Final Homework — Auction Theory (96573)
Note: No extensions will be given. I advise to start working on this as early as possible.
Question 1 (30 points)
In this question we will examine the revenue equivalence theorem, and its implications,
from a different perspective. Suppose we have risk-neutral players, with private values that
are drawn i.i.d. from some cumulative distribution F . We will consider a family F of auctions
in which each player bids a number, and the player with the highest bid wins. (for example
first-price and second-price are both such auctions).
Fix any such auction, A, with symmetric and non-decreasing equilibrium strategies b(x).
Let mA(z) denote the expected payment of a player that bids b(z), assuming all other players
play according to b() (the expectation is taken over the values of the other players). Let
Gi(y) = Pr(maxj
6=ivj ≤ y), i.e. the cumulative distribution to the highest value besides vi.
Since all values are drawn from the same distribution F , all the Gi’s are identical, and we
will denote Gi(y) = G(y) and g(y) = G
′(y).
1. Show that the fact that b() is an equilibrium implies that m′A(z) = g(z) · z. Guidance:
start by writing a formula to the expected utility, uA(x, z), for a player with value x
that bids b(z). Then, the equilibrium property implies that uA(x, z) is maximized for
z = x, and this will give you the result. Explain in details your answer.
2. Assume that the expected payment of a player with value zero is zero. Based on your
previous answer, what is the expected payment of a player with positive value? Prove
your answer.
3. In an all-pay auction, each player submits a bid, the player with the highest bid wins,
and every player pays her bid (both the winner and the losers). Notice that an all-pay
auction belongs to the family F . What will be the equilibrium strategy b(x) for this
auction? Hint: remember that a player always pays her bid.
4. Suppose that the probability distribution F is the uniform distribution on [0, 1]. What
will be the function b(x) from t