Consider an auction for a single good, with two bidders, each with a private value drawn independently from the uniform distribution on [0, 100]. The seller decides to hold a first-price sealed-bid auction, but bids are constrained to be taken from the set {0, 25, 50, 75}. If both bid 0, nobody will be awarded the object; otherwise if they bid the same, the winner will be chosen at random (with equal probabilities).
A. Show that in any equilibrium, neither bidder bids higher than his value. (Recall that weakly dominated strategies may be played in equilibrium, but that strategies which do not survive iterated elimination of strictly dominated strategies may not.)
B. Show that in any equilibrium, nobody bids 75.
C. Show that in any equilibrium, nobody with value above 25 bids 0.
D. Write down the symmetric equilibrium bidding function of this auction.