Some Further Results on the Winner s Rent in the Second-Price Business Auction

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1 Sankhyā : The Indian Journal of Statistics 28, Volume 7-A, Part 1, pp c 28, Indian Statistical Institute Some Further Results on the Winner s Rent in the Second-Price Business Auction Maochao Xu Portland State University, Portland, Oregon 9721, USA Xiaohu Li Lanzhou University, Lanzhou 73, China Abstract The winner s rent for a buyer s auction is shown to be decreasing in the number of bidders according to the mean residual life order if the common distribution of valuations is of decreasing mean residual life. Stochastic comparisons of the conditional winner s rent are conducted for both buyer s case and reverse case. AMS (2) subject classification. Primary 6E15; secondary 6K1, 62N5. Keywords and phrases. Auction rent, conditional auction rent, DMRL, stochastic ordering 1 Introduction and Preliminaries 1.1. Introduction. As is well-known, the second-price closed-seal auction is most frequently used in business auction. In this model, under the private values framework, it is optimal for a bidder to bid his or her true valuation, whatever other bidders do. A lot of attention has been paid to this kind of auction in the literature. Interested readers may refer to Vickrey (1961), Bulow and Klemperer (1996) and Klemperer (1999) for more details. Paul and Gutierrez (24) made further investigation on the auction rent, which is the difference between the winner s valuation and the final price. As is pointed out there that for a buyer s auction in which bidders bid to buy, the auction rent is in fact the difference between the largest and the second largest valuation; for a reverse auction in which bidders compete to sell a service, the auction rent is just the difference between the lowest and the second lowest valuation. Theorems 1 and 2 there assert that

2 Some further results on the winner s rent in the second-price business auction 125 in a reverse auction, the expected rent of the winner is decreasing in the number of the bidders if the valuations are i.i.d. samples with a common concave distribution on its interval support, and in a buyer s auction, the expected rent of the winner is decreasing in the number of the bidders if the valuations are i.i.d. samples with a common convex distribution on its interval support, respectively. Recently, these results have been extended by Li (25) under the weak conditions that in a reverse auction, if the valuations are i.i.d. samples with a common DRHR (decreasing reversed hazard rate) distribution, then the expected rent of the winner is decreasing in the number of the bidders, and in a buyer s auction, if the valuations are i.i.d. samples with a common IFR (increasing failure rate) distribution, then the expected rent of the winner is decreasing in the number of the bidders, respectively. This problem will be further pursued in this paper. It is shown that under the assumption that the valuations are i.i.d. with a common DMRL (decreasing mean residual life) distribution, the rent of a buyer s auction will be decreasing in the number of the bidders according to the mean residual life order, which extends the result in Li (25). Paul and Gutierrez (24) also investigated auction rent from two populations. Theorem 3 there states that, for two different buyers or reverse auctions A and B with their valuations independently distributed as the nonnegative random variables X and Y, respectively, if X is less than Y in the sense of star order and X, Y have the same mean, then the expected winning rent in auction B is higher than that in auction A. However, the proof has a flaw, which will be discussed in Section 3. From the view of bidders, we discuss both the rent of the buyer s auction and that of the reverse auction in Section Preliminaries. Let X and Y be two nonnegative continuous random variables with distribution functions F and G, density function f and g, and survival functions F = 1 F and Ḡ = 1 G, respectively. The corresponding residual life of X and Y at time t are X t = [X t X > t], Y t = [Y t Y > t]; their survival functions can be represented, for any t, x, as F t (x) = P(X t > x) = F(x + t), F(t) Ḡ t (x) = P(Y t > x) = Ḡ(x + t). Ḡ(t)

3 126 Maochao Xu and Xiaohu Li The inactivity time of X and Y at time t > are X (t) = [t X X t], Y (t) = [t Y Y t]; their survival functions can be represented, for any t x, as F (t) (x) = P(X (t) > x) = F(t x), F(t) Ḡ (t) (x) = P(Y (t) > x) = G(t x). G(t) Let X k:n (k = 1,2,,n) be the kth order statistic of an independent random sample X 1,...,X n, all identically distributed as X. Similarly, let Y k:n (k = 1,2,,n) be the kth order statistic of an independent random sample Y 1,...,Y n, all identically distributed as Y. The distribution function and the density function of X k:n are denoted by F k:n and f k:n, respectively. For buyer s auction A, the auction rent with n bidders is given by the last sample spacings, that is, R n = X n:n X n 1:n. Similarly, for buyer s auction B, R n = Y n:n Y n 1:n. In the case of a reverse auction, the auction rent with n bidders is given by the first sample spacings, that is, for reverse auction A, R n = X 2:n X 1:n, for reverse auction B, R n = Y 2:n Y 1:n. We will use stochastic orders as the main tool to investigate the auction rent of different auctions, since they are more informative than usual means. Readers may refer to two excellent books, Müller and Stoyan (22) and Shaked and Shanthikumar (27) for more details. We recall the following stochastic orders, which will be used in the sequel. Definition 1.1. X is said to be smaller than Y in the (a) likelihood ratio order, denoted by X lr Y, if g(x)/f(x) is increasing in x;

4 Some further results on the winner s rent in the second-price business auction 127 (b) hazard rate order, denoted by X hr Y, if Ḡ(x)/ F(x) is increasing in x; (c) reversed hazard rate order, denoted by X rh Y, if G(x)/F(x) is increasing in x; (d) usual stochastic order, denoted by X st Y, if F(x) Ḡ(x) is increasing in x; (e) mean residual life order, denoted by X mrl Y, if EX t EY t for all t, or equivalently, F(x)dx t t F(t) Ḡ(x)dx Ḡ(t). (1.1) It is known that (cf. Shaked and Shanthikumar (27)), X lr Y = X hr(rh) Y = X st Y = EX EY, and X hr Y = X mrl Y = EX EY. The following order will also be discussed in Section 3. Definition 1.2. X is said to be smaller than Y in the star order, denoted by X Y, if G 1 F(x)/x is increasing in x, where G 1 (p) = sup{x : F(x) p}. The star order is very useful in reliability theory. It compares the relative IFRA (increasing failure rate average) property of two probability distributions (cf. Barlow and Proschan (1981) and Shaked and Shanthikumar (27)). It is worth remarking that the star order implies the Lorenz order which plays an important role in economics. 2 Rent of a Buyer s Auction with DMRL Valuations Recall that a nonnegative random variable X is said to be DMRL if m(t) = E(X t ) is decreasing in t. The following result investigates the effect of the DMRL property of the valuation on the auction rent. Theorem 2.1. If X is DMRL, then, R n mrl R n+1, n 2.

5 128 Maochao Xu and Xiaohu Li Proof. Recall that the conditional distribution of X n+1:n+1 X n:n+1 given X n:n+1 = u is the same as the unconditional distribution F u, thus, for any x, and for any t, S n+1 (x) = P(R n+1 x) = t S n+1 (x)dx S n+1 (t) = = t where U n+1 has the following density g n+1 (u) = t F(x + u) df n:n+1 (u), F(x + u) df n:n+1 (u)dx F(t + u) df n:n+1 (u) F(x + u) dxdf n:n+1 (u) F(t + u) df n:n+1 (u) m(t + u) F(t + u) df n:n+1 (u) = F(t + u) df n:n+1 (u) = E[m(t + U n+1 )], (2.1) F(t + u) f n:n+1 (u). F(t + v) df n:n+1 (v) F(v) According to Raqab and Amin (1996), it holds that X n:n+1 lr X n 1:n. Note that g n+1 (u) g n (u) it follows immediately that = f n:n+1(u) f n 1:n (u) F(t + v) df n 1:n (v) F(v), F(t + v) df n:n+1 (v) F(v) U n+1 lr U n,

6 Some further results on the winner s rent in the second-price business auction 129 which implies t + U n+1 lr t + U n, for any t. The DMRL property of X implies that m(t) is decreasing in t. By Theorem 1.C.8 of Shaked and Shanthikumar (27), it holds that Thus, for any t, m(t + U n+1 ) lr m(t + U n ), for any t. E[m(t + U n+1 )] E[m(t + U n )]. Observing (1.1) and (2.1), it holds that, for n 2. R n+1 mrl R n, Remarks. (i) By a similar argument, it can be shown that if X is IMRL (increasing mean residual life), i.e., m(t) is increasing in t, then, R n mrl R n+1, for n 2. (ii) As pointed out by the referee, Theorem 2.1 is also true for the general random variable X, not restricted to the nonnegative case. The following result is a direct consequence of Theorem 2.1. Corollary 2.1. If the common distribution of valuations is DMRL, then, the expected rent in a buyer s auction is decreasing in the number of the bidders. Remark. Since DMRL class is larger than IFR class (cf. Barlow and Proschan (1981)), the above result extends Proposition 2.4 in Li (25). In practical situation, once valuation samples exhibit DMRL property (one may set up nonparametric testing for DMRL property of the collected data), the above result works.

7 13 Maochao Xu and Xiaohu Li 3 Auction Rent from Two Populations Theorem 3 of Paul and Gutierrez (24) claims that if X Y with EX = EY, then ER n ER n and E R n E R n, where R n (R n ) denotes the auction rent in buyer s auction A (B), and R n ( R n ) denotes the auction rent in reverse auction A (B). Unfortunately, the proof there has a flaw. In fact, the following two conditions are implicitly assumed in the proof. Emin{X 1,,X n } = Emin{Y 1,,Y n }, Emax{X 1,,X n } = Emax{Y 1,,Y n }. However, Theorem 7.6 in Barlow and Proschan (1981) asserts that X Y with EX = EY implies Emin{X 1,,X n } Emin{Y 1,,Y n } and Emax{X 1,,X n } Emax{Y 1,,Y n }. As a result, the above two equalities are not always simultaneously true. One may now wonder whether conclusions there are still valid? Li (25) pointed out that X Y with EX = EY implies X ew Y, where ew means the excess wealth order (cf. Shaked and Shanthikumar (27)), and Proposition 2.6 there states that under X ew Y, for a buyer s auction, the expected winner s rent in auction B remains larger than that in auction A. Thus, the conclusion for the buyer s auction in Theorem 3 of Paul and Gutierrez (24) remains valid. However, the following example reveals that the conclusion for the reverse auction in Theorem 3 there is invalid. Example 3.1. Consider a nonnegative random variable X with distribution { x, x 1, F(x) = 1, x 1. and the random variable Y having exponential distribution with distribution function G(x) = 1 exp{ 2x}, x.

8 Some further results on the winner s rent in the second-price business auction 131 It is easy to check that X Y and EX = EY = 1/2. However, for n = 4, E R 4 = F(x) F 3 (x)dx = 1 2, E R 4 = G(x)Ḡ3 (x)dx = Hence, the expected winner s rent in reverse auction A is larger than that in B. 4 Stochastic Comparison of Conditional Winner s Rent In this section, we will discuss the auction rent from the viewpoint of the bidders. In an auction, the winner only knows his or her own valuation v in a practical situation. Given that some bidder, say the i-th one, wins, then the conditional rent in buyer s auctions A and B should respectively be R n (v) = [v X n 1:n X i = v = X n:n ] = min j i (X j) (v) ; R n (v) = [v Y n 1:n Y i = v = Y n:n ] = min j i (Y j) (v). Similarly, the conditional rent of reverse auctions A and B should be, respectively, R n (v) = [X 2:n v X i = v = X 1:n ] = min j i (X j) v ; R n(v) = [Y 2:n v Y i = v = Y 1:n ] = min j i (Y j) v. The survival functions of R n (v) and R n (v) could be represented as [ ] F(v x) n 1 S n (x,v) = P (R n (v) x) = ; F(v) ( [ ] n 1 F(v + x) K n (x,v) = P Rn (v) x) =. F(v) It is interesting to note that R n (v) takes the form of RLS n 1,n,v in Belzunce et al. (1999) where they consider it as the residual life of a k- out-of-n system given the time of some failure. The following result, which can be easily derived, asserts that the auction rent, either in a buyer s auction or in a reverse auction, decreases in the number of bidders in the sense of the likelihood ratio order.

9 132 Maochao Xu and Xiaohu Li Proposition 4.1. If X has an absolutely continuous distribution, then (a) R n (v) lr R n+1 (v); (b) R n (v) lr R n+1 (v). Two propositions below investigate how variation of valuations will affect the auction rent. Since the proofs are simple, they are omitted for brevity. Proposition 4.2. For two absolutely continuous random variables X and Y, the following statements are equivalent. (a) X lr Y ; (b) R n (v) lr R n(v); (c) R n (v) lr R n (v). Proposition 4.3. For two random variables X and Y, the following statements are equivalent. (a) X hr ( rh )Y ; (b) R n (v) hr ( hr ) R n(v); (c) R n (v) st ( st ) R n (v). Acknowledgements. The authors are grateful to the anonymous referee and the Co-Editor for their constructive comments and suggestions which have led to an improved version of this paper. References Barlow, R. E. and Proschan, F. (1981). Statistical Theory of Reliability and Life Testing. To Begin with, Silver Spring, Maryland. Belzunce, F., Franco, M. and Ruiz, J. M. (1999). On aging properties based on the residual life of k-out-of-n systems. Probability in the Engineering and Informational Sciences, 13, Bulow, J. and Klemperer, P. (1996). Auctions versus negotiations. American Ecomonic Review 86, Klemperer, P. (1999). Auction theory: a guide to the literature. J. Economic surveys, 13, Li, X. (25). A note on expected rent in auction theory. Operations Research Letters, 33, Müller, A. and Stoyan, D. (22). Comparison Methods for Stochastic Models and Risks. John Wiley & Sons, New York.

10 Some further results on the winner s rent in the second-price business auction 133 Paul, A. and Gutierrez, G. (24). Mean sample spacings, sample size and variability in an auction-theoretic framework. Operations Research Letters, 32, Raqab, M. Z. and Amin, W. A. (1996). Some ordering results on order statistics and record values. IAPQR Transactions, 21, 1-8. Shaked, M. and Shanthikumar, J. G. (27). Stochastic Orders and Their Applications. Springer, New York. Vickrey, W. (1961). Counterspeculation, auctions, and competitive sealed tenders. J. Finance, 16, Maochao Xu Department of Mathematics and Statistics Portland State University Portland, Oregon 9721, USA Xiaohu Li School of Mathematics and Statistics Lanzhou University Lanzhou 73, China xhli@lzu.edu.cn Paper received January 28; revised September 28.