Consider a common value auction:

There are a total of n bidders.

Each bidder receives a signal ti that is uniformly distributed on [0, 1].

Bidders are bidding on an item with the value V = t1 + t2 + tn

Player i knows only their signal ti and that all other signals are uniformly distributed over [0, 1].

(a) What is the expected value of V to player 1 given t1.

(b) What is the expected value of V to player 1 given t1 and given that player 1 wins the auction bidding as if he has a signal r.

(c) If the auction is a first price auction, the equilibrium bidding function is B(t) = t. What is ? (To find this write an equation for expected payoff for player 1 if he has signal t1 but bids as if he received a signal r, and then maximize and find the that makes it so that the maximum is at r = t1.)

(d) If the auction is an all-pay auction, the equilibrium bidding function is B(t) = tn. What is ?