Unobserved Correlation in Ascending Auctions: Example And Extensions
|
|
- Lucinda Sparks
- 5 years ago
- Views:
Transcription
1 Unobseved Coelation in Ascending Auctions: Example And Extensions Daniel Quint Univesity of Wisconsin Novembe 2009 Intoduction In pivate-value ascending auctions, the winning bidde s willingness to pay is not obseved. Unde assumptions weake than independent pivate values, the joint distibution of bidde valuations is not identified see Athey and Haile 2002, so the expected evenue at a positive eseve pice, and the eseve pice that would maximize expected evenue, ae not uniquely pinned down. In a sepaate pape, Quint 2008, I calculate tight uppe and lowe bounds on these two measues fo the symmetic affiliated pivate values case; the uppe bounds coincide with the values achieved unde the special case of independent pivate values. Hee, I give an illustative example and seveal extensions. 2 Model A selle has one indivisible object to sell, and values it at. Thee ae n potential buyes, with pivate values v,..., v n whose joint distibution f is symmetic and has bounded suppot [v, v] n. Let v v 2 v n be the ode statistics of the values, and F i the cumulative distibution function 2 of v i. I conside a stylized vesion of an ascending auction: the selle announces a eseve pice, and as long as at least one buye s valuation exceeds this pice, the object is sold to the buye with the highest valuation, at a pice which is the geate of the eseve pice and the second-highest valuation. I assume that the distibution of this second-highest valuation F 2 is known exactly, but that no futhe infomation is available about F. 3 Note that expected selle pofit can be witten as π = F 2 F + Given the distibution F 2, define H implicitly by v df 2 v H F 2 = nn s n 2 sds Social Science Bldg., 80 Obsevatoy D., Madison WI 53706, United States; dquint@ssc.wisc.edu 2 Cumulative distibution functions in this pape exclude any mass at the point being consideed, that is, F i Pv i <, not Pv i. 3 The evenue assumption, and pecise knowledge of F 2, would hold exactly fo second-pice sealed-bid auctions and button auctions, and up to a bid incement fo fist-pice auctions with poxy bidding and any ascending auction without jump bids.
2 o, equivalently, F 2 = nh n n H n, and define π v df 2 v π F 2 H n + v df 2 v 3 The main esult fom Quint 2008: Theoem. Suppose biddes have pivate values which ae symmetic and affiliated.. Fo any >, expected evenue π is bounded above by π and below by π, and both bounds ae tight 2. Suppose in addition that π is continuous, diffeentiable, and stictly quasiconcave; let I be its maximize. Then the optimal eseve pice is bounded above by I and below by, and both bounds ae tight 3 An Example With A Paamete Fo Coelation Let ɛ, ɛ 2,..., ɛ n be i.i.d. daws fom the unifom distibution on [0, ], and let ɛ ɛ 2... ɛ n be thei ode statistics. Let biddes i s pivate value be v i = ɛ 2 + ɛ i 4 Since v 2 = ɛ 2, the obseved distibution F 2 does not depend on ; thus, this example allows us to paameteize the coelation between bidde values while holding fixed the data that would be obseved. = 0 coesponds to the case of independent pivate values, while = would be pefectly coelated values. Fo simplicity, let = 0. Result. Fo <, expected evenue is π = n n n+ n+ n+ + n n+ fo n n n+ n+ n+ + n n+ + + n n fo > 5 and the evenue-maximizing eseve pice is = n n + n+ 6 both of which ae stictly deceasing in. Since = 0 coesponds to independent pivate values, both π and ae bounded above by thei value unde IPV, and both ae deceasing in the degee of unobseved coelation. 2
3 4 Relaxing Affiliation Theoem 2. Suppose that v, v 2,..., v n ae conditionally independent 4 but not necessaily affiliated.. The same evenue bounds hold: π π π, with both bounds being tight. 2. The lowe bound on is still, and still tight. 3. It is not necessaily tue that I = ag max π. An uppe bound not tight on is povided by H n F 2 vdv 7 Thus, the fist pat of Theoem extends to conditionally independent values. In fact, a sufficient condition fo the evenue bounds is that fo any v [, v], Pv i < v is inceasing in the numbe of othe biddes with values v j < v. Howeve, the second pat of Theoem does not fully extend to conditionally independent values: in the appendix, we give an example whee > I. Equation 7 is still a nontivial uppe bound on, since as v appoaches v, v H n v appoaches v and F 2 v does not. 5 What If Losing Bids Ae Obseved Above, I assumed that the distibution of v 2 was known exactly, but that no othe infomation was available about the joint distibution f of values. Hee, I conside the infeences that can be made fom othe losing bids. Let b i denote bidde i s bid, and b i the i th highest bid. As in Haile and Tame 2003, I do not intepet a losing bids as an exact indication of that bidde s willingness to pay, only as a lowe bound on it. Thus, no obsevations will be able to falsify pefect coelation of bidde values, which is used to pove the lowe bounds on both π and. These lowe bounds, theefoe, ae unchanged if losing bids ae obseved. On the othe hand, if losing bids ae sufficiently high close enough to v 2, they may falsify the assumption of independence, in which case a tighte uppe bound on π will follow, which may in tun lead to a tighte uppe bound on. As a demonstation, conside the case of symmetic, affiliated pivate values when the distibution of the thid-highest bid b 3 is obseved along with F 2. Simila esults will hold fo othe losing bids. Let G 3 be the obseved distibution of b 3, and note that by assumption, v 3 b 3, and theefoe F 3 G 3. Then unde symmety and affiliation, F n F 2 F n F 2 F nc 2 F 3 F 2 n F 2 F nc 2 G 3 F 2 The fist inequality is fom the poof of Theoem in Quint Simplifying gives F F 2 F 2 n 2n G 3 F Conditionally independent values satisfy fv, v 2,..., v n = E θ {fv θfv 2 θ fv n θ} fo some family of distibutions f θ. 3
4 Since the left-hand side is stictly inceasing in F, Equation 9 gives a lowe bound on F, which gives an uppe bound on π. As we saw above, an uppe bound π π imposes an uppe bound π π on ; if the losing bids ae high enough, this bound may be lowe than I. 6 Auctions With Enty Fees Results fo auctions with enty fees ae simila to the esults fo auctions without. Fist, conside auctions whee potential biddes must pay an enty fee e befoe leaning thei pivate values and paticipating in the auction. That is, playes lean e and but not v i, decide simultaneously whethe to pay e and paticipate, lean v i, and then the auction is held. I efe to this as an ealy enty fee. It is easy to show that in such an auction with symmetic biddes, the selle maximizes expected evenue by setting = and using the enty fee to extact all expected suplus fom the selles by setting e = e v v df v π 0 n Let e I denote the value of e when bidde values ae independent that is, substituting H n v fo F v in Equation 0. Theoem 3. Suppose bidde values ae symmetic and affiliated o conditionally independent. In an auction with an ealy enty fee, the optimal eseve pice is ; the optimal enty fee is bounded below by 0 and above by e I, with both bounds being tight. Finally, conside the hade poblem of auctions with an enty fee which is paid afte biddes lean thei valuations. That is, biddes lean e,, and v i, decide simultaneously whethe to paticipate, and then the auction is held among those who ente. The esults ae not as complete, but I do offe the following bounds on the evenue-maximizing paametes: Theoem 4. Suppose bidde values ae symmetic and affiliated o conditionally independent. In an auction with a late enty fee, the optimal eseve pice and enty fee, e ae not bounded away fom, 0; an uppe bound on + e is given by Refeences +e v dh n v π, 0. Athey, S. and P. Haile 2002, Identification of Standad Auction Models, Econometica Haile, P. and E. Tame 2003, Infeence with an Incomplete Model of English Auctions, JPE. 3. Quint, D. 2008, Unobseved Coelation in Pivate-Value Ascending Auctions, Economics Lettes
5 Appendix Poof of Result We begin by calculating F, the cumulative density function of v : F = Pv < = Pɛ 2 + ɛ < Since we know by constuction the distibution of ɛ 2, F 2, we can ewite this as F = Pɛ 2 + ɛ < ɛ 2 = xdf 2 x = 0 Pɛ < ɛ2 ɛ 2 = xdf 2 x Now, the distibution of ɛ, conditioned on a given value of ɛ 2, is simply the distibution of ɛ conditional on being above that value. That is, knowing that ɛ 2 = x makes the conditional distibution of ɛ the unifom distibution on [x, ]. So P ɛ < ɛ2 ɛ2 = x = if x < + + x/ x if x [ x 0 if x >, ] Plugging this into the integals above gives 0 F = x x df 2x F / x x df 2x if if > Since v 2 = ɛ 2, F 2 x = nx n n x n ; plugging in, integating, and simplifying then gives Case : > F = n n + n n As noted in equation, when = 0, expected evenue is When >, this is π = F 2 F + xdf 2x = if if > π = F 2 F + xdf 2 x n n n n n + + n n + = n n n n+ n+ + + n n + nn = n n n n+ n+ + + n n + n xn x= x nn x n 2 x dx x n x n dx xn+ nn n+ = n n n n+ n+ + + n n + n n n nn n+ + nn n+ n+ = n n n n n n+ + nn n+ n+ n+ + + n n = n n n+ n+ n+ + + n n + n n+ 5 x= + n nn n+
6 and so π = n n n n n+ n + + n n + n+ n n = n n n n n + + n = + = + + n n + n n n n + n + + n + n n + n n n + n + + n n < n + n n + + n n + + n Since = + > 0, > + ; the inequality then follows, since n > + n and + n > + + n. Then π < n < n < n < n 2 n + + n n 2 + n n Since π < 0 fo >, we know that [0, ]. To show that π is deceasing in, ewite expected evenue as n n { } x π = F 2 min, df 2 x + xdf 2 x 0 x { } x which is stictly deceasing in by inspection. This simply equies that min, x on a positive-measue w..t. F 2 subset of [0, ]; this is the case on [ +, ]. Case 2 : Fo, F = n, and so π = n n n + n+ n+ + n n + which is deceasing in by inspection. Diffeentiating, π = n n n n n + n Note that π has the same sign as n n + n+, so π is stictly quasiconcave. Thus, the fist-ode condition gives us the maximize, which is = n n + n+ n 6
7 Poof of Theoem 2 As in the poof of Theoem in Quint 2008, we show F H n ; then π π fo. Define ψ, ψ 2 : [0, ] [0, ] by ψ x = x n and ψ 2 x = nx n n x n. Fo a given distibution H of one vaiable, then, ψ H and ψ 2 H ae the distibutions of the highest and second highest, espectively, of n independent daws on H. If values ae conditionally independent, let {H θ } be the set of distibutions fom which values may be independently dawn. It is easy to show that F 2 = E θ ψ 2 H θ and F = E θ ψ H θ Next, we show that ψ ψ 2 is convex. This is because ψ ψ 2 ψ s = ψ 2 s ntn = s nn t n 2 t = ψ 2 ψ 2 t n t whee t = ψ2 s; since this is inceasing in t, and theefoe s, ψ ψ2 is inceasing so ψ ψ2 is convex. Recall also that H was defined by F 2 = ψ 2 H. Applying Jensen s inequality, Fom Equations and 3, then, F = E θ ψ H θ = E θ ψ ψ2 ψ 2 H θ = E θ ψ ψ2 ψ 2 H θ = E θ ψ ψ2 ψ 2 H θ ψ ψ2 E θ ψ 2 H θ = ψ ψ2 F 2 = H n π π = H n F 0 and the bound is tight because IPV is a special case of conditionally independent pivate values. The lowe bound on π, as well as on, is poved the same way as in Quint To show that is not necessaily lowe than ag max π, we offe a counteexample. Let n = 3, = 0, and suppose that θ takes the values 0, with equal pobabilities and when θ = 0, bidde valuations ae i.i.d. U[3, 9] when θ =, bidde valuations ae i.i.d. U [[0, 3] [9, 0]]. Fo i {, 2}, F i x = 2 ψ x i 4 2 ψ 3 i 4 2 ψ x 6 i 4 fo x ψ x 3 i 6 fo x 3, fo x 9 This allows us to calculate a closed-fom if messy expession fo π. While we don t have a closed-fom expession fo H = ψ 2 F 2, we can calculate it, and theefoe π, numeically. 7
8 Figue : An example with CIPV whee π π but ag max π > ag max π. It tuns out that while π π eveywhee as equied by Theoem 2, π is maximized at = 6.09, and π at I = 5.37, as shown in Figue. In Figue, π is not quasi-concave, so this example is not an exact contadiction of Theoem in Quint 2008 when affiliation is elaxed. Howeve, we can eliminate the lip in π nea = 9 without changing the esult. If athe than the unifom distibution on [3, 9], the CDF of each bidde s value when θ = 0 is x 3 6 fo x between 3 and 9, then π is stictly quasiconcave, and ag max π is still stictly geate than ag max π. As fo the new uppe bound on, π π π π the middle inequality is the optimality of, the fist and thid ae simply the bounds on π. π π can be witten as F 2 H n Integating the ight-hand side by pats and simplifying gives Poof of Theoem 3 H n v df 2 v F 2 vdv An auction with a eseve pice = and enty fee e = max{v n E, } max{v 2, } 2 8
9 achieves efficiency and extacts all bidde suplus; thus, it must be optimal. It is not had to show that in nondegeneate cases, this auction is uniquely optimal. Since max{v, } is an inceasing function of v, its expectation, and theefoe e, ae inceasing functions in the distibution F with espect to fist-ode stochastic dominance. We agued above that F F 2 eveywhee, with equality being attained fo the pefectlycoelated joint distibution. Thus, e max{v n E 2, } max{v 2, } = 0 foms a tight lowe bound. Similaly, we showed that F F I eveywhee, and since independent values ae a special case of symmetic affiliated values and conditionally independent values, equality is attainable, so e v max{v, }df I v E max{v 2, } = e I n foms a tight uppe bound. v Poof of Theoem 4 Lowe Bound We again use the pefectly-coelated values example consistent with the obseved distibution F 2, and claim that fo any, e, 0, π, e < π, 0. If e = 0 and, all playes ente, and the expected evenue is v df 2 v < v df 2 v = π, 0 Suppose, theefoe, that e > 0 fo the est of the poof. Fo a given value of v, let π v, e be the expected evenue including enty fees fom the auction with eseve pice and enty fee e when all biddes have the pivate value v fo the good, so that π, e = E v π v, e = v π v, edf 2 v We assume that each playe has an independent enty stategy τ i : [v, v] [0, ] giving thei pobability of enteing fo each ealization of thei pivate value v. Note that no playe will eve ente when v < e +, so π v = 0 fo v < e +. Fo a given v, we conside two cases: when only one playe consides enteing τ i v = 0 fo all i but at most one, and when moe than one conside enteing. In the fist case, letting x be the playe who may ente, π v, e = τ x ve + ; since τ x is zeo when v < e +, we know that π v, e max{0, e + v e+ } In the second case, note that the evenue fom the auction, excluding the enty fees, is 0 when nobody entes, when one playe entes, and v when at least two ente; thus, we can 9
10 expess total expected evenue as π v, e = i τ iv e + i τ i v j i τ jv + i τ iv i τ i v j i τ jv v Now, enteing biddes get no suplus fom an auction if any othe biddes ente, since the pice paid is equal to thei pivate value; so equilibium play equies that fo each i, eithe τ i v = 0, o e + j i τ jvv 0. In eithe case, τ i ve τ i v j i τ j vv Plugging this into the expession fo π v, e and simplifying gives π v, e τ i v v i Now, if moe than one playe consides enteing, let y be the playe with the second-highest value of τ i v. By assumption, τ y v > 0, so equilibium play equies o j y τ jv e v know that τ y v τ x v biddes conside enteing, e e j y τ j vv v. Letting x again be the playe with the highest value of τ iv, we e v, so i τ iv e 2; v thus, when moe than two π v, e e 2 v v Thus, we have now shown that given equilibium play by the biddes, { } e 2 π v, e v e+ max 0, e +, v v This expession is eveywhee weakly less than max{0, v }, and stictly less than v on, v] {e + }. Thus, π, e = v π v, e < v df 2 v = π, 0 Since this agument holds fo any, e, 0, it follows that, e =, 0 must be optimal. Uppe Bound Fo the uppe bound on + e, note that fo any, e, the maximum possible suplus to both the selle and the buyes is +e v df v since nobody will ente when v < + e. Since in equilibium, biddes must have nonnegative expected payoff, π, e +e v df v +e v df I v The last inequality is because v v>+e is a nondeceasing function of v, so its expectation is inceasing with espect to fist-ode stochastic dominance, and we showed in the poof of Theoem that F F I eveywhee. Optimality of, e implies π, e π, 0 = π ; combining the inequalities gives +e v df I v π, completing the poof. 0
Chapter 10 Mechanism Design and Postcontractual Hidden Knowledge
Chapte 10 Mechanism Design and Postcontactual Hidden Knowledge 10.1 Mechanisms, Unavelling, Coss Checking, and the Revelation Pinciple A mechanism is a set of ules that one playe constucts and anothe feely
More informationEconomics Alumni Working Paper Series
Economics Alumni Woking Pape Seies Auctioning with Aspiations: Keep Them Low (Enough) Nicholas Shunda Univesity of Redlands Woking Pape 2009-02 July 2009 341 Mansfield Road, Unit 1063 Stos, CT 06269 1063
More informationCompactly Supported Radial Basis Functions
Chapte 4 Compactly Suppoted Radial Basis Functions As we saw ealie, compactly suppoted functions Φ that ae tuly stictly conditionally positive definite of ode m > do not exist The compact suppot automatically
More informationStrategic Information Acquisition in Auctions.
Stategic Infomation Acquisition in Auctions. Kai Hao Yang 12/19/217 Abstact We study a stategic infomation acquisition poblem in auctions in which the buyes have independent pivate valuations and can choose
More informationInternet Appendix for A Bayesian Approach to Real Options: The Case of Distinguishing Between Temporary and Permanent Shocks
Intenet Appendix fo A Bayesian Appoach to Real Options: The Case of Distinguishing Between Tempoay and Pemanent Shocks Steven R. Genadie Gaduate School of Business, Stanfod Univesity Andey Malenko Gaduate
More information3.1 Random variables
3 Chapte III Random Vaiables 3 Random vaiables A sample space S may be difficult to descibe if the elements of S ae not numbes discuss how we can use a ule by which an element s of S may be associated
More informationMath 301: The Erdős-Stone-Simonovitz Theorem and Extremal Numbers for Bipartite Graphs
Math 30: The Edős-Stone-Simonovitz Theoem and Extemal Numbes fo Bipatite Gaphs May Radcliffe The Edős-Stone-Simonovitz Theoem Recall, in class we poved Tuán s Gaph Theoem, namely Theoem Tuán s Theoem Let
More informationON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION. 1. Introduction. 1 r r. r k for every set E A, E \ {0},
ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION E. J. IONASCU and A. A. STANCU Abstact. We ae inteested in constucting concete independent events in puely atomic pobability
More informationMultiple Experts with Binary Features
Multiple Expets with Binay Featues Ye Jin & Lingen Zhang Decembe 9, 2010 1 Intoduction Ou intuition fo the poect comes fom the pape Supevised Leaning fom Multiple Expets: Whom to tust when eveyone lies
More informationTHE CONE THEOREM JOEL A. TROPP. Abstract. We prove a fixed point theorem for functions which are positive with respect to a cone in a Banach space.
THE ONE THEOEM JOEL A. TOPP Abstact. We pove a fixed point theoem fo functions which ae positive with espect to a cone in a Banach space. 1. Definitions Definition 1. Let X be a eal Banach space. A subset
More informationOnline Appendix Licensing Process Innovations when Losers Messages Determine Royalty Rates
Online Appendi Licensing Pocess Innovations when Loses Messages Detemine Royalty Rates Cuihong Fan Shanghai Univesity of Finance and Economics School of Economics Elma G. Wolfstette Humboldt-Univesity
More informationQIP Course 10: Quantum Factorization Algorithm (Part 3)
QIP Couse 10: Quantum Factoization Algoithm (Pat 3 Ryutaoh Matsumoto Nagoya Univesity, Japan Send you comments to yutaoh.matsumoto@nagoya-u.jp Septembe 2018 @ Tokyo Tech. Matsumoto (Nagoya U. QIP Couse
More informationNotes on McCall s Model of Job Search. Timothy J. Kehoe March if job offer has been accepted. b if searching
Notes on McCall s Model of Job Seach Timothy J Kehoe Mach Fv ( ) pob( v), [, ] Choice: accept age offe o eceive b and seach again next peiod An unemployed oke solves hee max E t t y t y t if job offe has
More informationStanford University CS259Q: Quantum Computing Handout 8 Luca Trevisan October 18, 2012
Stanfod Univesity CS59Q: Quantum Computing Handout 8 Luca Tevisan Octobe 8, 0 Lectue 8 In which we use the quantum Fouie tansfom to solve the peiod-finding poblem. The Peiod Finding Poblem Let f : {0,...,
More informationONE-POINT CODES USING PLACES OF HIGHER DEGREE
ONE-POINT CODES USING PLACES OF HIGHER DEGREE GRETCHEN L. MATTHEWS AND TODD W. MICHEL DEPARTMENT OF MATHEMATICAL SCIENCES CLEMSON UNIVERSITY CLEMSON, SC 29634-0975 U.S.A. E-MAIL: GMATTHE@CLEMSON.EDU, TMICHEL@CLEMSON.EDU
More informationPhysics 2A Chapter 10 - Moment of Inertia Fall 2018
Physics Chapte 0 - oment of netia Fall 08 The moment of inetia of a otating object is a measue of its otational inetia in the same way that the mass of an object is a measue of its inetia fo linea motion.
More informationA Multivariate Normal Law for Turing s Formulae
A Multivaiate Nomal Law fo Tuing s Fomulae Zhiyi Zhang Depatment of Mathematics and Statistics Univesity of Noth Caolina at Chalotte Chalotte, NC 28223 Abstact This pape establishes a sufficient condition
More informationac p Answers to questions for The New Introduction to Geographical Economics, 2 nd edition Chapter 3 The core model of geographical economics
Answes to questions fo The New ntoduction to Geogaphical Economics, nd edition Chapte 3 The coe model of geogaphical economics Question 3. Fom intoductoy mico-economics we know that the condition fo pofit
More informationMultiple Criteria Secretary Problem: A New Approach
J. Stat. Appl. Po. 3, o., 9-38 (04 9 Jounal of Statistics Applications & Pobability An Intenational Jounal http://dx.doi.og/0.785/jsap/0303 Multiple Citeia Secetay Poblem: A ew Appoach Alaka Padhye, and
More informationSurveillance Points in High Dimensional Spaces
Société de Calcul Mathématique SA Tools fo decision help since 995 Suveillance Points in High Dimensional Spaces by Benad Beauzamy Januay 06 Abstact Let us conside any compute softwae, elying upon a lage
More informationInference of Bidders Risk Attitudes in Ascending Auctions with Endogenous Entry
Infeence of Biddes Risk Attitudes in Ascending Auctions with Endogenous Enty Hanming Fang y Xun Tang z Fist Vesion: May 28, 2. This Vesion: Apil 7, 22 Abstact Biddes isk attitudes have impotant implications
More informationworking pages for Paul Richards class notes; do not copy or circulate without permission from PGR 2004/11/3 10:50
woking pages fo Paul Richads class notes; do not copy o ciculate without pemission fom PGR 2004/11/3 10:50 CHAPTER7 Solid angle, 3D integals, Gauss s Theoem, and a Delta Function We define the solid angle,
More informationDo Managers Do Good With Other People s Money? Online Appendix
Do Manages Do Good With Othe People s Money? Online Appendix Ing-Haw Cheng Haison Hong Kelly Shue Abstact This is the Online Appendix fo Cheng, Hong and Shue 2013) containing details of the model. Datmouth
More informationGoodness-of-fit for composite hypotheses.
Section 11 Goodness-of-fit fo composite hypotheses. Example. Let us conside a Matlab example. Let us geneate 50 obsevations fom N(1, 2): X=nomnd(1,2,50,1); Then, unning a chi-squaed goodness-of-fit test
More informationJournal of Inequalities in Pure and Applied Mathematics
Jounal of Inequalities in Pue and Applied Mathematics COEFFICIENT INEQUALITY FOR A FUNCTION WHOSE DERIVATIVE HAS A POSITIVE REAL PART S. ABRAMOVICH, M. KLARIČIĆ BAKULA AND S. BANIĆ Depatment of Mathematics
More informationf h = u, h g = v, we have u + v = f g. So, we wish
Answes to Homewok 4, Math 4111 (1) Pove that the following examples fom class ae indeed metic spaces. You only need to veify the tiangle inequality. (a) Let C be the set of continuous functions fom [0,
More informationEfficiency Loss in a Network Resource Allocation Game
Efficiency Loss in a Netwok Resouce Allocation Game Ramesh Johai johai@mit.edu) John N. Tsitsiklis jnt@mit.edu) June 11, 2004 Abstact We exploe the popeties of a congestion game whee uses of a congested
More informationCentral Coverage Bayes Prediction Intervals for the Generalized Pareto Distribution
Statistics Reseach Lettes Vol. Iss., Novembe Cental Coveage Bayes Pediction Intevals fo the Genealized Paeto Distibution Gyan Pakash Depatment of Community Medicine S. N. Medical College, Aga, U. P., India
More informationOn the ratio of maximum and minimum degree in maximal intersecting families
On the atio of maximum and minimum degee in maximal intesecting families Zoltán Lóánt Nagy Lale Özkahya Balázs Patkós Máté Vize Septembe 5, 011 Abstact To study how balanced o unbalanced a maximal intesecting
More informationKOEBE DOMAINS FOR THE CLASSES OF FUNCTIONS WITH RANGES INCLUDED IN GIVEN SETS
Jounal of Applied Analysis Vol. 14, No. 1 2008), pp. 43 52 KOEBE DOMAINS FOR THE CLASSES OF FUNCTIONS WITH RANGES INCLUDED IN GIVEN SETS L. KOCZAN and P. ZAPRAWA Received Mach 12, 2007 and, in evised fom,
More information10/04/18. P [P(x)] 1 negl(n).
Mastemath, Sping 208 Into to Lattice lgs & Cypto Lectue 0 0/04/8 Lectues: D. Dadush, L. Ducas Scibe: K. de Boe Intoduction In this lectue, we will teat two main pats. Duing the fist pat we continue the
More informationAustralian Intermediate Mathematics Olympiad 2017
Austalian Intemediate Mathematics Olympiad 207 Questions. The numbe x is when witten in base b, but it is 22 when witten in base b 2. What is x in base 0? [2 maks] 2. A tiangle ABC is divided into fou
More informationExploration of the three-person duel
Exploation of the thee-peson duel Andy Paish 15 August 2006 1 The duel Pictue a duel: two shootes facing one anothe, taking tuns fiing at one anothe, each with a fixed pobability of hitting his opponent.
More informationNew problems in universal algebraic geometry illustrated by boolean equations
New poblems in univesal algebaic geomety illustated by boolean equations axiv:1611.00152v2 [math.ra] 25 Nov 2016 Atem N. Shevlyakov Novembe 28, 2016 Abstact We discuss new poblems in univesal algebaic
More informationChapter 3: Theory of Modular Arithmetic 38
Chapte 3: Theoy of Modula Aithmetic 38 Section D Chinese Remainde Theoem By the end of this section you will be able to pove the Chinese Remainde Theoem apply this theoem to solve simultaneous linea conguences
More informationIdentification of Time and Risk Preferences in Buy Price Auctions
Identification of Time and Risk Pefeences in Buy Pice Auctions Daniel Ackebeg Univesity of Michigan ackebe@umich.edu Keisuke Hiano Univesity of Aizona hiano@u.aizona.edu Quazi Shahia San Diego State Univesity
More informationWeb-based Supplementary Materials for. Controlling False Discoveries in Multidimensional Directional Decisions, with
Web-based Supplementay Mateials fo Contolling False Discoveies in Multidimensional Diectional Decisions, with Applications to Gene Expession Data on Odeed Categoies Wenge Guo Biostatistics Banch, National
More informationarxiv: v1 [math.co] 6 Mar 2008
An uppe bound fo the numbe of pefect matchings in gaphs Shmuel Fiedland axiv:0803.0864v [math.co] 6 Ma 2008 Depatment of Mathematics, Statistics, and Compute Science, Univesity of Illinois at Chicago Chicago,
More informationarxiv: v1 [physics.pop-ph] 3 Jun 2013
A note on the electostatic enegy of two point chages axiv:1306.0401v1 [physics.pop-ph] 3 Jun 013 A C Tot Instituto de Física Univesidade Fedeal do io de Janeio Caixa Postal 68.58; CEP 1941-97 io de Janeio,
More information6 Matrix Concentration Bounds
6 Matix Concentation Bounds Concentation bounds ae inequalities that bound pobabilities of deviations by a andom vaiable fom some value, often its mean. Infomally, they show the pobability that a andom
More informationPublic vs. Private O ers in the Market for Lemons (PRELIMINARY AND INCOMPLETE)
Public vs. Pivate O es in the Maket fo Lemons (PRELIMINARY AND INCOMPLETE) Johannes Höne and Nicolas Vieille Febuay 2006 Abstact We analyze a vesion of Akelof s maket fo lemons in which a sequence of buyes
More informationTemporal-Difference Learning
.997 Decision-Making in Lage-Scale Systems Mach 17 MIT, Sping 004 Handout #17 Lectue Note 13 1 Tempoal-Diffeence Leaning We now conside the poblem of computing an appopiate paamete, so that, given an appoximation
More informationA generalization of the Bernstein polynomials
A genealization of the Benstein polynomials Halil Ouç and Geoge M Phillips Mathematical Institute, Univesity of St Andews, Noth Haugh, St Andews, Fife KY16 9SS, Scotland Dedicated to Philip J Davis This
More informationOn the ratio of maximum and minimum degree in maximal intersecting families
On the atio of maximum and minimum degee in maximal intesecting families Zoltán Lóánt Nagy Lale Özkahya Balázs Patkós Máté Vize Mach 6, 013 Abstact To study how balanced o unbalanced a maximal intesecting
More informationPROBLEM SET #1 SOLUTIONS by Robert A. DiStasio Jr.
POBLM S # SOLUIONS by obet A. DiStasio J. Q. he Bon-Oppenheime appoximation is the standad way of appoximating the gound state of a molecula system. Wite down the conditions that detemine the tonic and
More informationAuctioning Process Innovations when Losers Bids Determine Royalty Rates
Auctioning Pocess Innovations when Loses Bids Detemine Royalty Rates Cuihong Fan Shanghai Univesity of Finance and Economics School of Economics Elma G. Wolfstette Humboldt-Univesity at Belin and Koea
More informationA Comment on Increasing Returns and Spatial. Unemployment Disparities
The Society fo conomic Studies The nivesity of Kitakyushu Woking Pape Seies No.06-5 (accepted in Mach, 07) A Comment on Inceasing Retuns and Spatial nemployment Dispaities Jumpei Tanaka ** The nivesity
More informationOn the Poisson Approximation to the Negative Hypergeometric Distribution
BULLETIN of the Malaysian Mathematical Sciences Society http://mathusmmy/bulletin Bull Malays Math Sci Soc (2) 34(2) (2011), 331 336 On the Poisson Appoximation to the Negative Hypegeometic Distibution
More informationPhysics 121 Hour Exam #5 Solution
Physics 2 Hou xam # Solution This exam consists of a five poblems on five pages. Point values ae given with each poblem. They add up to 99 points; you will get fee point to make a total of. In any given
More informationNOTE. Some New Bounds for Cover-Free Families
Jounal of Combinatoial Theoy, Seies A 90, 224234 (2000) doi:10.1006jcta.1999.3036, available online at http:.idealibay.com on NOTE Some Ne Bounds fo Cove-Fee Families D. R. Stinson 1 and R. Wei Depatment
More informationOn the integration of the equations of hydrodynamics
Uebe die Integation de hydodynamischen Gleichungen J f eine u angew Math 56 (859) -0 On the integation of the equations of hydodynamics (By A Clebsch at Calsuhe) Tanslated by D H Delphenich In a pevious
More informationSTUDY OF SOLUTIONS OF LOGARITHMIC ORDER TO HIGHER ORDER LINEAR DIFFERENTIAL-DIFFERENCE EQUATIONS WITH COEFFICIENTS HAVING THE SAME LOGARITHMIC ORDER
UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA doi: 104467/20843828AM170027078 542017, 15 32 STUDY OF SOLUTIONS OF LOGARITHMIC ORDER TO HIGHER ORDER LINEAR DIFFERENTIAL-DIFFERENCE EQUATIONS WITH COEFFICIENTS
More informationIntroduction to Mathematical Statistics Robert V. Hogg Joeseph McKean Allen T. Craig Seventh Edition
Intoduction to Mathematical Statistics Robet V. Hogg Joeseph McKean Allen T. Caig Seventh Edition Peason Education Limited Edinbugh Gate Halow Essex CM2 2JE England and Associated Companies thoughout the
More informationA NEW VARIABLE STIFFNESS SPRING USING A PRESTRESSED MECHANISM
Poceedings of the ASME 2010 Intenational Design Engineeing Technical Confeences & Computes and Infomation in Engineeing Confeence IDETC/CIE 2010 August 15-18, 2010, Monteal, Quebec, Canada DETC2010-28496
More informationJENSEN S INEQUALITY FOR DISTRIBUTIONS POSSESSING HIGHER MOMENTS, WITH APPLICATION TO SHARP BOUNDS FOR LAPLACE-STIELTJES TRANSFORMS
J. Austal. Math. Soc. Se. B 40(1998), 80 85 JENSEN S INEQUALITY FO DISTIBUTIONS POSSESSING HIGHE MOMENTS, WITH APPLICATION TO SHAP BOUNDS FO LAPLACE-STIELTJES TANSFOMS B. GULJAŠ 1,C.E.M.PEACE 2 and J.
More informationTest 2, ECON , Summer 2013
Test, ECON 6090-9, Summe 0 Instuctions: Answe all questions as completely as possible. If you cannot solve the poblem, explaining how you would solve the poblem may ean you some points. Point totals ae
More informationWhat Form of Gravitation Ensures Weakened Kepler s Third Law?
Bulletin of Aichi Univ. of Education, 6(Natual Sciences, pp. - 6, Mach, 03 What Fom of Gavitation Ensues Weakened Keple s Thid Law? Kenzi ODANI Depatment of Mathematics Education, Aichi Univesity of Education,
More informationMATH 415, WEEK 3: Parameter-Dependence and Bifurcations
MATH 415, WEEK 3: Paamete-Dependence and Bifucations 1 A Note on Paamete Dependence We should pause to make a bief note about the ole played in the study of dynamical systems by the system s paametes.
More informationResearch Article On Alzer and Qiu s Conjecture for Complete Elliptic Integral and Inverse Hyperbolic Tangent Function
Abstact and Applied Analysis Volume 011, Aticle ID 697547, 7 pages doi:10.1155/011/697547 Reseach Aticle On Alze and Qiu s Conjectue fo Complete Elliptic Integal and Invese Hypebolic Tangent Function Yu-Ming
More information7.2. Coulomb s Law. The Electric Force
Coulomb s aw Recall that chaged objects attact some objects and epel othes at a distance, without making any contact with those objects Electic foce,, o the foce acting between two chaged objects, is somewhat
More informationEM-2. 1 Coulomb s law, electric field, potential field, superposition q. Electric field of a point charge (1)
EM- Coulomb s law, electic field, potential field, supeposition q ' Electic field of a point chage ( ') E( ) kq, whee k / 4 () ' Foce of q on a test chage e at position is ee( ) Electic potential O kq
More informationq i i=1 p i ln p i Another measure, which proves a useful benchmark in our analysis, is the chi squared divergence of p, q, which is defined by
CSISZÁR f DIVERGENCE, OSTROWSKI S INEQUALITY AND MUTUAL INFORMATION S. S. DRAGOMIR, V. GLUŠČEVIĆ, AND C. E. M. PEARCE Abstact. The Ostowski integal inequality fo an absolutely continuous function is used
More informationLET a random variable x follows the two - parameter
INTERNATIONAL JOURNAL OF MATHEMATICS AND SCIENTIFIC COMPUTING ISSN: 2231-5330, VOL. 5, NO. 1, 2015 19 Shinkage Bayesian Appoach in Item - Failue Gamma Data In Pesence of Pio Point Guess Value Gyan Pakash
More informationLecture 28: Convergence of Random Variables and Related Theorems
EE50: Pobability Foundations fo Electical Enginees July-Novembe 205 Lectue 28: Convegence of Random Vaiables and Related Theoems Lectue:. Kishna Jagannathan Scibe: Gopal, Sudhasan, Ajay, Swamy, Kolla An
More informationVoltage ( = Electric Potential )
V-1 of 10 Voltage ( = lectic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage
More informationGeometry of the homogeneous and isotropic spaces
Geomety of the homogeneous and isotopic spaces H. Sonoda Septembe 2000; last evised Octobe 2009 Abstact We summaize the aspects of the geomety of the homogeneous and isotopic spaces which ae most elevant
More informationTHE JEU DE TAQUIN ON THE SHIFTED RIM HOOK TABLEAUX. Jaejin Lee
Koean J. Math. 23 (2015), No. 3, pp. 427 438 http://dx.doi.og/10.11568/kjm.2015.23.3.427 THE JEU DE TAQUIN ON THE SHIFTED RIM HOOK TABLEAUX Jaejin Lee Abstact. The Schensted algoithm fist descibed by Robinson
More informationProblem Set #10 Math 471 Real Analysis Assignment: Chapter 8 #2, 3, 6, 8
Poblem Set #0 Math 47 Real Analysis Assignment: Chate 8 #2, 3, 6, 8 Clayton J. Lungstum Decembe, 202 xecise 8.2 Pove the convese of Hölde s inequality fo = and =. Show also that fo eal-valued f / L ),
More informationThe CENTRE for EDUCATION in MATHEMATICS and COMPUTING cemc.uwaterloo.ca Galois Contest. Wednesday, April 12, 2017
The ENTRE fo EDUATIN in MATHEMATIS and MPUTING cemc.uwateloo.ca 2017 Galois ontest Wednesday, Apil 12, 2017 (in Noth Ameica and South Ameica) Thusday, Apil 13, 2017 (outside of Noth Ameica and South Ameica)
More informationTo Feel a Force Chapter 7 Static equilibrium - torque and friction
To eel a oce Chapte 7 Chapte 7: Static fiction, toque and static equilibium A. Review of foce vectos Between the eath and a small mass, gavitational foces of equal magnitude and opposite diection act on
More informationSocial learning and monopoly pricing with forward looking buyers
Social leaning and monopoly picing with fowad looking buyes JOB MARKET PAPER Click hee fo the most ecent vesion Tuomas Laiho and Julia Salmi Januay 11, 217 Abstact We study monopoly picing in a dynamic
More informationQuasi-Randomness and the Distribution of Copies of a Fixed Graph
Quasi-Randomness and the Distibution of Copies of a Fixed Gaph Asaf Shapia Abstact We show that if a gaph G has the popety that all subsets of vetices of size n/4 contain the coect numbe of tiangles one
More informationMethod for Approximating Irrational Numbers
Method fo Appoximating Iational Numbes Eic Reichwein Depatment of Physics Univesity of Califonia, Santa Cuz June 6, 0 Abstact I will put foth an algoithm fo poducing inceasingly accuate ational appoximations
More information4/18/2005. Statistical Learning Theory
Statistical Leaning Theoy Statistical Leaning Theoy A model of supevised leaning consists of: a Envionment - Supplying a vecto x with a fixed but unknown pdf F x (x b Teache. It povides a desied esponse
More informationBoundary Layers and Singular Perturbation Lectures 16 and 17 Boundary Layers and Singular Perturbation. x% 0 Ω0æ + Kx% 1 Ω0æ + ` : 0. (9.
Lectues 16 and 17 Bounday Layes and Singula Petubation A Regula Petubation In some physical poblems, the solution is dependent on a paamete K. When the paamete K is vey small, it is natual to expect that
More informationAPPLICATION OF MAC IN THE FREQUENCY DOMAIN
PPLICION OF MC IN HE FREQUENCY DOMIN D. Fotsch and D. J. Ewins Dynamics Section, Mechanical Engineeing Depatment Impeial College of Science, echnology and Medicine London SW7 2B, United Kingdom BSRC he
More informationON THE INVERSE SIGNED TOTAL DOMINATION NUMBER IN GRAPHS. D.A. Mojdeh and B. Samadi
Opuscula Math. 37, no. 3 (017), 447 456 http://dx.doi.og/10.7494/opmath.017.37.3.447 Opuscula Mathematica ON THE INVERSE SIGNED TOTAL DOMINATION NUMBER IN GRAPHS D.A. Mojdeh and B. Samadi Communicated
More information1 Similarity Analysis
ME43A/538A/538B Axisymmetic Tubulent Jet 9 Novembe 28 Similaity Analysis. Intoduction Conside the sketch of an axisymmetic, tubulent jet in Figue. Assume that measuements of the downsteam aveage axial
More informationStrategic timing of adoption of new technologies under uncertainty: A note. Georg Götz
Stategic timing of adoption of new technologies unde uncetainty: A note Geog Götz Abstact: This note claifies the cicumstances unde which ex ante identical fims will choose diffeent dates fo the adoption
More information1 Notes on Order Statistics
1 Notes on Ode Statistics Fo X a andom vecto in R n with distibution F, and π S n, define X π by and F π by X π (X π(1),..., X π(n) ) F π (x 1,..., x n ) F (x π 1 (1),..., x π 1 (n)); then the distibution
More informationMath 124B February 02, 2012
Math 24B Febuay 02, 202 Vikto Gigoyan 8 Laplace s equation: popeties We have aleady encounteed Laplace s equation in the context of stationay heat conduction and wave phenomena. Recall that in two spatial
More informationAn Application of Fuzzy Linear System of Equations in Economic Sciences
Austalian Jounal of Basic and Applied Sciences, 5(7): 7-14, 2011 ISSN 1991-8178 An Application of Fuzzy Linea System of Equations in Economic Sciences 1 S.H. Nassei, 2 M. Abdi and 3 B. Khabii 1 Depatment
More informationExceptional regular singular points of second-order ODEs. 1. Solving second-order ODEs
(May 14, 2011 Exceptional egula singula points of second-ode ODEs Paul Gaett gaett@math.umn.edu http://www.math.umn.edu/ gaett/ 1. Solving second-ode ODEs 2. Examples 3. Convegence Fobenius method fo solving
More informationIntroduction to Nuclear Forces
Intoduction to Nuclea Foces One of the main poblems of nuclea physics is to find out the natue of nuclea foces. Nuclea foces diffe fom all othe known types of foces. They cannot be of electical oigin since
More informationSupplementary information Efficient Enumeration of Monocyclic Chemical Graphs with Given Path Frequencies
Supplementay infomation Efficient Enumeation of Monocyclic Chemical Gaphs with Given Path Fequencies Masaki Suzuki, Hioshi Nagamochi Gaduate School of Infomatics, Kyoto Univesity {m suzuki,nag}@amp.i.kyoto-u.ac.jp
More informationAlternative Tests for the Poisson Distribution
Chiang Mai J Sci 015; 4() : 774-78 http://epgsciencecmuacth/ejounal/ Contibuted Pape Altenative Tests fo the Poisson Distibution Manad Khamkong*[a] and Pachitjianut Siipanich [b] [a] Depatment of Statistics,
More informationInformation Retrieval Advanced IR models. Luca Bondi
Advanced IR models Luca Bondi Advanced IR models 2 (LSI) Pobabilistic Latent Semantic Analysis (plsa) Vecto Space Model 3 Stating point: Vecto Space Model Documents and queies epesented as vectos in the
More informationSection 8.2 Polar Coordinates
Section 8. Pola Coodinates 467 Section 8. Pola Coodinates The coodinate system we ae most familia with is called the Catesian coodinate system, a ectangula plane divided into fou quadants by the hoizontal
More informationarxiv: v2 [physics.data-an] 15 Jul 2015
Limitation of the Least Squae Method in the Evaluation of Dimension of Factal Bownian Motions BINGQIANG QIAO,, SIMING LIU, OUDUN ZENG, XIANG LI, and BENZONG DAI Depatment of Physics, Yunnan Univesity,
More informationAST 121S: The origin and evolution of the Universe. Introduction to Mathematical Handout 1
Please ead this fist... AST S: The oigin and evolution of the Univese Intoduction to Mathematical Handout This is an unusually long hand-out and one which uses in places mathematics that you may not be
More informationChapter 3 Optical Systems with Annular Pupils
Chapte 3 Optical Systems with Annula Pupils 3 INTRODUCTION In this chapte, we discuss the imaging popeties of a system with an annula pupil in a manne simila to those fo a system with a cicula pupil The
More informationIn the previous section we considered problems where the
5.4 Hydodynamically Fully Developed and Themally Developing Lamina Flow In the pevious section we consideed poblems whee the velocity and tempeatue pofile wee fully developed, so that the heat tansfe coefficient
More informationThe Omega Function -Not for Circulation-
The Omega Function -Not fo Ciculation- A. Cascon, C. Keating and W. F. Shadwick The Finance Development Cente London, England July 23 2002, Final Revision 11 Mach 2003 1 Intoduction In this pape we intoduce
More informationAsymptotically Lacunary Statistical Equivalent Sequence Spaces Defined by Ideal Convergence and an Orlicz Function
"Science Stays Tue Hee" Jounal of Mathematics and Statistical Science, 335-35 Science Signpost Publishing Asymptotically Lacunay Statistical Equivalent Sequence Spaces Defined by Ideal Convegence and an
More informationPsychometric Methods: Theory into Practice Larry R. Price
ERRATA Psychometic Methods: Theoy into Pactice Lay R. Pice Eos wee made in Equations 3.5a and 3.5b, Figue 3., equations and text on pages 76 80, and Table 9.1. Vesions of the elevant pages that include
More informationMEASURING CHINESE RISK AVERSION
MEASURING CHINESE RISK AVERSION --Based on Insuance Data Li Diao (Cental Univesity of Finance and Economics) Hua Chen (Cental Univesity of Finance and Economics) Jingzhen Liu (Cental Univesity of Finance
More informationHua Xu 3 and Hiroaki Mukaidani 33. The University of Tsukuba, Otsuka. Hiroshima City University, 3-4-1, Ozuka-Higashi
he inea Quadatic Dynamic Game fo Discete-ime Descipto Systems Hua Xu 3 and Hioai Muaidani 33 3 Gaduate School of Systems Management he Univesity of suuba, 3-9- Otsua Bunyo-u, oyo -0, Japan xuhua@gssm.otsua.tsuuba.ac.jp
More information(n 1)n(n + 1)(n + 2) + 1 = (n 1)(n + 2)n(n + 1) + 1 = ( (n 2 + n 1) 1 )( (n 2 + n 1) + 1 ) + 1 = (n 2 + n 1) 2.
Paabola Volume 5, Issue (017) Solutions 151 1540 Q151 Take any fou consecutive whole numbes, multiply them togethe and add 1. Make a conjectue and pove it! The esulting numbe can, fo instance, be expessed
More informationEfficiency Loss in a Network Resource Allocation Game: The Case of Elastic Supply
Efficiency Loss in a Netwok Resouce Allocation Game: The Case of Elastic Supply axiv:cs/0506054v1 [cs.gt] 14 Jun 2005 Ramesh Johai (johai@stanfod.edu) Shie Manno (shie@mit.edu) John N. Tsitsiklis (jnt@mit.edu)
More informationOn the Quasi-inverse of a Non-square Matrix: An Infinite Solution
Applied Mathematical Sciences, Vol 11, 2017, no 27, 1337-1351 HIKARI Ltd, wwwm-hikaicom https://doiog/1012988/ams20177273 On the Quasi-invese of a Non-squae Matix: An Infinite Solution Ruben D Codeo J
More information