ECE 586GT: Problem Set 2: Problems and Solutions Uniqueness of Nash equilibria, zero sum games, evolutionary dynamics
|
|
- Rafe Tyler
- 5 years ago
- Views:
Transcription
1 Unversty of Illnos Fall 08 ECE 586GT: Problem Set : Problems and Solutons Unqueness of Nash equlbra, zero sum games, evolutonary dynamcs Due: Tuesday, Sept. 5, at begnnng of class Readng: Course notes, Sectons.5-.7 and Chapter. [Exstence and unqueness of NE for games wth contnuous type strateges] Ths problem concerns an n player game wth strategy space S = (0, + ) for all players, and space of strategy profles S = S S n. (a) Consder the payoff functons u (s) = (ln s ) s s tot, where s tot = s + + s n. Does there exst a pure strategy Nash equlbrum? If so, s t unque? Can you fnd an explct expresson for t? Soluton: For [n], for any s fxed, u s a strctly concave functon of s wth lmt at 0 and +. Hence, for each s fxed there s a unque best response whch s obtaned by settng u s = 0. Note that s tot = s + s, where s = [n], s. Wrtng u (s) = ln s s s s we fnd u s = 0 s equvalent to s s = 0. () s (Although t sn t needed, we note () mples that the best response functon for any player s B (s ) = s + s +8 4.) Usng the fact s tot = s + s, () can be rewrtten as whch can be used to determne s as a functon of s tot : s s s tot = 0, () s = s tot + s tot + 4 We can conclude that any Nash equlbrum (n pure strateges) s symmetrc. Therefore, for a Nash equlbrum, s tot = ns and so s = ns + n s + 4 Thus, there s a unque Nash equlbrum gven by s = n+ for all [n]. (b) Consder the payoff functons u (s) = (ln s ) s s tot + ɛ cos(s ) for some ɛ > 0. Fnd a postve constant ɛ such that there exsts a Nash equlbrum f 0 ɛ < ɛ. To avod some tedous steps, you may assume that the strategy space for each player s restrcted to the closed nterval [δ, /δ], such that δ can be chosen so small that any best response for any player s n the nteror of the nterval. Soluton: Note that u (s) ( s = ɛ cos(s ) s ) < 0 f ɛ. so u (s) s a concave functon of s for s fxed, f ɛ. Also, the payoff functons are contnuous and the strategy spaces [δ, /δ] are compact, so there exsts a pure strategy NE f ɛ by the Debreu, Glcksberg, and Fan exstence theorem.
2 (c) For the payoff functons n part (b), fnd a postve constant ɛ such that there exsts a unque pure strategy Nash equlbrum f 0 ɛ < ɛ. (Hnt: A dagonal matrx wth negatve entres on the dagonal s negatve defnte, and the sum of a negatve defnte matrx and a negatve sem-defnte matrx s negatve defnte.) Soluton: We use the suffcent condtons for unqueness n the notes based on the followng matrx: U(s) = u (s) ( s ) u (s)... s s. u (s) s s By drect ( calculaton we) fnd U(s) = D(s) J, where D(s) s the dagonal matrx dag ɛ cos(s s ) and J s the all ones n n matrx. In partcular, U s a symmetrc matrx so we don t need to consder the symmetrzaton U(s) + U T (s). If 0 ɛ then D(s) s negatve defnte. Also, J s negatve semdefnte. Thus, f 0 ɛ, U(s) s negatve defnte, and there exsts a unque Nash equlbrum,. [Market power n terrtory control game] Recall the terrtory control game for a graph G = (V, E) and n players, defned n problem set. The game s symmetrc n the sense that the payoff functons are nvarant under permutaton of the players. A strategy profle (n ether pure or mxed strateges) s symmetrc f every player uses the same strategy. (a) Prove that for any symmetrc fnte player game, there exsts a symmetrc Nash equlbrum n mxed strateges. Thus, for the terrtory control game based on a connected graph G = (V, E), there s a Nash equlbrum such that the expected payoff of each player s V n. Soluton: Ths follows from the same proof as Nash s theorem. (b) Show that a two-player game such that the sum of payoffs s constant for all strategy pars s equvalent to a zero sum two-player game. In partcular, there exsts a saddle pont and a value of the game for each player. Soluton: Let u tot denote the sum of payoffs, whch s assumed to be ndependent of the strategy profle. We could mpose a tax n the amount u tot to player, or mpose a tax of u tot /n to every player, to make the sum of payoffs always equal to zero. (c) Suppose there are three players, but players and 3 team up and splt ther payoffs wth each other. Thus, players and 3 together can be vewed as a super player; from the perspectve of player, the game becomes a two-player game, wth constant sum of payoffs V. Player could be placng a Burger Kng and players and 3 together place two McDonalds. Show that the value of the modfed game for player s greater than or equal to V /4 for any connected graph G. (Hnt: What f there were two super players, each selectng nodes for two restaurants?) Soluton: If there were two super players, each selectng two nodes usng a mxed strategy, then by symmetry the value of the game would be V / for each of the players. Transform the strategy of one of those super players by havng the player frst select ts two nodes (possbly the same one twce), and then flppng a far con to determne whch of those choces to keep, and whch to dscard. The mean payoff generated by the randomly retaned node was at least V /4 before the other randomly selected node
3 was dropped. The droppng of the other choce by the same player cannot decrease the expected ponts s generated by the retaned node, whch s thus at least V /4. For example, f the player had two Burger Kng restaurants, then closng one of them could generate more traffc for the other. (d) Consder the scenaro of part (c) for the specal case of a lne graph V = {,,..., m}, such that m = 4b + for some nonnegatve nteger b, and such that E = {[, + ] : m }. Show that the value of the modfed game for player s equal to V /4. (Hnt: By part(c) only an upper bound on the average payoff of player s needed.) Soluton: Suppose player selects b + and player 3 selects 3b +. Then the payoff of player s less than or equal to m/4 for all possbltes. Specfcally, t ranges from (k + )/ to V /4 as s ncreases from 0 to b +. It remans flat at V /4 for b + s 3b [Dual of a transport problem] Let n and let W be an n n matrx wth nonnegatve elements. Consder the followng lnear optmzaton problem: mn X, [n] X, W, subect to: X, = [n] X, = [n] X, 0 for [n] for [n] for, [n] An nterpretaton s that there are n sources of some good, n destnatons, W, s the cost of transportng a unt of good from to, and X, s the amount of good transported from to. Each source provdes one unt of good and each destnaton receves one unt of good. The problem n vector matrx form can be wrtten as: mn X X, W subect to: X =, X T =, X 0. (a) Derve the dual problem by frst fndng the Lagrangan functon. Fnd a smple verson of the dual problem. (Hnt: You can ether elmnate the Lagrange multplers for the constrants X, 0, or don t handle those constrants wth Lagrange multplers n the frst place. 3
4 Soluton: Usng strong dualty for lnear programs, we fnd mn X, W X:X=, X T =, X 0 = mn X:X 0 µ (),µ () mn X:X 0 max X, W + ( X) T µ () + ( X T ) T µ () µ (),µ () X, W + ( X) T µ () + ( X T ) T µ () µ (),µ () mn X:X 0 T (µ () + µ () ) + X, W µ () T (µ () ) T µ (),µ () :µ () T +(µ () ) T W In other words the dual problem s max µ () µ (),µ () + µ () [n] [n] T (µ () + µ () ) subect to: µ () + µ () W, for, [n] (An nterpretaton of the dual varables s that µ () s a transport charge per unt good at source and µ () s a transport charge per unt good at destnaton. The dual problem entals maxmzng the sum of payments subect to a constrant showng the charges are ustfed by the transport costs. The equalty constrants n the prmal problem could be replaced by nequalty constrants: X, X T, and that would lead to nonnegatvty constrants on the multplers n the dual problem. ( ) 3 8 (b) Fnd the common value of the prmal and dual, and solutons, for W =. 4 ( ) ( ) ( ) Soluton: X =, µ 0 () =, µ () =, wth value 7 each. (The ( ) soluton to the dual problem s not unque. Addng any constant multple of to µ () and subtractng t from µ () yelds another soluton.) 4. [Evolutonarly stable strateges and states] Consder the followng symmetrc, two-player game::,,, 0,0 (a) Does ether player have a (weakly or strongly) domnant strategy? Soluton: No. (b) Identfy all the pure strategy and mxed strategy Nash equlbra. Soluton: (,) and (,) are pure strategy NE. As usual, there s no NE n whch only one strategy s pure. If (p, q) s an NE such that both p and q are nondegenerate mxed ( strateges, ether acton of player one must be a best response to q, so = q, or q =, ) (. Smlarly, p =, ) ((. Thus,, ) (,, )) s the unque NE n nondegenerate mxed strateges. 4
5 (c) Identfy all evolutonarly stable pure strateges and all evolutonarly stable mxed strateges. Soluton: Recall that f p s an ESS, then (p, p) s an NE. The pure strategy NEs found n part (b) are not symmetrc, so there are no pure ESSs. It remans to see whether the mxed strategy NE p gven by p = (, ) s an ESS. By defnton, we need to check whether for any p p, ether () u(p, p) < u(p, p), or () (u(p, p) = u(p, p) and u(p, p ) > u(p, p )). Snce u(p, p) = u(p, p) for all choces of p, the queston comes down to whether u(p, p ) > u(p, p ) for all p p. That s, whether + ()p > p + ( p )(p ) for all p p. Or, equvalently, whether ( p ( ) > 0 for p p. Ths condton s true, so, ) s a mxed ESS. (d) The replcator dynamcs based on ths game represents a large populaton consstng of type and type ndvduals. Show that the evoluton of the populaton share vector θ(t) under the replcator dynamcs for ths model reduces to a one dmensonal ordnary dfferental equaton for θ t (), the fracton of the populaton that s type. Soluton: The replcator dynamcs for the populaton share vector θ t are gven by θ t (a) = θ t (a)(u(a, θ t ) u(θ t, θ t )) for a {, }. Although there are two equatons, ths system s actually one dmensonal because θ t () = θ t (). Let x t = θ t (). Then u(, θ t ), and u(θ t, θ t ) = x t + 3x t ( x t ) = 3x t x t. So the replcator dynamcs become ẋ t = x t ( 3x t + x t ), or, equvalently, ẋ t = x t ( x t )( x t ). (3) (e) Identfy the steady states of the replcator dynamcs. Soluton: The rght hand sde of (3) s zero for x t { 0,, }, so there are three steady states for the replcator dynamcs: (,0), (, ), and (,0). (f) Of the steady states dentfed n the prevous part, whch are asymptotcally stable states of the replcator dynamcs? Justfy your answer. Soluton: Of the three steady states, only (, ) s asymptotcally stable. If x0 = ɛ for an arbtrarly small but postve ɛ, then ẋ t > 0 and x wll converge monotoncally up to, so 0 s not even a stable steady state (so t s not asymptotcally stable). Smlarly, s not a stable steady state. However, s an asymptotcally stable steady state of x because the rght hand sde of (3) has a down crossng of zero at. Hence (, ) s an asymptotcally stable state for the replcator dynamcs. (Another ustfcaton for ths problem can be gven by applyng general facts about ESSs and replcator dynamcs. States (, 0) and (0, ) can t be asymptotcally stable, or even stable, states for the replcator dynamcs, because, when played aganst themselves, they don t gve NEs. The mxed state (, ) s an asymptotcally stable state of the replcator dynamcs because t s an ESS.) 5
Lecture 7. We now use Brouwer s fixed point theorem to prove Nash s theorem.
Topcs on the Border of Economcs and Computaton December 11, 2005 Lecturer: Noam Nsan Lecture 7 Scrbe: Yoram Bachrach 1 Nash s Theorem We begn by provng Nash s Theorem about the exstance of a mxed strategy
More informationOPERATIONS RESEARCH. Game Theory
OPERATIONS RESEARCH Chapter 2 Game Theory Prof. Bbhas C. Gr Department of Mathematcs Jadavpur Unversty Kolkata, Inda Emal: bcgr.umath@gmal.com 1.0 Introducton Game theory was developed for decson makng
More informationProblem Set 6 Finance 1,
Carnege Mellon Unversty Graduate School of Industral Admnstraton Chrs Telmer Wnter 2006 Problem Set 6 Fnance, 47-720. (representatve agent constructon) Consder the followng two-perod, two-agent economy.
More informationEconomics 1410 Fall Section 7 Notes 1. Define the tax in a flexible way using T (z), where z is the income reported by the agent.
Economcs 1410 Fall 2017 Harvard Unversty Yaan Al-Karableh Secton 7 Notes 1 I. The ncome taxaton problem Defne the tax n a flexble way usng T (), where s the ncome reported by the agent. Retenton functon:
More informationTCOM501 Networking: Theory & Fundamentals Final Examination Professor Yannis A. Korilis April 26, 2002
TO5 Networng: Theory & undamentals nal xamnaton Professor Yanns. orls prl, Problem [ ponts]: onsder a rng networ wth nodes,,,. In ths networ, a customer that completes servce at node exts the networ wth
More informationCS 286r: Matching and Market Design Lecture 2 Combinatorial Markets, Walrasian Equilibrium, Tâtonnement
CS 286r: Matchng and Market Desgn Lecture 2 Combnatoral Markets, Walrasan Equlbrum, Tâtonnement Matchng and Money Recall: Last tme we descrbed the Hungaran Method for computng a maxmumweght bpartte matchng.
More informationII. Random Variables. Variable Types. Variables Map Outcomes to Numbers
II. Random Varables Random varables operate n much the same way as the outcomes or events n some arbtrary sample space the dstncton s that random varables are smply outcomes that are represented numercally.
More informationProduction and Supply Chain Management Logistics. Paolo Detti Department of Information Engeneering and Mathematical Sciences University of Siena
Producton and Supply Chan Management Logstcs Paolo Dett Department of Informaton Engeneerng and Mathematcal Scences Unversty of Sena Convergence and complexty of the algorthm Convergence of the algorthm
More informationIntroduction to game theory
Introducton to game theory Lectures n game theory ECON5210, Sprng 2009, Part 1 17.12.2008 G.B. Ashem, ECON5210-1 1 Overvew over lectures 1. Introducton to game theory 2. Modelng nteractve knowledge; equlbrum
More informationElements of Economic Analysis II Lecture VI: Industry Supply
Elements of Economc Analyss II Lecture VI: Industry Supply Ka Hao Yang 10/12/2017 In the prevous lecture, we analyzed the frm s supply decson usng a set of smple graphcal analyses. In fact, the dscusson
More informationAppendix for Solving Asset Pricing Models when the Price-Dividend Function is Analytic
Appendx for Solvng Asset Prcng Models when the Prce-Dvdend Functon s Analytc Ovdu L. Caln Yu Chen Thomas F. Cosmano and Alex A. Hmonas January 3, 5 Ths appendx provdes proofs of some results stated n our
More informationAppendix - Normally Distributed Admissible Choices are Optimal
Appendx - Normally Dstrbuted Admssble Choces are Optmal James N. Bodurtha, Jr. McDonough School of Busness Georgetown Unversty and Q Shen Stafford Partners Aprl 994 latest revson September 00 Abstract
More informationPrice and Quantity Competition Revisited. Abstract
rce and uantty Competton Revsted X. Henry Wang Unversty of Mssour - Columba Abstract By enlargng the parameter space orgnally consdered by Sngh and Vves (984 to allow for a wder range of cost asymmetry,
More informationLinear Combinations of Random Variables and Sampling (100 points)
Economcs 30330: Statstcs for Economcs Problem Set 6 Unversty of Notre Dame Instructor: Julo Garín Sprng 2012 Lnear Combnatons of Random Varables and Samplng 100 ponts 1. Four-part problem. Go get some
More informationGames and Decisions. Part I: Basic Theorems. Contents. 1 Introduction. Jane Yuxin Wang. 1 Introduction 1. 2 Two-player Games 2
Games and Decsons Part I: Basc Theorems Jane Yuxn Wang Contents 1 Introducton 1 2 Two-player Games 2 2.1 Zero-sum Games................................ 3 2.1.1 Pure Strateges.............................
More information15-451/651: Design & Analysis of Algorithms January 22, 2019 Lecture #3: Amortized Analysis last changed: January 18, 2019
5-45/65: Desgn & Analyss of Algorthms January, 09 Lecture #3: Amortzed Analyss last changed: January 8, 09 Introducton In ths lecture we dscuss a useful form of analyss, called amortzed analyss, for problems
More informationUNIVERSITY OF NOTTINGHAM
UNIVERSITY OF NOTTINGHAM SCHOOL OF ECONOMICS DISCUSSION PAPER 99/28 Welfare Analyss n a Cournot Game wth a Publc Good by Indraneel Dasgupta School of Economcs, Unversty of Nottngham, Nottngham NG7 2RD,
More informationStatic (or Simultaneous- Move) Games of Complete Information
Statc (or Smultaneous- Move) Games of Complete Informaton Nash Equlbrum Best Response Functon F. Valognes - Game Theory - Chp 3 Outlne of Statc Games of Complete Informaton Introducton to games Normal-form
More informationMechanisms for Efficient Allocation in Divisible Capacity Networks
Mechansms for Effcent Allocaton n Dvsble Capacty Networks Antons Dmaks, Rahul Jan and Jean Walrand EECS Department Unversty of Calforna, Berkeley {dmaks,ran,wlr}@eecs.berkeley.edu Abstract We propose a
More informationProblems to be discussed at the 5 th seminar Suggested solutions
ECON4260 Behavoral Economcs Problems to be dscussed at the 5 th semnar Suggested solutons Problem 1 a) Consder an ultmatum game n whch the proposer gets, ntally, 100 NOK. Assume that both the proposer
More information- contrast so-called first-best outcome of Lindahl equilibrium with case of private provision through voluntary contributions of households
Prvate Provson - contrast so-called frst-best outcome of Lndahl equlbrum wth case of prvate provson through voluntary contrbutons of households - need to make an assumpton about how each household expects
More informationAn Efficient Nash-Implementation Mechanism for Divisible Resource Allocation
SUBMITTED TO IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS 1 An Effcent Nash-Implementaton Mechansm for Dvsble Resource Allocaton Rahul Jan IBM T.J. Watson Research Center Hawthorne, NY 10532 rahul.jan@us.bm.com
More informationEquilibrium in Prediction Markets with Buyers and Sellers
Equlbrum n Predcton Markets wth Buyers and Sellers Shpra Agrawal Nmrod Megddo Benamn Armbruster Abstract Predcton markets wth buyers and sellers of contracts on multple outcomes are shown to have unque
More informationEDC Introduction
.0 Introducton EDC3 In the last set of notes (EDC), we saw how to use penalty factors n solvng the EDC problem wth losses. In ths set of notes, we want to address two closely related ssues. What are, exactly,
More informationApplications of Myerson s Lemma
Applcatons of Myerson s Lemma Professor Greenwald 28-2-7 We apply Myerson s lemma to solve the sngle-good aucton, and the generalzaton n whch there are k dentcal copes of the good. Our objectve s welfare
More informationA MODEL OF COMPETITION AMONG TELECOMMUNICATION SERVICE PROVIDERS BASED ON REPEATED GAME
A MODEL OF COMPETITION AMONG TELECOMMUNICATION SERVICE PROVIDERS BASED ON REPEATED GAME Vesna Radonć Đogatovć, Valentna Radočć Unversty of Belgrade Faculty of Transport and Traffc Engneerng Belgrade, Serba
More informationMgtOp 215 Chapter 13 Dr. Ahn
MgtOp 5 Chapter 3 Dr Ahn Consder two random varables X and Y wth,,, In order to study the relatonshp between the two random varables, we need a numercal measure that descrbes the relatonshp The covarance
More informationSIMPLE FIXED-POINT ITERATION
SIMPLE FIXED-POINT ITERATION The fed-pont teraton method s an open root fndng method. The method starts wth the equaton f ( The equaton s then rearranged so that one s one the left hand sde of the equaton
More informationCompetitive Rumor Spread in Social Networks
Compettve Rumor Spread n Socal Networks Yongwhan Lm Operatons Research Center, Massachusetts Insttute of Technology yongwhan@mt.edu Asuman Ozdaglar EECS, Massachusetts Insttute of Technology asuman@mt.edu
More informationFinance 402: Problem Set 1 Solutions
Fnance 402: Problem Set 1 Solutons Note: Where approprate, the fnal answer for each problem s gven n bold talcs for those not nterested n the dscusson of the soluton. 1. The annual coupon rate s 6%. A
More informationEconomic Design of Short-Run CSP-1 Plan Under Linear Inspection Cost
Tamkang Journal of Scence and Engneerng, Vol. 9, No 1, pp. 19 23 (2006) 19 Economc Desgn of Short-Run CSP-1 Plan Under Lnear Inspecton Cost Chung-Ho Chen 1 * and Chao-Yu Chou 2 1 Department of Industral
More informationTests for Two Correlations
PASS Sample Sze Software Chapter 805 Tests for Two Correlatons Introducton The correlaton coeffcent (or correlaton), ρ, s a popular parameter for descrbng the strength of the assocaton between two varables.
More informationSurvey of Math Test #3 Practice Questions Page 1 of 5
Test #3 Practce Questons Page 1 of 5 You wll be able to use a calculator, and wll have to use one to answer some questons. Informaton Provded on Test: Smple Interest: Compound Interest: Deprecaton: A =
More information2) In the medium-run/long-run, a decrease in the budget deficit will produce:
4.02 Quz 2 Solutons Fall 2004 Multple-Choce Questons ) Consder the wage-settng and prce-settng equatons we studed n class. Suppose the markup, µ, equals 0.25, and F(u,z) = -u. What s the natural rate of
More informationParallel Prefix addition
Marcelo Kryger Sudent ID 015629850 Parallel Prefx addton The parallel prefx adder presented next, performs the addton of two bnary numbers n tme of complexty O(log n) and lnear cost O(n). Lets notce the
More informationreferences Chapters on game theory in Mas-Colell, Whinston and Green
Syllabus. Prelmnares. Role of game theory n economcs. Normal and extensve form of a game. Game-tree. Informaton partton. Perfect recall. Perfect and mperfect nformaton. Strategy.. Statc games of complete
More informationFast Laplacian Solvers by Sparsification
Spectral Graph Theory Lecture 19 Fast Laplacan Solvers by Sparsfcaton Danel A. Spelman November 9, 2015 Dsclamer These notes are not necessarly an accurate representaton of what happened n class. The notes
More informationLecture Note 1: Foundations 1
Economcs 703 Advanced Mcroeconomcs Prof. Peter Cramton ecture Note : Foundatons Outlne A. Introducton and Examples B. Formal Treatment. Exstence of Nash Equlbrum. Exstence wthout uas-concavty 3. Perfect
More informationSOCIETY OF ACTUARIES FINANCIAL MATHEMATICS. EXAM FM SAMPLE SOLUTIONS Interest Theory
SOCIETY OF ACTUARIES EXAM FM FINANCIAL MATHEMATICS EXAM FM SAMPLE SOLUTIONS Interest Theory Ths page ndcates changes made to Study Note FM-09-05. January 14, 014: Questons and solutons 58 60 were added.
More informationS yi a bx i cx yi a bx i cx 2 i =0. yi a bx i cx 2 i xi =0. yi a bx i cx 2 i x
LEAST-SQUARES FIT (Chapter 8) Ft the best straght lne (parabola, etc.) to a gven set of ponts. Ths wll be done by mnmzng the sum of squares of the vertcal dstances (called resduals) from the ponts to the
More informationSingle-Item Auctions. CS 234r: Markets for Networks and Crowds Lecture 4 Auctions, Mechanisms, and Welfare Maximization
CS 234r: Markets for Networks and Crowds Lecture 4 Auctons, Mechansms, and Welfare Maxmzaton Sngle-Item Auctons Suppose we have one or more tems to sell and a pool of potental buyers. How should we decde
More informationGeneral Examination in Microeconomic Theory. Fall You have FOUR hours. 2. Answer all questions
HARVARD UNIVERSITY DEPARTMENT OF ECONOMICS General Examnaton n Mcroeconomc Theory Fall 2010 1. You have FOUR hours. 2. Answer all questons PLEASE USE A SEPARATE BLUE BOOK FOR EACH QUESTION AND WRITE THE
More informationStill Simpler Way of Introducing Interior-Point method for Linear Programming
Stll Smpler Way of Introducng Interor-Pont method for Lnear Programmng Sanjeev Saxena Dept. of Computer Scence and Engneerng, Indan Insttute of Technology, Kanpur, INDIA-08 06 October 9, 05 Abstract Lnear
More information3: Central Limit Theorem, Systematic Errors
3: Central Lmt Theorem, Systematc Errors 1 Errors 1.1 Central Lmt Theorem Ths theorem s of prme mportance when measurng physcal quanttes because usually the mperfectons n the measurements are due to several
More informationFoundations of Machine Learning II TP1: Entropy
Foundatons of Machne Learnng II TP1: Entropy Gullaume Charpat (Teacher) & Gaétan Marceau Caron (Scrbe) Problem 1 (Gbbs nequalty). Let p and q two probablty measures over a fnte alphabet X. Prove that KL(p
More informationMaximum Likelihood Estimation of Isotonic Normal Means with Unknown Variances*
Journal of Multvarate Analyss 64, 183195 (1998) Artcle No. MV971717 Maxmum Lelhood Estmaton of Isotonc Normal Means wth Unnown Varances* Nng-Zhong Sh and Hua Jang Northeast Normal Unversty, Changchun,Chna
More informationConsumption Based Asset Pricing
Consumpton Based Asset Prcng Mchael Bar Aprl 25, 208 Contents Introducton 2 Model 2. Prcng rsk-free asset............................... 3 2.2 Prcng rsky assets................................ 4 2.3 Bubbles......................................
More informationAn Efficient Mechanism for Network Bandwidth Auction
1 An Effcent Mechansm for Network Bandwdth Aucton Rahul Jan IBM T.J. Watson Research Center Hawthorne, NY 10532 rahul.jan@us.bm.com Jean Walrand EECS Department, Unversty of Calforna, Berkeley wlr@eecs.berkeley.edu
More informationQuadratic Games. First version: February 24, 2017 This version: August 3, Abstract
Quadratc Games Ncolas S. Lambert Gorgo Martn Mchael Ostrovsky Frst verson: February 24, 2017 Ths verson: August 3, 2018 Abstract We study general quadratc games wth multdmensonal actons, stochastc payoff
More informationA DUAL EXTERIOR POINT SIMPLEX TYPE ALGORITHM FOR THE MINIMUM COST NETWORK FLOW PROBLEM
Yugoslav Journal of Operatons Research Vol 19 (2009), Number 1, 157-170 DOI:10.2298/YUJOR0901157G A DUAL EXTERIOR POINT SIMPLEX TYPE ALGORITHM FOR THE MINIMUM COST NETWORK FLOW PROBLEM George GERANIS Konstantnos
More informationQuadratic Games. First version: February 24, 2017 This version: December 12, Abstract
Quadratc Games Ncolas S. Lambert Gorgo Martn Mchael Ostrovsky Frst verson: February 24, 2017 Ths verson: December 12, 2017 Abstract We study general quadratc games wth mult-dmensonal actons, stochastc
More informationTwo Period Models. 1. Static Models. Econ602. Spring Lutz Hendricks
Two Perod Models Econ602. Sprng 2005. Lutz Hendrcks The man ponts of ths secton are: Tools: settng up and solvng a general equlbrum model; Kuhn-Tucker condtons; solvng multperod problems Economc nsghts:
More informationPREFERENCE DOMAINS AND THE MONOTONICITY OF CONDORCET EXTENSIONS
PREFERECE DOMAIS AD THE MOOTOICITY OF CODORCET EXTESIOS PAUL J. HEALY AD MICHAEL PERESS ABSTRACT. An alternatve s a Condorcet wnner f t beats all other alternatves n a parwse majorty vote. A socal choce
More information332 Mathematical Induction Solutions for Chapter 14. for every positive integer n. Proof. We will prove this with mathematical induction.
33 Mathematcal Inducton. Solutons for Chapter. Prove that 3 n n n for every postve nteger n. Proof. We wll prove ths wth mathematcal nducton. Observe that f n, ths statement s, whch s obvously true. Consder
More informationMathematical Thinking Exam 1 09 October 2017
Mathematcal Thnkng Exam 1 09 October 2017 Name: Instructons: Be sure to read each problem s drectons. Wrte clearly durng the exam and fully erase or mark out anythng you do not want graded. You may use
More informationAn annuity is a series of payments made at equal intervals. There are many practical examples of financial transactions involving annuities, such as
2 Annutes An annuty s a seres of payments made at equal ntervals. There are many practcal examples of fnancal transactons nvolvng annutes, such as a car loan beng repad wth equal monthly nstallments a
More informationc slope = -(1+i)/(1+π 2 ) MRS (between consumption in consecutive time periods) price ratio (across consecutive time periods)
CONSUMPTION-SAVINGS FRAMEWORK (CONTINUED) SEPTEMBER 24, 2013 The Graphcs of the Consumpton-Savngs Model CONSUMER OPTIMIZATION Consumer s decson problem: maxmze lfetme utlty subject to lfetme budget constrant
More informationIntroduction to PGMs: Discrete Variables. Sargur Srihari
Introducton to : Dscrete Varables Sargur srhar@cedar.buffalo.edu Topcs. What are graphcal models (or ) 2. Use of Engneerng and AI 3. Drectonalty n graphs 4. Bayesan Networks 5. Generatve Models and Samplng
More informationTHE ECONOMICS OF TAXATION
THE ECONOMICS OF TAXATION Statc Ramsey Tax School of Economcs, Xamen Unversty Fall 2015 Overvew of Optmal Taxaton Combne lessons on ncdence and effcency costs to analyze optmal desgn of commodty taxes.
More informationMultifactor Term Structure Models
1 Multfactor Term Structure Models A. Lmtatons of One-Factor Models 1. Returns on bonds of all maturtes are perfectly correlated. 2. Term structure (and prces of every other dervatves) are unquely determned
More informationOption pricing and numéraires
Opton prcng and numérares Daro Trevsan Unverstà degl Stud d Psa San Mnato - 15 September 2016 Overvew 1 What s a numerare? 2 Arrow-Debreu model Change of numerare change of measure 3 Contnuous tme Self-fnancng
More informationQuiz on Deterministic part of course October 22, 2002
Engneerng ystems Analyss for Desgn Quz on Determnstc part of course October 22, 2002 Ths s a closed book exercse. You may use calculators Grade Tables There are 90 ponts possble for the regular test, or
More informationOCR Statistics 1 Working with data. Section 2: Measures of location
OCR Statstcs 1 Workng wth data Secton 2: Measures of locaton Notes and Examples These notes have sub-sectons on: The medan Estmatng the medan from grouped data The mean Estmatng the mean from grouped data
More informationUnderstanding Annuities. Some Algebraic Terminology.
Understandng Annutes Ma 162 Sprng 2010 Ma 162 Sprng 2010 March 22, 2010 Some Algebrac Termnology We recall some terms and calculatons from elementary algebra A fnte sequence of numbers s a functon of natural
More informationTests for Two Ordered Categorical Variables
Chapter 253 Tests for Two Ordered Categorcal Varables Introducton Ths module computes power and sample sze for tests of ordered categorcal data such as Lkert scale data. Assumng proportonal odds, such
More informationMode is the value which occurs most frequency. The mode may not exist, and even if it does, it may not be unique.
1.7.4 Mode Mode s the value whch occurs most frequency. The mode may not exst, and even f t does, t may not be unque. For ungrouped data, we smply count the largest frequency of the gven value. If all
More informationCh Rival Pure private goods (most retail goods) Non-Rival Impure public goods (internet service)
h 7 1 Publc Goods o Rval goods: a good s rval f ts consumpton by one person precludes ts consumpton by another o Excludable goods: a good s excludable f you can reasonably prevent a person from consumng
More informationTaxation and Externalities. - Much recent discussion of policy towards externalities, e.g., global warming debate/kyoto
Taxaton and Externaltes - Much recent dscusson of polcy towards externaltes, e.g., global warmng debate/kyoto - Increasng share of tax revenue from envronmental taxaton 6 percent n OECD - Envronmental
More informationLecture Note 2 Time Value of Money
Seg250 Management Prncples for Engneerng Managers Lecture ote 2 Tme Value of Money Department of Systems Engneerng and Engneerng Management The Chnese Unversty of Hong Kong Interest: The Cost of Money
More informationJeffrey Ely. October 7, This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.
October 7, 2012 Ths work s lcensed under the Creatve Commons Attrbuton-NonCommercal-ShareAlke 3.0 Lcense. Recap We saw last tme that any standard of socal welfare s problematc n a precse sense. If we want
More informationLikelihood Fits. Craig Blocker Brandeis August 23, 2004
Lkelhood Fts Crag Blocker Brandes August 23, 2004 Outlne I. What s the queston? II. Lkelhood Bascs III. Mathematcal Propertes IV. Uncertantes on Parameters V. Mscellaneous VI. Goodness of Ft VII. Comparson
More informationMacroeconomic Theory and Policy
ECO 209 Macroeconomc Theory and Polcy Lecture 7: The Open Economy wth Fxed Exchange Rates Gustavo Indart Slde 1 Open Economy under Fxed Exchange Rates Let s consder an open economy wth no captal moblty
More informationFinal Examination MATH NOTE TO PRINTER
Fnal Examnaton MATH 329 2005 01 1 NOTE TO PRINTER (These nstructons are for the prnter. They should not be duplcated.) Ths examnaton should be prnted on 8 1 2 14 paper, and stapled wth 3 sde staples, so
More informationiii) pay F P 0,T = S 0 e δt when stock has dividend yield δ.
Fnal s Wed May 7, 12:50-2:50 You are allowed 15 sheets of notes and a calculator The fnal s cumulatve, so you should know everythng on the frst 4 revews Ths materal not on those revews 184) Suppose S t
More informationA Constant-Factor Approximation Algorithm for Network Revenue Management
A Constant-Factor Approxmaton Algorthm for Networ Revenue Management Yuhang Ma 1, Paat Rusmevchentong 2, Ma Sumda 1, Huseyn Topaloglu 1 1 School of Operatons Research and Informaton Engneerng, Cornell
More informationMacroeconomic equilibrium in the short run: the Money market
Macroeconomc equlbrum n the short run: the Money market 2013 1. The bg pcture Overvew Prevous lecture How can we explan short run fluctuatons n GDP? Key assumpton: stcky prces Equlbrum of the goods market
More informationCyclic Scheduling in a Job shop with Multiple Assembly Firms
Proceedngs of the 0 Internatonal Conference on Industral Engneerng and Operatons Management Kuala Lumpur, Malaysa, January 4, 0 Cyclc Schedulng n a Job shop wth Multple Assembly Frms Tetsuya Kana and Koch
More informationWages as Anti-Corruption Strategy: A Note
DISCUSSION PAPER November 200 No. 46 Wages as Ant-Corrupton Strategy: A Note by dek SAO Faculty of Economcs, Kyushu-Sangyo Unversty Wages as ant-corrupton strategy: A Note dek Sato Kyushu-Sangyo Unversty
More informationRandom Variables. 8.1 What is a Random Variable? Announcements: Chapter 8
Announcements: Quz starts after class today, ends Monday Last chance to take probablty survey ends Sunday mornng. Next few lectures: Today, Sectons 8.1 to 8. Monday, Secton 7.7 and extra materal Wed, Secton
More informationGlobal Optimization in Multi-Agent Models
Global Optmzaton n Mult-Agent Models John R. Brge R.R. McCormck School of Engneerng and Appled Scence Northwestern Unversty Jont work wth Chonawee Supatgat, Enron, and Rachel Zhang, Cornell 11/19/2004
More informationAnswers to exercises in Macroeconomics by Nils Gottfries 2013
. a) C C b C C s the ntercept o the consumpton uncton, how much consumpton wll be at zero ncome. We can thnk that, at zero ncome, the typcal consumer would consume out o hs assets. The slope b s the margnal
More informationFinancial mathematics
Fnancal mathematcs Jean-Luc Bouchot jean-luc.bouchot@drexel.edu February 19, 2013 Warnng Ths s a work n progress. I can not ensure t to be mstake free at the moment. It s also lackng some nformaton. But
More informationBargaining over Strategies of Non-Cooperative Games
Games 05, 6, 73-98; do:0.3390/g603073 Artcle OPEN ACCESS games ISSN 073-4336 www.mdp.com/ournal/games Barganng over Strateges of Non-Cooperatve Games Guseppe Attanas, *, Aurora García-Gallego, Nkolaos
More informationChapter 5 Student Lecture Notes 5-1
Chapter 5 Student Lecture Notes 5-1 Basc Busness Statstcs (9 th Edton) Chapter 5 Some Important Dscrete Probablty Dstrbutons 004 Prentce-Hall, Inc. Chap 5-1 Chapter Topcs The Probablty Dstrbuton of a Dscrete
More informationA characterization of intrinsic reciprocity
Int J Game Theory (2008) 36:571 585 DOI 10.1007/s00182-007-0085-2 ORIGINAL PAPER A characterzaton of ntrnsc recprocty Uz Segal Joel Sobel Accepted: 20 March 2007 / Publshed onlne: 1 May 2007 Sprnger-Verlag
More informationContests with Group-Specific Public-Good Prizes
Contests wth Group-Specfc Publc-Good Przes Kyung Hwan ak * Department of Economcs Sungkyunkwan Unversty Seoul 110-745 South Korea September 2005 Abstract I examne the equlbrum effort levels of ndvdual
More informationOn the Moments of the Traces of Unitary and Orthogonal Random Matrices
Proceedngs of Insttute of Mathematcs of NAS of Ukrane 2004 Vol. 50 Part 3 1207 1213 On the Moments of the Traces of Untary and Orthogonal Random Matrces Vladmr VASILCHU B. Verkn Insttute for Low Temperature
More informationMicroeconomics: BSc Year One Extending Choice Theory
mcroeconomcs notes from http://www.economc-truth.co.uk by Tm Mller Mcroeconomcs: BSc Year One Extendng Choce Theory Consumers, obvously, mostly have a choce of more than two goods; and to fnd the favourable
More informationarxiv: v1 [math.nt] 29 Oct 2015
A DIGITAL BINOMIAL THEOREM FOR SHEFFER SEQUENCES TOUFIK MANSOUR AND HIEU D. NGUYEN arxv:1510.08529v1 [math.nt] 29 Oct 2015 Abstract. We extend the dgtal bnomal theorem to Sheffer polynomal sequences by
More informationJean-Paul Murara, Västeras, 26-April Mälardalen University, Sweden. Pricing EO under 2-dim. B S PDE by. using the Crank-Nicolson Method
Prcng EO under Mälardalen Unversty, Sweden Västeras, 26-Aprl-2017 1 / 15 Outlne 1 2 3 2 / 15 Optons - contracts that gve to the holder the rght but not the oblgaton to buy/sell an asset sometmes n the
More information4.4 Doob s inequalities
34 CHAPTER 4. MARTINGALES 4.4 Doob s nequaltes The frst nterestng consequences of the optonal stoppng theorems are Doob s nequaltes. If M n s a martngale, denote M n =max applen M. Theorem 4.8 If M n s
More informationEfficiency of a Two-Stage Market for a Fixed-Capacity Divisible Resource
Effcency of a Two-Stage Market for a Fxed-Capacty Dvsble Resource Amar Prakash Azad, and John Musaccho Abstract Two stage markets allow partcpants to trade resources lke power both n a forward market (so
More informationMacroeconomic Theory and Policy
ECO 209 Macroeconomc Theory and Polcy Lecture 7: The Open Economy wth Fxed Exchange Rates Gustavo Indart Slde 1 Open Economy under Fxed Exchange Rates Let s consder an open economy wth no captal moblty
More informationAutomatica. An efficient Nash-implementation mechanism for network resource allocation
Automatca 46 (2010 1276 1283 Contents lsts avalable at ScenceDrect Automatca ournal homepage: www.elsever.com/locate/automatca An effcent Nash-mplementaton mechansm for networ resource allocaton Rahul
More informationON THE COMPLEMENTARITY BETWEEN LAND REFORMS AND TRADE REFORMS
ON HE COMPLEMENARIY BEWEEN LAND REFORMS AND RADE REFORMS Abhrup Sarkar Indan Statstcal Insttute Calcutta January, 1 December, 1 (Revsed) Abstract he purpose of the paper s to look at the welfare effects
More informationWhen is the lowest equilibrium payoff in a repeated game equal to the min max payoff?
JID:YJETH AID:3744 /FLA [m1+; v 1.113; Prn:21/08/2009; 11:31] P.1 (1-22) Journal of Economc Theory ( ) www.elsever.com/locate/jet When s the lowest equlbrum payoff n a repeated game equal to the mn max
More informationSupplementary material for Non-conjugate Variational Message Passing for Multinomial and Binary Regression
Supplementary materal for Non-conjugate Varatonal Message Passng for Multnomal and Bnary Regresson October 9, 011 1 Alternatve dervaton We wll focus on a partcular factor f a and varable x, wth the am
More informationWHEN IS THE LOWEST EQUILIBRIUM PAYOFF IN A REPEATED GAME EQUAL TO THE MINMAX PAYOFF? OLIVIER GOSSNER and JOHANNES HÖRNER
WHEN IS THE LOWEST EQUILIBRIUM PAYOFF IN A REPEATED GAME EQUAL TO THE MINMAX PAYOFF? BY OLIVIER GOSSNER and JOHANNES HÖRNER COWLES FOUNDATION PAPER NO. 1294 COWLES FOUNDATION FOR RESEARCH IN ECONOMICS
More informationEvaluating Performance
5 Chapter Evaluatng Performance In Ths Chapter Dollar-Weghted Rate of Return Tme-Weghted Rate of Return Income Rate of Return Prncpal Rate of Return Daly Returns MPT Statstcs 5- Measurng Rates of Return
More informationDynamic Analysis of Knowledge Sharing of Agents with. Heterogeneous Knowledge
Dynamc Analyss of Sharng of Agents wth Heterogeneous Kazuyo Sato Akra Namatame Dept. of Computer Scence Natonal Defense Academy Yokosuka 39-8686 JAPAN E-mal {g40045 nama} @nda.ac.jp Abstract In ths paper
More information