Expectations & Randomization Normal Form Games Dominance Iterated Dominance. Normal Form Games & Dominance
|
|
- Buck Barton
- 5 years ago
- Views:
Transcription
1 Normal Form Games & Dominance
2 Let s play the quarters game again We each have a quarter. Let s put them down on the desk at the same time. If they show the same side (HH or TT), you take my quarter. If they show opposite sides (HT or TH), I take yours. Q: What is your expected payoff for playing H? A: It depends on what you believe I will choose. Suppose that you think I will choose H with probability p. If you play H, you expect to win (50) with probability p and lose with probability 1 p So your expected value from H is EV [H] = p 50 + (1 p) 0 = 50p Modeling choice under uncertainty: beliefs form, expected payoff calculated, outcomes weighted by perceived likelihood
3 Let s play the quarters game again We each have a quarter. Let s put them down on the desk at the same time. If they show the same side (HH or TT), you take my quarter. If they show opposite sides (HT or TH), I take yours. But this time you have a third option (besides H and T): let your neighbor flip the coin for you. What do you choose? (H,T, or Flip) Notice: Uncertainty can come from randomizing own strategy (mixed strategy), not just strategic uncertainty (or external randomness)
4 Part I: Games in Normal Form Definition A strategy is a complete contingent plan for a player in the game. Example: Pick the highest integer. What are the strategies of each player? General notation: individual strategy space = S i, individual strategy = s i S i. A strategy profile: s S, where S = S 1 S 2 S n Notation: i and i. s = (s i, s i ). Another element: Payoff function u i : S R Definition A game in normal form (also called strategic form) consists of a set of players, along with strategy spaces and payoff functions for each player.
5 Matrix Games Two-player, finite (as in S is finite) normal form games can be described succinctly with bi-matrices. Take a look at your handout. Let s verify that the first matrix game satisfies the requirements of the normal form. Ok, before starting to analyze this game, let s try playing one of these...
6 Let s Play Let s play (g) for real money (see handout) For (g): odd perm number choose A, B, or C; even perm number choose X, Y, or Z. Two separate clicker votes: first row players, then column players. Now: I will select participants by drawing two cards for each game, pay in $ according to strategies chosen Next time: I will select two more people and pay according to clicker vote from today. What kind of reasoning did you use?
7 How do clickers benefit us? Make it easy to play (some) in-class games Can help me see what you ve learned Can help you see whether or not you understand Can help you see how far you ve come Allow you to give me feedback/opinions Also: attendance/participation Make sure your clicker is registered, bring it every class.
8 Back to game a) How do you win at this game? Ask Dilbert What s going on in this game?
9 Three concepts at play Concepts Beliefs: What Player i thinks all the other players will do (θ i S i ). Best response: A strategy is a best response if it is at least as good as any other strategy, given beliefs. A strategy is dominated if there is another strategy that is better for every set of beliefs. In other words, if Y is better than X no matter what you think other players will choose, then X is dominated by Y. No rational person would ever choose a strategy that is dominated
10 Solution Concept: Dominance Assumes rationality: no one would ever choose an action this is dominated by another. Apply to PD by crossing out all rows or columns that are dominated Now let s try to solve b) Does each player in game (b) have a dominant strategy? NO! b) is not dominance solvable
11 What s the problem? Need to further narrow down choices Rationality is too weak an a assumption Stronger: assume rationality is common knowledge Rational, know you are, know you know, know you know you know, etc. Eliminate strategies through iterative process Our second solution concept: Iterated Elimination of Dominated Strategies
12 Iterated Dominance Also known as Rationalizability Strategies that survived iterated elmination are called rationalizable Try on b) Now we get a unique outcome. Yay!
13 Iterated Dominance: Weaknesses? Is iterated dominance THE ULTIMATE SOLUTION CONCEPT? To see, let s return to game c). Is c) dominance solvable? NO! We can get more predictive power with stronger assumptions. Do we want this?
14 Iterated Dominance: Are we missing anything? Let s try d) Seems like no dominance going on. Are we missing anything? Free your mind and dominance will follow If we enrich the strategy space to allow mixed strategies, then B is dominated By what? (1/2, 0, 1/2) Now we continue iterated elimination A strategy may be dominated by a combination of other strategies
15 Player 1 s expected payoff, as a function of beliefs about Player 2 s strategy
16 Mixed strategies and best responses Let s look at e) now Is B dominated? No, not even by mixed strategies Is it ever a best response? Not to X or Y But it is a BR to (1/2, 1/2) (one example) If a strategy is a best response to some beliefs, it can t be dominated A strategy may be a best response (therefore undominated) even if it is not the best response to any pure strategy
Using the Maximin Principle
Using the Maximin Principle Under the maximin principle, it is easy to see that Rose should choose a, making her worst-case payoff 0. Colin s similar rationality as a player induces him to play (under
More informationECE 586GT: Problem Set 1: Problems and Solutions Analysis of static games
University of Illinois Fall 2018 ECE 586GT: Problem Set 1: Problems and Solutions Analysis of static games Due: Tuesday, Sept. 11, at beginning of class Reading: Course notes, Sections 1.1-1.4 1. [A random
More informationEconomics 109 Practice Problems 1, Vincent Crawford, Spring 2002
Economics 109 Practice Problems 1, Vincent Crawford, Spring 2002 P1. Consider the following game. There are two piles of matches and two players. The game starts with Player 1 and thereafter the players
More informationEconomics 171: Final Exam
Question 1: Basic Concepts (20 points) Economics 171: Final Exam 1. Is it true that every strategy is either strictly dominated or is a dominant strategy? Explain. (5) No, some strategies are neither dominated
More informationMS&E 246: Lecture 2 The basics. Ramesh Johari January 16, 2007
MS&E 246: Lecture 2 The basics Ramesh Johari January 16, 2007 Course overview (Mainly) noncooperative game theory. Noncooperative: Focus on individual players incentives (note these might lead to cooperation!)
More informationHandout on Rationalizability and IDSDS 1
EconS 424 - Strategy and Game Theory Handout on Rationalizability and ISS 1 1 Introduction In this handout, we will discuss an extension of best response functions: Rationalizability. Best response: As
More informationSI 563 Homework 3 Oct 5, Determine the set of rationalizable strategies for each of the following games. a) X Y X Y Z
SI 563 Homework 3 Oct 5, 06 Chapter 7 Exercise : ( points) Determine the set of rationalizable strategies for each of the following games. a) U (0,4) (4,0) M (3,3) (3,3) D (4,0) (0,4) X Y U (0,4) (4,0)
More informationNotes for Section: Week 7
Economics 160 Professor Steven Tadelis Stanford University Spring Quarter, 004 Notes for Section: Week 7 Notes prepared by Paul Riskind (pnr@stanford.edu). spot errors or have questions about these notes.
More informationMicroeconomics II. CIDE, MsC Economics. List of Problems
Microeconomics II CIDE, MsC Economics List of Problems 1. There are three people, Amy (A), Bart (B) and Chris (C): A and B have hats. These three people are arranged in a room so that B can see everything
More informationMath 135: Answers to Practice Problems
Math 35: Answers to Practice Problems Answers to problems from the textbook: Many of the problems from the textbook have answers in the back of the book. Here are the answers to the problems that don t
More information6.254 : Game Theory with Engineering Applications Lecture 3: Strategic Form Games - Solution Concepts
6.254 : Game Theory with Engineering Applications Lecture 3: Strategic Form Games - Solution Concepts Asu Ozdaglar MIT February 9, 2010 1 Introduction Outline Review Examples of Pure Strategy Nash Equilibria
More informationCHAPTER 9 Nash Equilibrium 1-1
. CHAPTER 9 Nash Equilibrium 1-1 Rationalizability & Strategic Uncertainty In the Battle of Sexes, uncertainty about other s strategy can lead to poor payoffs, even if both players rational Rationalizability
More informationIntroduction to Multi-Agent Programming
Introduction to Multi-Agent Programming 10. Game Theory Strategic Reasoning and Acting Alexander Kleiner and Bernhard Nebel Strategic Game A strategic game G consists of a finite set N (the set of players)
More informationPAULI MURTO, ANDREY ZHUKOV
GAME THEORY SOLUTION SET 1 WINTER 018 PAULI MURTO, ANDREY ZHUKOV Introduction For suggested solution to problem 4, last year s suggested solutions by Tsz-Ning Wong were used who I think used suggested
More informationIntroduction to Game Theory
Chapter G Notes 1 Epstein, 2013 Introduction to Game Theory G.1 Decision Making Game theory is a mathematical model that provides a atic way to deal with decisions under circumstances where the alternatives
More informationRationalizable Strategies
Rationalizable Strategies Carlos Hurtado Department of Economics University of Illinois at Urbana-Champaign hrtdmrt2@illinois.edu Jun 1st, 2015 C. Hurtado (UIUC - Economics) Game Theory On the Agenda 1
More informationGames of Incomplete Information
Games of Incomplete Information EC202 Lectures V & VI Francesco Nava London School of Economics January 2011 Nava (LSE) EC202 Lectures V & VI Jan 2011 1 / 22 Summary Games of Incomplete Information: Definitions:
More informationPrisoner s dilemma with T = 1
REPEATED GAMES Overview Context: players (e.g., firms) interact with each other on an ongoing basis Concepts: repeated games, grim strategies Economic principle: repetition helps enforcing otherwise unenforceable
More informationIterated Dominance and Nash Equilibrium
Chapter 11 Iterated Dominance and Nash Equilibrium In the previous chapter we examined simultaneous move games in which each player had a dominant strategy; the Prisoner s Dilemma game was one example.
More informationCS 798: Homework Assignment 4 (Game Theory)
0 5 CS 798: Homework Assignment 4 (Game Theory) 1.0 Preferences Assigned: October 28, 2009 Suppose that you equally like a banana and a lottery that gives you an apple 30% of the time and a carrot 70%
More informationLecture 5 Leadership and Reputation
Lecture 5 Leadership and Reputation Reputations arise in situations where there is an element of repetition, and also where coordination between players is possible. One definition of leadership is that
More informationMixed Strategies. In the previous chapters we restricted players to using pure strategies and we
6 Mixed Strategies In the previous chapters we restricted players to using pure strategies and we postponed discussing the option that a player may choose to randomize between several of his pure strategies.
More informationCS711 Game Theory and Mechanism Design
CS711 Game Theory and Mechanism Design Problem Set 1 August 13, 2018 Que 1. [Easy] William and Henry are participants in a televised game show, seated in separate booths with no possibility of communicating
More informationGame Theory Notes: Examples of Games with Dominant Strategy Equilibrium or Nash Equilibrium
Game Theory Notes: Examples of Games with Dominant Strategy Equilibrium or Nash Equilibrium Below are two different games. The first game has a dominant strategy equilibrium. The second game has two Nash
More informationThursday, March 3
5.53 Thursday, March 3 -person -sum (or constant sum) game theory -dimensional multi-dimensional Comments on first midterm: practice test will be on line coverage: every lecture prior to game theory quiz
More informationMATH 4321 Game Theory Solution to Homework Two
MATH 321 Game Theory Solution to Homework Two Course Instructor: Prof. Y.K. Kwok 1. (a) Suppose that an iterated dominance equilibrium s is not a Nash equilibrium, then there exists s i of some player
More informationStochastic Games and Bayesian Games
Stochastic Games and Bayesian Games CPSC 532l Lecture 10 Stochastic Games and Bayesian Games CPSC 532l Lecture 10, Slide 1 Lecture Overview 1 Recap 2 Stochastic Games 3 Bayesian Games 4 Analyzing Bayesian
More informationGame Theory. VK Room: M1.30 Last updated: October 22, 2012.
Game Theory VK Room: M1.30 knightva@cf.ac.uk www.vincent-knight.com Last updated: October 22, 2012. 1 / 33 Overview Normal Form Games Pure Nash Equilibrium Mixed Nash Equilibrium 2 / 33 Normal Form Games
More informationMIDTERM 1 SOLUTIONS 10/16/2008
4. Game Theory MIDTERM SOLUTIONS 0/6/008 Prof. Casey Rothschild Instructions. Thisisanopenbookexam; you canuse anywritten material. You mayuse a calculator. You may not use a computer or any electronic
More information10.1 Elimination of strictly dominated strategies
Chapter 10 Elimination by Mixed Strategies The notions of dominance apply in particular to mixed extensions of finite strategic games. But we can also consider dominance of a pure strategy by a mixed strategy.
More informationFDPE Microeconomics 3 Spring 2017 Pauli Murto TA: Tsz-Ning Wong (These solution hints are based on Julia Salmi s solution hints for Spring 2015.
FDPE Microeconomics 3 Spring 2017 Pauli Murto TA: Tsz-Ning Wong (These solution hints are based on Julia Salmi s solution hints for Spring 2015.) Hints for Problem Set 2 1. Consider a zero-sum game, where
More informationMath 167: Mathematical Game Theory Instructor: Alpár R. Mészáros
Math 167: Mathematical Game Theory Instructor: Alpár R. Mészáros Midterm #1, February 3, 2017 Name (use a pen): Student ID (use a pen): Signature (use a pen): Rules: Duration of the exam: 50 minutes. By
More informationHW Consider the following game:
HW 1 1. Consider the following game: 2. HW 2 Suppose a parent and child play the following game, first analyzed by Becker (1974). First child takes the action, A 0, that produces income for the child,
More informationTest 1. ECON3161, Game Theory. Tuesday, September 25 th
Test 1 ECON3161, Game Theory Tuesday, September 2 th Directions: Answer each question completely. If you cannot determine the answer, explaining how you would arrive at the answer may earn you some points.
More informationpreferences of the individual players over these possible outcomes, typically measured by a utility or payoff function.
Leigh Tesfatsion 26 January 2009 Game Theory: Basic Concepts and Terminology A GAME consists of: a collection of decision-makers, called players; the possible information states of each player at each
More informationWeek 8: Basic concepts in game theory
Week 8: Basic concepts in game theory Part 1: Examples of games We introduce here the basic objects involved in game theory. To specify a game ones gives The players. The set of all possible strategies
More informationAn introduction on game theory for wireless networking [1]
An introduction on game theory for wireless networking [1] Ning Zhang 14 May, 2012 [1] Game Theory in Wireless Networks: A Tutorial 1 Roadmap 1 Introduction 2 Static games 3 Extensive-form games 4 Summary
More informationExercises Solutions: Game Theory
Exercises Solutions: Game Theory Exercise. (U, R).. (U, L) and (D, R). 3. (D, R). 4. (U, L) and (D, R). 5. First, eliminate R as it is strictly dominated by M for player. Second, eliminate M as it is strictly
More informationm 11 m 12 Non-Zero Sum Games Matrix Form of Zero-Sum Games R&N Section 17.6
Non-Zero Sum Games R&N Section 17.6 Matrix Form of Zero-Sum Games m 11 m 12 m 21 m 22 m ij = Player A s payoff if Player A follows pure strategy i and Player B follows pure strategy j 1 Results so far
More informationNow we return to simultaneous-move games. We resolve the issue of non-existence of Nash equilibrium. in pure strategies through intentional mixing.
Econ 221 Fall, 2018 Li, Hao UBC CHAPTER 7. SIMULTANEOUS-MOVE GAMES: MIXED STRATEGIES Now we return to simultaneous-move games. We resolve the issue of non-existence of Nash equilibrium in pure strategies
More informationECON 803: MICROECONOMIC THEORY II Arthur J. Robson Fall 2016 Assignment 9 (due in class on November 22)
ECON 803: MICROECONOMIC THEORY II Arthur J. Robson all 2016 Assignment 9 (due in class on November 22) 1. Critique of subgame perfection. 1 Consider the following three-player sequential game. In the first
More informationRegret Minimization and Security Strategies
Chapter 5 Regret Minimization and Security Strategies Until now we implicitly adopted a view that a Nash equilibrium is a desirable outcome of a strategic game. In this chapter we consider two alternative
More informationSolution to Tutorial 1
Solution to Tutorial 1 011/01 Semester I MA464 Game Theory Tutor: Xiang Sun August 4, 011 1 Review Static means one-shot, or simultaneous-move; Complete information means that the payoff functions are
More informationMA300.2 Game Theory 2005, LSE
MA300.2 Game Theory 2005, LSE Answers to Problem Set 2 [1] (a) This is standard (we have even done it in class). The one-shot Cournot outputs can be computed to be A/3, while the payoff to each firm can
More informationECO 5341 (Section 2) Spring 2016 Midterm March 24th 2016 Total Points: 100
Name:... ECO 5341 (Section 2) Spring 2016 Midterm March 24th 2016 Total Points: 100 For full credit, please be formal, precise, concise and tidy. If your answer is illegible and not well organized, if
More informationSolution to Tutorial /2013 Semester I MA4264 Game Theory
Solution to Tutorial 1 01/013 Semester I MA464 Game Theory Tutor: Xiang Sun August 30, 01 1 Review Static means one-shot, or simultaneous-move; Complete information means that the payoff functions are
More informationPAULI MURTO, ANDREY ZHUKOV. If any mistakes or typos are spotted, kindly communicate them to
GAME THEORY PROBLEM SET 1 WINTER 2018 PAULI MURTO, ANDREY ZHUKOV Introduction If any mistakes or typos are spotted, kindly communicate them to andrey.zhukov@aalto.fi. Materials from Osborne and Rubinstein
More informationCSE 316A: Homework 5
CSE 316A: Homework 5 Due on December 2, 2015 Total: 160 points Notes There are 8 problems on 5 pages below, worth 20 points each (amounting to a total of 160. However, this homework will be graded out
More informationName. Answers Discussion Final Exam, Econ 171, March, 2012
Name Answers Discussion Final Exam, Econ 171, March, 2012 1) Consider the following strategic form game in which Player 1 chooses the row and Player 2 chooses the column. Both players know that this is
More informationS 2,2-1, x c C x r, 1 0,0
Problem Set 5 1. There are two players facing each other in the following random prisoners dilemma: S C S, -1, x c C x r, 1 0,0 With probability p, x c = y, and with probability 1 p, x c = 0. With probability
More informationIntroduction to Game Theory
Chapter G Notes 1 Epstein, 2013 Introduction to Game Theory G.1 Decision Making Game theory is a mathematical model that provides a systematic way to deal with decisions under circumstances where the alternatives
More informationYao s Minimax Principle
Complexity of algorithms The complexity of an algorithm is usually measured with respect to the size of the input, where size may for example refer to the length of a binary word describing the input,
More informationStochastic Games and Bayesian Games
Stochastic Games and Bayesian Games CPSC 532L Lecture 10 Stochastic Games and Bayesian Games CPSC 532L Lecture 10, Slide 1 Lecture Overview 1 Recap 2 Stochastic Games 3 Bayesian Games Stochastic Games
More informationIntroduction to Game Theory
Introduction to Game Theory 3a. More on Normal-Form Games Dana Nau University of Maryland Nau: Game Theory 1 More Solution Concepts Last time, we talked about several solution concepts Pareto optimality
More informationName. FINAL EXAM, Econ 171, March, 2015
Name FINAL EXAM, Econ 171, March, 2015 There are 9 questions. Answer any 8 of them. Good luck! Remember, you only need to answer 8 questions Problem 1. (True or False) If a player has a dominant strategy
More information6.207/14.15: Networks Lecture 9: Introduction to Game Theory 1
6.207/14.15: Networks Lecture 9: Introduction to Game Theory 1 Daron Acemoglu and Asu Ozdaglar MIT October 13, 2009 1 Introduction Outline Decisions, Utility Maximization Games and Strategies Best Responses
More informationG5212: Game Theory. Mark Dean. Spring 2017
G5212: Game Theory Mark Dean Spring 2017 Modelling Dynamics Up until now, our games have lacked any sort of dynamic aspect We have assumed that all players make decisions at the same time Or at least no
More informationAdvanced Microeconomics
Advanced Microeconomics ECON5200 - Fall 2014 Introduction What you have done: - consumers maximize their utility subject to budget constraints and firms maximize their profits given technology and market
More informationChapter 10: Mixed strategies Nash equilibria, reaction curves and the equality of payoffs theorem
Chapter 10: Mixed strategies Nash equilibria reaction curves and the equality of payoffs theorem Nash equilibrium: The concept of Nash equilibrium can be extended in a natural manner to the mixed strategies
More informationMicroeconomics Comprehensive Exam
Microeconomics Comprehensive Exam June 2009 Instructions: (1) Please answer each of the four questions on separate pieces of paper. (2) When finished, please arrange your answers alphabetically (in the
More informationJanuary 26,
January 26, 2015 Exercise 9 7.c.1, 7.d.1, 7.d.2, 8.b.1, 8.b.2, 8.b.3, 8.b.4,8.b.5, 8.d.1, 8.d.2 Example 10 There are two divisions of a firm (1 and 2) that would benefit from a research project conducted
More informationPROBLEM SET 6 ANSWERS
PROBLEM SET 6 ANSWERS 6 November 2006. Problems.,.4,.6, 3.... Is Lower Ability Better? Change Education I so that the two possible worker abilities are a {, 4}. (a) What are the equilibria of this game?
More informationCS 331: Artificial Intelligence Game Theory I. Prisoner s Dilemma
CS 331: Artificial Intelligence Game Theory I 1 Prisoner s Dilemma You and your partner have both been caught red handed near the scene of a burglary. Both of you have been brought to the police station,
More informationB w x y z a 4,4 3,3 5,1 2,2 b 3,6 2,5 6,-3 1,4 A c -2,0 2,-1 0,0 2,1 d 1,4 1,2 1,1 3,5
Econ 414, Exam 1 Name: There are three questions taken from the material covered so far in the course. All questions are equally weighted. If you have a question, please raise your hand and I will come
More informationEpistemic Experiments: Utilities, Beliefs, and Irrational Play
Epistemic Experiments: Utilities, Beliefs, and Irrational Play P.J. Healy PJ Healy (OSU) Epistemics 2017 1 / 62 Motivation Question: How do people play games?? E.g.: Do people play equilibrium? If not,
More informationBasic Game-Theoretic Concepts. Game in strategic form has following elements. Player set N. (Pure) strategy set for player i, S i.
Basic Game-Theoretic Concepts Game in strategic form has following elements Player set N (Pure) strategy set for player i, S i. Payoff function f i for player i f i : S R, where S is product of S i s.
More informationMAT 4250: Lecture 1 Eric Chung
1 MAT 4250: Lecture 1 Eric Chung 2Chapter 1: Impartial Combinatorial Games 3 Combinatorial games Combinatorial games are two-person games with perfect information and no chance moves, and with a win-or-lose
More informationG5212: Game Theory. Mark Dean. Spring 2017
G5212: Game Theory Mark Dean Spring 2017 Why Game Theory? So far your microeconomic course has given you many tools for analyzing economic decision making What has it missed out? Sometimes, economic agents
More information1 R. 2 l r 1 1 l2 r 2
4. Game Theory Midterm I Instructions. This is an open book exam; you can use any written material. You have one hour and 0 minutes. Each question is 35 points. Good luck!. Consider the following game
More informationGame theory and applications: Lecture 1
Game theory and applications: Lecture 1 Adam Szeidl September 20, 2018 Outline for today 1 Some applications of game theory 2 Games in strategic form 3 Dominance 4 Nash equilibrium 1 / 8 1. Some applications
More informationECON 459 Game Theory. Lecture Notes Auctions. Luca Anderlini Spring 2017
ECON 459 Game Theory Lecture Notes Auctions Luca Anderlini Spring 2017 These notes have been used and commented on before. If you can still spot any errors or have any suggestions for improvement, please
More informationIn the Name of God. Sharif University of Technology. Microeconomics 2. Graduate School of Management and Economics. Dr. S.
In the Name of God Sharif University of Technology Graduate School of Management and Economics Microeconomics 2 44706 (1394-95 2 nd term) - Group 2 Dr. S. Farshad Fatemi Chapter 8: Simultaneous-Move Games
More informationGAME THEORY. (Hillier & Lieberman Introduction to Operations Research, 8 th edition)
GAME THEORY (Hillier & Lieberman Introduction to Operations Research, 8 th edition) Game theory Mathematical theory that deals, in an formal, abstract way, with the general features of competitive situations
More informationGAME THEORY. Game theory. The odds and evens game. Two person, zero sum game. Prototype example
Game theory GAME THEORY (Hillier & Lieberman Introduction to Operations Research, 8 th edition) Mathematical theory that deals, in an formal, abstract way, with the general features of competitive situations
More informationMath 180A. Lecture 5 Wednesday April 7 th. Geometric distribution. The geometric distribution function is
Geometric distribution The geometric distribution function is x f ( x) p(1 p) 1 x {1,2,3,...}, 0 p 1 It is the pdf of the random variable X, which equals the smallest positive integer x such that in a
More information6.3: The Binomial Model
6.3: The Binomial Model The Normal distribution is a good model for many situations involving a continuous random variable. For experiments involving a discrete random variable, where the outcome of the
More informationStrategies and Nash Equilibrium. A Whirlwind Tour of Game Theory
Strategies and Nash Equilibrium A Whirlwind Tour of Game Theory (Mostly from Fudenberg & Tirole) Players choose actions, receive rewards based on their own actions and those of the other players. Example,
More informationOveruse of a Common Resource: A Two-player Example
Overuse of a Common Resource: A Two-player Example There are two fishermen who fish a common fishing ground a lake, for example Each can choose either x i = 1 (light fishing; for example, use one boat),
More informationODD. Answers to Odd-Numbered Problems, 4th Edition of Games and Information, Rasmusen PROBLEMS FOR CHAPTER 1
ODD Answers to Odd-Numbered Problems, 4th Edition of Games and Information, Rasmusen PROBLEMS FOR CHAPTER 1 26 March 2005. 12 September 2006. 29 September 2012. Erasmuse@indiana.edu. Http://www.rasmusen
More informationIn the Name of God. Sharif University of Technology. Graduate School of Management and Economics
In the Name of God Sharif University of Technology Graduate School of Management and Economics Microeconomics (for MBA students) 44111 (1393-94 1 st term) - Group 2 Dr. S. Farshad Fatemi Game Theory Game:
More informationGames of Incomplete Information ( 資訊不全賽局 ) Games of Incomplete Information
1 Games of Incomplete Information ( 資訊不全賽局 ) Wang 2012/12/13 (Lecture 9, Micro Theory I) Simultaneous Move Games An Example One or more players know preferences only probabilistically (cf. Harsanyi, 1976-77)
More informationSimon Fraser University Fall Econ 302 D200 Final Exam Solution Instructor: Songzi Du Wednesday December 16, 2015, 8:30 11:30 AM
Simon Fraser University Fall 2015 Econ 302 D200 Final Exam Solution Instructor: Songzi Du Wednesday December 16, 2015, 8:30 11:30 AM NE = Nash equilibrium, SPE = subgame perfect equilibrium, PBE = perfect
More informationIntroduction to Industrial Organization Professor: Caixia Shen Fall 2014 Lecture Note 5 Games and Strategy (Ch. 4)
Introduction to Industrial Organization Professor: Caixia Shen Fall 2014 Lecture Note 5 Games and Strategy (Ch. 4) Outline: Modeling by means of games Normal form games Dominant strategies; dominated strategies,
More information15.053/8 February 28, person 0-sum (or constant sum) game theory
15.053/8 February 28, 2013 2-person 0-sum (or constant sum) game theory 1 Quotes of the Day My work is a game, a very serious game. -- M. C. Escher (1898-1972) Conceal a flaw, and the world will imagine
More informationMicroeconomic Theory III Final Exam March 18, 2010 (80 Minutes)
4. Microeconomic Theory III Final Exam March 8, (8 Minutes). ( points) This question assesses your understanding of expected utility theory. (a) In the following pair of games, check whether the players
More informationChapter 2 Strategic Dominance
Chapter 2 Strategic Dominance 2.1 Prisoner s Dilemma Let us start with perhaps the most famous example in Game Theory, the Prisoner s Dilemma. 1 This is a two-player normal-form (simultaneous move) game.
More informationThe Nash equilibrium of the stage game is (D, R), giving payoffs (0, 0). Consider the trigger strategies:
Problem Set 4 1. (a). Consider the infinitely repeated game with discount rate δ, where the strategic fm below is the stage game: B L R U 1, 1 2, 5 A D 2, 0 0, 0 Sketch a graph of the players payoffs.
More informationCUR 412: Game Theory and its Applications, Lecture 12
CUR 412: Game Theory and its Applications, Lecture 12 Prof. Ronaldo CARPIO May 24, 2016 Announcements Homework #4 is due next week. Review of Last Lecture In extensive games with imperfect information,
More informationTheir opponent will play intelligently and wishes to maximize their own payoff.
Two Person Games (Strictly Determined Games) We have already considered how probability and expected value can be used as decision making tools for choosing a strategy. We include two examples below for
More informationG5212: Game Theory. Mark Dean. Spring 2017
G5212: Game Theory Mark Dean Spring 2017 Bargaining We will now apply the concept of SPNE to bargaining A bit of background Bargaining is hugely interesting but complicated to model It turns out that the
More informationGame Theory I. Author: Neil Bendle Marketing Metrics Reference: Chapter Neil Bendle and Management by the Numbers, Inc.
Game Theory I This module provides an introduction to game theory for managers and includes the following topics: matrix basics, zero and non-zero sum games, and dominant strategies. Author: Neil Bendle
More informationBest-Reply Sets. Jonathan Weinstein Washington University in St. Louis. This version: May 2015
Best-Reply Sets Jonathan Weinstein Washington University in St. Louis This version: May 2015 Introduction The best-reply correspondence of a game the mapping from beliefs over one s opponents actions to
More informationProblem Set 2 Answers
Problem Set 2 Answers BPH8- February, 27. Note that the unique Nash Equilibrium of the simultaneous Bertrand duopoly model with a continuous price space has each rm playing a wealy dominated strategy.
More informationCS711: Introduction to Game Theory and Mechanism Design
CS711: Introduction to Game Theory and Mechanism Design Teacher: Swaprava Nath Domination, Elimination of Dominated Strategies, Nash Equilibrium Domination Normal form game N, (S i ) i N, (u i ) i N Definition
More information5.2 Random Variables, Probability Histograms and Probability Distributions
Chapter 5 5.2 Random Variables, Probability Histograms and Probability Distributions A random variable (r.v.) can be either continuous or discrete. It takes on the possible values of an experiment. It
More informationOut of equilibrium beliefs and Refinements of PBE
Refinements of PBE Out of equilibrium beliefs and Refinements of PBE Requirement 1 and 2 of the PBE say that no player s strategy can be strictly dominated beginning at any information set. The problem
More informationSequential-move games with Nature s moves.
Econ 221 Fall, 2018 Li, Hao UBC CHAPTER 3. GAMES WITH SEQUENTIAL MOVES Game trees. Sequential-move games with finite number of decision notes. Sequential-move games with Nature s moves. 1 Strategies in
More informationPreliminary Notions in Game Theory
Chapter 7 Preliminary Notions in Game Theory I assume that you recall the basic solution concepts, namely Nash Equilibrium, Bayesian Nash Equilibrium, Subgame-Perfect Equilibrium, and Perfect Bayesian
More informationContinuing game theory: mixed strategy equilibrium (Ch ), optimality (6.9), start on extensive form games (6.10, Sec. C)!
CSC200: Lecture 10!Today Continuing game theory: mixed strategy equilibrium (Ch.6.7-6.8), optimality (6.9), start on extensive form games (6.10, Sec. C)!Next few lectures game theory: Ch.8, Ch.9!Announcements
More informationNot 0,4 2,1. i. Show there is a perfect Bayesian equilibrium where player A chooses to play, player A chooses L, and player B chooses L.
Econ 400, Final Exam Name: There are three questions taken from the material covered so far in the course. ll questions are equally weighted. If you have a question, please raise your hand and I will come
More information