Reveue Equivalece Theorem Felix Muoz-Garcia Advaced Microecoomics II Washigto State Uiversity
So far, several di eret auctio types have bee cosidered. The questio remais: How does the expected reveue for the seller vary across the di eret types of auctio? Vickrey (1961) ad Myerso (1981) were about to prove what is ow kow as the reveue equivalece theorem, statig that uder certai, geeral coditios, all auctio types produce the same expected reveue.
Four coditios for reveue equivalece 1 Each bidder s type is draw from a "well behaved" distributio. (The CDF must be strictly icreasig ad cotiuous) 2 Bidders are risk eutral. 3 The bidder with the highest type wis. 4 The bidder with the lowest possible type (θ) has a expected payo of zero.
Before we start a example, a bit of backgroud. Imagie that we are drawig a series of idepedet realizatios from some distributio F (.). Give the sample of draws (x 1, x 2,..., x ), we ca cosider the rakig of the realized values ad rak them from highest to lowest. The highest draw would be x [1] = maxfx 1, x 2,..., x g, also kow as the rst-order statistic. Similarly, the secod highest draw, x [2] would be the secod order statistic, ad so o.
Let s see if the expected reveue from a rst-price ad secod-price auctio are equivalet. Startig with the rst-price auctio, we ll assume that there are symmetric bidders whose types are draw from a uiform [0, 1] distributio. All bidders use the same biddig strategy s(θ). This implies that the reveue received by the seller would be the rst-order statistic of some biddig distributio G (.). Hece, the expected reveue of the seller is E (b [1] ), which is the expected value of the rst-order statistic of the bid fuctio.
As previously show, the symmetric Bayesia Nash equilibrium with bidders whose types are uiformly distributed over a iterval [θ, θ] is give by s(θ) = θ 1 (Bidders all bid a share 1 of their actual valuatio, θ). The wiig bid will be θ1 ( 1) max, θ 2( 1),..., θ ( 1), which is just the rst-order statistic from draws o the uiform distributio [0, 1 ].
We the eed to compare: FPA: The expected reveue from the rst-order statistic of the biddig fuctio, i.e., the rst-order statistic from draws o the uiform distributio [0, 1 ], agaist SPA: The expected reveue from the secod-order statistic of the value fuctio, i.e., the secod-order statistic from draws o the uiform distributio [0, 1].
We ll eed to derive the CDF of the rst-order statistic for the uiform distributio i order to calculate the expected value. F [1] (x) is equal to the probability that all the bids that were draw are less tha or equal to x, ad hece F [1] (x) = Prfmaxfb 1, b 2,..., b g xg = (F (x)) Ituitively, the last equality follows from the fact that values are draw idepedetly. For the uiform distributio, the bid of each aget b i is uiformly draw from [0, 1 ], so that F (b i ) = 1 b i.
Thus, the distributio of the rst-order statistic is F [1] x (x) = [F (x)] = for 0 x 1 1 We ca obtai the PDF by di eretiatig with respect to x f [1] (x) = x 1 1 1 = x x 1
Lastly, we ca calculate the expected rst-order statistic from Bayesia Nash equilibrium bids i a rst-price auctio E (b [1] ) = = Z 1 0 + 1 Z xf [1] 1 x (x)dx = 0 1 1 x +1 1 0 = 1 + 1 dx Thus, the seller s expected reveue from the rst-price sealed-bid auctio with bidders whose valuatios are idepedetly draw from the uiform distributio over [0, 1] is equal to 1 +1.
Now let s do the same thig for a secod-price auctio. I this case, all bidders will bid their valuatio, ad thus the expected reveue will be equal to the expected secod-order statistic of a series of draws from the uiform distributio [0, 1] (Sice the wier pays the secod highest bid). Let s derive the distributio agai.
The CDF follows F [2] (x) = Prfx [2] xg. The evet fx [2] xg ca occur i oe of two distict (mutually exclusive) ways. I the rst evet, all the draws x i are below x, or x i x for all i = 1, 2,...,. I the secod evet, for oly oe draw j, x j > x. 1 of the draws, x i x ad for This evet ca occur i di eret ways: (1) x 1 > x ad for all i 6= 1, x i x, (2) x 2 > x ad for all i 6= 2, x i x, ad so o up to.
We ca therefore de e the CDF as follows F [2] (x) = First Evet z } { Prfmaxfb 1, b 2,..., b g xg + i=1 Prfx i > x ad for all j 6= i, x j xg {z } = [F (x)] + i=1 Secod Evet (1 F (x))[f (x)] 1
Rearragig F [2] (x) = [F (x)] + (1 F (x))[f (x)] 1 = [F (x)] 1 ( 1)[F (x)]
The bid of each aget b i i the secod-price sealed-bid auctio is his valuatio, which we assumed is uiformly draw from [0, 1], so that F (b i ) = b i, ad we obtai F [2] (x) = [F (x)] 1 ( 1)[F (x)] = x 1 ( 1)x Like before, we obtai the PDF by di eretiatig with respect to x, obtaiig f [2] (x) = ( 1)x 2 ( 1)x 1
Lastly, the expected secod-order statistic is E (θ [2] ) = = Z 1 Z 1 xf [2] dx = [( 1)x 1 ( 1)x ]dx 0 0 ( 1)x ( 1)x +1 1 + 1 = = ( 1) ( 1) + 1 = 1 + 1 0
Thus, the expected reveue for the seller is the same i either a rst or secod-price sealed bid auctio. Books by Krisha (2002) ad Milgrom (2004) both explore this topic i greater detail. You ca try the same process with other types of auctio.