Revenue equivalence theorem

As revenue equivalence (revenue equivalence theorem) is called a key result of the auction theory . In a nutshell, it means that the expected proceeds of the seller (and also that of the bidders) are always identical across a whole class of auction formats under certain standard conditions. The theorem goes back to the American economist William Vickrey (1961) and generalizations of the results by Roger Myerson (1981) and John Riley and William Samuelson (1981).

presentation

Let the estimates of all n bidders be a random variable independently and identically distributed (iid) and let the bidders all be risk-neutral . Now consider any two auction formats A ₁ and A ₂ that meet these requirements, have a symmetrical equilibrium and also meet the following conditions: On the one hand, the bidder who makes the highest bid wins the auction and, on the other hand, the expected revenue is one The bidder with the lowest possible rating (usually 0 ) is identical in A ₁ and A ₂ . Then each bidder in may A ₁ and A ₂ expect the same revenue and the seller / auctioneer according to the same in both formats expected profit.

Proof of the auction of an object

Either by , ( given an auction, the symmetrical balance: maximum appreciation). Furthermore, one describes the expected costs resulting in equilibrium for a bidder with an appreciation of v . It is and is strictly monotonously increasing in its argument (higher values → higher commandments). ${\ displaystyle \ beta (v)}$ ${\ displaystyle \ beta: [0, {\ bar {v}}] \ rightarrow \ mathbb {R} _ {+}}$ ${\ displaystyle {\ bar {v}}}$ ${\ displaystyle m (v)}$ ${\ displaystyle m (0) = 0}$ ${\ displaystyle \ beta (v)}$

Assume now that all bidders play the equilibrium strategy (symmetrical equilibrium). Now consider player i . Imagine that i now does not necessarily offer the optimal bid for its appreciation (which we denote by v _i ) , but instead some offer . As usual, let Y ₁ denote the highest valuation of all other bidders. i wins with his bid if and only if , which in turn implies because of the monotony of the bidding function . Let G (z) be the distribution function of Y ₁ . Then the expected profit for i is even : With probability G (z) the highest bid of the others is actually lower (then he wins and realizes his appreciation v _i ), but in any case the expected costs m (z) arise. ${\ displaystyle j = 1, \ ldots, i-1, i + 1, \ ldots n}$ ${\ displaystyle \ beta}$ ${\ displaystyle \ beta (v_ {i})}$ ${\ displaystyle \ beta (z)}$ ${\ displaystyle \ beta (z)> \ beta (Y_ {1})}$ ${\ displaystyle z> Y_ {1}}$ ${\ displaystyle \ Pi (z, v_ {i}) = G (z) v_ {i} -m (z)}$

Maximizing the gain function with respect to z yields the first order condition

{\ displaystyle {\ frac {\ partial \ Pi (z, v)} {\ partial z}} = G '(z) v_ {i} -m' (z) = 0 \ Leftrightarrow G '(z) v_ { i} = m '(z)}

It is referred to as the density function . In the optimum i should just put, from which it follows that there . Since the equation will be used as an integral limit in the next step, we will use the variable y instead of v _i in the following . Integrate on each side delivers ${\ displaystyle G '(z) =: g (z)}$ ${\ displaystyle z = v_ {i}}$ ${\ displaystyle g (v_ {i}) v_ {i} = m '(v_ {i})}$

{\ displaystyle m (0) + \ int _ {0} ^ {v_ {i}} yg (y) \ mathrm {d} y = m (v_ {i})}

However, according to the laws of calculating with conditional probabilities, the integral is exactly equal to the antiderivative of the density function (i.e. the distribution function) evaluated at the upper integration limit times the conditional expected value (given that Y ₁ <v _i ) and thus

{\ displaystyle m (v_ {i}) = G (v_ {i}) \ cdot \ mathbb {E} (Y_ {1} | Y_ {1} <v_ {i})}

,

which concludes the proof, since the right-hand side is only dependent on one's own appreciation and the distribution of the highest appreciation of the other bidders, but regardless of the auction format used.

literature

Vijay Krishna: Auction Theory. 2nd ed. Academic Press, San Diego et al. a. 2010, ISBN 978-0-12-374507-1 .

Remarks

^ William Vickrey: Counterspeculation, Auctions, and Competitive Sealed Tenders. In: Journal of Finance. 16, No. 1, 1961, pp. 8-37, doi : 10.1111 / j.1540-6261.1961.tb02789.x (also JSTOR 2977633 ).
^ Roger B. Myerson: Optimal Auction Design. In: Mathematics of Operation Research. 6, No. 1, 1981, pp. 58-73 ( JSTOR 3689266 ).
^ John G. Riley and William F. Samuelson: Optimal Auctions. In: The American Economic Review. 71, No. 3, 1981, pp. 381-392 ( JSTOR 1802786 , EBSCOhost ).
↑ Note: It is not necessary that he has to pay this at the end as in a first price auction.
↑ See also Lawrence M. Ausubel: Auctions (theory). In: Steven N. Durlauf and Lawrence E. Blume (Eds.): The New Palgrave Dictionary of Economics. 2nd Edition. Palgrave Macmillan 2008, Internet http://www.dictionaryofeconomics.com/article?id=pde2008_A000217 (online edition).
↑ The following proof follows Krishna 2010, p. 27 f.

[1] William Vickrey: Counterspeculation, Auctions, and Competitive Sealed Tenders. In: Journal of Finance. 16, No. 1, 1961, pp. 8-37, doi : 10.1111 / j.1540-6261.1961.tb02789.x (also JSTOR 2977633 ).

[2] Roger B. Myerson: Optimal Auction Design. In: Mathematics of Operation Research. 6, No. 1, 1981, pp. 58-73 ( JSTOR 3689266 ).

[3] John G. Riley and William F. Samuelson: Optimal Auctions. In: The American Economic Review. 71, No. 3, 1981, pp. 381-392 ( JSTOR 1802786 , EBSCOhost ).

[4] Note: It is not necessary that he has to pay this at the end as in a first price auction.

[5] See also Lawrence M. Ausubel: Auctions (theory). In: Steven N. Durlauf and Lawrence E. Blume (Eds.): The New Palgrave Dictionary of Economics. 2nd Edition. Palgrave Macmillan 2008, Internet http://www.dictionaryofeconomics.com/article?id=pde2008_A000217 (online edition).

[6] The following proof follows Krishna 2010, p. 27 f.