PLEASE HOW TO THINK ABOUT IT AND SOLVE 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, thenplayer 3 does. If both accept the proposal, it is implemented and each player i receives payoff₁. If either rejects the proposal, then we move to the second round of bargaining, in whichplayer 2 proposes a division y = (y1, 92, 93) with y₁ +92 +93 = 1, which player 3 first decideswhether to accept or reject, followed by player 1. If both 3 and 1 accept, the proposal isimplemented, and each player i receives payoff dy,, where 8 € (0,1) is a common discountfactor. If either rejects the proposal, we move to the third round, in which player 3 makesa proposal z = (21, 22, 23) with 21 +22 +23 = 1 that player 1 accepts or rejects, followed byplayer 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 haveto 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 givenby 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 2will therefore offer y = (0,1-8,6). Given this, in the first round, Player 2 will acceptany offer of at least 8(1-5) and Player 3 will accept any offer of at least 8². Thus thereis a unique SPE outcome, which involves Player 1 offering ax = (1 - 8,6(1-6), 6²) andboth other players accepting.(b) Now suppose that, in each round of bargaining, a proposal is adopted if at least oneof the non-proposing players agrees to it (instead of both having to agree). Find allsubgame perfect equilibrium outcomes of this game.Solution: Proceed by backward induction. By similar logic to that of the UltimatumGame, the subgame beginning with Player 3's proposal in the third round has manysubgame perfect equilibria-differing in whether Player 1 accepts proposals that Player2 will accept, and in which proposals Players 1 and 2 choose to accept when they areoffered 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 asimilar 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 appliesto the first round, resulting in Player 1 offering x = (1,0,0) and at least one of the othertwo players accepting. (In particular, there are three equilibrium outcomes, one in whichonly Player 2 accepts, one in which only Player 3 accepts, and one in which both accept).I=

The technique of determining the appropriate course of action by working backwards in time from the…

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,

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter8: Game Theory

Section: Chapter Questions

Problem 8.7P

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