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

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

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

Transcription

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 Paper received January 28; revised September 28.

Optimal Auctions Continued

Optimal Auctions Continued Lecture 6 Optimal Auctions Continued 1 Recap Last week, we... Set up the Myerson auction environment: n risk-neutral bidders independent types t i F i with support [, b i ] residual valuation of t 0 for

More information

Chapter 7. Sealed-bid Auctions

Chapter 7. Sealed-bid Auctions Chapter 7 Sealed-bid Auctions An auction is a procedure used for selling and buying items by offering them up for bid. Auctions are often used to sell objects that have a variable price (for example oil)

More information

Hybrid Auctions Revisited

Hybrid Auctions Revisited Hybrid Auctions Revisited Dan Levin and Lixin Ye, Abstract We examine hybrid auctions with affiliated private values and risk-averse bidders, and show that the optimal hybrid auction trades off the benefit

More information

Next Tuesday: Amit Gandhi guest lecture on empirical work on auctions Next Wednesday: first problem set due

Next Tuesday: Amit Gandhi guest lecture on empirical work on auctions Next Wednesday: first problem set due Econ 805 Advanced Micro Theory I Dan Quint Fall 2007 Lecture 6 Sept 25 2007 Next Tuesday: Amit Gandhi guest lecture on empirical work on auctions Next Wednesday: first problem set due Today: the price-discriminating

More information

Nonparametric adaptive age replacement with a one-cycle criterion

Nonparametric adaptive age replacement with a one-cycle criterion Nonparametric adaptive age replacement with a one-cycle criterion P. Coolen-Schrijner, F.P.A. Coolen Department of Mathematical Sciences University of Durham, Durham, DH1 3LE, UK e-mail: Pauline.Schrijner@durham.ac.uk

More information

5. Convergence of sequences of random variables

5. Convergence of sequences of random variables 5. Convergence of sequences of random variables Throughout this chapter we assume that {X, X 2,...} is a sequence of r.v. and X is a r.v., and all of them are defined on the same probability space (Ω,

More information

Intrinsic Aging and Classes of Nonparametric Distributions Rhonda Righter Department of Industrial Engineering and Operations Research University of

Intrinsic Aging and Classes of Nonparametric Distributions Rhonda Righter Department of Industrial Engineering and Operations Research University of Intrinsic Aging and Classes of Nonparametric Distributions Rhonda Righter Department of Industrial Engineering and Operations Research University of California Berkeley, CA 94720 U.S.A. Moshe Shaked Department

More information

Dynamic Auction: Mechanisms and Applications

Dynamic Auction: Mechanisms and Applications Dynamic Auction: Mechanisms and Applications Lin Gao IERG 6280 Network Economics Spring 2011 Preliminary Benchmark Model Extensions Applications Conclusion Outline What is an Auction? An auction is a market

More information

Principle of Data Reduction

Principle of Data Reduction Chapter 6 Principle of Data Reduction 6.1 Introduction An experimenter uses the information in a sample X 1,..., X n to make inferences about an unknown parameter θ. If the sample size n is large, then

More information

5. Continuous Random Variables

5. Continuous Random Variables 5. Continuous Random Variables Continuous random variables can take any value in an interval. They are used to model physical characteristics such as time, length, position, etc. Examples (i) Let X be

More information

Estadística. An Introduction to the Theory of Stochastic Orders

Estadística. An Introduction to the Theory of Stochastic Orders Boletín de Estadística e Investigación Operativa Vol. 26, No. 1, Febrero 2010, pp. 4-18 Estadística An Introduction to the Theory of Stochastic Orders Félix Belzunce Departamento de Estadística e Investigación

More information

A Theory of Auction and Competitive Bidding

A Theory of Auction and Competitive Bidding A Theory of Auction and Competitive Bidding Paul Milgrom and Rober Weber Econometrica 1982 Presented by Yifang Guo Duke University November 3, 2011 Yifang Guo (Duke University) Envelope Theorems Nov. 3,

More information

Sets and functions. {x R : x > 0}.

Sets and functions. {x R : x > 0}. Sets and functions 1 Sets The language of sets and functions pervades mathematics, and most of the important operations in mathematics turn out to be functions or to be expressible in terms of functions.

More information

Managerial Economics

Managerial Economics Managerial Economics Unit 8: Auctions Rudolf Winter-Ebmer Johannes Kepler University Linz Winter Term 2012 Managerial Economics: Unit 8 - Auctions 1 / 40 Objectives Explain how managers can apply game

More information

A Note on the Ruin Probability in the Delayed Renewal Risk Model

A Note on the Ruin Probability in the Delayed Renewal Risk Model Southeast Asian Bulletin of Mathematics 2004 28: 1 5 Southeast Asian Bulletin of Mathematics c SEAMS. 2004 A Note on the Ruin Probability in the Delayed Renewal Risk Model Chun Su Department of Statistics

More information

Exponential Distribution

Exponential Distribution Exponential Distribution Definition: Exponential distribution with parameter λ: { λe λx x 0 f(x) = 0 x < 0 The cdf: F(x) = x Mean E(X) = 1/λ. f(x)dx = Moment generating function: φ(t) = E[e tx ] = { 1

More information

ON LIMIT LAWS FOR CENTRAL ORDER STATISTICS UNDER POWER NORMALIZATION. E. I. Pancheva, A. Gacovska-Barandovska

ON LIMIT LAWS FOR CENTRAL ORDER STATISTICS UNDER POWER NORMALIZATION. E. I. Pancheva, A. Gacovska-Barandovska Pliska Stud. Math. Bulgar. 22 (2015), STUDIA MATHEMATICA BULGARICA ON LIMIT LAWS FOR CENTRAL ORDER STATISTICS UNDER POWER NORMALIZATION E. I. Pancheva, A. Gacovska-Barandovska Smirnov (1949) derived four

More information

Characterizations of classes of lifetime distributions generalizing the NBUE class.

Characterizations of classes of lifetime distributions generalizing the NBUE class. Characterizations of classes of lifetime distributions generalizing the NBUE class. Bernhard Klar Institut für Mathematische Stochastik, Universität Karlsruhe, Kaiserstr. 12, D-76128 Karlsruhe, Germany

More information

Overview of Monte Carlo Simulation, Probability Review and Introduction to Matlab

Overview of Monte Carlo Simulation, Probability Review and Introduction to Matlab Monte Carlo Simulation: IEOR E4703 Fall 2004 c 2004 by Martin Haugh Overview of Monte Carlo Simulation, Probability Review and Introduction to Matlab 1 Overview of Monte Carlo Simulation 1.1 Why use simulation?

More information

Continuous random variables

Continuous random variables Continuous random variables So far we have been concentrating on discrete random variables, whose distributions are not continuous. Now we deal with the so-called continuous random variables. A random

More information

On Compulsory Per-Claim Deductibles in Automobile Insurance

On Compulsory Per-Claim Deductibles in Automobile Insurance The Geneva Papers on Risk and Insurance Theory, 28: 25 32, 2003 c 2003 The Geneva Association On Compulsory Per-Claim Deductibles in Automobile Insurance CHU-SHIU LI Department of Economics, Feng Chia

More information

Sealed Bid Second Price Auctions with Discrete Bidding

Sealed Bid Second Price Auctions with Discrete Bidding Sealed Bid Second Price Auctions with Discrete Bidding Timothy Mathews and Abhijit Sengupta August 16, 2006 Abstract A single item is sold to two bidders by way of a sealed bid second price auction in

More information

Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions

Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions Natalia Lazzati Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions Note 5 is based on Madden (1986, Ch. 1, 2, 4 and 7) and Simon and Blume (1994, Ch. 13 and 21). Concave functions

More information

Properties of moments of random variables

Properties of moments of random variables Properties of moments of rom variables Jean-Marie Dufour Université de Montréal First version: May 1995 Revised: January 23 This version: January 14, 23 Compiled: January 14, 23, 1:5pm This work was supported

More information

Chapter 4 Expected Values

Chapter 4 Expected Values Chapter 4 Expected Values 4. The Expected Value of a Random Variables Definition. Let X be a random variable having a pdf f(x). Also, suppose the the following conditions are satisfied: x f(x) converges

More information

Regret Minimization for Reserve Prices in Second-Price Auctions

Regret Minimization for Reserve Prices in Second-Price Auctions Regret Minimization for Reserve Prices in Second-Price Auctions Nicolò Cesa-Bianchi Università degli Studi di Milano Joint work with: Claudio Gentile (Varese) and Yishay Mansour (Tel-Aviv) N. Cesa-Bianchi

More information

A Uniform Asymptotic Estimate for Discounted Aggregate Claims with Subexponential Tails

A Uniform Asymptotic Estimate for Discounted Aggregate Claims with Subexponential Tails 12th International Congress on Insurance: Mathematics and Economics July 16-18, 2008 A Uniform Asymptotic Estimate for Discounted Aggregate Claims with Subexponential Tails XUEMIAO HAO (Based on a joint

More information

Math 370/408, Spring 2008 Prof. A.J. Hildebrand. Actuarial Exam Practice Problem Set 3 Solutions

Math 370/408, Spring 2008 Prof. A.J. Hildebrand. Actuarial Exam Practice Problem Set 3 Solutions Math 37/48, Spring 28 Prof. A.J. Hildebrand Actuarial Exam Practice Problem Set 3 Solutions About this problem set: These are problems from Course /P actuarial exams that I have collected over the years,

More information

Lecture 6: Discrete & Continuous Probability and Random Variables

Lecture 6: Discrete & Continuous Probability and Random Variables Lecture 6: Discrete & Continuous Probability and Random Variables D. Alex Hughes Math Camp September 17, 2015 D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September

More information

Random variables P(X = 3) = P(X = 3) = 1 8, P(X = 1) = P(X = 1) = 3 8.

Random variables P(X = 3) = P(X = 3) = 1 8, P(X = 1) = P(X = 1) = 3 8. Random variables Remark on Notations 1. When X is a number chosen uniformly from a data set, What I call P(X = k) is called Freq[k, X] in the courseware. 2. When X is a random variable, what I call F ()

More information

Basics of Statistical Machine Learning

Basics of Statistical Machine Learning CS761 Spring 2013 Advanced Machine Learning Basics of Statistical Machine Learning Lecturer: Xiaojin Zhu jerryzhu@cs.wisc.edu Modern machine learning is rooted in statistics. You will find many familiar

More information

A Comparison of the Optimal Costs of Two Canonical Inventory Systems

A Comparison of the Optimal Costs of Two Canonical Inventory Systems A Comparison of the Optimal Costs of Two Canonical Inventory Systems Ganesh Janakiraman 1, Sridhar Seshadri 2, J. George Shanthikumar 3 First Version: December 2005 First Revision: July 2006 Subject Classification:

More information

governments and firms, and in particular ebay on market design; the views expressed are our own.

governments and firms, and in particular ebay on market design; the views expressed are our own. On the Design of Simple Multi-unit Online Auctions Thomas Kittsteiner (London School of Economics) and Axel Ockenfels (University of Cologne) 1 The increased use of online market places (like ebay) by

More information

Lecture 7: Continuous Random Variables

Lecture 7: Continuous Random Variables Lecture 7: Continuous Random Variables 21 September 2005 1 Our First Continuous Random Variable The back of the lecture hall is roughly 10 meters across. Suppose it were exactly 10 meters, and consider

More information

Properties of BMO functions whose reciprocals are also BMO

Properties of BMO functions whose reciprocals are also BMO Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a non-negative BMO-function w, whose reciprocal is also in BMO, belongs to p> A p,and

More information

Gambling Systems and Multiplication-Invariant Measures

Gambling Systems and Multiplication-Invariant Measures Gambling Systems and Multiplication-Invariant Measures by Jeffrey S. Rosenthal* and Peter O. Schwartz** (May 28, 997.. Introduction. This short paper describes a surprising connection between two previously

More information

Optimal slot restriction and slot supply strategy in a keyword auction

Optimal slot restriction and slot supply strategy in a keyword auction WIAS Discussion Paper No.21-9 Optimal slot restriction and slot supply strategy in a keyword auction March 31, 211 Yoshio Kamijo Waseda Institute for Advanced Study, Waseda University 1-6-1 Nishiwaseda,

More information

MTH4100 Calculus I. Lecture notes for Week 8. Thomas Calculus, Sections 4.1 to 4.4. Rainer Klages

MTH4100 Calculus I. Lecture notes for Week 8. Thomas Calculus, Sections 4.1 to 4.4. Rainer Klages MTH4100 Calculus I Lecture notes for Week 8 Thomas Calculus, Sections 4.1 to 4.4 Rainer Klages School of Mathematical Sciences Queen Mary University of London Autumn 2009 Theorem 1 (First Derivative Theorem

More information

61. REARRANGEMENTS 119

61. REARRANGEMENTS 119 61. REARRANGEMENTS 119 61. Rearrangements Here the difference between conditionally and absolutely convergent series is further refined through the concept of rearrangement. Definition 15. (Rearrangement)

More information

On characterization of a class of convex operators for pricing insurance risks

On characterization of a class of convex operators for pricing insurance risks On characterization of a class of convex operators for pricing insurance risks Marta Cardin Dept. of Applied Mathematics University of Venice e-mail: mcardin@unive.it Graziella Pacelli Dept. of of Social

More information

Structural Econometric Modeling in Industrial Organization Handout 1

Structural Econometric Modeling in Industrial Organization Handout 1 Structural Econometric Modeling in Industrial Organization Handout 1 Professor Matthijs Wildenbeest 16 May 2011 1 Reading Peter C. Reiss and Frank A. Wolak A. Structural Econometric Modeling: Rationales

More information

Probability Generating Functions

Probability Generating Functions page 39 Chapter 3 Probability Generating Functions 3 Preamble: Generating Functions Generating functions are widely used in mathematics, and play an important role in probability theory Consider a sequence

More information

Vickrey-Dutch Procurement Auction for Multiple Items

Vickrey-Dutch Procurement Auction for Multiple Items Vickrey-Dutch Procurement Auction for Multiple Items Debasis Mishra Dharmaraj Veeramani First version: August 2004, This version: March 2006 Abstract We consider a setting where there is a manufacturer

More information

1 Sufficient statistics

1 Sufficient statistics 1 Sufficient statistics A statistic is a function T = rx 1, X 2,, X n of the random sample X 1, X 2,, X n. Examples are X n = 1 n s 2 = = X i, 1 n 1 the sample mean X i X n 2, the sample variance T 1 =

More information

Asymptotics for ruin probabilities in a discrete-time risk model with dependent financial and insurance risks

Asymptotics for ruin probabilities in a discrete-time risk model with dependent financial and insurance risks 1 Asymptotics for ruin probabilities in a discrete-time risk model with dependent financial and insurance risks Yang Yang School of Mathematics and Statistics, Nanjing Audit University School of Economics

More information

ECON 459 Game Theory. Lecture Notes Auctions. Luca Anderlini Spring 2015

ECON 459 Game Theory. Lecture Notes Auctions. Luca Anderlini Spring 2015 ECON 459 Game Theory Lecture Notes Auctions Luca Anderlini Spring 2015 These notes have been used before. If you can still spot any errors or have any suggestions for improvement, please let me know. 1

More information

Availability of a system with gamma life and exponential repair time under a perfect repair policy

Availability of a system with gamma life and exponential repair time under a perfect repair policy Statistics & Probability Letters 43 (1999) 189 196 Availability of a system with gamma life and exponential repair time under a perfect repair policy Jyotirmoy Sarkar, Gopal Chaudhuri 1 Department of Mathematical

More information

CHAPTER 2. Inequalities

CHAPTER 2. Inequalities CHAPTER 2 Inequalities In this section we add the axioms describe the behavior of inequalities (the order axioms) to the list of axioms begun in Chapter 1. A thorough mastery of this section is essential

More information

Tail inequalities for order statistics of log-concave vectors and applications

Tail inequalities for order statistics of log-concave vectors and applications Tail inequalities for order statistics of log-concave vectors and applications Rafał Latała Based in part on a joint work with R.Adamczak, A.E.Litvak, A.Pajor and N.Tomczak-Jaegermann Banff, May 2011 Basic

More information

Section 5.1 Continuous Random Variables: Introduction

Section 5.1 Continuous Random Variables: Introduction Section 5. Continuous Random Variables: Introduction Not all random variables are discrete. For example:. Waiting times for anything (train, arrival of customer, production of mrna molecule from gene,

More information

Analyzing the Demand for Deductible Insurance

Analyzing the Demand for Deductible Insurance Journal of Risk and Uncertainty, 18:3 3 1999 1999 Kluwer Academic Publishers. Manufactured in The Netherlands. Analyzing the emand for eductible Insurance JACK MEYER epartment of Economics, Michigan State

More information

Nash Equilibrium. Ichiro Obara. January 11, 2012 UCLA. Obara (UCLA) Nash Equilibrium January 11, 2012 1 / 31

Nash Equilibrium. Ichiro Obara. January 11, 2012 UCLA. Obara (UCLA) Nash Equilibrium January 11, 2012 1 / 31 Nash Equilibrium Ichiro Obara UCLA January 11, 2012 Obara (UCLA) Nash Equilibrium January 11, 2012 1 / 31 Best Response and Nash Equilibrium In many games, there is no obvious choice (i.e. dominant action).

More information

160 CHAPTER 4. VECTOR SPACES

160 CHAPTER 4. VECTOR SPACES 160 CHAPTER 4. VECTOR SPACES 4. Rank and Nullity In this section, we look at relationships between the row space, column space, null space of a matrix and its transpose. We will derive fundamental results

More information

17.6.1 Introduction to Auction Design

17.6.1 Introduction to Auction Design CS787: Advanced Algorithms Topic: Sponsored Search Auction Design Presenter(s): Nilay, Srikrishna, Taedong 17.6.1 Introduction to Auction Design The Internet, which started of as a research project in

More information

Geometrical Characterization of RN-operators between Locally Convex Vector Spaces

Geometrical Characterization of RN-operators between Locally Convex Vector Spaces Geometrical Characterization of RN-operators between Locally Convex Vector Spaces OLEG REINOV St. Petersburg State University Dept. of Mathematics and Mechanics Universitetskii pr. 28, 198504 St, Petersburg

More information

Conditional Tail Expectations for Multivariate Phase Type Distributions

Conditional Tail Expectations for Multivariate Phase Type Distributions Conditional Tail Expectations for Multivariate Phase Type Distributions Jun Cai Department of Statistics and Actuarial Science University of Waterloo Waterloo, ON N2L 3G1, Canada jcai@math.uwaterloo.ca

More information

Sensitivity analysis of utility based prices and risk-tolerance wealth processes

Sensitivity analysis of utility based prices and risk-tolerance wealth processes Sensitivity analysis of utility based prices and risk-tolerance wealth processes Dmitry Kramkov, Carnegie Mellon University Based on a paper with Mihai Sirbu from Columbia University Math Finance Seminar,

More information

Lecture 8. Confidence intervals and the central limit theorem

Lecture 8. Confidence intervals and the central limit theorem Lecture 8. Confidence intervals and the central limit theorem Mathematical Statistics and Discrete Mathematics November 25th, 2015 1 / 15 Central limit theorem Let X 1, X 2,... X n be a random sample of

More information

1. (First passage/hitting times/gambler s ruin problem:) Suppose that X has a discrete state space and let i be a fixed state. Let

1. (First passage/hitting times/gambler s ruin problem:) Suppose that X has a discrete state space and let i be a fixed state. Let Copyright c 2009 by Karl Sigman 1 Stopping Times 1.1 Stopping Times: Definition Given a stochastic process X = {X n : n 0}, a random time τ is a discrete random variable on the same probability space as

More information

Math 370, Spring 2008 Prof. A.J. Hildebrand. Practice Test 1 Solutions

Math 370, Spring 2008 Prof. A.J. Hildebrand. Practice Test 1 Solutions Math 70, Spring 008 Prof. A.J. Hildebrand Practice Test Solutions About this test. This is a practice test made up of a random collection of 5 problems from past Course /P actuarial exams. Most of the

More information

COMPLETE MARKETS DO NOT ALLOW FREE CASH FLOW STREAMS

COMPLETE MARKETS DO NOT ALLOW FREE CASH FLOW STREAMS COMPLETE MARKETS DO NOT ALLOW FREE CASH FLOW STREAMS NICOLE BÄUERLE AND STEFANIE GRETHER Abstract. In this short note we prove a conjecture posed in Cui et al. 2012): Dynamic mean-variance problems in

More information

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

More information

MOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao yufeiz@mit.edu

MOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao yufeiz@mit.edu Integer Polynomials June 9, 007 Yufei Zhao yufeiz@mit.edu We will use Z[x] to denote the ring of polynomials with integer coefficients. We begin by summarizing some of the common approaches used in dealing

More information

x if x 0, x if x < 0.

x if x 0, x if x < 0. Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the

More information

Some Research Problems in Uncertainty Theory

Some Research Problems in Uncertainty Theory Journal of Uncertain Systems Vol.3, No.1, pp.3-10, 2009 Online at: www.jus.org.uk Some Research Problems in Uncertainty Theory aoding Liu Uncertainty Theory Laboratory, Department of Mathematical Sciences

More information

Definition: Suppose that two random variables, either continuous or discrete, X and Y have joint density

Definition: Suppose that two random variables, either continuous or discrete, X and Y have joint density HW MATH 461/561 Lecture Notes 15 1 Definition: Suppose that two random variables, either continuous or discrete, X and Y have joint density and marginal densities f(x, y), (x, y) Λ X,Y f X (x), x Λ X,

More information

Optimal base-stock policy for the inventory system with periodic review, backorders and sequential lead times

Optimal base-stock policy for the inventory system with periodic review, backorders and sequential lead times 44 Int. J. Inventory Research, Vol. 1, No. 1, 2008 Optimal base-stock policy for the inventory system with periodic review, backorders and sequential lead times Søren Glud Johansen Department of Operations

More information

General theory of stochastic processes

General theory of stochastic processes CHAPTER 1 General theory of stochastic processes 1.1. Definition of stochastic process First let us recall the definition of a random variable. A random variable is a random number appearing as a result

More information

F. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)

F. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein) Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p -CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p

More information

3. Continuous Random Variables

3. Continuous Random Variables 3. Continuous Random Variables A continuous random variable is one which can take any value in an interval (or union of intervals) The values that can be taken by such a variable cannot be listed. Such

More information

Permanents, Order Statistics, Outliers, and Robustness

Permanents, Order Statistics, Outliers, and Robustness Permanents, Order Statistics, Outliers, and Robustness N. BALAKRISHNAN Department of Mathematics and Statistics McMaster University Hamilton, Ontario, Canada L8S 4K bala@mcmaster.ca Received: November

More information

A Farkas-type theorem for interval linear inequalities. Jiri Rohn. Optimization Letters. ISSN Volume 8 Number 4

A Farkas-type theorem for interval linear inequalities. Jiri Rohn. Optimization Letters. ISSN Volume 8 Number 4 A Farkas-type theorem for interval linear inequalities Jiri Rohn Optimization Letters ISSN 1862-4472 Volume 8 Number 4 Optim Lett (2014) 8:1591-1598 DOI 10.1007/s11590-013-0675-9 1 23 Your article is protected

More information

Proof: A logical argument establishing the truth of the theorem given the truth of the axioms and any previously proven theorems.

Proof: A logical argument establishing the truth of the theorem given the truth of the axioms and any previously proven theorems. Math 232 - Discrete Math 2.1 Direct Proofs and Counterexamples Notes Axiom: Proposition that is assumed to be true. Proof: A logical argument establishing the truth of the theorem given the truth of the

More information

Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1

Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1 Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1 J. Zhang Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing

More information

Jointly Distributed Random Variables

Jointly Distributed Random Variables Jointly Distributed Random Variables COMP 245 STATISTICS Dr N A Heard Contents 1 Jointly Distributed Random Variables 1 1.1 Definition......................................... 1 1.2 Joint cdfs..........................................

More information

A SURVEY ON CONTINUOUS ELLIPTICAL VECTOR DISTRIBUTIONS

A SURVEY ON CONTINUOUS ELLIPTICAL VECTOR DISTRIBUTIONS A SURVEY ON CONTINUOUS ELLIPTICAL VECTOR DISTRIBUTIONS Eusebio GÓMEZ, Miguel A. GÓMEZ-VILLEGAS and J. Miguel MARÍN Abstract In this paper it is taken up a revision and characterization of the class of

More information

Section 6.1 Joint Distribution Functions

Section 6.1 Joint Distribution Functions Section 6.1 Joint Distribution Functions We often care about more than one random variable at a time. DEFINITION: For any two random variables X and Y the joint cumulative probability distribution function

More information

Data Modeling & Analysis Techniques. Probability & Statistics. Manfred Huber 2011 1

Data Modeling & Analysis Techniques. Probability & Statistics. Manfred Huber 2011 1 Data Modeling & Analysis Techniques Probability & Statistics Manfred Huber 2011 1 Probability and Statistics Probability and statistics are often used interchangeably but are different, related fields

More information

Schatten van de extreme value index voor afgeronde data (Engelse titel: Estimation of the extreme value index for imprecise data)

Schatten van de extreme value index voor afgeronde data (Engelse titel: Estimation of the extreme value index for imprecise data) Technische Universiteit Delft Faculteit Elektrotechniek, Wiskunde en Informatica Delft Institute of Applied Mathematics Schatten van de extreme value index voor afgeronde data (Engelse titel: Estimation

More information

Joint Exam 1/P Sample Exam 1

Joint Exam 1/P Sample Exam 1 Joint Exam 1/P Sample Exam 1 Take this practice exam under strict exam conditions: Set a timer for 3 hours; Do not stop the timer for restroom breaks; Do not look at your notes. If you believe a question

More information

Anomaly detection for Big Data, networks and cyber-security

Anomaly detection for Big Data, networks and cyber-security Anomaly detection for Big Data, networks and cyber-security Patrick Rubin-Delanchy University of Bristol & Heilbronn Institute for Mathematical Research Joint work with Nick Heard (Imperial College London),

More information

1. Prove that the empty set is a subset of every set.

1. Prove that the empty set is a subset of every set. 1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since

More information

VERTICES OF GIVEN DEGREE IN SERIES-PARALLEL GRAPHS

VERTICES OF GIVEN DEGREE IN SERIES-PARALLEL GRAPHS VERTICES OF GIVEN DEGREE IN SERIES-PARALLEL GRAPHS MICHAEL DRMOTA, OMER GIMENEZ, AND MARC NOY Abstract. We show that the number of vertices of a given degree k in several kinds of series-parallel labelled

More information

A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS. In memory of Rou-Huai Wang

A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS. In memory of Rou-Huai Wang A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS XAVIER CABRÉ, MANEL SANCHÓN, AND JOEL SPRUCK In memory of Rou-Huai Wang 1. Introduction In this note we consider semistable

More information

Practice with Proofs

Practice with Proofs Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using

More information

How to Sell a (Bankrupt) Company

How to Sell a (Bankrupt) Company How to Sell a (Bankrupt) Company Francesca Cornelli (London Business School and CEPR) Leonardo Felli (London School of Economics) March 2000 Abstract. The restructuring of a bankrupt company often entails

More information

An Introduction to Basic Statistics and Probability

An Introduction to Basic Statistics and Probability An Introduction to Basic Statistics and Probability Shenek Heyward NCSU An Introduction to Basic Statistics and Probability p. 1/4 Outline Basic probability concepts Conditional probability Discrete Random

More information

Lecture 13: Martingales

Lecture 13: Martingales Lecture 13: Martingales 1. Definition of a Martingale 1.1 Filtrations 1.2 Definition of a martingale and its basic properties 1.3 Sums of independent random variables and related models 1.4 Products of

More information

C2922 Economics Utility Functions

C2922 Economics Utility Functions C2922 Economics Utility Functions T.C. Johnson October 30, 2007 1 Introduction Utility refers to the perceived value of a good and utility theory spans mathematics, economics and psychology. For example,

More information

3 Multiple Discrete Random Variables

3 Multiple Discrete Random Variables 3 Multiple Discrete Random Variables 3.1 Joint densities Suppose we have a probability space (Ω, F,P) and now we have two discrete random variables X and Y on it. They have probability mass functions f

More information

Auctions with Endogenous Selling

Auctions with Endogenous Selling Auctions with Endogenous Selling Nicolae Gârleanu University of Pennsylvania and NBER Lasse Heje Pedersen New York University, CEPR, and NBER Current Version: June 20, 2006 Abstract The seminal paper by

More information

Chapter 1: The binomial asset pricing model

Chapter 1: The binomial asset pricing model Chapter 1: The binomial asset pricing model Simone Calogero April 17, 2015 Contents 1 The binomial model 1 2 1+1 dimensional stock markets 4 3 Arbitrage portfolio 8 4 Implementation of the binomial model

More information

No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics

No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results

More information

The sum of digits of polynomial values in arithmetic progressions

The sum of digits of polynomial values in arithmetic progressions The sum of digits of polynomial values in arithmetic progressions Thomas Stoll Institut de Mathématiques de Luminy, Université de la Méditerranée, 13288 Marseille Cedex 9, France E-mail: stoll@iml.univ-mrs.fr

More information

Auctioning Keywords in Online Search

Auctioning Keywords in Online Search Auctioning Keywords in Online Search Jianqing Chen The Uniersity of Calgary iachen@ucalgary.ca De Liu Uniersity of Kentucky de.liu@uky.edu Andrew B. Whinston Uniersity of Texas at Austin abw@uts.cc.utexas.edu

More information

Notes from Week 1: Algorithms for sequential prediction

Notes from Week 1: Algorithms for sequential prediction CS 683 Learning, Games, and Electronic Markets Spring 2007 Notes from Week 1: Algorithms for sequential prediction Instructor: Robert Kleinberg 22-26 Jan 2007 1 Introduction In this course we will be looking

More information

Sharing Online Advertising Revenue with Consumers

Sharing Online Advertising Revenue with Consumers Sharing Online Advertising Revenue with Consumers Yiling Chen 2,, Arpita Ghosh 1, Preston McAfee 1, and David Pennock 1 1 Yahoo! Research. Email: arpita, mcafee, pennockd@yahoo-inc.com 2 Harvard University.

More information

Solution Using the geometric series a/(1 r) = x=1. x=1. Problem For each of the following distributions, compute

Solution Using the geometric series a/(1 r) = x=1. x=1. Problem For each of the following distributions, compute Math 472 Homework Assignment 1 Problem 1.9.2. Let p(x) 1/2 x, x 1, 2, 3,..., zero elsewhere, be the pmf of the random variable X. Find the mgf, the mean, and the variance of X. Solution 1.9.2. Using the

More information

Chapters 5. Multivariate Probability Distributions

Chapters 5. Multivariate Probability Distributions Chapters 5. Multivariate Probability Distributions Random vectors are collection of random variables defined on the same sample space. Whenever a collection of random variables are mentioned, they are

More information

INTRODUCTORY SET THEORY

INTRODUCTORY SET THEORY M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H-1088 Budapest, Múzeum krt. 6-8. CONTENTS 1. SETS Set, equal sets, subset,

More information