Game TheoryGame TheoryGame TheoryGame Theory

BBA3274 / DBS1084 QUANTITATIVE METHODS for BUSINESSBBA3274 / DBS1084 QUANTITATIVE METHODS for BUSINESS

byStephen Ong

Visiting Fellow, Birmingham City University Business School, UK

Visiting Professor, Shenzhen University

Today’s Overview Today’s Overview

Learning ObjectivesLearning Objectives

1.1. Understand the importance and use Understand the importance and use of game theory in decision making.of game theory in decision making.

2.2. Understand the principles of zero-Understand the principles of zero-sum, two person games.sum, two person games.

3.3. Analyse pure strategy games and Analyse pure strategy games and use dominance to reduce the size of use dominance to reduce the size of the game.the game.

4.4. Solve mixed strategy games when Solve mixed strategy games when there is no saddle point.there is no saddle point.

After this lecture, students will be able to:After this lecture, students will be able to:

OutlineOutline

44.1.1 IntroductionIntroduction

44..22 Language of GamesLanguage of Games

44..33 The Minimax The Minimax CriterionCriterion

44..44 Pure Strategy GamesPure Strategy Games

44..55 Mixed Strategy Mixed Strategy GamesGames

44..66 DominanceDominance

Game Theory ModelsGame Theory Models

A set of mathematical tools for A set of mathematical tools for analyzing situations in which players analyzing situations in which players make various strategic moves and have make various strategic moves and have different outcomes or payoffs different outcomes or payoffs associated with those moves.associated with those moves.

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Dominant Strategies and the Dominant Strategies and the Prisoner’s DilemmaPrisoner’s Dilemma

This payoff matrix This payoff matrix shows the various shows the various prison terms for Bonnie prison terms for Bonnie and Clyde that would and Clyde that would result from the result from the combination of combination of strategies chosen when strategies chosen when questioned about a questioned about a crime spree.crime spree.

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Prisoner’s Dilemma – Prisoner’s Dilemma – Dominant StrategyDominant Strategy

A dominant strategy A dominant strategy is one that results in is one that results in the best outcome or the best outcome or highest payoff to a highest payoff to a given player no given player no matter what action matter what action or choice the other or choice the other player makes.player makes.

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Nash EquilibriumNash Equilibrium

Nash equilibrium is Nash equilibrium is a set of strategies a set of strategies from which all from which all players are players are choosing their best choosing their best strategy, given the strategy, given the actions of the other actions of the other players.players.

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Game Model ClassificationGame Model Classification

Number of Number of playersplayers

Sum of all Sum of all payoffspayoffs

Number of Number of strategies strategies employedemployed

Two person Two person (X, Y)(X, Y)

Zero sum, Zero sum, where sum of losses where sum of losses by one player = sum by one player = sum of gains by other of gains by other playerplayer

Example : Duopoly of 2 StoresExample : Duopoly of 2 Stores There are only 2 There are only 2

lighting fixture stores, X lighting fixture stores, X and Y.and Y.

Owner of store X has 2 Owner of store X has 2 advertising strategies – advertising strategies – radio spots and radio spots and newspaper ads.newspaper ads.

Owner of store Y Owner of store Y prepares to respond prepares to respond with radio spots and with radio spots and newspaper ads.newspaper ads.

The 2x2 payoff matrix The 2x2 payoff matrix shows the effect on shows the effect on market shares when market shares when both stores advertise.both stores advertise.10

Example : Duopoly Game Example : Duopoly Game OutcomesOutcomes

Store X Store X StrategyStrategy

Store Y Store Y StrategyStrategy

Outcome Outcome (% Change in Market (% Change in Market

Share)Share)

X1X1 (Use Radio)(Use Radio)

Y1 Y1 (Use Radio)(Use Radio)

X wins 3X wins 3And Y loses 3And Y loses 3

X1 X1 (Use Radio)(Use Radio)

Y2 Y2 (Use Newspaper)(Use Newspaper)

X wins 5X wins 5And Y loses 5And Y loses 5

X2 X2 (Use Newspaper)(Use Newspaper)

Y1 Y1 (Use Radio)(Use Radio)

X wins 1X wins 1And Y loses 1And Y loses 1

X2 X2 (Use Newspaper)(Use Newspaper)

Y2 Y2 (Use Newspaper)(Use Newspaper)

X loses 2X loses 2And Y wins 2And Y wins 2

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Minimax CriterionMinimax CriterionUsed to find the strategy that minimises Used to find the strategy that minimises

the maximum loss, or maximizes the the maximum loss, or maximizes the minimum payoff (maximin approach).minimum payoff (maximin approach).

Locate the minimum payoff for each strategy.Locate the minimum payoff for each strategy. Select the strategy with the maximum number.Select the strategy with the maximum number. The upper value of the game equal to the The upper value of the game equal to the

minimum of the maximum values in the minimum of the maximum values in the columns.columns.

The lower value of the game is equal to the The lower value of the game is equal to the maximum of the minimum values in the rows.maximum of the minimum values in the rows.

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Minimax SolutionMinimax Solution

STRATEGIESSTRATEGIES Y1Y1 Y2Y2 MINIMUMMINIMUM

X1X1 33 55 33

X2X2 11 -2-2 -2-2

MAXIMUMMAXIMUM 33 55

Saddle PointSaddle Point

An equilibrium or saddle point An equilibrium or saddle point condition exists if the upper value of the condition exists if the upper value of the game is equal to the lower value of the game is equal to the lower value of the game. This is called the value of the game. This is called the value of the game.game.

Minimum of maximumsMinimum of maximums Maximum of minimumsMaximum of minimums

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Pure Strategy GamePure Strategy Game

STRATEGIESSTRATEGIES Y1Y1 Y2Y2 MINIMUMMINIMUM

X1X1 1010 66 66

X2X2 -12-12 22 -12-12

MAXIMUMMAXIMUM 1010 66

Saddle PointSaddle Point

When a saddle point is present, the When a saddle point is present, the strategy each player should follow will strategy each player should follow will always be the same regardless of the always be the same regardless of the other player’s strategy.other player’s strategy.

Minimum of maximumsMinimum of maximums Maximum of minimumsMaximum of minimums

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Mixed Strategy GameMixed Strategy Game

STRATEGIESSTRATEGIESY1Y1(P)(P)

Y2Y2(1 - P)(1 - P)

Expected Expected GainGain

X1 (Q)X1 (Q) 44 22 4P + 2(1-P)4P + 2(1-P)

X2 (1 - Q)X2 (1 - Q) 11 1010 1P + 10(1-P)1P + 10(1-P)

Expected Expected gaingain 4Q + 1(1-Q)4Q + 1(1-Q) 2Q + 10(1-Q)2Q + 10(1-Q)

When there is no saddle point, players When there is no saddle point, players will play each strategy for a certain will play each strategy for a certain percentage of the time (P, Q). To solve a percentage of the time (P, Q). To solve a mixed strategy game, use the expected mixed strategy game, use the expected gain or loss approach.gain or loss approach.

Mixed Strategy GameMixed Strategy GameThe goal of this approach is for a player to play each The goal of this approach is for a player to play each

strategy a particular percentage of the time so that strategy a particular percentage of the time so that the expected value of the game does not depend the expected value of the game does not depend upon what the opponent does. This will only occur if upon what the opponent does. This will only occur if the expected value of each strategy is the same.the expected value of each strategy is the same.

For player Y, For player Y,

4P + 2 (1 – P) = 1P + 10(1 – P)4P + 2 (1 – P) = 1P + 10(1 – P)

P = P = 88//1111

For player X,For player X,

4Q + 1(1 – Q) = 2Q + 10(1 – Q)4Q + 1(1 – Q) = 2Q + 10(1 – Q)

Q = Q = 99//11113-16

DominanceDominance

The principle of The principle of dominance can be used dominance can be used to reduce the size of the to reduce the size of the games by eliminating games by eliminating strategies that would strategies that would never be played.never be played.

A strategy can be A strategy can be eliminatedeliminated if all its if all its game’s outcomes are game’s outcomes are the same or worse than the same or worse than the corresponding the corresponding game outcomes of game outcomes of another strategy.another strategy.

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Example : DominanceExample : Dominance

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Y1Y1 Y2Y2 Y3Y3 Y4Y4

X1X1 -5-5 44 66 -3-3

X2X2 -2-2 66 22 -20-20

TutorialTutorial

Lab Practical : Spreadsheet Lab Practical : Spreadsheet

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Further ReadingFurther Reading

Render, B., Stair Jr.,R.M. & Hanna, M.E. (2013) Quantitative Analysis for Management, Pearson, 11th Edition

Waters, Donald (2007) Quantitative Methods for Business, Prentice Hall, 4th Edition.

Anderson D, Sweeney D, & Williams T. (2006) Quantitative Methods For Business Thompson Higher Education, 10th Ed.

QUESTIONS?QUESTIONS?