Please be aware that all the information (text and picture) is provided. Do not just say the incomplete information, then do not solve it.
Just answer any one of the following question that you know how to do, but 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.
- [5 points] Find all SPE of the game
- [5 points] Suppose now that dollars are indivisible, so x must be an integer number ? 1, 2, ...M . Find all SPE of the game.
- [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)
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 E [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)6M 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)\fQuestion 3. [30 points] (Bayesian game) Consider the Bayesian game represented by the following table. Ygritte Hate (Prob=1) Love (Prob=) Indifferent (Prob=) CB BW CB BW CB BW CB 3.-1 2.3 CB 4.2 1.-1 CB 1.0 1.0 Jon BW 0.2 1.-1 BW 0.-2 3.3 BW 3.0 There are three different states: i) Ygritte hates Jon with prior probability 1/4 ii) Ygritte loves Jon with prior probability 1/2 and iii) Ygritte is indifferent for Jon with probability 1/4 . Suppose Ygritte knows her attitude perfectly. However, Jon is only fully informed if Ygritte hates him but cannot tell whether Ygritte loves him or is indifferent. 1. [4 points] How many different signals could Jon receive and how many different signals could Ygritte receive? Please write down both players' signal set and signal function. 2. [3 points] Suppose Jon receives a signal that tells him either state Love or state Indifferent will be played. Using Bayes rule calculate Jon's belief about the probability of state Love after he receives such a signal. 3. [6 points] What are Jon and Ygritte's pure strategy sets respectively? (Hint: a player's strategy in Bayesian game is a plan of actions for each signal/type.) 4. [10 points] For each possible pure strategy taken by Ygritte, please calculate Jon's expected payoffs of each of his pure strategy when he receives a signal that tells him either state Love or state Indifferent will be played 2 5. [7 points] Find the set of all pure strategy Nash Equilibria of this gameQuestion 4. [25 points] Consider a variant of the sealed-bid auctions with imperfect information considered in class in which the players are risk-averse. Specifically, assume that the payoff of a player with valuation v who wins the object and pays the price p is (v -p) /", 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 second-price 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 first- price auction. (Hint: Assume that the player strategy is a linear function of their evaluation, i.e. when player 2's valuation is vz she bids Buz, where / is a constant. Find the best response of player 1 to this strategy of player 2 when player 1's valuation is v1.) 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 second-price and a first-price auction. How does the auctioneer's revenue differ between the two auctions
