Preferences over valuation distributions in auctions

Economics Letters 68 (2000) 55–59 www.elsevier.com / locate / econbase

Preferences over valuation distributions in auctions David A. Hennessy* Department of Economics, 478 B Heady Hall, Iowa State University, Ames, IA 50011 -1070, USA Received 5 August 1999; accepted 22 December 1999

Abstract Sellers auctioning items use advertisements and location to generate interest. They seek a rightward shift in the ex-ante distribution of bidder valuations. We characterize distribution pairs such that all more-is-better expected utility maximizing sellers in preference-revealing auctions prefer the same distribution.  2000 Elsevier Science S.A. All rights reserved. Keywords: Convex cone; Multivariate distribution; Order statistics; Preference-revealing auctions JEL classification: D44; G12

1. Introduction In preference-revealing auctions, such as the English and Second Price Sealed Bid (SPSB) auctions, a rational bidder will bid her valuation because to do otherwise would reduce the level of private surplus. Suppose a multi-unit auction involves selling s identical items in either an English or a SPSB auction to n interested parties. Each bidder can purchase at most one unit, and all units are auctioned off simultaneously. It is well known that, in these circumstances, the equilibrium bid for a set of rational bidders is the (n 2 s)th smallest valuation. In statistics terminology, this is the (n 2 s)th order statistic in the vector of proffered bids at a preference-revealing auction. For an owner contemplating whether to auction an asset, it is of interest to understand the statistical attributes of this order statistic. Consider a seller, U, in the class, U1 , of von-Neumann and Morgenstern (vN&M) expected utility maximizers possessed of an increasing utility function and seeking to sell one unit of an item. This seller may be deciding between auctioning the item in New York or San Francisco or via an internet site. Alternatively, the problem for U may involve choices in media advertisement of the forthcoming auction. Each choice generates an n-variate distribution of bidder valuations. The optimal decisions *Tel.: 11-515-294-6171; fax: 11-515-294-0221. E-mail address: [email protected] (D.A. Hennessy) 0165-1765 / 00 / $ – see front matter PII: S0165-1765( 00 )00232-9

 2000 Elsevier Science S.A. All rights reserved.

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will be clear if it is believed that the ex-ante random (n21)th order statistic arising from one set of choices dominates the others in the first order sense.1 It is the problem of preparing to auction an asset that we intend to analyze. For the U1 class of decision makers, we will identify a pair of equivalent characterizations of the types of multivariate valuation distribution function shifts that will be preferred by the seller.2

2. Analysis n Define by Ln the set of all random vectors in R¯ 1 . For a given non-negative realization in this random vector space, z [ R¯ n1 , define as z (i ) 5 (z 1 ,z 2 , . . . ,z n ) (i ) the ith order statistic, i.e. the ith smallest ordinate in vector z [ R¯ n1 . We will assume a vector representation of distributions, and upper case Z represents the distribution from which z [ R¯ n1 is drawn. The distribution of the (n 2 s)th order statistic of this multivariate distribution is denoted by Z(n 2s) . n n For a pair of random vectors X [ R¯ 1 and Y [ R¯ 1 , a stochastic partial ordering of concern to us is FSD when X(n 2s) # Y(n 2s) , i.e. when the (n 2 s)th order statistic of distribution Y dominates that of X in the first order sense. Assuming that the expectation exists, let Ehw (z)j represent the expectation of function w ( ? ):R¯ n1 → R with respect to distribution Z. We will identify a convex cone of functions F such that for all w ( ? ) [ F we have [X(n 2s) # FSDY(n 2s) ]⇔[Ehw (x)j # Ehw ( y)j]. Our notation will generally follow that of Scarsini and Shaked (1987) who study a larger class of partially ordered distributions. Define by ct the set of all subsets of h1,2, . . . ,nj that have cardinality t. Choose one subset, I 5 hi 1 ,i 2 , . . . ,i t j [ ct . Let z I 5 hz i 1 ,z i 2 , . . . ,z i t j, and define by I c the complement of I in the universal index set h1,2, . . . ,nj. Therefore, the vector realization z from distribution Z may be partitioned into FSD (z I ,z I c ). In arriving at cone F that defines the order [X(n 2s) # Y(n2s) ], we will need to pay particular attention to realizations where the realizations of z I c are at the subspace boundaries, z I c 5 0 n 2t and z I c 5 ` n2t . Where it is convenient not to explicitly identify vector dimensions, we will denote the n origin for the complement subspace by 0 and the extended largest point by `. For z [ R¯ 1 , define by k [A] the indicator function that equals 1 when z [ A and equals 0 when z [ ⁄ A. For set function ¯ ¯ k [A]:R¯ n1 → h0,1j, the expression k [(z),(z ] )] operates on the rectangular set of all z such that z] # z # z, z] [ R¯ n1 , z] [ R¯ n1 , where order relation # has the usual coordinate-wise interpretation. We denote ¯ When the z I c coordinates of z] recede to 0 with direction of approach Lim z¯ →` k [(z),(z ] n 2t )] by k [(z),(`)]. ] from z I c [ R 1 , then we write k [(z I ,0),(z¯ )]. For s [ h1, . . . ,nj, we use indicator functions to generate p n the function set F s,n with elements h(z; p):R¯ 1 3 R¯ 1 → h0,1j such that

O (21)

Sts 2 1D O k [( pe ,0),(`)].

n

h(z; p) 5

t5s11

S

t 2s21

I

(1)

I [ ct

D

Here, ts 2 1 5 (t 2 1)! / [(s)!(t 2 1 2 s)!]. And all n! / [t!(n 2 t)!] elements of index set ct enter the summation o I [ ct . Vector ]z 5 ( pe I ,0) is a vector with p [ R¯ 1 for each of the ordinates in coordinate set 1

We assume that n is known. A larger problem would be to randomize the number of bidders as well as the draws from bidder valuation distributions. We also ignore disparities in taxes and commissions across auction venues. 2 The property ‘increasing’ is to be interpreted in the nonstrict sense throughout this letter.

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I and with, at the limit, 0 for the ordinates in coordinate set I c . That h(z; p) is an indicator function is immediate from work by David (1970, p. 75). For example if n 5 4 and s 5 2, then h(z; p)5 k [z 1 $p,z 2 $p,z 3 $p]1 k [z 1 $p,z 2 $p,z 4 $p]1 k [z 1 $p,z 3 $p,z 4 $p]1 k [z 2 $p,z 3 $p,z 4 $p]23k [z 1 $p,z 2 $p,z 3 $p,z 4 $p].

(2)

And we have h(z; p) 5 0 for z ( 2 ) , p with h(z; p) 5 1 for z (2 ) $ p. The function is precisely tailored for our needs. n Theorem 1. Let X [ R¯ n1 and Y [ R¯ 1 . Then all U [ U1 sellers in preference-revealing auctions where s items are sold to n bidders prefer valuation distribution Y over valuation distribution X if and only if Ehh( y; p)j $ Ehh(x; p)j ; p [ [0,`) such that h( ? ; p) [ F ps,n as provided in Eq. (1).

Proof. David (1970) has shown that

O (21)

Sts 2 1D O k[( pe ,0),(`)].

n

k [z (n 2s) $ p] 5

t 2s21

t5s11

I

(3)

I [ ct

And so Ehh(z; p)j ; Prob[z (n 2s) $ p].

(4)

It follows from Hadar and Russell (1969) that all U [ U1 comparing random payoffs x (n 2s) and y (n2s) will prefer distribution Y(n 2s) if and only if Ehh( y; p)j $ Ehh(x; p)j ; p [ [0,`). h Integration by parts and Riemann sum decompositions show that functions h(z; p), p $ 0, comprise a complete set of extreme points in the closed convex cone C of functions such that if g( ? ):R¯ n1 → R FSD 3 is in C then Ehg( y)j $ Ehg(x)j whenever X(n 2s) # Y(n 2s) . By this it is, loosely conveyed, meant that FSD any function g( ? ) for which Ehg( y)j $ Ehg(x)j whenever X(n 2s) # Y(n 2s) can be generated by integration with respect to some positive weighting w( p) on the relevant domain of p. That is, if g( ? ) [ C then there exists a w( p):R¯ 1 → R¯ 1 such that g(z) 5 e0` h(z; p)w( p)dp.4 Thus, the function set F that we seek can be generated from the function ‘basis’ h(z; p):R¯ n1 3 R¯ 1 → h0,1j as p varies on FSD

[0,`). Placing X(n 2s) # Y(n 2s) in the stochastic order literature, it is a symmetry restricted version of the scaled ordering studied by Scarsini and Shaked (1987). Whereas the scaled ordering considers a FSD n n vector p [ R¯ 1 , the X(n 2s) # Y(n 2s) order restricts p to the line in R¯ 1 with parametric representation p1 5 p2 5 ? ? ? 5 pn 5 p. This means that the convex cone of functions that our stochastic order must rank upon integration is smaller than the cone that the scaled ordering must rank. And so the set of multivariate distributions that satisfy our order is larger than the set which satisfy their scaled ordering. We now turn to an equivalent representation of Theorem 1. Note first that the (n 2 s)th order statistic of distribution Z with realization z [ R¯ n1 is also the (n 2 s)th order statistic of distribution V Z with realizations v Z 5 (min[z 1 ,z n 2s11 ,z n 2s12 , . . . ,z n ], min[z 2 ,z n 2s11 ,z n2s12 , . . . ,z n ], . . . , 3 4

See Athey (1998) for a comprehensive treatise on the closed convex cone method for analyzing dominance. But one needs to be careful in handling topological issues. The reader is again referred to Athey (1998).

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min[z n 2s ,z n2s 11 ,z n2s12 , . . . ,z n ]) [ R¯ n12s . Here, the superscripted Z denotes that distribution Z generZ ated distribution V . The advantage of employing this marginalization procedure is that the order statistic we now seek is the largest order statistic. And the largest order statistic is simpler to analyze FSD Y X than an arbitrary order statistic. We seek to ascertain when V (n2s) # V (n 2s) . That is, we seek to Y X n 2s ascertain when Prob[v (n 2s) # pe n 2s ] # Prob[v (n2s) # pe n2s ] where e n 2s [ R¯ 1 is the unit vector. This stochastic ordering is the symmetric lower orthant order as discussed in Shaked and Shanthikumar (1997, pp. 90–91). In contrast with the complexity of general expression (1) above, the linear combination of indicator functions that is now of interest is the much simpler expression h(v Z ; p) 5 k [(0),( pe n2s )].

(5) Z

This is the indicator function that equals 1 when v (n 2s) # p and equals zero otherwise. It is decreasing in v Z . Theorem 19. Let X [ R¯ n1 and Y [ R¯ n1 . Then all U [ U1 sellers in truth-telling auctions where s items are sold to n bidders prefer valuation distribution Y over valuation distribution X if and only if Ehh(v Y ; p)j # Ehh(v X ; p)j ; p [ [0,`) where h( ? ):R n2s 3 R → h0,1j is provided in Eq. (5) above.

3. Concluding remark It would be of some interest to identify the convex cone of functions for which comparison of expectations would exactly identify preferences over valuation distributions for risk averse sellers in preference-revealing auctions. This is not an easy problem because while the z (n2s) , s 5 0,1, . . . ,n 2 1 are increasing in (z 1 , . . . ,z n ), only z (n) is convex in z 5 (z 1 , . . . ,z n ) and only z (1) is concave in z. And neither of the extreme order statistics are of interest in preference-revealing auctions. However, in converting interior order statistics to extreme order statistics, the marginalization procedure applied in Theorem 19 might be relevant for future work. Then the facts that z (n) is convex in z and z (1) is concave in z might provide an approach for comparing valuation distributions when the seller is risk averse. Perhaps a restricted variant of the lower orthant concave order, a stochastic relation studied in Shaked and Shanthikumar (1994, p. 160), might provide a starting point.

Acknowledgements The helpful comments of a referee are appreciated. Journal paper No. J-18743 of the Iowa Agriculture and Home Economics Experiment Station, Ames, Iowa. Project No. 3463, and supported by Hatch Act and State of Iowa funds.

References Athey, S., 1998. Characterizing properties of stochastic objective functions, Massachussetts Institute of Technology, Working paper. David, H.A., 1970. Order Statistics, Wiley, New York.

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Hadar, J., Russell, W.R., 1969. Rules of ordering uncertain prospects. American Economic Review 59 (1), 25–34. ¨ Operations Research 31, Scarsini, M., Shaked, M., 1987. Ordering distributions by scaled order statistics. Zeitschrift fur A1–A13. Shaked, M., Shanthikumar, J.G., 1994. Stochastic Orders and their Applications, Academic Press, San Diego. Shaked, M., Shanthikumar, J.G., 1997. Supermodular stochastic orders and positive dependence of random variables. Journal of Multivariate Analysis 61, 86–101.