Consider the following two period alternating offers game. Both players have $1. Player 1 makes an offer such as the following to Player 2: "with probability p you pay me your dollar and with probability 1p, I'll pay you my dollar," where Player 1 selects p. If Player 2 accepts the offer, the game ends and they receive the expected payoff according to the probability selected by Player 1. If player 2 rejects, he gets to offer a similar bet by selecting q where Player 1 gives her dollar to Player 2 with probability q and with probability 1 q Player 2 gives his dollar to Player 1. If Player 1 rejects the offer, the game ends and both keep their $1. Think of these bets as a toss of a biased coin: the player offering the bet can control the odds.

In addition, assume that

1. both players discount second period's payoff using the same discount factor .

2. for the first two parts of the question that each player's payoff from ending up with $x is x. So that the expected payoff from ending up with $x with probability r and $y with probability 1r is rx +(1r)y.

(a) Draw the extensive form.

(b) Compute the SPE.

(c) Now assume that Player 1's payoff from ending up with $x is x2 so that Player 1's expected 22

payoff from ending up with $x with probability r and $y with probability 1 r is rx + (1 r )y . (Player 2's preferences is as it was for the earlier parts of this question.)

Compute the SPE, and compare the bargaining weights with those you found in (b).