects, all players receive payoffs of 0. (a) Find all subgame perfect equilibrium outcomes of this game. (Note that you do not have to describe the full equilibrium strategies.) Solution: Proceed by backward induction. By the same logic as in the Ultimatum Game, the subgame beginning with Player 3's proposal in the third round has a unique SPE given by z = (0, 0, 1) and the other players accepting all offers. Given this, in the second round, Player 1 will accept any offer and Player 3 will accept any offer of at least 8. Player 2 will therefore offer y = (0,1-6,8). Given this, in the first round, Player 2 will accept any offer of at least 8(1-5) and Player 3 will accept any offer of at least 62. Thus there is a unique SPE outcome, which involves Player 1 offering x = (1-8, 8(1-6), 6²) and both other players accepting. (b) Now suppose that, in each round of bargaining, a proposal is adopted if at least one

Three players bargain over the division of 1 dollar. There are at most three rounds of bargain- ing. In the first round, player 1 proposes a division = (#₁, #2, #3), with x₁ + x2 + x3 = 1. After observing the proposed x, first player 2 chooses whether to accept or reject it, then player 3 does. If both accept the proposal, it is implemented and each player i receives payoff ₁. If either rejects the proposal, then we move to the second round of bargaining, in which player 2 proposes a division y = (y1, 92, 93) with y₁ + y2 + y3 = 1, which player 3 first decides whether to accept or reject, followed by player 1. If both 3 and 1 accept, the proposal is implemented, and each player i receives payoff dyi, where & € (0,1) is a common discount factor. If either rejects the proposal, we move to the third round, in which player 3 makes a proposal z = (21, 22, 23) with z1+z2+z3 = 1 that player 1 accepts or rejects, followed by player 2. If both 1 and 2 accept, each player i receives payoff 6²%, whereas if either rejects, all players receive payoffs of 0. (a) Find all subgame perfect equilibrium outcomes of this game. (Note that you do not have to describe the full equilibrium strategies.) Solution: Proceed by backward induction. By the same logic as in the Ultimatum Game, the subgame beginning with Player 3's proposal in the third round has a unique SPE given by z = (0, 0, 1) and the other players accepting all offers. Given this, in the second round, Player 1 will accept any offer and Player 3 will accept any offer of at least 8. Player 2 will therefore offer y = (0,1-6,8). Given this, in the first round, Player 2 will accept any offer of at least 8(1-5) and Player 3 will accept any offer of at least 62. Thus there is a unique SPE outcome, which involves Player 1 offering x = (1-8, 8(1-6), 6²) and both other players accepting. (b) Now suppose that, in each round of bargaining, a proposal is adopted if at least one

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Three players bargain over the division of 1 dollar. There are at most three rounds of bargain- ing. In the first round, player 1 proposes a division = (#₁, #2, #3), with x₁ + x2 + x3 = 1. After observing the proposed x, first player 2 chooses whether to accept or reject it, then player 3 does. If both accept the proposal, it is implemented and each player i receives payoff ₁. If either rejects the proposal, then we move to the second round of bargaining, in which player 2 proposes a division y = (y1, 92, 93) with y₁ + y2 + y3 = 1, which player 3 first decides whether to accept or reject, followed by player 1. If both 3 and 1 accept, the proposal is implemented, and each player i receives payoff dyi, where 8 € (0,1) is a common discount factor. If either rejects the proposal, we move to the third round, in which player 3 makes a proposal z = (21, 22, 23) with 21+ 22 + 23 = 1 that player 1 accepts or rejects, followed by player 2. If both 1 and 2 accept, each player i receives payoff 6²%;, whereas if either rejects, all players receive payoffs of 0. (a) Find all subgame perfect equilibrium outcomes of this game. (Note that you do not have to describe the full equilibrium strategies.) Solution: Proceed by backward induction. By the same logic as in the Ultimatum Game, the subgame beginning with Player 3's proposal in the third round has a unique SPE given by z = (0, 0, 1) and the other players accepting all offers. Given this, in the second round, Player 1 will accept any offer and Player 3 will accept any offer of at least 8. Player 2 will therefore offer y = (0,1-6,8). Given this, in the first round, Player 2 will accept any offer of at least 8(1-5) and Player 3 will accept any offer of at least 62. Thus there is a unique SPE outcome, which involves Player 1 offering x = (1-5, 8(1-6), 6²) and both other players accepting. (b) Now suppose that, in each round of bargaining, a proposal is adopted if at least one
Transcribed Image Text:Three players bargain over the division of 1 dollar. There are at most three rounds of bargain- ing. In the first round, player 1 proposes a division = (#₁, #2, #3), with x₁ + x2 + x3 = 1. After observing the proposed x, first player 2 chooses whether to accept or reject it, then player 3 does. If both accept the proposal, it is implemented and each player i receives payoff ₁. If either rejects the proposal, then we move to the second round of bargaining, in which player 2 proposes a division y = (y1, 92, 93) with y₁ + y2 + y3 = 1, which player 3 first decides whether to accept or reject, followed by player 1. If both 3 and 1 accept, the proposal is implemented, and each player i receives payoff dyi, where 8 € (0,1) is a common discount factor. If either rejects the proposal, we move to the third round, in which player 3 makes a proposal z = (21, 22, 23) with 21+ 22 + 23 = 1 that player 1 accepts or rejects, followed by player 2. If both 1 and 2 accept, each player i receives payoff 6²%;, whereas if either rejects, all players receive payoffs of 0. (a) Find all subgame perfect equilibrium outcomes of this game. (Note that you do not have to describe the full equilibrium strategies.) Solution: Proceed by backward induction. By the same logic as in the Ultimatum Game, the subgame beginning with Player 3's proposal in the third round has a unique SPE given by z = (0, 0, 1) and the other players accepting all offers. Given this, in the second round, Player 1 will accept any offer and Player 3 will accept any offer of at least 8. Player 2 will therefore offer y = (0,1-6,8). Given this, in the first round, Player 2 will accept any offer of at least 8(1-5) and Player 3 will accept any offer of at least 62. Thus there is a unique SPE outcome, which involves Player 1 offering x = (1-5, 8(1-6), 6²) and both other players accepting. (b) Now suppose that, in each round of bargaining, a proposal is adopted if at least one
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