Question

What does the revenue equivalence theorem say? Given an example to illustrate the revenue equivalence theorem.

Answer 1

The revenue equivalence theorem states that for certain economic environments, the expected revenue and bidder profits for a broad class of auctions will be the same provided that bidders use equilibrium strategies.

The Revenue Equivalence Theorem

Theorem 1

If there are two bidders with values drawn from U[0,1], then any standard auction has an expected revenue 1/3 and gives a bidder with value v and expected profit of v^2 / 2, the same as the second-price auction.

Theorem 2

If there are N bidders with values drawn from a continuous distribution (e.g. uniform on [a, b]), then any standard auction leads to the same expected revenue, and same expected bidder profit, as a second-price auction.

Example: Second Price Auction

The second price auction is a standard auction. The payment rule has t(bi,bj ) equal to zero if bi < bj , and equal to bj if bi > bj . In equilibrium, each bidder bids his value, so the equilibrium strategy is b(v) = v. In equilibrium the bidder with the higher value will win, and pay the bid (or equivalently value) of the lower-valued bidder. The expected revenue is 1/3. Why 1/3? If we take two draws from a uniform distribution on [0,1], the higher draw will be on average 2/3 and the lower draw will be on average 1/3. Notice also that if a bidder has value v, he expects to win whenever the other bidder has a value less than v; which happens with probability equal to v. If he does win, he expects to pay v/2. So the expected profit of a bidder with value v is U(v) = v.(v - v/2) = v ^2 / 2.