All pay auctions with certain and uncertain prizes a comment

Size: px
Start display at page:

Download "All pay auctions with certain and uncertain prizes a comment"

Transcription

1 CENTER FOR RESEARC IN ECONOMICS AND MANAGEMENT CREAM Publiction No All py uctions with certin nd uncertin prizes comment Christin Riis

2 All py uctions with certin nd uncertin prizes comment Christin Riis Norwegin Business School April 24, 2015 Abstrct In the importnt contribution "All py uctions with certin nd uncertin prizes" published in Gmes nd Economic Behvior My 2014, Minchu nd nd Sel nlyze n ll py uction with multiple prizes. The specific feture of the model is tht ll vlutions re common except for the vlution of one of the prizes, for which contestnts hve privte vlutions. owever, the equilibrium chrcteriztion derived in the pper is incorrect. This note derives the correct equilibrium of the model. Key words: All py uctions, uncertin prizes JE codes: D 44, D82, J31, J41 Minchu nd Sel (2014) (herefter MS) consider n ll py uction with multiple prizes. The specific feture of their model is tht ll vlutions re common except for the vlution of one of the prizes. For this prticulr prize contestnts hve privte vlutions, independently drwn from common distribution. The uthors clim tht the equilibrium bid function is symmetric nd monotone in the vlution of the uncertin prize. owever this is only the cse if the uncertin prize hs the highest or lowest vlue. It is not if the uncertin prize hs n intermedite vlue, which is MS min cse. et me first provide intuition for why MS result fils. I follow MS nottion. There re n contestnts competing for m different prizes. The highest bidder wins the most vluble prize, with common vlue of v n. 1

3 The second highest bidder wins the second most vluble prize v n 1 nd so on. The uncertin prize is indexed n j +1, with privte vlue, denoted by, drwn independently from distribution F ( ) with support on [, +2 ]. Suppose now tht the equilibrium bid function β() is strictly incresing. The probbility tht plyer who bids ccording to vlution s will win the uncertin prize is then (n 1)! (j 1)! (n j)! F (s)n j (1 F (s)) j 1. Obviously this probbility is non-monotonic function of s, strictly incresing in s for low bid levels, nd strictly decresing for high bid levels. The probbility of winning the uncertin prize reches mximum t vlution â implicitly determined by F (â) = n j n 1 It follows from stndrd single crossing conditions 1 tht strictly incresing bid function on bid segment is prt of seprting equilibrium only if the win probbility is incresing in vlution s the plyer with high vlution hs stronger incentive to bid ggressively thn bidder with lower vlution. In other words, strictly incresing bid function is incomptible with optiml bidding behvior if the probbility of winning the uncertin prize declines with the bid level in which cse bidder with prticulrly high vlution for the uncertin prize will lower her bid. Thus, seprting equilibrium cnnot be monotone in the vlution of the uncertin prize in our setting. I will now by construction derive the seprting equilibrium. Suppose the number of plyers strictly exceeds the number of prizes, ming generliztion of the result strightforwrd. A stndrd chrcteristic of seprting equilibrium in ll py uctions with ex nte identicl contestnts is the following: The plyer with the lowest possible vlution for the uncertin prize obtins zero pyoff. An impliction of this is tht the bid function must hve upper support equl to v n, the vlue of the highest prize. Otherwise the plyer with the 1 For t detiled exposition see Athey (2001) 2

4 lowest possible vlution obtins strictly positive rent by jumping to the upper support. As the lower support must be zero, seprting equilibrium will be mpping from vlution ɛ [, +2 ] to bids β on [0, v n ]. 2 Observe tht the equilibrium cn be nchored in the following observtion: For high bids, the probbility of winning the uncertin prize declines s the bid level increses. Therefore, single crossing indictes tht the bid function must hve declining segment t high vlutions. For low vlutions the equilibrium bid function must be incresing. A conjecture would be tht the equilibrium bid function hs the shpe illustrted in figure 1, consisting of two bid segments: declining segment, β (), for those bid levels t which the win probbility (for the uncertin prize) declines with the bid; nd n incresing segment β () for bid levels t which the win probbility increses. A contestnt with vlution then rndomizes between the two bid levels, nd chooses β () with probbility q(). With one exception: the contestnt with the highest possible vlution for the uncertin prize, = +2, bids certin mount, corresponding to the bid level tht mximizes the probbility of winning the uncertin prize. Note tht the plyer with the lowest possible vlution for the uncertin prize, =, rndomizes between bidding zero nd v n. Figure 1 I will now show tht the seprting equilibrium indeed stisfies this pt- 2 It is lso stndrd tht the bid distribution must be tomless. 3

5 tern. Denote by i () the equilibrium probbility tht plyer with vlution wins the prize indexed i, given tht she bids ccording to bid segment β (), =,. The probbility tht this plyer will win the uncertin prize, indexed n j + 1, is then () = (1) ( ) n j ( (n 1)! ) j 1 q(z)f(z)dz 1 q(z)f(z)dz (i 1)! (n i)! if she bids β () 3, nd () = (2) ( ) n j ( (n 1)! ) j 1 1 (1 q(z)) f(z)dz (1 q(z)) f(z)dz (j 1)! (n j)! if she bids β (). et us chrcterize the equilibrium bid functions following MS procedure, which is lso the stndrd procedure. A plyer with vlution behves s plyer with vlution s in order to mximize ( =, ) mx s i i (s)v i + (s) β (s) The necessry conditions yields the following pir of differentil equtions β (s) = i d i(s) v i + ds d ds (s) 3 To see this, note tht bid s wins the prize n j + 1 if n exct number j 1 of the bidder s contestnts outbid her (note tht there is probbility 1 s q(z)f(z)dz tht single contestnt will choose bid bove s), nd the remining n j contestnts will bid below s. 4

6 The first order condition evluted t s = yields the cndidte bid functions 4 d β () = β ( ) + i (x) dx v d (x) i + x dx dx = β ( ) + i i i ()v i i i ( )v i + [ d If = we hve β ( ) = 0 nd thus i () = 0, s the plyer loses with certinty. If = we hve β ( ) = v 1 nd thus n() = 1, s the plyer in this cse is certin to win the most vluble prize. In both cses, the cndidte bid function is (where the second equlity follows from integrtion by prts) β () = = i i This yields ssocited utilities i ()v i + i ()v i + [ d dx () ] (x) xdx [ ] (x) dx dx ] (x) xdx U () = (x)dx Thus the bidder s rent is ssocited with vlution of the uncertin prize bove the lower support, exctly s described by MS. Note tht β (+2 ) = β (+2 ) since i (+2) = i (+2) for ll i = n m + 1,.., n 5, which confirms tht the two bid segments meet t = +2. It remins to determine q(). The equilibrium probbility function, q( ), mes ech contestnt indifferent between choosing β () nd β (). This 4 Note tht i () = i ( ) + d i (x) dx dx. 5 This cn esily be checed by inserting +2 in the integrl limits in (1) nd (2), nd generlizing to ny i. 5

7 mens tht we hve to find function q( ) such tht U () = U () for every type, thus (x)dx = (x)dx (3) must lwys hold. Obviously this is equivlent to the following condition: for ny ɛ[, +2 ] we hve () = () (4) ence, in seprting equilibrium, for ech type, the probbility of winning the uncertin price is independent of the choice between β () nd β (). Accordingly, the net cost of submitting high bid, β () β (), cncels out with the net gin from the higher probbility of winning one of the certin nd more vluble prizes. Substituting from (1) nd (2) condition (4) cn be written = ( ( q(z)f(z)dz 1 ) j 1 ( (1 q(z)) f(z)dz 1 q(z)f(z)dz ) j 1 ( ) n j (5) (1 q(z)) f(z)dz ) n j which must hold for ll ɛ[, +2 ]. q( ) is the solution to the functionl eqution given by (5). One cse is prticulrly simple. If the uncertin prize is the medin prize, j = (n + 1)/2, the equilibrium q is constnt, q() = 1/2 for ll. To see this, insert q() = 1/2 in (5) nd solve the integrls, which yields 6 ( ) (n 1)/2 ( 1 2 F () 1 1 ) (n 1)/2 2 F () = (1 12 ) (n 1)/2 ( ) (n 1)/2 12 F () F () 6 An intuitive rgument for existence more generlly goes s follows: note tht the equilibrium condition hs recursive structure in tht (5) determines q( ) up to ny rbitrry, independent of the shpe of q( ) bove. Furthermore since () = () is n identity, their derivtives re equl, d ()/d = d ()/d, nd since d ()/d is proportionl to q() nd d ()/d is proportionl to 1 q(), nd given the equilibrium function q( ) up to, q() is uniquely determined by the condition d ()/d = d ()/d. 6

8 Note tht the equilibrium is fully reveling, despite the fct tht plyers rndomize bid levels, s ech pir of possible bids is unique for the plyer s vlution. 1 References Athey, Susn. "Single crossing properties nd the existence of pure strtegy equilibri in gmes of incomplete informtion." Econometric 69.4 (2001): Minchu, Yizhq, nd Aner Sel. "All-py uctions with certin nd uncertin prizes." Gmes nd Economic Behvior 88 (2014):

9 Center for Reserch in Economics nd Mngement (CREAM) ndelshøysolen BI / BI Norwegin Business School 0442 Oslo, Norwy The objective of CREAM is to provide reserch nd nlysis in the re of industril economics nd lbor economics with pplictions to mngement, nd provide reserch-bsed nlysis for decision mers in public nd privte sector. ISSN X

Reasoning to Solve Equations and Inequalities

Reasoning to Solve Equations and Inequalities Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing

More information

Lecture 3 Gaussian Probability Distribution

Lecture 3 Gaussian Probability Distribution Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike

More information

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered: Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you

More information

Econ 4721 Money and Banking Problem Set 2 Answer Key

Econ 4721 Money and Banking Problem Set 2 Answer Key Econ 472 Money nd Bnking Problem Set 2 Answer Key Problem (35 points) Consider n overlpping genertions model in which consumers live for two periods. The number of people born in ech genertion grows in

More information

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding 1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde

More information

Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur

Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur Module Anlysis of Stticlly Indeterminte Structures by the Mtrix Force Method Version CE IIT, Khrgpur esson 9 The Force Method of Anlysis: Bems (Continued) Version CE IIT, Khrgpur Instructionl Objectives

More information

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( ) Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +

More information

9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes

9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is so-clled becuse when the sclr product of two vectors

More information

9 CONTINUOUS DISTRIBUTIONS

9 CONTINUOUS DISTRIBUTIONS 9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete

More information

Lower Bound for Envy-Free and Truthful Makespan Approximation on Related Machines

Lower Bound for Envy-Free and Truthful Makespan Approximation on Related Machines Lower Bound for Envy-Free nd Truthful Mespn Approximtion on Relted Mchines Lis Fleischer Zhenghui Wng July 14, 211 Abstrct We study problems of scheduling jobs on relted mchines so s to minimize the mespn

More information

UNIVERSITY OF NOTTINGHAM. Discussion Papers in Economics STRATEGIC SECOND SOURCING IN A VERTICAL STRUCTURE

UNIVERSITY OF NOTTINGHAM. Discussion Papers in Economics STRATEGIC SECOND SOURCING IN A VERTICAL STRUCTURE UNVERSTY OF NOTTNGHAM Discussion Ppers in Economics Discussion Pper No. 04/15 STRATEGC SECOND SOURCNG N A VERTCAL STRUCTURE By Arijit Mukherjee September 004 DP 04/15 SSN 10-438 UNVERSTY OF NOTTNGHAM Discussion

More information

MATH 150 HOMEWORK 4 SOLUTIONS

MATH 150 HOMEWORK 4 SOLUTIONS MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive

More information

4.11 Inner Product Spaces

4.11 Inner Product Spaces 314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces

More information

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one. 5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous rel-vlued

More information

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of

More information

Integration by Substitution

Integration by Substitution Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is

More information

College Admissions with Entrance Exams: Centralized versus Decentralized

College Admissions with Entrance Exams: Centralized versus Decentralized Is E. Hflir Rustmdjn Hkimov Dorothe Kübler Morimitsu Kurino College Admissions with Entrnce Exms: Centrlized versus Decentrlized Discussion Pper SP II 2014 208 October 2014 (WZB Berlin Socil Science Center

More information

Vectors 2. 1. Recap of vectors

Vectors 2. 1. Recap of vectors Vectors 2. Recp of vectors Vectors re directed line segments - they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms

More information

CHAPTER 11 Numerical Differentiation and Integration

CHAPTER 11 Numerical Differentiation and Integration CHAPTER 11 Numericl Differentition nd Integrtion Differentition nd integrtion re bsic mthemticl opertions with wide rnge of pplictions in mny res of science. It is therefore importnt to hve good methods

More information

Basic Analysis of Autarky and Free Trade Models

Basic Analysis of Autarky and Free Trade Models Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently

More information

Week 7 - Perfect Competition and Monopoly

Week 7 - Perfect Competition and Monopoly Week 7 - Perfect Competition nd Monopoly Our im here is to compre the industry-wide response to chnges in demnd nd costs by monopolized industry nd by perfectly competitive one. We distinguish between

More information

and thus, they are similar. If k = 3 then the Jordan form of both matrices is

and thus, they are similar. If k = 3 then the Jordan form of both matrices is Homework ssignment 11 Section 7. pp. 249-25 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If

More information

Graphs on Logarithmic and Semilogarithmic Paper

Graphs on Logarithmic and Semilogarithmic Paper 0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl

More information

COMPONENTS: COMBINED LOADING

COMPONENTS: COMBINED LOADING LECTURE COMPONENTS: COMBINED LOADING Third Edition A. J. Clrk School of Engineering Deprtment of Civil nd Environmentl Engineering 24 Chpter 8.4 by Dr. Ibrhim A. Asskkf SPRING 2003 ENES 220 Mechnics of

More information

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers. 2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this

More information

COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE. Skandza, Stockholm ABSTRACT

COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE. Skandza, Stockholm ABSTRACT COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE Skndz, Stockholm ABSTRACT Three methods for fitting multiplictive models to observed, cross-clssified

More information

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100 hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by

More information

Euler Euler Everywhere Using the Euler-Lagrange Equation to Solve Calculus of Variation Problems

Euler Euler Everywhere Using the Euler-Lagrange Equation to Solve Calculus of Variation Problems Euler Euler Everywhere Using the Euler-Lgrnge Eqution to Solve Clculus of Vrition Problems Jenine Smllwood Principles of Anlysis Professor Flschk My 12, 1998 1 1. Introduction Clculus of vritions is brnch

More information

How To Understand The Theory Of Inequlities

How To Understand The Theory Of Inequlities Ostrowski Type Inequlities nd Applictions in Numericl Integrtion Edited By: Sever S Drgomir nd Themistocles M Rssis SS Drgomir) School nd Communictions nd Informtics, Victori University of Technology,

More information

MODULE 3. 0, y = 0 for all y

MODULE 3. 0, y = 0 for all y Topics: Inner products MOULE 3 The inner product of two vectors: The inner product of two vectors x, y V, denoted by x, y is (in generl) complex vlued function which hs the following four properties: i)

More information

Physics 43 Homework Set 9 Chapter 40 Key

Physics 43 Homework Set 9 Chapter 40 Key Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nm-wide region t x

More information

Math 135 Circles and Completing the Square Examples

Math 135 Circles and Completing the Square Examples Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for

More information

Warm-up for Differential Calculus

Warm-up for Differential Calculus Summer Assignment Wrm-up for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:

More information

SPECIAL PRODUCTS AND FACTORIZATION

SPECIAL PRODUCTS AND FACTORIZATION MODULE - Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come

More information

The Impact of Oligopolistic Competition in Networks

The Impact of Oligopolistic Competition in Networks OPERATIONS RESEARCH Vol. 57, No. 6, November December 2009, pp. 1421 1437 issn 0030-364X eissn 1526-5463 09 5706 1421 informs doi 10.1287/opre.1080.0653 2009 INFORMS The Impct of Oligopolistic Competition

More information

6.2 Volumes of Revolution: The Disk Method

6.2 Volumes of Revolution: The Disk Method mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine so-clled volumes of

More information

EQUATIONS OF LINES AND PLANES

EQUATIONS OF LINES AND PLANES EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in point-direction nd twopoint

More information

g(y(a), y(b)) = o, B a y(a)+b b y(b)=c, Boundary Value Problems Lecture Notes to Accompany

g(y(a), y(b)) = o, B a y(a)+b b y(b)=c, Boundary Value Problems Lecture Notes to Accompany Lecture Notes to Accompny Scientific Computing An Introductory Survey Second Edition by Michel T Heth Boundry Vlue Problems Side conditions prescribing solution or derivtive vlues t specified points required

More information

Distributions. (corresponding to the cumulative distribution function for the discrete case).

Distributions. (corresponding to the cumulative distribution function for the discrete case). Distributions Recll tht n integrble function f : R [,] such tht R f()d = is clled probbility density function (pdf). The distribution function for the pdf is given by F() = (corresponding to the cumultive

More information

Factoring Polynomials

Factoring Polynomials Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles

More information

Regular Sets and Expressions

Regular Sets and Expressions Regulr Sets nd Expressions Finite utomt re importnt in science, mthemtics, nd engineering. Engineers like them ecuse they re super models for circuits (And, since the dvent of VLSI systems sometimes finite

More information

The Velocity Factor of an Insulated Two-Wire Transmission Line

The Velocity Factor of an Insulated Two-Wire Transmission Line The Velocity Fctor of n Insulted Two-Wire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the

More information

Optiml Control of Seril, Multi-Echelon Inventory (E&I) & Mixed Erlng demnds

Optiml Control of Seril, Multi-Echelon Inventory (E&I) & Mixed Erlng demnds Optiml Control of Seril, Multi-Echelon Inventory/Production Systems with Periodic Btching Geert-Jn vn Houtum Deprtment of Technology Mngement, Technische Universiteit Eindhoven, P.O. Box 513, 56 MB, Eindhoven,

More information

How To Set Up A Network For Your Business

How To Set Up A Network For Your Business Why Network is n Essentil Productivity Tool for Any Smll Business TechAdvisory.org SME Reports sponsored by Effective technology is essentil for smll businesses looking to increse their productivity. Computer

More information

INTERCHANGING TWO LIMITS. Zoran Kadelburg and Milosav M. Marjanović

INTERCHANGING TWO LIMITS. Zoran Kadelburg and Milosav M. Marjanović THE TEACHING OF MATHEMATICS 2005, Vol. VIII, 1, pp. 15 29 INTERCHANGING TWO LIMITS Zorn Kdelburg nd Milosv M. Mrjnović This pper is dedicted to the memory of our illustrious professor of nlysis Slobodn

More information

Research Article Competition with Online and Offline Demands considering Logistics Costs Based on the Hotelling Model

Research Article Competition with Online and Offline Demands considering Logistics Costs Based on the Hotelling Model Mthemticl Problems in Engineering Volume 4, Article ID 678, pges http://dx.doi.org/.55/4/678 Reserch Article Competition with Online nd Offline Demnds considering Logistics Costs Bsed on the Hotelling

More information

RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS

RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS Known for over 500 yers is the fct tht the sum of the squres of the legs of right tringle equls the squre of the hypotenuse. Tht is +b c. A simple proof is

More information

Pure C4. Revision Notes

Pure C4. Revision Notes Pure C4 Revision Notes Mrch 0 Contents Core 4 Alger Prtil frctions Coordinte Geometry 5 Prmetric equtions 5 Conversion from prmetric to Crtesin form 6 Are under curve given prmetriclly 7 Sequences nd

More information

On the Robustness of Most Probable Explanations

On the Robustness of Most Probable Explanations On the Robustness of Most Probble Explntions Hei Chn School of Electricl Engineering nd Computer Science Oregon Stte University Corvllis, OR 97330 chnhe@eecs.oregonstte.edu Adnn Drwiche Computer Science

More information

15.6. The mean value and the root-mean-square value of a function. Introduction. Prerequisites. Learning Outcomes. Learning Style

15.6. The mean value and the root-mean-square value of a function. Introduction. Prerequisites. Learning Outcomes. Learning Style The men vlue nd the root-men-squre vlue of function 5.6 Introduction Currents nd voltges often vry with time nd engineers my wish to know the verge vlue of such current or voltge over some prticulr time

More information

Space Vector Pulse Width Modulation Based Induction Motor with V/F Control

Space Vector Pulse Width Modulation Based Induction Motor with V/F Control Interntionl Journl of Science nd Reserch (IJSR) Spce Vector Pulse Width Modultion Bsed Induction Motor with V/F Control Vikrmrjn Jmbulingm Electricl nd Electronics Engineering, VIT University, Indi Abstrct:

More information

2005-06 Second Term MAT2060B 1. Supplementary Notes 3 Interchange of Differentiation and Integration

2005-06 Second Term MAT2060B 1. Supplementary Notes 3 Interchange of Differentiation and Integration Source: http://www.mth.cuhk.edu.hk/~mt26/mt26b/notes/notes3.pdf 25-6 Second Term MAT26B 1 Supplementry Notes 3 Interchnge of Differentition nd Integrtion The theme of this course is bout vrious limiting

More information

The Relative Advantages of Flexible versus Designated Manufacturing Technologies

The Relative Advantages of Flexible versus Designated Manufacturing Technologies The Reltive Advntges of Flexible versus Designted Mnufcturing Technologies George Normn Cummings Professor of Entrepreneurship nd Business Economics Tufts University Medford MA 055 USA e-mil: george.normn@tufts.edu

More information

Integration. 148 Chapter 7 Integration

Integration. 148 Chapter 7 Integration 48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology

More information

How To Network A Smll Business

How To Network A Smll Business Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology

More information

Algebra Review. How well do you remember your algebra?

Algebra Review. How well do you remember your algebra? Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then

More information

Operations with Polynomials

Operations with Polynomials 38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply

More information

3 The Utility Maximization Problem

3 The Utility Maximization Problem 3 The Utility Mxiiztion Proble We hve now discussed how to describe preferences in ters of utility functions nd how to forulte siple budget sets. The rtionl choice ssuption, tht consuers pick the best

More information

Online Multicommodity Routing with Time Windows

Online Multicommodity Routing with Time Windows Konrd-Zuse-Zentrum für Informtionstechnik Berlin Tkustrße 7 D-14195 Berlin-Dhlem Germny TOBIAS HARKS 1 STEFAN HEINZ MARC E. PFETSCH TJARK VREDEVELD 2 Online Multicommodity Routing with Time Windows 1 Institute

More information

Babylonian Method of Computing the Square Root: Justifications Based on Fuzzy Techniques and on Computational Complexity

Babylonian Method of Computing the Square Root: Justifications Based on Fuzzy Techniques and on Computational Complexity Bbylonin Method of Computing the Squre Root: Justifictions Bsed on Fuzzy Techniques nd on Computtionl Complexity Olg Koshelev Deprtment of Mthemtics Eduction University of Texs t El Pso 500 W. University

More information

2. Transaction Cost Economics

2. Transaction Cost Economics 3 2. Trnsction Cost Economics Trnsctions Trnsctions Cn Cn Be Be Internl Internl or or Externl Externl n n Orgniztion Orgniztion Trnsctions Trnsctions occur occur whenever whenever good good or or service

More information

Unleashing the Power of Cloud

Unleashing the Power of Cloud Unleshing the Power of Cloud A Joint White Pper by FusionLyer nd NetIQ Copyright 2015 FusionLyer, Inc. All rights reserved. No prt of this publiction my be reproduced, stored in retrievl system, or trnsmitted,

More information

AREA OF A SURFACE OF REVOLUTION

AREA OF A SURFACE OF REVOLUTION AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.

More information

Education and Optimal Dynamic Taxation: The Role of Income-Contingent Student Loans

Education and Optimal Dynamic Taxation: The Role of Income-Contingent Student Loans University of Zurich Deprtment of Economics Center for Institutions, Policy nd Culture in the Development Process Working Pper Series Working Pper No. 421 Eduction nd Optiml Dynmic Txtion: The Role of

More information

** Dpt. Chemical Engineering, Kasetsart University, Bangkok 10900, Thailand

** Dpt. Chemical Engineering, Kasetsart University, Bangkok 10900, Thailand Modelling nd Simultion of hemicl Processes in Multi Pulse TP Experiment P. Phnwdee* S.O. Shekhtmn +. Jrungmnorom** J.T. Gleves ++ * Dpt. hemicl Engineering, Ksetsrt University, Bngkok 10900, Thilnd + Dpt.hemicl

More information

TITLE THE PRINCIPLES OF COIN-TAP METHOD OF NON-DESTRUCTIVE TESTING

TITLE THE PRINCIPLES OF COIN-TAP METHOD OF NON-DESTRUCTIVE TESTING TITLE THE PRINCIPLES OF COIN-TAP METHOD OF NON-DESTRUCTIVE TESTING Sung Joon Kim*, Dong-Chul Che Kore Aerospce Reserch Institute, 45 Eoeun-Dong, Youseong-Gu, Dejeon, 35-333, Kore Phone : 82-42-86-231 FAX

More information

Value Function Approximation using Multiple Aggregation for Multiattribute Resource Management

Value Function Approximation using Multiple Aggregation for Multiattribute Resource Management Journl of Mchine Lerning Reserch 9 (2008) 2079-2 Submitted 8/08; Published 0/08 Vlue Function Approximtion using Multiple Aggregtion for Multittribute Resource Mngement Abrhm George Wrren B. Powell Deprtment

More information

Probability m odels on horse-race outcomes

Probability m odels on horse-race outcomes Jour nl of Applied Sttistics, Vol. 25, No. 2, 1998, 221± 229 Probbility m odels on horse-rce outcomes M UKHTAR M. ALI, Deprtment of Economics, University of Kentucy, USA SUMMARY A number of models hve

More information

Review guide for the final exam in Math 233

Review guide for the final exam in Math 233 Review guide for the finl exm in Mth 33 1 Bsic mteril. This review includes the reminder of the mteril for mth 33. The finl exm will be cumultive exm with mny of the problems coming from the mteril covered

More information

Health insurance marketplace What to expect in 2014

Health insurance marketplace What to expect in 2014 Helth insurnce mrketplce Wht to expect in 2014 33096VAEENBVA 06/13 The bsics of the mrketplce As prt of the Affordble Cre Act (ACA or helth cre reform lw), strting in 2014 ALL Americns must hve minimum

More information

Physics 6010, Fall 2010 Symmetries and Conservation Laws: Energy, Momentum and Angular Momentum Relevant Sections in Text: 2.6, 2.

Physics 6010, Fall 2010 Symmetries and Conservation Laws: Energy, Momentum and Angular Momentum Relevant Sections in Text: 2.6, 2. Physics 6010, Fll 2010 Symmetries nd Conservtion Lws: Energy, Momentum nd Angulr Momentum Relevnt Sections in Text: 2.6, 2.7 Symmetries nd Conservtion Lws By conservtion lw we men quntity constructed from

More information

Section 5-4 Trigonometric Functions

Section 5-4 Trigonometric Functions 5- Trigonometric Functions Section 5- Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form

More information

Experiment 6: Friction

Experiment 6: Friction Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht

More information

Treatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.

Treatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3. The nlysis of vrince (ANOVA) Although the t-test is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the t-test cn be used to compre the mens of only

More information

Applications to Physics and Engineering

Applications to Physics and Engineering Section 7.5 Applictions to Physics nd Engineering Applictions to Physics nd Engineering Work The term work is used in everydy lnguge to men the totl mount of effort required to perform tsk. In physics

More information

The Riemann Integral. Chapter 1

The Riemann Integral. Chapter 1 Chpter The Riemnn Integrl now of some universities in Englnd where the Lebesgue integrl is tught in the first yer of mthemtics degree insted of the Riemnn integrl, but now of no universities in Englnd

More information

Or more simply put, when adding or subtracting quantities, their uncertainties add.

Or more simply put, when adding or subtracting quantities, their uncertainties add. Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re

More information

DlNBVRGH + Sickness Absence Monitoring Report. Executive of the Council. Purpose of report

DlNBVRGH + Sickness Absence Monitoring Report. Executive of the Council. Purpose of report DlNBVRGH + + THE CITY OF EDINBURGH COUNCIL Sickness Absence Monitoring Report Executive of the Council 8fh My 4 I.I...3 Purpose of report This report quntifies the mount of working time lost s result of

More information

Small Business Cloud Services

Small Business Cloud Services Smll Business Cloud Services Summry. We re thick in the midst of historic se-chnge in computing. Like the emergence of personl computers, grphicl user interfces, nd mobile devices, the cloud is lredy profoundly

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology

More information

Lecture 5. Inner Product

Lecture 5. Inner Product Lecture 5 Inner Product Let us strt with the following problem. Given point P R nd line L R, how cn we find the point on the line closest to P? Answer: Drw line segment from P meeting the line in right

More information

2 DIODE CLIPPING and CLAMPING CIRCUITS

2 DIODE CLIPPING and CLAMPING CIRCUITS 2 DIODE CLIPPING nd CLAMPING CIRCUITS 2.1 Ojectives Understnding the operting principle of diode clipping circuit Understnding the operting principle of clmping circuit Understnding the wveform chnge of

More information

Health insurance exchanges What to expect in 2014

Health insurance exchanges What to expect in 2014 Helth insurnce exchnges Wht to expect in 2014 33096CAEENABC 02/13 The bsics of exchnges As prt of the Affordble Cre Act (ACA or helth cre reform lw), strting in 2014 ALL Americns must hve minimum mount

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology

More information

This paper considers two independent firms that invest in resources such as capacity or inventory based on

This paper considers two independent firms that invest in resources such as capacity or inventory based on MANAGEMENT SCIENCE Vol. 5, No., December 006, pp. 93 99 issn 005-909 eissn 56-550 06 5 93 informs doi 0.87/mnsc.060.0574 006 INFORMS Strtegic Investments, Trding, nd Pricing Under Forecst Updting Jiri

More information

Helicopter Theme and Variations

Helicopter Theme and Variations Helicopter Theme nd Vritions Or, Some Experimentl Designs Employing Pper Helicopters Some possible explntory vribles re: Who drops the helicopter The length of the rotor bldes The height from which the

More information

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive

More information

SUBSTITUTION I.. f(ax + b)

SUBSTITUTION I.. f(ax + b) Integrtion SUBSTITUTION I.. f(x + b) Grhm S McDonld nd Silvi C Dll A Tutoril Module for prctising the integrtion of expressions of the form f(x + b) Tble of contents Begin Tutoril c 004 g.s.mcdonld@slford.c.uk

More information

Math 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1.

Math 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1. Mth 4, Homework Assignment. Prove tht two nonverticl lines re perpendiculr if nd only if the product of their slopes is. Proof. Let l nd l e nonverticl lines in R of slopes m nd m, respectively. Suppose

More information

EE247 Lecture 4. For simplicity, will start with all pole ladder type filters. Convert to integrator based form- example shown

EE247 Lecture 4. For simplicity, will start with all pole ladder type filters. Convert to integrator based form- example shown EE247 Lecture 4 Ldder type filters For simplicity, will strt with ll pole ldder type filters Convert to integrtor bsed form exmple shown Then will ttend to high order ldder type filters incorporting zeros

More information

Week 11 - Inductance

Week 11 - Inductance Week - Inductnce November 6, 202 Exercise.: Discussion Questions ) A trnsformer consists bsiclly of two coils in close proximity but not in electricl contct. A current in one coil mgneticlly induces n

More information

FAULT TREES AND RELIABILITY BLOCK DIAGRAMS. Harry G. Kwatny. Department of Mechanical Engineering & Mechanics Drexel University

FAULT TREES AND RELIABILITY BLOCK DIAGRAMS. Harry G. Kwatny. Department of Mechanical Engineering & Mechanics Drexel University SYSTEM FAULT AND Hrry G. Kwtny Deprtment of Mechnicl Engineering & Mechnics Drexel University OUTLINE SYSTEM RBD Definition RBDs nd Fult Trees System Structure Structure Functions Pths nd Cutsets Reliility

More information

Lectures 8 and 9 1 Rectangular waveguides

Lectures 8 and 9 1 Rectangular waveguides 1 Lectures 8 nd 9 1 Rectngulr wveguides y b x z Consider rectngulr wveguide with 0 < x b. There re two types of wves in hollow wveguide with only one conductor; Trnsverse electric wves

More information

Module Summary Sheets. C3, Methods for Advanced Mathematics (Version B reference to new book) Topic 2: Natural Logarithms and Exponentials

Module Summary Sheets. C3, Methods for Advanced Mathematics (Version B reference to new book) Topic 2: Natural Logarithms and Exponentials MEI Mthemtics in Ection nd Instry Topic : Proof MEI Structured Mthemtics Mole Summry Sheets C, Methods for Anced Mthemtics (Version B reference to new book) Topic : Nturl Logrithms nd Eponentils Topic

More information

Quasi-log concavity conjecture and its applications in statistics

Quasi-log concavity conjecture and its applications in statistics Wen et l. Journl of Inequlities nd Applictions 2014, 2014:339 http://www.journlofinequlitiesndpplictions.com/content/2014/1/339 R E S E A R C H Open Access Qusi-log concvity conjecture nd its pplictions

More information

QUADRATURE METHODS. July 19, 2011. Kenneth L. Judd. Hoover Institution

QUADRATURE METHODS. July 19, 2011. Kenneth L. Judd. Hoover Institution QUADRATURE METHODS Kenneth L. Judd Hoover Institution July 19, 2011 1 Integrtion Most integrls cnnot be evluted nlyticlly Integrls frequently rise in economics Expected utility Discounted utility nd profits

More information

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Nme Chpter Eponentil nd Logrithmic Functions Section. Eponentil Functions nd Their Grphs Objective: In this lesson ou lerned how to recognize, evlute, nd grph eponentil functions. Importnt Vocbulr Define

More information

Project 6 Aircraft static stability and control

Project 6 Aircraft static stability and control Project 6 Aircrft sttic stbility nd control The min objective of the project No. 6 is to compute the chrcteristics of the ircrft sttic stbility nd control chrcteristics in the pitch nd roll chnnel. The

More information