Revenue equivalence is a concept in auction theory that states that given certain conditions, any mechanism that results in the same outcomes (i.e. allocates items to the same bidders) also has the same expected revenue.
Notation
There is a set of possible outcomes.
There are agents which have different valuations for each outcome. The valuation of agent (also called its "type") is represented as a function:
which expresses the value it has for each alternative, in monetary terms.
The agents have quasilinear utility functions; this means that, if the outcome is and in addition the agent receives a payment (positive or negative), then the total utility of agent is:
The vector of all value-functions is denoted by .
For every agent , the vector of all value-functions of the other agents is denoted by . So .
A mechanism is a pair of functions: An function, that takes as input the value-vector and returns an outcome (it is also called a social choice function); A function, that takes as input the value-vector and returns a vector of payments, , determining how much each player should receive (a negative payment means that the player should pay a positive amount).
The agents' types are independent identically-distributed random variables. Thus, a mechanism induces a Bayesian game in which a player's strategy is his reported type as a function of his true type. A mechanism is said to be Bayesian-Nash incentive compatible if there is a Bayesian Nash equilibrium in which all players report their true type.
Statement
Under these assumptions, the revenue equivalence theorem then says the following.
For any two Bayesian-Nash incentive compatible mechanisms, if: The function is the same in both mechanisms, and: For some type , the expected payment of player (averaged on the types of the other players) is the same in both mechanisms; The valuation of each player is drawn from a path-connected set,
then: The expected payments of all types are the same in both mechanisms, and hence: The expected revenue (- sum of payments) is the same in both mechanisms.
Example
A classic example is the pair of auction mechanisms: first price auction and second price auction. First-price auction has a variant which is Bayesian-Nash incentive compatible; second-price auction is dominant-strategy-incentive-compatible, which is even stronger than Bayesian-Nash incentive compatible. The two mechanisms fulfill the conditions of the theorem because: The function is the same in both mechanisms - the highest bidder wins the item; and: A player who values the item as 0 always pays 0 in both mechanisms.
Indeed, the expected payment for each player is the same in both auctions, and the auctioneer's revenue is the same; see the page on first-price sealed-bid auction for details.
Equivalence of auction mechanisms in single item auctions
In fact, we can use revenue equivalence to prove that many types of auctions are revenue equivalent. For example, the first price auction, second price auction, and the all-pay auction are all revenue equivalent when the bidders are symmetric (that is, their valuations are independent and identically distributed).
Second price auction
Consider the second price single item auction, in which the player with the highest bid pays the second highest bid. It is optimal for each player to bid its own value .
Suppose wins the auction, and pays the second highest bid, or . The revenue from this auction is simply .
First price auction
In the first price auction, where the player with the highest bid simply pays its bid, if all players bid using a bidding function this is a Nash equilibrium.
In other words, if each player bids such that they bid the expected value of second highest bid, assuming that theirs was the highest, then no player has any incentive to deviate. If this were true, then it is easy to see that the expected revenue from this auction is also if wins the auction.
Proof
To prove this, suppose that a player 1 bids where , effectively bluffing that its value is rather than . We want to find a value of such that the player's expected payoff is maximized.
The probability of winning is then . The expected cost of this bid is . Then a player's expected payoff is
Let , a random variable. Then we can rewrite the above as
.
Using the general fact that , we can rewrite the above as
.
Taking derivatives with respect to , we obtain
.
Thus bidding with your value maximizes the player's expected payoff. Since is monotone increasing, we verify that this is indeed a maximum point.