Revenue equivalence

As revenue equivalence theorem ( revenue equivalence theorem ) is called a central result from the auction theory. It states shortened, that the expected revenue of the seller (and the tenderer ) over a whole class of auction formats of time under certain standard conditions is always the same. The theorem can be traced back to the American economist William Vickrey (1961) and generalization of the results by Roger Myerson ( 1981) and John Riley and William Samuelson (1981).

Representation

Let be a random variable, the value estimates of all bidders n independent and identically distributed ( iid) and are all the bidders are risk neutral. Now consider any two auction formats A1 and A2 which satisfy these conditions, have a symmetrical balance and also meet the following conditions: Firstly, one bidder wins the auction, makes the highest bid, and on the other is the expected proceeds of a bidder with the lowest possible value estimate (usually 0 ) is identical to A1 and A2. Then all tenderers to be in A1 and A2 expect the same revenue and the seller / auctioneer has accordingly in both formats the same expected profit.

Evidence for the auction of an object

Be by, (: maximum appreciation ), where the symmetric equilibrium auction. Next we denote by its perfect balance resulting expected cost of a bidder with appreciation v. It is and is strictly increasing in its argument (higher value estimates → higher bids ).

Now assume that all bidders play the equilibrium strategy ( symmetric equilibrium). Consider now the player i Just imagine, i now offers not necessary for the his appreciation ( which we denote by vi) optimal bid, but instead either. Now let, as usual, with Y1 denotes the highest esteem of all other bidders. i won with his bid if and only if, which implies because of the monotonicity of the bidding function. Is more G ( z) is the distribution function of Y1. Then the expected profit of i is just: With probability G (z) is the highest bid of the other actually lower ( then he wins and implements his appreciation vi), but in any case fall on the expected cost of m ( z).

Maximize the profit function with respect to z yields the first-order condition

This is referred to as a density function. The optimum i should just set, it follows that there. Since the equation is already in use in the next step as an integral limit, we use instead of vi in the following part, the variable name y. Integrate delivers on every page

The integral is but according to the laws governing the computation with conditional probabilities just equal to the primitive function of the density function ( ie, the distribution function ) evaluated at the upper limit of integration times the conditional expectation ( given that Y1 < vi) and thus

Which completes the proof, since the right-hand side depends only on its own appreciation and the distribution of the highest appreciation of the other bidders, but regardless of the auction format used.