Solve it mathematically by using first order conditions etc.

1. Auctions. Consider the following common value auction. There are two bidders, 1 and 2, whose types I91- for both i E [1? 2}. are independently drawn from a uniform distribution on [[1, 200]. The value of the object to both bidders is the sum of the types, Le. 61 + 62. The object is offered for sale in a rst price auction. Hence the payoffs depend on the bids bi and types as follows (assume a coin toss if bi = bj): a+aj 151. isz- > bi, Hi (bl-.bjhj) = %(6i+6jibt) if bi :55), 0 otherwise. (a) Show that strategies 31(91) = 61' 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 13's expected payoff when player 3' bids ()3- : 01 and player 33 bids b1- and has a type 9,; (remember that player i does not know 93; but uses that it is drawn from U [0, 200]), ii) Show that 5;- : +9;- maximizes the expected payoff. (b) If 31: = 1, the equilibrium bid is l, but it might seem that the expected value of the object is 1 + 100 = 101. Why 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 t' is 261' 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)