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.
Contents
Notation
There is a set
There are
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
The vector of all value-functions is denoted by
For every agent
A mechanism is a pair of functions:
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 (BNIC) 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 BNIC mechanisms, if:
then:
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 BNIC; second-price auction is dominant-strategy-incentive-compatible (DSIC), which is even stronger than BNIC. The two mechanisms fulfill the conditions of the theorem because:
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.
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
Suppose
First Price Auction
In the first price auction, where the player with the highest bid simply pays her bid, if all players bid using a bidding function
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
Proof
To prove this, suppose that a player 1 bids
The probability of winning is then
Let
Using the general fact that
Taking derivatives with respect to
Thus bidding with your value
English auction
In the open ascending price auction (aka English auction), a buyer’s dominant strategy is to remain in the auction until the asking price is equal to his value. Then, if he is the last one remaining in the arena, he wins and pays the second-highest bid.
Consider the case of two buyers, each with a value that is an independent draw from a distribution with support [0,1], cumulative distribution function F(v) and probability density function f(v). If buyers behave according to their dominant strategies, then a buyer with value v wins if his opponent’s value x is lower. Thus his win probability is
and his expected payment is
The expected payment conditional upon winning is therefore
Multiplying both sides by F(v) and differentiating by v yields the following differential equation for e(v).
Rearranging this equation,
Let B(v) be the equilibrium bid function in the sealed first-price auction. We establish revenue equivalence by showing that B(v)=e(v), that is, the equilibrium payment by the winner in one auction is equal to the equilibrium expected payment by the winner in the other.
Suppose that a buyer has value v and bids b. His opponent bids according to the equilibrium bidding strategy. The support of the opponent’s bid distribution is [0,B(1)]. Thus any bid of at least B(1) wins with probability 1. Therefore, the best bid b lies in the interval [0,B(1)] and so we can write this bid as b = B(x) where x lies in [0,1]. If the opponent has value y he bids B(y). Therefore, the win probability is
The buyer’s expected payoff is his win probability times his net gain if he wins, that is,
Differentiating, the necessary condition for a maximum is
That is if B(x) is the buyer’s best response it must satisfy this first order condition. Finally we note that for B(v) to be the equilibrium bid function, the buyer’s best response must be B(v). Thus x=v. Substituting for x in the necessary condition,
Note that this differential equation is identical to that for e(v). Since e(0)=B(0)=0 it follows that
Using Revenue Equivalence to Predict Bidding Functions
We can use revenue equivalence to predict the bidding function of a player in a game. Consider the two player version of the second price auction and the first price auction, where each player's value is drawn uniformly from
Second Price Auction
The expected payment of the first player in the second price auction can be computed as follows:
Since players bid truthfully in a second price auction, we can replace all prices with players' values. If player 1 wins, he pays what player 2 bids, or
Since
First Price Auction
We can use revenue equivalence to generate the correct symmetric bidding function in the first price auction. Suppose that in the first price auction, each player has the bidding function
The expected payment of player 1 in this game is then
Now, a player simply pays what the player bids, and let's assume that players with higher values still win, so that the probability of winning is simply a player's value, as in the second price auction. We will later show that this assumption was correct. Again, a player pays nothing if he loses the auction. We then obtain
By the Revenue Equivalence principle, we can equate this expression to the revenue of the second-price auction that we calculated above:
From this, we can infer the bidding function:
Note that with this bidding function, the player with the higher value still wins. We can show that this is the correct equilibrium bidding function in an additional way, by thinking about how a player should maximize his bid given that all other players are bidding using this bidding function. See the page on First-price sealed-bid auction.
All-pay Auctions
Similarly, we know that the expected payment of player 1 in the second price auction is
Thus, the bidding function for each player in the all-pay auction is
Implications
An important implication of the theorem is that any single-item auction which unconditionally gives the item to the highest bidder is going to have the same expected revenue. This means that, if we want to increase the auctioneer's revenue, we must change the Outcome function. One way to do this is to set a Reservation price on the item. This changes the Outcome function since now the item is not always given to the highest bidder. Interestingly, by carefully selecting the reservation price, an auctioneer can get a substantially higher expected revenue.
Limitations
The revenue-equivalence theorem breaks in some important cases: