DOI: 10.1007/s00199-004-0549-6Economic Theory 26, 983–987 (2005)

Exposita Notes

Asymptotic pricesin uniform-price multi-unit auctions�

Indranil Chakraborty1 and Richard Engelbrecht-Wiggans2

1 Department of Economics, University of Oklahoma, Norman, OK 73019, USA(e-mail: indro@ou.edu)

2 Department of Business Administration, University of Illinois at Urbana-Champaign,Champaign, IL 61820, USA (e-mail: eplus17@uiuc.edu)

Received: April 26, 2004; revised version: August 3, 2004

Summary. This paper considers a uniform-price auction in which each of n sym-metric bidders can place, say, M bids. Each bidder has privately known, decreas-ing marginal values from an arbitrary M -dimensional distribution. We providea quantile-type description of the asymptotic price that appropriately generalizesthe characterization of the unit-demand asymptotic price. Specifically, the limitingprice equals the (1 − α)-th quantile of the “average” of the marginal distributionsif a fraction α of the demand is met asymptotically. The result also implies that theexpected price in the limit as n becomes large depends only on the aggregate of themarginal distributions of each bidder’s marginal values (and not on the correlationbetween the marginal values).

Keywords and Phrases: Multi-unit auctions, Uniform price.

JEL Classification Numbers: D44.

1 Introduction

Prices play a central role in the analysis of auctions. Precise expressions for expectedprices make comparisons of revenues from different auction formats and econo-metric estimation of auction models possible. Moreover, certain characterizationsof prices provide interesting insights into various aspects of auctions.

� We thank George Deltas, N. D. Shyamal Kumar, and Jeroen Swinkels for valuable discussions,and an anonymous referee for helpful comments. Richard Engelbrecht-Wiggans’ research was fundedin part by the National Science Foundation under the NSF projects ECS-0000577 and SES-03-38994.Correspondence to: I. Chakraborty

984 I. Chakraborty and R. Engelbrecht-Wiggans

On the other hand, auctions play a central role in the analysis of competitivepricing. Agents in large auctions have been often shown to display a price-takingbehavior. A price established in the limit when the demand and supply can bothgrow indefinitely in the auction is then a competitive market-clearing price, aswell. Characterization of this limiting price, therefore, serves as a description ofthe competitive market outcome. Moreover, strategic agent behavior in auctionsprovide interesting insights into various aspects of price formation.

One interesting description of expected prices can be obtained by considering,for example, the standard Vickrey or second-price auction (a single-unit, sealed-bidauction with the price set equal to the second-highest bid) when bidder valuationsare uniformly distributed (F (v) = v

v ) on [0, v]. When there are n bidders, theexpected price n−1

n+1v = F−1(n−1n+1 ) is the n−1

n+1 -th quantile of the value distribution.Such a characterization of the expected price does not hold generally for otherdistributions.

A similar description, again, is obtained asymptotically when the number ofbidders and the number of units is allowed to increase, but each bidder may stillwin at most one unit. In particular, suppose {An}∞

n=1 is a sequence of auctions withauction An having n (symmetric and unit-demand) bidders and Kn units on sale.Consider the uniform-price generalization of the second-price auction where eachbidder submits a bid, the highest bids are awarded the units, and the price in theauction is set equal to the highest losing bid. The price in the auction converges to the(1−α)-th quantile F−1(1−α) of the value distribution where α = limn→∞ Kn

n ∈(0, 1), whenever the limit exists. This description of prices is intuitively easy: Asthe auction becomes large, the empirical distribution of the bidder valuations isexpected to resemble F (·) more and more closely. The (1 − α)-th quantile of theempirical distribution which is roughly equal to the price then converges to thecorresponding quantile of the distribution F (·).1 In the context of an infinitelylarge market (say, a continuum between 0 and 1), where the quantity demandedat price p (by price-taking buyers) is 1 − F (p) and a fixed quantity α is suppliedthen the market clearing price p∗ is given by equating the demand and supply atp∗ = F−1(1 − α) which is precisely the above limiting price from the auction.

Neither the result nor the intuition extend in any straightforward way to auctionswhere bidders (have values for, and) can submit bids on more than one unit. The firstdifficulty is that while the highest bids are still awarded the units and the price is stillset by the highest losing bid, bidders do not bid the value of the units (except on thefirst unit) in uniform-price auctions (see, for instance, Engelbrecht-Wiggans andKahn, 1998). However, the problem is resolved by the fact that bidders tend to bidtheir actual values asymptotically (see Swinkels, 1997, 2001). The next difficultyis that the value distributions are multivariate in nature, and although due to theabove result the price tends to be the highest losing value, this value is the Kn +1-sthighest value out of a pool of n independent M -vectors. While this price is knownto converge to a limit, there is no known result that can express this limiting pricein a way similar to the unit demand case.

1 See also Arnold, Balakrishnan and Nagaraja (1992).

Asymptotic prices in uniform-price multi-unit auctions 985

Nonetheless, one could still hope to extend the asymptotic characterization ofthe price in uniform-price auctions. Consider the special case where each biddercan bid on up to M units and the bidder’s marginal value of the m-th unit is them-th highest order statistic from M independent draws from F (·). Suppose thatin the limit the auctioneer meets a fraction α of the total demand nM . It is notdifficult to see that in this case the value corresponding to the price setting bid isthe αnM + 1-st highest order statistic from nM independent draws from F (·).This price, in the limit, converges to the (1 − α)-th quantile F−1(1 − α) of thedistribution. Unfortunately, this line of argument is limited only to this specialexample, and does not extend to the general value-distribution setting.

We show below that in the general setting where the vector of diminishingmarginal values v of each bidder has a multivariate distribution F (·) the priceconverges to the (1−α)-th quantile for the “average” of the multivariate distribution.Specifically, if Fr(·) denotes the marginal distribution of the marginal value vr forthe r-th unit then the price in the auction converges to the (1 − α)-th quantile( 1

M

∑Mr=1 Fr)−1(1 − α) of the average marginal distribution 1

M

∑Mr=1 Fr(·).

2 Asymptotic price

Suppose that auction An has n symmetric bidders and Kn units with lim Kn

nM =α ∈ (0, 1). Thus a proportion α of the total demand is met asymptotically. Eachbidder has diminishing (nonnegative) marginal values for M units of the item witha joint distribution F (·) on a support V = [v = (v1, .., vM ) ∈ [0, v]M |v1 ≥ v2 ≥· · · ≥ vM ]. The price in an auction is set to be the Kn + 1-st bid in the auction.2

Swinkels (1997, 2001) showed that the relevant bids in the auction converge tothe corresponding values, hence the price is the Kn + 1-st order statistic of thevalues in the limit. Moreover, the distribution of this order statistic tends to becomedegenerate in the limit. Thus the problem is really to find the probability limit ofthe Kn + 1-st value.

Denote by Fr(·) the marginal distribution of the r-th highest value (i.e., themarginal value of a bidder’s r-th unit) of a bidder and define x∗ implicitly by thefollowing equation

M∑r=0

Fr(x∗) = (1 − α)M .

(It is straightforward to check that there exists a unique x∗ satisfying this equation.)Suppose that the nM bidder values, M from each of the n bidders, are ranked. LetF̃r:nM (·) denote the distribution of the r-th highest of these nM values.

Proposition. For any fixed integer r, we have

limn→∞ F̃Kn+1:Mn(x) = 1 if x > x∗

= 0 if x < x∗

2 Our result, in fact, generalizes straightforwardly to a more general uniform-price rule.

986 I. Chakraborty and R. Engelbrecht-Wiggans

Proof. In order to prove the result we start with some preliminary notations. Fora given x let {Xl(x)}∞

l=1 be an i.i.d. sequence of random variables on the integersupport {0, 1, ..., M} with probability distribution

P [Xl(x) = k] = pk(x)

where pk(x) = FM−k+1(x) − FM−k(x), and by convention FM+1(x) ≡ 1 andF0(x) ≡ 0. (Thus Xl(x) = k is the event that bidder l has exactly M − k valuesabove x.)

Define Sn(x) �∑n

j=1 Xj(x) and S̄n(x) � 1nSn(x). We have

F̃Kn+1:Mn(x) = P [Sn(x) ≥ Mn − Kn]

= P [S̄n(x) ≥ (1 − α)M + αM − Kn

n].

Also,

E[X1(x)] =M∑

j=0

jpj(x)

=M∑

j=0

j(FM−j+1(x) − FM−j(x))

=M∑

r=1

Fr(x)

Recall that∑M

r=1 Fr(x) < or > (1 − α)M depending on whether x < or > x∗.

Therefore, for x < x∗, defining δx ≡ (1−α)M−E[X1(x)]2 we have

P [S̄n(x) ≥ (1 − α)M + αM − Kn

n]

≤ P [|S̄n(x) − E[X1]| > δx]

for all large n since αM − Kn

n → 0 as n → ∞. Finally, P [|S̄n(x) − E[X1]| >δx] → 0 as n −→ ∞ by the Weak Law of Large Numbers (WLLN).3

For x > x∗, defining δx ≡ E[X1]−(1−α)M2 we have for all large n (so that

αM − Kn

n is small)

P [S̄n(x) ≥ (1 − α)M + αM − Kn

n]

≥ P [|S̄n(x) − E[X1]| ≤ δx] −→ 1

as n −→ ∞, again, by the WLLN. ��It follows that in the uniform-price auction where each bidder bids on M

units the price in the auction converges to the (1 − α)-th quantile x∗ =(1M

∑Mr=0 Fr

)−1(1 − α) of the average of the marginal distributions where α

is the proportion of the total demand met in the limit.3 See Billingsley (1995).

Asymptotic prices in uniform-price multi-unit auctions 987

3 Concluding remarks

We have expressed the limiting price in the multi-unit uniform-price auctions interms of a quantile-type expression. This is also an interesting description of thecompetitive market-clearing price when buyers have multi-unit demands. The de-scription involves simply averaging of the marginal distributions of the multivariatepopulation from which the values are drawn. Our observation takes a special sig-nificance since it means that given the marginal distributions, the asymptotic pricedoes not depend on the degree of correlation between the marginal valuations of abuyer. The result also has econometric implications. It means that price data fromauctions with large participations may be limited in their ability to identify theprimitives of the auction. Thus an identification of the auction model may have aminimum requirement of information on the individual bids.

References

Arnold, B.C., Balakrishnan, N., Nagaraja, H.N.: A first course in order statistics. NewYork: Wiley 1992Billingsley, P.: Probability and measure, 3rd edn. New York: Wiley 1995Deltas, G.: Asymptotic and small sample analysis of the stochastic properties and certainty equivalents

of winning bids in independent private value auctions. Economic Theory 23, 715–738 (2004)Engelbrecht-Wiggans, R., Kahn, C.M.: Multi-unit auctions with uniform prices. Economic Theory 12,

227–258 (1998)Swinkels, J.: Asymptotic efficiency for uniform private value auctions. Manuscript, Washington Uni-

versity, St. Louis (1997)Swinkels, J.: Efficiency of large private value auctions. Econometrica 69, 37–68 (2001)