Exercise 4 -- Three Problems

1. We are talking about a sealed-bid second-price auction with two bidders. Values are independently and uniformly distributed on [0,1].

Bidder 1 is certain of her value; bidder 2 is completely uncertain of his value.

Find the expected surplus of each bidder if:

(a) Bidder 1 bids truthfully and bidder 2 bids randomly (uniformly on [0,1]).

(b) Bidder 1 bids truthfully and bidder 2 always bids his expected value of 1/2.

(c) How much can bidder 1 hurt bidder 2 in part (b) without hurting herself?

(d) Compare these surpluses with the usual private-value Vickrey auction, where both bidders know their values.

2. Find the equilibrium bidding function for the standard first-price auction where the iid values are exponentially distributed over [0,∞] and there are two bidders. That is, the cdf of the value distribution is F(v) = 1 - e ^{-λ v}, where v can be any nonnegative real number and λ > 0 . Sketch and interpret.

3. Consider the standard first-price auction with n bidders where the iid values are distributed on [0,1] according to a general cdf F. As usual, call the equilibrium bidding function b. Suppose bidder 1 bids b(z) when her value is actually v. We know that the derivative of her expected surplus with respect to z at z = v is zero, by definition of equilibrium. Check this from the known equilibrium bidding function b. Then show that the second derivative of her expected surplus with respect to z at z = v is negative. What does this mean? What does the magnitude of this second derivative mean? When will this magnitude be very small and what might you expect in that case?