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Three players bargain over the division of 1 dollar. There are at most three rounds of bargain- ng. In the first round, player 1 proposes a division 1 = (₁, 12, 13), with 2₁ +22₂ + x3 = 1. After observing the proposed r, first player 2 chooses whether to accept or reject it, then layer 3 does. If both accept the proposal, it is implemented and each player i receives payoff If either rejects the proposal, then we move to the second round of bargaining, in which layer 2 proposes a division y = (31, 32, 33) with y₁ +92 +93 = 1, which player 3 first decides whether to accept or reject, followed by player 1. If both 3 and 1 accept, the proposal is mplemented, and each player i receives payoff dy, where & € (0,1) is a common discount actor. If either rejects the proposal, we move to the third round, in which player 3 makes proposal z = (21, 22, 23) with 2₁ +2₂ +23 = 1 that player 1 accepts or rejects, followed by layer 2. If both 1 and 2 accept, each player i receives payoff ²;, whereas if either rejects,

Three players bargain over the division of 1 dollar. There are at most three rounds of bargain- ng. In the first round, player 1 proposes a division 1 = (₁, 12, 13), with 2₁ +22₂ + x3 = 1. After observing the proposed r, first player 2 chooses whether to accept or reject it, then layer 3 does. If both accept the proposal, it is implemented and each player i receives payoff If either rejects the proposal, then we move to the second round of bargaining, in which layer 2 proposes a division y = (31, 32, 33) with y₁ +92 +93 = 1, which player 3 first decides whether to accept or reject, followed by player 1. If both 3 and 1 accept, the proposal is mplemented, and each player i receives payoff dy, where & € (0,1) is a common discount actor. If either rejects the proposal, we move to the third round, in which player 3 makes proposal z = (21, 22, 23) with 2₁ +2₂ +23 = 1 that player 1 accepts or rejects, followed by layer 2. If both 1 and 2 accept, each player i receives payoff ²;, whereas if either rejects,

ENGR.ECONOMIC ANALYSIS
ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN:9780190931919
Author:NEWNAN
Publisher:NEWNAN
Chapter1: Making Economics Decisions
Section: Chapter Questions
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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 21 + 2 + 3 = 1. After observing the proposed r, 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 +92 +93 = 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 dy,, 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 822₁, 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 2= (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 = (0,1-8,6). 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 8². Thus there is a unique SPE outcome, which involves Player 1 offering ax = (1 - 8,6(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 of the non-proposing players agrees to it (instead of both having to agree). Find all subgame perfect equilibrium outcomes of this game. Solution: Proceed by backward induction. By similar logic to that of the Ultimatum Game, the subgame beginning with Player 3's proposal in the third round has many subgame perfect equilibria-differing in whether Player 1 accepts proposals that Player 2 will accept, and in which proposals Players 1 and 2 choose to accept when they are offered nothing but the outcome of every equilibrium is that Player 3 offers z = (0,0,1) and at least one of other players accepts. Given this, in the second round, following a similar logic, while there are many equilibria, in each one Player 2 offers y = (0,1,0) which is accepted by at least one of the other players. The same argument now applies to the first round, resulting in Player 1 offering x = (1,0,0) and at least one of the other two players accepting. (In particular, there are three equilibrium outcomes, one in which only Player 2 accepts, one in which only Player 3 accepts, and one in which both accept). I=
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 21 + 2 + 3 = 1. After observing the proposed r, 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 +92 +93 = 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 dy,, 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 822₁, 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 2= (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 = (0,1-8,6). 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 8². Thus there is a unique SPE outcome, which involves Player 1 offering ax = (1 - 8,6(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 of the non-proposing players agrees to it (instead of both having to agree). Find all subgame perfect equilibrium outcomes of this game. Solution: Proceed by backward induction. By similar logic to that of the Ultimatum Game, the subgame beginning with Player 3's proposal in the third round has many subgame perfect equilibria-differing in whether Player 1 accepts proposals that Player 2 will accept, and in which proposals Players 1 and 2 choose to accept when they are offered nothing but the outcome of every equilibrium is that Player 3 offers z = (0,0,1) and at least one of other players accepts. Given this, in the second round, following a similar logic, while there are many equilibria, in each one Player 2 offers y = (0,1,0) which is accepted by at least one of the other players. The same argument now applies to the first round, resulting in Player 1 offering x = (1,0,0) and at least one of the other two players accepting. (In particular, there are three equilibrium outcomes, one in which only Player 2 accepts, one in which only Player 3 accepts, and one in which both accept). I=
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