PLEASE CHECK THIS HOW TO 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 x = (₁, 72, 73), with x₁ + x2 + x3 = 1.After observing the proposed z, 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 payoffI. 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 + y2+y3 = 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 dyi, 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+z3 = 1 that player 1 accepts or rejects, followed byplayer 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 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 = (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 d. Player 2will therefore offer y = (0,1-6,8). Given this, in the first round, Player 2 will acceptany offer of at least 6(1-5) and Player 3 will accept any offer of at least 82. Thus thereis a unique SPE outcome, which involves Player 1 offering x = (1-6, 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)

Backward induction is a technique for figuring out the best course of action by going backward in…

ing. In the first round, player 1 proposes a division z = (₁, 72, 73), with 7₁ +1₂ + 13 = 1. After observing the proposed z, 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 I¡. If either rejects the proposal, then we move to the second round of bargaining, in which player 2 proposes a division y = (y₁, 32, 33) 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 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 2

ing. In the first round, player 1 proposes a division z = (₁, 72, 73), with 7₁ +1₂ + 13 = 1. After observing the proposed z, 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 I¡. If either rejects the proposal, then we move to the second round of bargaining, in which player 2 proposes a division y = (y₁, 32, 33) 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 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 2

ENGR.ECONOMIC ANALYSIS

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ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Question

PLEASE CHECK THIS HOW TO 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 x = (₁, 72, 73), with x₁ + x2 + x3 = 1.
After observing the proposed z, 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
I. 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+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 d. 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 6(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 x = (1-6, 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)

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