Game Theory. “If you don’t think the math matters, then you don’t know the right math.” Chris Ferguson 2002 World Series of Poker Champion.

Slides:

Advertisements

Similar presentations

Introduction to Game Theory

Advertisements

NON - zero sum games.

Module 4 Game Theory To accompany Quantitative Analysis for Management, Tenth Edition, by Render, Stair, and Hanna Power Point slides created by Jeff Heyl.

Game Theory S-1.

APPENDIX An Alternative View of the Payoff Matrix n Assume total maximum profits of all oligopolists is constant at 200 units. n Alternative policies.

Chapter Twenty-Eight Game Theory. u Game theory models strategic behavior by agents who understand that their actions affect the actions of other agents.

Two-Player Zero-Sum Games

Chapter 10 Game Theory and Strategic Behavior

Operations Research Assistant Professor Dr. Sana’a Wafa Al-Sayegh 2 nd Semester ITGD4207 University of Palestine.

1 Game Theory. By the end of this section, you should be able to…. ► In a simultaneous game played only once, find and define:  the Nash equilibrium.

Chapter 6 Game Theory © 2006 Thomson Learning/South-Western.

1 Chapter 4: Minimax Equilibrium in Zero Sum Game SCIT1003 Chapter 4: Minimax Equilibrium in Zero Sum Game Prof. Tsang.

Chapter 6 Game Theory © 2006 Thomson Learning/South-Western.

An Introduction to... Evolutionary Game Theory

MIT and James Orlin © Game Theory 2-person 0-sum (or constant sum) game theory 2-person game theory (e.g., prisoner’s dilemma)

Game Theory Advertising Example 1. Game Theory What is the optimal strategy for Firm A if Firm B chooses to advertise? 2.

Game Theory: introduction and applications to computer networks Game Theory: introduction and applications to computer networks Zero-Sum Games (follow-up)

Part 3: The Minimax Theorem

An Introduction to Game Theory Part I: Strategic Games

Game Theory Part 5: Nash’s Theorem.

Chapter 6 © 2006 Thomson Learning/South-Western Game Theory.

5/16/20151 Game Theory Game theory was developed by John Von Neumann and Oscar Morgenstern in Economists! One of the fundamental principles of.

Rational Choice Sociology Lecture 5 Game theory I: the concept and classification of games.

Principles of Microeconomics November 26 th, 2013.

Decision Analysis April 11, Game Theory Frame Work Players ◦ Decision maker: optimizing agent ◦ Opponent  Nature: offers uncertain outcome  Competition:

1 Game Theory Here we study a method for thinking about oligopoly situations. As we consider some terminology, we will see the simultaneous move, one shot.

Game Theory Here we study a method for thinking about oligopoly situations. As we consider some terminology, we will see the simultaneous move, one shot.

1 Section 2d Game theory Game theory is a way of thinking about situations where there is interaction between individuals or institutions. The parties.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 1 of 68 Chapter 9 The Theory of Games.

Introduction to Game Theory Yale Braunstein Spring 2007.

Game Theory Objectives:

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna S-1 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ Supplement 1.

UNIT II: The Basic Theory Zero-sum Games Nonzero-sum Games Nash Equilibrium: Properties and Problems Bargaining Games Bargaining and Negotiation Review.

Game Theory Statistics 802. Lecture Agenda Overview of games 2 player games representations 2 player zero-sum games Render/Stair/Hanna text CD QM for.

Starbucks, Beating The Prisoners Dillema Brooke Hatcher Econ 202 April 28 th, 2012.

Chapter 12 & Module E Decision Theory & Game Theory.

To accompany Quantitative Analysis for Management,9e by Render/Stair/Hanna M4-1 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ Module 4 Game.

Game Theory Warin Chotekorakul MD 1/2004. Introduction A game is a contest involving to or more players, each of whom wants to win. Game theory is the.

Dynamic Games of complete information: Backward Induction and Subgame perfection - Repeated Games -

Standard and Extended Form Games A Lesson in Multiagent System Based on Jose Vidal’s book Fundamentals of Multiagent Systems Henry Hexmoor, SIUC.

Game Theory Part 2: Zero Sum Games. Zero Sum Games The following matrix defines a zero-sum game. Notice the sum of the payoffs to each player, at every.

The Design & Analysis of the Algorithms Lecture by me M. Sakalli Download two pdf files..

Game theory & Linear Programming Steve Gu Mar 28, 2008.

THE “CLASSIC” 2 x 2 SIMULTANEOUS CHOICE GAMES Topic #4.

Game Theory: introduction and applications to computer networks Game Theory: introduction and applications to computer networks Lecture 2: two-person non.

Chapter 12 - Imperfect Competition: A Game-Theoretic Approach Copyright © 2015 The McGraw-Hill Companies, Inc. All rights reserved.

Game Theory: introduction and applications to computer networks Game Theory: introduction and applications to computer networks Introduction Giovanni Neglia.

John Locke Second Treatise on Government. Locke’s Second Treatise I.Property and Labor II.State of Nature 2 III.Mutual Advantage & the Social Contract.

Game theory is the study of the best play–safe strategy to adopt if 2 players are playing a game where decisions affect each other. Ex Two men are caught.

Part 3 Linear Programming

1 GAME THEORY AND OLIGOPOLY l Principles of Microeconomic Theory, ECO 284 l John Eastwood l CBA 247 l l address:

1 What is Game Theory About? r Analysis of situations where conflict of interests is present r Goal is to prescribe how conflicts can be resolved 2 2 r.

Strategic Behavior in Business and Econ Static Games of complete information: Dominant Strategies and Nash Equilibrium in pure and mixed strategies.

1. 2 You should know by now… u The security level of a strategy for a player is the minimum payoff regardless of what strategy his opponent uses. u A.

Prisoners’ Dilemma Scenario: You and an accomplice are arrested on suspicion of committing some nasty crime The District Attorney and the police have been.

Statistics Overview of games 2 player games representations 2 player zero-sum games Render/Stair/Hanna text CD QM for Windows software Modeling.

INDEX Introduction of game theory Introduction of game theory Significance of game theory Significance of game theory Essential features of game theory.

GAME THEORY Day 5. Minimax and Maximin Step 1. Write down the minimum entry in each row. Which one is the largest? Maximin Step 2. Write down the maximum.

Lec 23 Chapter 28 Game Theory.

1 Fahiem Bacchus, University of Toronto CSC384: Intro to Artificial Intelligence Game Tree Search I ● Readings ■ Chapter 6.1, 6.2, 6.3, 6.6 ● Announcements:

Previous example incorporated a feature that is present in many oligopolies: Conflict between collective and individual interests. Duopolists’ collective.

Introduction to Game Theory

Rational Choice Sociology

Game Theory and Strategic Play

Presentation transcript:

Game Theory

“If you don’t think the math matters, then you don’t know the right math.” Chris Ferguson 2002 World Series of Poker Champion

Prisoners You and I have been arrested for a bank robbery (of which we are guilty!), and they have evidence to convict us, but not enough for the maximum sentence. They decide to separate us and try to get each of us to confess and implicate the other one.

A Dilemma If we both stonewall (i.e. keep our mouths shut) then we will get 2 years in prison. If we both confess, we’ll each get 4 years in prison. But if only one person finks (confesses), that person will get 1 year in prison, while the person who stayed quiet gets 5 years!

Summary of Your Outcomes I Stonewall I Fink You Stonewall 2 years prison 5 years prison You Fink 1 year prison 4 years prison

A Dominant Strategy  No matter what I choose, you’ll be better off having finked.  But the same reasoning applies for me!  So even though we’d both be better off had we both stonewalled (2 years prison each instead of 4), unless we can cooperate (i.e. make binding agreements), we are both forced to fink.

I Stonewall I Fink You Stonewall (-2,-2)(-5,-1) You Fink (-1,-5)(-4,-4) Summary of Both of Our Outcomes Equilibrium Outcome

A ‘game’ is any situation in which:  There are at least two players  Each player has a number of possible strategies  The strategies chosen by each player determine the outcome of the game  Associated to each outcome is a collection of numerical payoffs to each player

Payoffs in the Prisoner’s Dilemma: Me You The first number in each ordered pair indicates the payoff to the first player, whose strategy options correspond to the rows of the matrix. (-2,-2)(-5,-1) (-1,-5)(-4,-4) SFSF SFSF

Domination A strategy A dominates a strategy B if every outcome in A is at least as good (for that player) as the corresponding outcome in strategy B. Dominance Principle: A rational player should never play a dominated strategy.

A game of total conflict: The following table indicates payoffs for a game. Notice that the payoffs are the negative of each other, so what is good for you is bad for me, and vice versa! Me You (5,-5)(2,-2) (0,0)(-4,4) ABAB ABAB

 Strategy A is dominant for You.  Strategy B is dominant for Me.  By the Dominance Principle, the outcome of this game should be AB for rational players. Me You (5,-5)(2,-2) (0,0)(-4,4) ABAB ABAB

Zero-Sum Games In the previous game, since we know that the payoffs for the second player are negatives of the payoffs for the first player, we could have described the outcomes with a matrix where each entry has payoffs for the first player only. Me You ABAB ABAB

For the game below, what is your rational strategy? ABCABC Me ABCABC

A B C Me ABCABC You The rational outcome is BB.

A Game without domination A B C ABCABC Me You Goal: Guarantee yourself the least-worst result. Minimum 4 -2 maximin maximum minimax

Saddle Points  An outcome in a matrix game is called a saddle point if the entry at that outcome is both less than or equal to any entry in its row, and greater than or equal to any entry in its column.  If the minimax column and maximin row are equal in payoff, the corresponding outcome is a saddle point.  That payoff is called the value of the game.

Saddle Points

Expected value This game does not have a dominant strategy or saddle point. But you happen to know that I am going to play strategy A half the time and Strategy B the other half of the time. What should you do? Me You ABAB A B

Expected Value  If you choose strategy A, your expected payoff will be  If you choose strategy B, your expected payoff will be

Mixed Strategy What happens if I play strategy A one fourth of the time and strategy B the remaining three fourths of the time? Me You ABAB A B

Mixed Strategy  Your expected payoff for playing strategy A will be:  Your expected payoff for playing strategy B will be:

Mixed Strategy  By playing randomly with the correct weights for each strategy, I can guarantee that your strategy choice makes no difference. This is called a mixed-strategy on my part.  You can even know exactly what my mixed strategy is, and it gives you no advantage!

Computing a Mixed Strategy Equilibrium  I want to play strategy A with some weight p, and strategy B with weight 1-p.  I want the expected payoffs to you to be independent of your strategy.  So I need to solve the equation:  The solution is p=1/4.

Utility Theory  Payoffs (also called utilities by economists) can be selected by ranking outcomes in order of preference.  The magnitude of these rankings need not matter, just the order. Thus the payoffs are called ordinal utilities.

Utility Theory  If the magnitudes can be compared, then the payoffs are called cardinal utilities.  The most common situation with cardinal utilities is when the payoffs correspond to money.  Only cardinal utilities make sense with mixed strategies!

Multiple-move games  It is X’s turn to act in Tic-Tac-Toe:  What are X’s strategies? OOA BXC DXO

Bibliography  Game Theory and Strategy, Philip D. Straffin  For All Practical Purposes, COMAP  The Mathematics of Poker, Bill Chen and Jerrod Ankenman  Microeconomic Theory, Walter Nicholson  Theory of Games and Economic Behavior, John von Neumann, et. al.