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.
(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
(positive or negative), then the total utility of agent
A mechanism is a pair of functions: The agents' types are independent identically-distributed random variables.
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.
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.
wins the auction, and pays the second highest bid, or
If this were true, then it is easy to see that the expected revenue from this auction is also
is monotone increasing, we verify that this is indeed a maximum point.
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).
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.
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).
Substituting for x in the necessary condition, Note that this differential equation is identical to that for e(v).
We can use revenue equivalence to predict the bidding function of a player in a game.
come from a uniform distribution, we can simplify this to 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
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.
Similarly, we know that the expected payment of player 1 in the second price auction is
, and this must be equal to the expected payment in the all-pay auction, i.e.
Thus, the bidding function for each player in the all-pay auction is
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, the outcome function must be changed.
This changes the Outcome function since now the item is not always given to the highest bidder.
By carefully selecting the reservation price, an auctioneer can get a substantially higher expected revenue.
[1]: 237 The revenue-equivalence theorem breaks in some important cases:[1]: 238–239