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 = (31, 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 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)