Exercise 4 -- Three Problems

You may discuss this assignment with classmates, but all your code and written work must be your own. Reference all sources. Please hand in hard copy.

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?