SF2972 Game Theory Problem set on extensive form games
|
|
- Reginald Price
- 6 years ago
- Views:
Transcription
1 SF2972 Game Theor Problem set on etensive form games Mark Voorneveld There are five eercises, to be handed in at the final lecture (March 0). For a bonus point, please answer all questions; at least half of our answers must be correct. Do not hesitate to contact me with questions. If ou want to draw game trees, I recommend using the ps-tree commands in the PSTricks package for L A TEX. Good luck! It was fun teaching ou! Eercise. (a) Make eercise 03.2 from Osborne and Rubinstein. (b) Find the subgame perfect equilibria if there is an upper bound M N on the numbers that the plaers can announce in the second round. Eercise 2. (a) Find all (pure) Nash and subgame perfect equilibria of the game below. U 2 D U 2 2 U 4 D 4 (4, 6) (7, 5) (8, 7) D (6, 5) U 3 2 D 3 (9, 5) (b) Determine the corresponding strategic game. For each subgame perfect equilibrium, specif an order of iterated elimination of weakl dominated strategies that preserves the corresponding equilibrium. Can the same be done for the game s Nash equilibria?
2 Eercise 3. Consider the two games below. Out In Out 2 (4, 0) (4, 0) 2 (5, ) (3, 4) (5, ) (3, 4) (a) Do the games have the same (i) strategic form, (ii) reduced strategic form, (iii) agent strategic form? (b) For both games, compute the set of sequential equilibria. (c) Discuss the difference between the two games. What equilibrium (if an) do ou find most appealing?
3 Eercise 4. In this eercise, we compare consistenc with two other requirements on beliefs which I discussed informall during lecture. A belief sstem µ in an etensive form game with perfect recall is structurall consistent if for each information set I there is a strateg profile β such that: I is reached with positive probabilit under β, µ(i) is derived from β using Baes rule. (a) In Osborne and Rubinstein, Figure 228., specif the set of structurall consistent belief sstems. (b) One could define a new equilibrium notion: an assessment (β, µ) that is sequentiall rational and structurall consistent. Would that make sense? An assessment (β, µ) is weakl consistent if for each information set I that is reached with positive probabilit under β, the beliefs µ(i) over its histories are derived from β using Baes rule. Notice the difference with consistenc: weak consistenc imposes no constraints on beliefs over information sets that are reached with probabilit zero. (c) Consider the information set I of plaer 2 in the game below. The are meant to indicate that whatever else happens in the game tree is irrelevant. c 2/3 /3 2 Which beliefs µ(i) are feasible under (i) consistenc, (ii) structural consistenc, (iii) weak consistenc. Do ou find all beliefs under weak consistenc reasonable? See Osborne and Rubinstein, def. 228.
4 Eercise 5. I skipped Section 2.3: it introduces unnecessar notation for special cases of etensive form games. This eercise goes through some of the essential parts. (, 3) u u (2, ) L R d d (4, 0) /2 A 2 c 2 (2, 4) u /2 B u (, 0) L R d d (0, ) (, 2) A signalling game is plaed as follows (see the game above for a special case): Chance starts b assigning to plaer, the sender, a tpe. In our eample, the set of tpes is {A, B} and each tpe has equal probabilit. Plaer observes his tpe and sends a message to plaer 2, the receiver. In our eample, the set of messages is {L, R}. Plaer 2 observes the message of plaer (but not s tpe) and chooses an action. In our eample, the set of actions is {u, d}. The game ends. Paoffs can depend on tpes, messages, and actions. This models a range of economic situations in which one plaer controls the information, but another plaer controls the actions. Sometimes the sender might benefit from tring to inform the receiver about his tpe, sometimes it is more beneficial to tr to mislead the receiver. Rather than searching for all sequential equilibria of a signalling game, it is common to make a number of simplifications: firstl, it is common to look at weakl perfect Baesian equilibria in pure strategies (or, more precisel, behavioral strateg profiles where each plaer chooses in each information set a feasible action with probabilit one). Formall, an assessment (β, µ) is a weakl perfect Baesian equilibrium if it is sequentiall rational and weakl consistent. Secondl, the sender ma or ma not be interested in sending information about his tpe: a weakl perfect Baesian equilibrium in pure strategies is called a pooling equilibrium if he assigns to each tpe the same message and a separating equilibrium if he assigns to each tpe a different message. For instance, in a pooling equilibrium of our eample, β (A) = β (B), whereas in a separating equilibrium, β (A) β (B). (a) Find all separating and pooling equilibria of the game above. (b) Which of these equilibria are sequential equilibria?
5 Eercise a. Game: The game N, H, P, ( i ) i N has N = {, 2}, H = {, Stop, Continue, {(Continue, (, )) :, Z + }}, where (Continue, (, )) denotes the histor where plaer continues and plaers and 2 announce nonnegative integers and, respectivel. P( ) =, P(Continue) = {, 2}, and preferences i over terminal histories represented b the utilit function u i (Stop) =, u i (Continue, (, )) =. Subgame perfect equilibrium: Consider the subgame after Continue. There is no best response to a positive integer; ever nonnegative integer is a best response to 0. It follows that the unique equilibrium of the subgame is with associated paoffs. Anticipating this, plaer chooses Stop to reach paoff vector (, ). Conclude: the subgame perfect equilibrium has plaer plaing (Stop, 0) and plaer 2 plaing 0. Eercise b. Consider the subgame after Continue. The unique best response to a positive integer is M; ever nonnegative integer is a best response to 0. It follows that the subgame has two equilibria: with paoff vector and (M, M) with paoff vector (M 2, M 2 ). If plaer anticipates equilibrium, he prefers to Stop, if he anticipates equilibrium (M, M), he prefers to Continue. So in addition to the subgame perfect equilibrium in a, there is a new subgame perfect equilibrium where plaer plas (Continue, M) and plaer 2 plas M. Eercise 2. The strategic game and its equilibria: The corresponding strategic game is U 2 U 3 U 2 D 3 D 2 U 3 D 2 D 3 U U 4 7, 5 7, 5 4, 6 4, 6 U D 4 8, 7 8, 7 4, 6 4, 6 D U 4 6, 5 9, 5 6, 5 9, 5 D D 4 6, 5 9, 5 6, 5 9, 5 There are seven pure Nash equilibria, marked with (one or two) stars. The subgame perfect equilibria are marked with two stars. Iterated elimination and subgame perfect equilibria: Consider the subgame perfect equilibrium (U D 4, U 2 U 3 ). In the subgame after histor (U, U 2 ), plaer prefers D 4 over U 4 : U U 4 can be eliminated, as it is dominated b U D 4. Knowing that plaer prefers D 4 over U 4, in the subgame after histor U, now prefers U 2 over D 2. So now U 2 U 3 weakl dominates D 2 U 3 and U 2 D 3 weakl dominates D 2 D 3. In the remaining game (where pl. has 3 and pl. 2 has 2 strategies left), no more actions are weakl dominated, so the elimination process stops. Notice that both subgame perfect equilibria have survived the process. Iterated elimination and Nash equilibria: Nash equilibria in which plaer 2 chooses D 2 are alwas eliminated. Eercise 3a(i). No: in the left game, plaer has 4 pure strategies, in the right game onl 3.
6 Eercise 3a(ii). Yes (up to an arbitrar relabeling of the strategies): in the left game, the pure strategies (Out, ) and (Out, ) of plaer are paoff equivalent and can be replaced b a single one, Out, as in the game on the right. Eercise 3a(iii). No: in the left game, there are 3 information sets, hence 3 plaers in the agent strategic form, in the right game there are 2 information sets, hence 2 plaers in the agent strategic form. Eercise 3b. Let us start with the game on the right (Uhm, wh? Because it s better practice for the eam). The corresponding strategic game is Out 4, 0 4, 0 5, 0, 0 0, 0 3, 4 Since plaer s strateg is strictl dominated b Out, sequential rationalit requires that it is plaed with zero probabilit in a sequential equilibrium: β ( )() = 0. () Distinguish two cases: Case : sequential equilibria with β ( )() > 0: Consistenc requires that in information set {, }, which is reached with positive probabilit, the beliefs are derived from β ( ) using Baes rule. Together with (), this gives that µ({, })() = : plaer 2 believes to be in node with probabilit. Plaer 2 s unique best response to this belief is to choose : β 2 ({, })() =. Consequentl, plaer prefers, giving paoff 5, to Out, giving paoff 4: β ( )() =. We found a candidate sequential equilibrium: (β, β 2, µ) = ((0,, 0), (, 0), (, 0)). To verif consistenc, notice that (β, β 2, µ) is the limit of the sequence of assessments ( ( )) 2/n (/n, 2/n, /n), ( /n, /n), /n, /n. /n Case 2: sequential equilibria with β ( )() = 0: Together with (), this implies that β ( )(Out) =. For notational convenience, denote plaer 2 s strateg β 2 b (p, p) and the belief µ over {, } b (q, q). Sequential rationalit requires that Out is a best response of plaer. He won t pla (see ()) and gives epected paoff 5p, so Out is a best response as long as 5p 4, i.e., as long as p [0, 4/5]. But is ever such p sequentiall rational? If 0 < p 4/5, plaer 2 chooses both actions and with positive probabilit. Both and therefore have to best responses to 2 s beliefs in his information set. This is true onl if q = 4( q), i.e., if q = 4/5. If p = 0, onl action is chosen with positive probabilit. For this to be a best response to plaer 2 s beliefs in his information set, we need that q 4( q), i.e., that q [0, 4/5].
7 We found the following candidates for sequential equilibria: {(β, β 2, µ) = ((, 0, 0), (p, p), (4/5, /5)) 0 < p 4/5}, and {(β, β 2, µ) = ((, 0, 0), (0, ), (q, q)) 0 q 4/5}. It remains to verif that these assessments are consistent. For the first class of equilibria, let 0 < p 4/5. The assessment is a limit of (β n, β2 n, µ n ) = (( /n, 4/5n, /5n), (p, p), (4/5, /5)). For the second class of equilibria, let 0 q 4/5. The assessment is a limit of 0 + ε 0 q + ε,(ε, ε), (q + ε, q ε), 0 < ε 0. q ε For the game on the left, a similar analsis applies, but it s a bit simpler: conditional on reaching information set {In}, sequential rationalit requires the plaers to choose a Nash equilibrium of the 2 2 game 5, 0, 0 0, 0 3, 4 This game has three Nash equilibria (in the obvious notation): ((, 0), (, 0)) (both choose ), ((0, ), (0, )) (both choose ), and ((4/5, /5), (3/8, 5/8)). Comparing the corresponding equilibrium paoffs, plaer can choose between In and Out. We find three sequential equilibria:. Plaer chooses In with probabilit, with probabilit, plaer 2 chooses with probabilit, and the belief over {(In, ), (In, )} assigns prob. to (In, ). 2. Plaer chooses Out with probabilit, with probabilit, plaer 2 chooses with probabilit, and the belief over {(In, ), (In, )} assigns prob. to (In, ). 3. Plaer chooses Out with probabilit, miture (4/5, /5) over (, ), plaer 2 chooses miture (3/8, 5/8) over (, ), and the belief over {(In, ), (In, )} assigns prob. (4/5, /5) over histories ((In, ), (In, )). Eercise 3c. The games have the same reduced strategic form: if plaer chooses In, he knows that he immediatel has to choose between and. In the second game, this two-step choice is modelled as a single choice. For sequential equilibria, this matters: recall that the are trembling-hand perfect equilibria of the agent strategic form, allowing for mistakes in each information set. This makes it harder to sustain equilibria in the game on the left: there, plaer has two information sets and consequentl two opportunities for mistakes. Which equilibrium ou like most, is a matter of taste. I like the one ending in paoffs (5, ) because of a forward-induction argument. If plaer consciousl gives up secure paoff 4 b not choosing Out, that
8 must be because he s aiming to get the higher paoff 5 b choosing. Then it makes sense for plaer 2 to choose as well. Eercise 4a. Still to do. Eercise 4b. No, that onl requires plaers to choose rational responses to structurall consistent beliefs. Those beliefs require onl that there is some behavioral strateg from which it can be derived using Baes rule. That behavioral strateg profile need not be related to what plaers actuall do. Eercise 4c. If I is reached with positive probabilit under behavioral strategies β, it follows that plaer chose with positive probabilit, sa p > 0. So the upper node is reached with probabilit 2 3p and the lower node with probabilit 3p. B Baes rule, both consistent and structurall consistent beliefs must assign probabilit 2/3 to the upper and /3 to the lower node. Weak consistenc is more liberal : if plaer chooses with probabilit, there are no constraints on the beliefs over plaer 2 s information set. This seems unrealistic: conditional upon being in that information set, must have chosen (even if that was onl b mistake or as a thought eperiment), so onl the beliefs (2/3, /3) make sense. Eercise 5. Still to do.
MA Game Theory 2005, LSE
MA. Game Theor, LSE Problem Set The problems in our third and final homework will serve two purposes. First, the will give ou practice in studing games with incomplete information. Second (unless indicated
More informationSF2972 Game Theory Exam with Solutions March 15, 2013
SF2972 Game Theory Exam with s March 5, 203 Part A Classical Game Theory Jörgen Weibull and Mark Voorneveld. (a) What are N, S and u in the definition of a finite normal-form (or, equivalently, strategic-form)
More informationSF2972 Game Theory Written Exam with Solutions June 10, 2011
SF97 Game Theory Written Exam with Solutions June 10, 011 Part A Classical Game Theory Jörgen Weibull and Mark Voorneveld 1. Finite normal-form games. (a) What are N, S and u in the definition of a finite
More informationEquilibrium Refinements
Equilibrium Refinements Mihai Manea MIT Sequential Equilibrium In many games information is imperfect and the only subgame is the original game... subgame perfect equilibrium = Nash equilibrium Play starting
More informationGame Theory. Wolfgang Frimmel. Perfect Bayesian Equilibrium
Game Theory Wolfgang Frimmel Perfect Bayesian Equilibrium / 22 Bayesian Nash equilibrium and dynamic games L M R 3 2 L R L R 2 2 L R L 2,, M,2, R,3,3 2 NE and 2 SPNE (only subgame!) 2 / 22 Non-credible
More informationPerfect Bayesian Equilibrium
Perfect Bayesian Equilibrium For an important class of extensive games, a solution concept is available that is simpler than sequential equilibrium, but with similar properties. In a Bayesian extensive
More information1 The General Definition
MS&E 336 Lecture 1: Dynamic games Ramesh Johari April 4, 2007 1 The General Definition A dynamic game (or extensive game, or game in extensive form) consists of: A set of players N; A set H of sequences
More informationEconomics 3012 Strategic Behavior Andy McLennan October 20, 2006
Economics 301 Strategic Behavior Andy McLennan October 0, 006 Lecture 11 Topics Problem Set 9 Extensive Games of Imperfect Information An Example General Description Strategies and Nash Equilibrium Beliefs
More informationFebruary 11, JEL Classification: C72, D43, D44 Keywords: Discontinuous games, Bertrand game, Toehold, First- and Second- Price Auctions
Eistence of Mied Strateg Equilibria in a Class of Discontinuous Games with Unbounded Strateg Sets. Gian Luigi Albano and Aleander Matros Department of Economics and ELSE Universit College London Februar
More informationGame Theory Lecture 10+11: Knowledge
Game Theory Lecture 10+11: Knowledge Christoph Schottmüller University of Copenhagen November 13 and 20, 2014 1 / 36 Outline 1 (Common) Knowledge The hat game A model of knowledge Common knowledge Agree
More informationEconomics 201A Economic Theory (Fall 2009) Extensive Games with Perfect and Imperfect Information
Economics 201A Economic Theory (Fall 2009) Extensive Games with Perfect and Imperfect Information Topics: perfect information (OR 6.1), subgame perfection (OR 6.2), forward induction (OR 6.6), imperfect
More informationWe have examined power functions like f (x) = x 2. Interchanging x
CHAPTER 5 Eponential and Logarithmic Functions We have eamined power functions like f =. Interchanging and ields a different function f =. This new function is radicall different from a power function
More informationRefinements - change set of equilibria to find "better" set of equilibria by eliminating some that are less plausible
efinements efinements - change set of equilibria to find "better" set of equilibria by eliminating some that are less plausible Strategic Form Eliminate Weakly Dominated Strategies - Purpose - throwing
More informationExtensive games (with perfect information)
Extensive games (with perfect information) (also referred to as extensive-form games or dynamic games) DEFINITION An extensive game with perfect information has the following components A set N (the set
More information6.4 graphs OF logarithmic FUnCTIOnS
SECTION 6. graphs of logarithmic functions 9 9 learning ObjeCTIveS In this section, ou will: Identif the domain of a logarithmic function. Graph logarithmic functions. 6. graphs OF logarithmic FUnCTIOnS
More informationVECTORS IN THREE DIMENSIONS
1 CHAPTER 2. BASIC TRIGONOMETRY 1 INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW VECTORS IN THREE DIMENSIONS 1 Vectors in Two Dimensions A vector is an object which has magnitude
More informationON FORWARD INDUCTION
Econometrica, Submission #6956, revised ON FORWARD INDUCTION SRIHARI GOVINDAN AND ROBERT WILSON Abstract. A player s pure strategy is called relevant for an outcome of a game in extensive form with perfect
More information11.4 Polar Coordinates
11. Polar Coordinates 917 11. Polar Coordinates In Section 1.1, we introduced the Cartesian coordinates of a point in the plane as a means of assigning ordered pairs of numbers to points in the plane.
More informationLESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II
LESSON #4 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART COMMON CORE ALGEBRA II You will recall from unit 1 that in order to find the inverse of a function, ou must switch and and solve for. Also,
More informationMAT 1275: Introduction to Mathematical Analysis. Graphs and Simplest Equations for Basic Trigonometric Functions. y=sin( x) Function
MAT 275: Introduction to Mathematical Analsis Dr. A. Rozenblum Graphs and Simplest Equations for Basic Trigonometric Functions We consider here three basic functions: sine, cosine and tangent. For them,
More information6.207/14.15: Networks Lecture 11: Introduction to Game Theory 3
6.207/14.15: Networks Lecture 11: Introduction to Game Theory 3 Daron Acemoglu and Asu Ozdaglar MIT October 19, 2009 1 Introduction Outline Existence of Nash Equilibrium in Infinite Games Extensive Form
More informationSection 3.1. ; X = (0, 1]. (i) f : R R R, f (x, y) = x y
Paul J. Bruillard MATH 0.970 Problem Set 6 An Introduction to Abstract Mathematics R. Bond and W. Keane Section 3.1: 3b,c,e,i, 4bd, 6, 9, 15, 16, 18c,e, 19a, 0, 1b Section 3.: 1f,i, e, 6, 1e,f,h, 13e,
More information3.7 InveRSe FUnCTIOnS
CHAPTER functions learning ObjeCTIveS In this section, ou will: Verif inverse functions. Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one.
More informationGovernment 2005: Formal Political Theory I
Government 2005: Formal Political Theory I Lecture 11 Instructor: Tommaso Nannicini Teaching Fellow: Jeremy Bowles Harvard University November 9, 2017 Overview * Today s lecture Dynamic games of incomplete
More informationEngineering Mathematics I
Engineering Mathematics I_ 017 Engineering Mathematics I 1. Introduction to Differential Equations Dr. Rami Zakaria Terminolog Differential Equation Ordinar Differential Equations Partial Differential
More informationBiostatistics in Research Practice - Regression I
Biostatistics in Research Practice - Regression I Simon Crouch 30th Januar 2007 In scientific studies, we often wish to model the relationships between observed variables over a sample of different subjects.
More information10 Back to planar nonlinear systems
10 Back to planar nonlinear sstems 10.1 Near the equilibria Recall that I started talking about the Lotka Volterra model as a motivation to stud sstems of two first order autonomous equations of the form
More informationCh 3 Alg 2 Note Sheet.doc 3.1 Graphing Systems of Equations
Ch 3 Alg Note Sheet.doc 3.1 Graphing Sstems of Equations Sstems of Linear Equations A sstem of equations is a set of two or more equations that use the same variables. If the graph of each equation =.4
More informationP1 Chapter 4 :: Graphs & Transformations
P1 Chapter 4 :: Graphs & Transformations jfrost@tiffin.kingston.sch.uk www.drfrostmaths.com @DrFrostMaths Last modified: 14 th September 2017 Use of DrFrostMaths for practice Register for free at: www.drfrostmaths.com/homework
More informationGame Theory Lecture 7 Refinements: Perfect equilibrium
Game Theory Lecture 7 Refinements: Perfect equilibrium Christoph Schottmüller University of Copenhagen October 23, 2014 1 / 18 Outline 1 Perfect equilibrium in strategic form games 2 / 18 Refinements I
More informationExtensive Form Games with Perfect Information
Extensive Form Games with Perfect Information Levent Koçkesen 1 Extensive Form Games The strategies in strategic form games are speci ed so that each player chooses an action (or a mixture of actions)
More informationSequential Games with Incomplete Information
Sequential Games with Incomplete Information Debraj Ray, November 6 For the remaining lectures we return to extensive-form games, but this time we focus on imperfect information, reputation, and signalling
More information: Cryptography and Game Theory Ran Canetti and Alon Rosen. Lecture 8
0368.4170: Cryptography and Game Theory Ran Canetti and Alon Rosen Lecture 8 December 9, 2009 Scribe: Naama Ben-Aroya Last Week 2 player zero-sum games (min-max) Mixed NE (existence, complexity) ɛ-ne Correlated
More informationProperties of Limits
33460_003qd //04 :3 PM Page 59 SECTION 3 Evaluating Limits Analticall 59 Section 3 Evaluating Limits Analticall Evaluate a it using properties of its Develop and use a strateg for finding its Evaluate
More informationIndustrial Organization Lecture 3: Game Theory
Industrial Organization Lecture 3: Game Theory Nicolas Schutz Nicolas Schutz Game Theory 1 / 43 Introduction Why game theory? In the introductory lecture, we defined Industrial Organization as the economics
More informationBounded Rationality, Strategy Simplification, and Equilibrium
Bounded Rationality, Strategy Simplification, and Equilibrium UPV/EHU & Ikerbasque Donostia, Spain BCAM Workshop on Interactions, September 2014 Bounded Rationality Frequently raised criticism of game
More informationEC319 Economic Theory and Its Applications, Part II: Lecture 7
EC319 Economic Theory and Its Applications, Part II: Lecture 7 Leonardo Felli NAB.2.14 27 February 2014 Signalling Games Consider the following Bayesian game: Set of players: N = {N, S, }, Nature N strategy
More informationQUADRATIC FUNCTION REVIEW
Name: Date: QUADRATIC FUNCTION REVIEW Linear and eponential functions are used throughout mathematics and science due to their simplicit and applicabilit. Quadratic functions comprise another ver important
More informationSummer Math Packet (revised 2017)
Summer Math Packet (revised 07) In preparation for Honors Math III, we have prepared a packet of concepts that students should know how to do as these concepts have been taught in previous math classes.
More informationExtensive Form Games I
Extensive Form Games I Definition of Extensive Form Game a finite game tree X with nodes x X nodes are partially ordered and have a single root (minimal element) terminal nodes are z Z (maximal elements)
More informationSpace frames. b) R z φ z. R x. Figure 1 Sign convention: a) Displacements; b) Reactions
Lecture notes: Structural Analsis II Space frames I. asic concepts. The design of a building is generall accomplished b considering the structure as an assemblage of planar frames, each of which is designed
More information6.207/14.15: Networks Lecture 10: Introduction to Game Theory 2
6.207/14.15: Networks Lecture 10: Introduction to Game Theory 2 Daron Acemoglu and Asu Ozdaglar MIT October 14, 2009 1 Introduction Outline Mixed Strategies Existence of Mixed Strategy Nash Equilibrium
More information1.7 Inverse Functions
71_0107.qd 1/7/0 10: AM Page 17 Section 1.7 Inverse Functions 17 1.7 Inverse Functions Inverse Functions Recall from Section 1. that a function can be represented b a set of ordered pairs. For instance,
More informationChapter 18 Quadratic Function 2
Chapter 18 Quadratic Function Completed Square Form 1 Consider this special set of numbers - the square numbers or the set of perfect squares. 4 = = 9 = 3 = 16 = 4 = 5 = 5 = Numbers like 5, 11, 15 are
More informationINTRODUCTION TO DIOPHANTINE EQUATIONS
INTRODUCTION TO DIOPHANTINE EQUATIONS In the earl 20th centur, Thue made an important breakthrough in the stud of diophantine equations. His proof is one of the first eamples of the polnomial method. His
More informationLinear Programming. Maximize the function. P = Ax + By + C. subject to the constraints. a 1 x + b 1 y < c 1 a 2 x + b 2 y < c 2
Linear Programming Man real world problems require the optimization of some function subject to a collection of constraints. Note: Think of optimizing as maimizing or minimizing for MATH1010. For eample,
More informationAPPENDIX D Rotation and the General Second-Degree Equation
APPENDIX D Rotation and the General Second-Degree Equation Rotation of Aes Invariants Under Rotation After rotation of the - and -aes counterclockwise through an angle, the rotated aes are denoted as the
More information15. Eigenvalues, Eigenvectors
5 Eigenvalues, Eigenvectors Matri of a Linear Transformation Consider a linear ( transformation ) L : a b R 2 R 2 Suppose we know that L and L Then c d because of linearit, we can determine what L does
More informationLecture 1.1: The Minimax Theorem
Algorithmic Game Theor, Compleit and Learning Jul 17, 2017 Lecture 1.1: The Mini Theorem Lecturer: Constantinos Daskalakis Scribe: Constantinos Daskalakis In these notes, we show that a Nash equilibrium
More informationSEQUENTIAL EQUILIBRIA IN BAYESIAN GAMES WITH COMMUNICATION. Dino Gerardi and Roger B. Myerson. December 2005
SEQUENTIAL EQUILIBRIA IN BAYESIAN GAMES WITH COMMUNICATION By Dino Gerardi and Roger B. Myerson December 2005 COWLES FOUNDATION DISCUSSION AER NO. 1542 COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE
More informationIterated Strict Dominance in Pure Strategies
Iterated Strict Dominance in Pure Strategies We know that no rational player ever plays strictly dominated strategies. As each player knows that each player is rational, each player knows that his opponents
More information14.1 Systems of Linear Equations in Two Variables
86 Chapter 1 Sstems of Equations and Matrices 1.1 Sstems of Linear Equations in Two Variables Use the method of substitution to solve sstems of equations in two variables. Use the method of elimination
More informationBasics of Game Theory
Basics of Game Theory Giacomo Bacci and Luca Sanguinetti Department of Information Engineering University of Pisa, Pisa, Italy {giacomo.bacci,luca.sanguinetti}@iet.unipi.it April - May, 2010 G. Bacci and
More informationDefinitions and Proofs
Giving Advice vs. Making Decisions: Transparency, Information, and Delegation Online Appendix A Definitions and Proofs A. The Informational Environment The set of states of nature is denoted by = [, ],
More information4: Dynamic games. Concordia February 6, 2017
INSE6441 Jia Yuan Yu 4: Dynamic games Concordia February 6, 2017 We introduce dynamic game with non-simultaneous moves. Example 0.1 (Ultimatum game). Divide class into two groups at random: Proposers,
More information2.2 SEPARABLE VARIABLES
44 CHAPTER FIRST-ORDER DIFFERENTIAL EQUATIONS 6 Consider the autonomous DE 6 Use our ideas from Problem 5 to find intervals on the -ais for which solution curves are concave up and intervals for which
More informationEconomics 209A Theory and Application of Non-Cooperative Games (Fall 2013) Extensive games with perfect information OR6and7,FT3,4and11
Economics 209A Theory and Application of Non-Cooperative Games (Fall 2013) Extensive games with perfect information OR6and7,FT3,4and11 Perfect information A finite extensive game with perfect information
More informationMethods for Advanced Mathematics (C3) Coursework Numerical Methods
Woodhouse College 0 Page Introduction... 3 Terminolog... 3 Activit... 4 Wh use numerical methods?... Change of sign... Activit... 6 Interval Bisection... 7 Decimal Search... 8 Coursework Requirements on
More informationEDEXCEL ANALYTICAL METHODS FOR ENGINEERS H1 UNIT 2 - NQF LEVEL 4 OUTCOME 4 - STATISTICS AND PROBABILITY TUTORIAL 3 LINEAR REGRESSION
EDEXCEL AALYTICAL METHODS FOR EGIEERS H1 UIT - QF LEVEL 4 OUTCOME 4 - STATISTICS AD PROBABILITY TUTORIAL 3 LIEAR REGRESSIO Tabular and graphical form: data collection methods; histograms; bar charts; line
More informationExtensive Form Games with Perfect Information
Extensive Form Games with Perfect Information Pei-yu Lo 1 Introduction Recap: BoS. Look for all Nash equilibria. show how to nd pure strategy Nash equilibria. Show how to nd mixed strategy Nash equilibria.
More informationFeedforward Neural Networks
Chapter 4 Feedforward Neural Networks 4. Motivation Let s start with our logistic regression model from before: P(k d) = softma k =k ( λ(k ) + w d λ(k, w) ). (4.) Recall that this model gives us a lot
More informationSystems of Linear Equations: Solving by Graphing
8.1 Sstems of Linear Equations: Solving b Graphing 8.1 OBJECTIVE 1. Find the solution(s) for a set of linear equations b graphing NOTE There is no other ordered pair that satisfies both equations. From
More informationLecture December 2009 Fall 2009 Scribe: R. Ring In this lecture we will talk about
0368.4170: Cryptography and Game Theory Ran Canetti and Alon Rosen Lecture 7 02 December 2009 Fall 2009 Scribe: R. Ring In this lecture we will talk about Two-Player zero-sum games (min-max theorem) Mixed
More information2.3 Quadratic Functions
88 Linear and Quadratic Functions. Quadratic Functions You ma recall studing quadratic equations in Intermediate Algebra. In this section, we review those equations in the contet of our net famil of functions:
More information2.1 Rates of Change and Limits AP Calculus
. Rates of Change and Limits AP Calculus. RATES OF CHANGE AND LIMITS Limits Limits are what separate Calculus from pre calculus. Using a it is also the foundational principle behind the two most important
More informationMicroeconomics. 2. Game Theory
Microeconomics 2. Game Theory Alex Gershkov http://www.econ2.uni-bonn.de/gershkov/gershkov.htm 18. November 2008 1 / 36 Dynamic games Time permitting we will cover 2.a Describing a game in extensive form
More informationRoberto s Notes on Integral Calculus Chapter 3: Basics of differential equations Section 3. Separable ODE s
Roberto s Notes on Integral Calculus Chapter 3: Basics of differential equations Section 3 Separable ODE s What ou need to know alread: What an ODE is and how to solve an eponential ODE. What ou can learn
More informationPower in voting games: axiomatic and probabilistic approaches
: axiomatic and probabilistic approaches a a Department of Mathematics Technical Universit of Catalonia Summer School, Campione d Italia Game Theor and Voting Sstems Outline Power indices, several interpretations
More informationReview Topics for MATH 1400 Elements of Calculus Table of Contents
Math 1400 - Mano Table of Contents - Review - page 1 of 2 Review Topics for MATH 1400 Elements of Calculus Table of Contents MATH 1400 Elements of Calculus is one of the Marquette Core Courses for Mathematical
More informationFunctions. Introduction
Functions,00 P,000 00 0 70 7 80 8 0 000 00 00 Figure Standard and Poor s Inde with dividends reinvested (credit "bull": modification of work b Praitno Hadinata; credit "graph": modification of work b MeasuringWorth)
More information4 Inverse function theorem
Tel Aviv Universit, 2013/14 Analsis-III,IV 53 4 Inverse function theorem 4a What is the problem................ 53 4b Simple observations before the theorem..... 54 4c The theorem.....................
More information6.896 Topics in Algorithmic Game Theory February 8, Lecture 2
6.896 Topics in Algorithmic Game Theor Februar 8, 2010 Lecture 2 Lecturer: Constantinos Daskalakis Scribe: Yang Cai, Debmala Panigrahi In this lecture, we focus on two-plaer zero-sum games. Our goal is
More informationWars of Attrition with Budget Constraints
Wars of Attrition with Budget Constraints Gagan Ghosh Bingchao Huangfu Heng Liu October 19, 2017 (PRELIMINARY AND INCOMPLETE: COMMENTS WELCOME) Abstract We study wars of attrition between two bidders who
More informationMA123, Chapter 1: Equations, functions and graphs (pp. 1-15)
MA123, Chapter 1: Equations, functions and graphs (pp. 1-15) Date: Chapter Goals: Identif solutions to an equation. Solve an equation for one variable in terms of another. What is a function? Understand
More informationOn Decentralized Incentive Compatible Mechanisms for Partially Informed Environments
On Decentralized Incentive Compatible Mechanisms for Partially Informed Environments by Ahuva Mu alem June 2005 presented by Ariel Kleiner and Neil Mehta Contributions Brings the concept of Nash Implementation
More informationFinding Limits Graphically and Numerically. An Introduction to Limits
8 CHAPTER Limits and Their Properties Section Finding Limits Graphicall and Numericall Estimate a it using a numerical or graphical approach Learn different was that a it can fail to eist Stud and use
More informationResearch Article Chaos and Control of Game Model Based on Heterogeneous Expectations in Electric Power Triopoly
Discrete Dnamics in Nature and Societ Volume 29, Article ID 469564, 8 pages doi:.55/29/469564 Research Article Chaos and Control of Game Model Based on Heterogeneous Epectations in Electric Power Triopol
More informationReview of topics since what was covered in the midterm: Topics that we covered before the midterm (also may be included in final):
Review of topics since what was covered in the midterm: Subgame-perfect eqms in extensive games with perfect information where players choose a number (first-order conditions, boundary conditions, favoring
More informationSEPARABLE EQUATIONS 2.2
46 CHAPTER FIRST-ORDER DIFFERENTIAL EQUATIONS 4. Chemical Reactions When certain kinds of chemicals are combined, the rate at which the new compound is formed is modeled b the autonomous differential equation
More informationProbability Models of Information Exchange on Networks Lecture 5
Probability Models of Information Exchange on Networks Lecture 5 Elchanan Mossel (UC Berkeley) July 25, 2013 1 / 22 Informational Framework Each agent receives a private signal X i which depends on S.
More informationRealization Plans for Extensive Form Games without Perfect Recall
Realization Plans for Extensive Form Games without Perfect Recall Richard E. Stearns Department of Computer Science University at Albany - SUNY Albany, NY 12222 April 13, 2015 Abstract Given a game in
More informationINTRODUCTION TO DIFFERENTIAL EQUATIONS
INTRODUCTION TO DIFFERENTIAL EQUATIONS. Definitions and Terminolog. Initial-Value Problems.3 Differential Equations as Mathematical Models CHAPTER IN REVIEW The words differential and equations certainl
More informationSymmetry Arguments and the Role They Play in Using Gauss Law
Smmetr Arguments and the Role The la in Using Gauss Law K. M. Westerberg (9/2005) Smmetr plas a ver important role in science in general, and phsics in particular. Arguments based on smmetr can often simplif
More informationThe Force Table Introduction: Theory:
1 The Force Table Introduction: "The Force Table" is a simple tool for demonstrating Newton s First Law and the vector nature of forces. This tool is based on the principle of equilibrium. An object is
More informationCommon Knowledge of Rationality is Self-Contradictory. Herbert Gintis
Common Knowledge of Rationality is Self-Contradictory Herbert Gintis February 25, 2012 Abstract The conditions under which rational agents play a Nash equilibrium are extremely demanding and often implausible.
More information5.4 dividing POlynOmIAlS
SECTION 5.4 dividing PolNomiAls 3 9 3 learning ObjeCTIveS In this section, ou will: Use long division to divide polnomials. Use snthetic division to divide polnomials. 5.4 dividing POlnOmIAlS Figure 1
More informationA Polynomial-time Nash Equilibrium Algorithm for Repeated Games
A Polynomial-time Nash Equilibrium Algorithm for Repeated Games Michael L. Littman mlittman@cs.rutgers.edu Rutgers University Peter Stone pstone@cs.utexas.edu The University of Texas at Austin Main Result
More informationLESSON #12 - FORMS OF A LINE COMMON CORE ALGEBRA II
LESSON # - FORMS OF A LINE COMMON CORE ALGEBRA II Linear functions come in a variet of forms. The two shown below have been introduced in Common Core Algebra I and Common Core Geometr. TWO COMMON FORMS
More information4 The Cartesian Coordinate System- Pictures of Equations
The Cartesian Coordinate Sstem- Pictures of Equations Concepts: The Cartesian Coordinate Sstem Graphs of Equations in Two Variables -intercepts and -intercepts Distance in Two Dimensions and the Pthagorean
More informationEvolutionary Dynamics and Extensive Form Games by Ross Cressman. Reviewed by William H. Sandholm *
Evolutionary Dynamics and Extensive Form Games by Ross Cressman Reviewed by William H. Sandholm * Noncooperative game theory is one of a handful of fundamental frameworks used for economic modeling. It
More information17. f(x) = x 2 + 5x f(x) = x 2 + x f(x) = x 2 + 3x f(x) = x 2 + 3x f(x) = x 2 16x f(x) = x 2 + 4x 96
Section.3 Zeros of the Quadratic 473.3 Eercises In Eercises 1-8, factor the given quadratic polnomial. 1. 2 + 9 + 14 2. 2 + 6 + 3. 2 + + 9 4. 2 + 4 21. 2 4 6. 2 + 7 8 7. 2 7 + 12 8. 2 + 24 In Eercises
More informationFinding Limits Graphically and Numerically. An Introduction to Limits
60_00.qd //0 :05 PM Page 8 8 CHAPTER Limits and Their Properties Section. Finding Limits Graphicall and Numericall Estimate a it using a numerical or graphical approach. Learn different was that a it can
More informationWorksheet #1. A little review.
Worksheet #1. A little review. I. Set up BUT DO NOT EVALUATE definite integrals for each of the following. 1. The area between the curves = 1 and = 3. Solution. The first thing we should ask ourselves
More informationLESSON #48 - INTEGER EXPONENTS COMMON CORE ALGEBRA II
LESSON #8 - INTEGER EXPONENTS COMMON CORE ALGEBRA II We just finished our review of linear functions. Linear functions are those that grow b equal differences for equal intervals. In this unit we will
More informationAn axiomatization of minimal curb sets. 1. Introduction. Mark Voorneveld,,1, Willemien Kets, and Henk Norde
An axiomatization of minimal curb sets Mark Voorneveld,,1, Willemien Kets, and Henk Norde Department of Econometrics and Operations Research, Tilburg University, The Netherlands Department of Economics,
More information3.2 LOGARITHMIC FUNCTIONS AND THEIR GRAPHS
Section. Logarithmic Functions and Their Graphs 7. LOGARITHMIC FUNCTIONS AND THEIR GRAPHS Ariel Skelle/Corbis What ou should learn Recognize and evaluate logarithmic functions with base a. Graph logarithmic
More information4.3 Exercises. local maximum or minimum. The second derivative is. e 1 x 2x 1. f x x 2 e 1 x 1 x 2 e 1 x 2x x 4
SECTION 4.3 HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH 297 local maimum or minimum. The second derivative is f 2 e 2 e 2 4 e 2 4 Since e and 4, we have f when and when 2 f. So the curve is concave downward
More information3.0 PROBABILITY, RANDOM VARIABLES AND RANDOM PROCESSES
3.0 PROBABILITY, RANDOM VARIABLES AND RANDOM PROCESSES 3.1 Introduction In this chapter we will review the concepts of probabilit, rom variables rom processes. We begin b reviewing some of the definitions
More informationGame Theory and Algorithms Lecture 2: Nash Equilibria and Examples
Game Theory and Algorithms Lecture 2: Nash Equilibria and Examples February 24, 2011 Summary: We introduce the Nash Equilibrium: an outcome (action profile) which is stable in the sense that no player
More informationGeometry and Honors Geometry Summer Review Packet 2014
Geometr and Honors Geometr Summer Review Packet 04 This will not be graded. It is for our benefit onl. The problems in this packet are designed to help ou review topics from previous mathematics courses
More information