# Revenue equivalence theorem

Revenue equivalence is a concept in auction theory that states that given certain conditions, any auction mechanism that results in the same outcomes (i.e. allocates items to the same bidders) also has the same expected revenue.

An auction is a special case of a mechanism. In this case, the mechanism takes buyers’ bids and decides an outcome of the auction: who gets the object and what are the transfers for each buyer. The set of outcomes can be denoted by

$\{ (x,t)\in \mathbb{R}^I_+ \times \mathbb{R}^I | x_i \in \{0,1\} , \sum_{i=1}^I x_i = 1\}.$

The x component describes the allocation of the object and t the transfers.

The buyer’s types, or valuations of the object, are independent identically distributed random variables. A buyer of type $\theta_i$ has linear utility function ui over the set of outcomes (the theorem also holds for the more general quasilinear utility functions):

$u_i(x, t, \theta_i) = \theta_i x_i + t_i.$

Thus an auction is a Bayesian game in which a player’s strategy is his bid as a function of his type. An auction (more generally, a mechanism) is said to be Bayesian incentive compatible if all players bidding their true type is a Bayesian Nash equilibrium strategy profile.

Under these assumptions, the Revenue Equivalence Theorem then says the following:

Theorem For any two Bayesian incentive compatible auctions, if under their respective Bayesian Nash equilibria where all players bid their type,

1. a buyer of type θi has the same probability of getting the object across auctions, and
2. a buyer of lowest type has the same expected utility across auctions,

then the total expected transfers Eθti), i.e. the auctioneer’s expected revenue, is the same for the two auctions.

In other words, if a buyer of given type has the same expected utility in the two auctions in the interim stage, then the seller’s expected revenues are the same. However, ex post, the two mechanisms need not implement the same social choice functions. Two such examples are the second price auction and first price auction. Assume the types are drawn independently from the uniform distribution on [0,1]. In the second price auction, bidding one’s own type is a dominant strategy, therefore a fortiori the auction is Bayesian incentive compatible. For the first price auction, it can be shown that the bid functions

$b_i(\theta) = \frac{I-1}{I}(\theta^{I-1})$

form a Bayesian Nash equilibrium (a simple argument via the revelation principle shows it can be made Bayesian incentive compatible). Thus the Revenue Equivalence Theorem applies: in both auctions, the highest types get the object and a buyer of type 0 has zero expected interim utility. They do not implement the same social choice functions.