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 1−p, 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 δ.
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 1−r is rx +(1−r)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).

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