1. Auctions. Consider the following common value auction. There are two bidders, 1 and 2, whose types 9, for both i E {1, 2}, are independently drawn from a uniform distribution on [0,200]. The value of the object to both bidders is the sum of the types, is. 61 + 62. The object is offered for sale in a rst price auction. Hence the payoffs depend on the bids 6, and types as follows (assume a coin toss if b, = bj): 93463; bi1fbt > bji Hi (biibjaaisgj) = 092- +63 bi) if bi = by 0 otherwise. (a) Show that strategies 3,-(8,-) : 3, for i = 1,2 (i.e. the players bid their own type) form a Bayes Nash equilibrium in this game. To show that this is a BNE, do the following: i) formulate player *i's expected payoff when player j bids b,- = 9:,- and player i bids l),- aud has a type (9,; (remember that player i does not know 6,- but uses that it is drawn from U [0, 200]), ii) show that b, = 6',- maximizes the expected payoff. (b) If 6,- = 1, the equilibrium bid is 1, but it might seem that the expected value of the object is 1 + 100 = 101. 1Why doesn't the bidder behave more aggressively (informal discussion is enough)? (c) Assume now that the values are private: the value of the object to bidder 1' is 26.,- for i = 1,2. Solve the BNE where players use linear strategies (hint. we covered FPA with private values in the lectures and in PS 9). ((1) Compare the equilibria in (a) and in (c). Are the bidders better off when values are common or when they are private? What is the intuition? (informal discussion is enough)