Students / Subjects

# Revenue Equivalence for One-sided Markets

A question commonly addressed in the economic analysis of auctions is whether any two auction mechanisms are "revenue equivalent". Two auctions are said to be "revenue equivalent" if they result in the same expected sales price.

This is an important issue to a seller who wants to hold an auction to sell her item for the highest possible price. If one type of auction is found to generate higher average sales revenue, then that auction type will obviously be preferred by the seller.

The revenue equivalence theorem states that,if all bidders are risk-neutral bidder and have independent private value for the auctioned items, then all four of the standard single unit auctions have the same expected sales price (or seller's revenue).The four standard single unit auctions are the English auction, the Dutch auction,first-price sealed-bid auction,and the second-price sealed-bid auction.

## A more formal description of revenue equivalence

Paul Klemperer gives the following more formal statement (and a complete treatment in Appendix A) in his paper "Auction Theory: A Guide to the Literature". (Journal of Economic Surveys v13, n3 (July 1999): 227-86):

Assume each of a given number of risk-neutral potential buyers of an object has a privately-known signal independently drawn from a common strictly-increasing, atomless distribution. Then any auction mechanism in which

1. the object always goes to the buyer with the highest signal, and
2. any bidder with the lowest-feasible signal expects zero surplus

yields the same expected revenue (and results in each bidder making the same expected payment as a function of her signal).

The theorem applies very broadly to auction types, beyond the English, Dutch, First- and Second-Price auctions, to include nearly any "reasonable" auction mechanism, and even some peculiar mechanisms such as the All-Pay auction.

Klemperer also points out that the theorem can apply in common value auctions when the bidders' signals are independent, and that, "The theorem extends to the case of k > 1 indivisible objects being sold, provided bidders want no more than one object each; all auctions that give the objects to the k highest-value bidders are revenue equivalent."

## research focus on the issue of revenue equivalence

Tests of the revenue equivalence theorem involve two separate issues. The more basic issue concerns the strategic equivalence (or isomorphism) of (a) the first-price and Dutch auctions and (b) the second-price and English auctions. Strategic equivalence of the auctions in the pair(s)have the same revenue irrespective of bidders' risk attitudes.Assuming that strategic equivalence is satisfied for both auction pairs, a second issue concerns the possible revenue equivalence between auction pairs (a) and (b)