Consider the following variant of the alternating-offer bargaining model that differs from the standard model in that players have outside options giving them positive payoffs if they fail to reach agreement. Specifically, two players bargain over the division of 1 dollar. There are T rounds of bargaining, where T≥ 1. In the first round, player 1 chooses an amount r € [0, 1] to offer to player 2. After observing the value of r, player 2 decides whether to accept or reject the offer. If player 2 accepts, the game ends, he receives a payoff of r, and player 1 receives a payoff of 1-2. If player 2 rejects and T = 1, the game ends and both players receive payoffs of r, where r € (0,1/2). If player 2 rejects and T> 1, they enter the second round of bargaining, in which player 2 chooses an amount y € [0, 1] to offer to player 1, which player 1 observes before choosing whether to accept or reject it. If player 1 accepts, the game ends, she receives a payoff of dy, and player 2 receives a payoff of 6(1-y), where 8 € (0, 1) is the discount factor. Play proceeds in this way until either an offer is accepted, or the offers in all T rounds have been rejected. If agreement is reached in period t, both players' payoffs are discounted by 6-1. If the game ends with all offers having been rejected, each player receives a payoff of 6T-1r. (a) Suppose T = 1. Find all subgame perfect equilibria. Solution: There is a unique SPE: player 1 offers r and player 2 accepts any offer of at least r. (b) Suppose T = 3. Find all subgame perfect equilibrium outcomes. (Note that you do not have to describe the full equilibrium strategies.) Solution: If bargaining reaches the third round, by the answer to part (a), 1 offers r and 2 accepts any offer of at least r. Thus in the second round, 1 accepts any offer of at least 8(1-r). Knowing this, player 2 can either offer 8(1-r), which will be accepted giving him a payoff of 8(1-8(1-r)), or make an unacceptable offer and end up getting a payoff of 8²r; the former is preferable since 8 € (0,1). Thus if 2 rejects I's offer in the first round, he receives a payoff of d(1-8(1-r)). Hence he accepts any offer of at least 8(18(1-r)) in the first round, and 1 makes exactly this offer (giving her a payoff of 1-8(1-8(1-r)), as opposed to the payoff of 82(1-r) she would obtain by making a rejected offer).