Just answer any one of the following question that you know how to do. You are free to answer all of them.

Question 1. [15 points] Consider the following variation of the ultimatum game we have seen in class. Two player, Vince and Jules have to split a suitcase with M $ in it. Vince moves first and makes an offer x ? [0, 1]. Then Jules takes the suitcase and can burn a fraction ? ? [0, 1]of all the money in it. Then money are split according to Vince's suggestion. Vince's payoff is (1 ? x)?M and Jules is ?xM.

1. [5 points] Find all SPE of the game
2. [5 points] Suppose now that dollars are indivisible, so x must be an integer number ? 1, 2, ...M . Find all SPE of the game.
3. [5 points] Is there any NE in which Vince is indifferent between any of his actions? (Hint: it doesn't have to be subgame perfect)

\fQuestion 4. [25 points] Consider a variant of the sealedbid auctions with imperfect information censidered in class in which the players are riskaverse. Specically, assume that the payoff of a player with valuation v who wins the object and pays the price p is (v p)1/m, where m > 1. (In class we considered the case m = 1, in which the bidders are \"risk neutral\". For m > 1, the bidders are \"risk averse\".) 1. [7 points] Find an equilibrium of the secOndprice auction. 2. [12 points] Suppose that there are two players and each player's valuation is drawn independently from a uniform distribution on [0, 1] (as we assumed in class). Find an equilibrium of the rst price auction. (Hint: Assume that the player strateg is a linear function of their evaluation, i.e. when player 2's valuation is \"02 she bids 6'02, where B is a constant. Find the best response of player 1 to this strategy of player 2 when player 1's valuation is 121.) 3. [6 points] Following the environment of part (2) (two players, uniform distribution of valuations), compare the expected value of the price paid by a Winner with valuation v in the equilibrium of a secondprice and a rstprice auction. How does the auctioneer's revenue differ between the two auctions?. Question 1. [15 points] Consider the following variation of the ultimatum game we have seen in class. Two player, Vince and Jules have to split a suitcase with M $ in it. Vince moves first and makes an offer x 6 0, 1]. Then Jules takes the suitcase and can burn a fraction o E [0, 1]of all the money in it. Then money are split according to Vince's suggestion. Vince's payoff is (1 - x)SM and Jules is SxM. 1. [5 points] Find all SPE of the game 2. [5 points] Suppose now that dollars are indivisible, so x must be an integer number E 1, 2, ...M. Find all SPE of the game. 3. [5 points] Is there any NE in which Vince is indifferent between any of his actions? (Hint: it doesn't have to be subgame perfect)