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Preferences over valuation distributions in auctions
*
David A. Hennessy
Department of Economics, 478B 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 (n2s)th smallest valuation. In statistics terminology, this is the (n2s)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, U , of von-Neumann and Morgenstern (vN&M) expected utility1 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: hennessy@iastate.edu (D.A. Hennessy)
will be clear if it is believed that the ex-ante random (n21)th order statistic arising from one set of 1
choices dominates the others in the first order sense. It is the problem of preparing to auction an asset that we intend to analyze. For the U class of decision makers, we will identify a pair of equivalent1 characterizations of the types of multivariate valuation distribution function shifts that will be
2 preferred by the seller.
2. Analysis
n
¯
Define by Ln the set of all random vectors in R1. For a given non-negative realization in this
n
¯
random vector space, z[R1, define as z(i )5(z ,z , . . . ,z )1 2 n (i ) the ith order statistic, i.e. the ith
n
¯
smallest ordinate in vector z[R1. We will assume a vector representation of distributions, and upper
n
¯
case Z represents the distribution from which z[R1is drawn. The distribution of the (n2s)th order
statistic of this multivariate distribution is denoted by Z(n2s).
n n
¯ ¯
For a pair of random vectors XFSD [R1 and Y[R1, a stochastic partial ordering of concern to us is when X(n2s)#Y(n2s), i.e. when the (n2s)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
n
¯
function w(?):R1→Rwith respect to distribution Z. We will identify a convex cone of functions F FSD
such that for all w(?)[F we have [X(n2s)# Y(n2s)]⇔[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,
c
I5hi ,i , . . . ,i1 2 tj[ct. Let zI5hz ,z , . . . ,zi i ij, and define by I the complement of I in the universal
1 2 t
index set h1,2, . . . ,nj. Therefore, the vector realization z from distribution Z may be partitioned into
FSD
(z ,z ). In arriving at coneI Ic F that defines the order [X(n2s)#Y(n2s)], we will need to pay particular
n2t
attention to realizations where the realizations of zIc are at the subspace boundaries, zIc50 and
n2t
zIc5 ` . 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[R1, define by
k[A] the indicator function that equals 1 when z[A and equals 0 when z[⁄ A. For set function
n
¯ ¯ ¯
k[A]:R1→h0,1j, the expressionk[(z),(z )] operates on the rectangular set of all z such that z] ]#z#z,
n ] n
¯ ¯
z[R1, z[R1, where order relation # has the usual coordinate-wise interpretation. We denote
]
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
c
I and with, at the limit, 0 for the ordinates in coordinate set I . That h(z; p) is an indicator function is
immediate from work by David (1970, p. 75). For example if n54 and s52, then
where s items are sold to n bidders prefer valuation distribution Y over valuation distribution X if
p
It follows from Hadar and Russell (1969) that all U[U comparing random payoffs x1 (n2s) and y(n2s) will prefer distribution Y(n2s) 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$0, comprise
n
¯
a complete set of extreme points in the closed convex cone C of functions such that if g(?):R1→R
FSD 3
is in C then Ehg( y)j$Ehg(x)j whenever X(n2s)#Y(n2s). By this it is, loosely conveyed, meant that
FSD
any function g(?) for which Ehg( y)j$Ehg(x)j whenever X(n2s)#Y(n2s) can be generated by integration with respect to some positive weighting w( p) on the relevant domain of p. That is, if
` 4
[0,`). Placing X(n2s)#Y(n2s) 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[R1, the X(n2s)#Y(n2s) order restricts p to the line in R1 with parametric representation
p15p25 ? ? ? 5pn5p. 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 (n2s)th order
n
¯
statistic of distribution Z with realization z[R1 is also the (n2s)th order statistic of distribution
Z Z
V with realizations v 5(min[z ,z ,z , . . . ,z ], min[z ,z ,z , . . . ,z ], . . . ,
1 n2s11 n2s12 n 2 n2s11 n2s12 n
3
See Athey (1998) for a comprehensive treatise on the closed convex cone method for analyzing dominance. 4
n2s
¯
min[zn2s,zn2s11,zn2s12, . . . ,z ])n [R1 . Here, the superscripted Z denotes that distribution Z
gener-Z
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
X Y
than an arbitrary order statistic. We seek to ascertain when V(n2s)#V(n2s). That is, we seek to
Y X ¯n2s
ascertain when Prob[v #pe ]#Prob[v #pe ] where e [R is the unit vector. This
(n2s) n2s (n2s) n2s n2s 1
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
Z
h(v ; p)5k[(0),( pe )]. (5)
n2s
Z
This is the indicator function that equals 1 whenv #p and equals zero otherwise. It is decreasing
(n2s) Z
in v .
n n
¯ ¯
Theorem 19. Let X[R1and Y[R1. Then all U[U sellers in truth-telling auctions where s items1
are sold to n bidders prefer valuation distribution Y over valuation distribution X if and only if
Y X n2s
Ehh(v ; p)j#Ehh(v ; p)j ;p[[0,`) where h(?):R 3R→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), s50,1, . . . ,n21 are increasing in (z , . . . ,z ), only z1 n (n) is convex in z5(z , . . . ,z ) and only z1 n ( 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.
Hadar, J., Russell, W.R., 1969. Rules of ordering uncertain prospects. American Economic Review 59 (1), 25–34. ¨
Scarsini, M., Shaked, M., 1987. Ordering distributions by scaled order statistics. Zeitschrift fur Operations Research 31, A1–A13.
Shaked, M., Shanthikumar, J.G., 1994. Stochastic Orders and their Applications, Academic Press, San Diego.
