Game Theory Optimal Strategies Formulated in Conflict MGMT E-5070.

Game Theory

OptimalStrategies

Formulatedin

Conflict

MGMT E-5070

Game TheoryGame TheoryINTRODUCTIONINTRODUCTION

Strategies taken by otherfirms or individuals candramatically affect the

outcome of our decisions

Consequently, businesscannot make importantdecisions today withoutconsidering what otherfirms or individuals are

doing or might do

Game TheoryGame TheoryINTRODUCTORY TERMINOLOGYINTRODUCTORY TERMINOLOGY

Game TheoryGame Theory is the study of how optimal strategies are formulated in conflict.

A gamegame is a contest involving two or more decision makers, each of whom wants to win.

Game TheoryGame Theory considers the impact of the strategies of others on our strategies and outcomes.

Janos (John) von NeumannJanos (John) von Neumann( 1903 – 1957 )( 1903 – 1957 )

Formally introduced game theory in his book, Theory of Games and Economic Behavior ( 1944 )

University of Berlin ( 1926 – 1933 )

Princeton University ( 1930 – 1957 )

Los Alamos Scientific Laboratory ( 1943 – 1955 )

Foremost mathematician of the 20th century

John NashJohn Nash ( 1928 - )( 1928 - )

PhD (1950) Princeton University equilibrium points in N-person games the bargaining problem two-person cooperative games Nobel Prize in economics ( 1994 ) Senior research mathematician at Princeton at present

ON THE SET OF “A BEAUTIFUL MIND” WITH RUSSELL CROWEON THE SET OF “A BEAUTIFUL MIND” WITH RUSSELL CROWE

Game TheoryGame TheoryINTRODUCTORY TERMINOLOGYINTRODUCTORY TERMINOLOGY

Two – Person GameTwo – Person Game : A game in which only two parties ( ‘X’ and ‘Y’ ) can play.

Zero – Sum GameZero – Sum Game : A game where the sum of losses for one player must equal the sum of gains for the other player. In other words, every time one player wins, the other player loses.

Two-Person Zero-Sum GameTwo-Person Zero-Sum GameEXAMPLEEXAMPLE

ALL PAYOFFSSHOWN IN TERMS

OF PLAYER ‘X’

Players &Players &

StrategiesStrategies

( use radio )( use radio )

( use newspaper )( use newspaper )

( use radio )( use radio )3 5

( use newspaper )( use newspaper )1 - 2

Two Important ValuesTwo Important Values

The Lower Value ( LV )

Find the smallestsmallest number in each row.

Select the largestlargest of these numbers.

This is the LV.

The Upper Value ( UV )

Find the largestlargest number in each column.

Select the smallestsmallest of these numbers.

This is the UV.

Two-Person Zero-Sum GameTwo-Person Zero-Sum GameEXAMPLEEXAMPLE

Players &Players &

MAXIMUM 3 5

M 3 - 2

THE UPPER VALUE

THE LOWER VALUE

Maxi-Min Maxi-Min CrCriterion for Player ‘X’iterion for Player ‘X’

A player using the maxi-min criterion will select the strategy that maximizes the minimum possible gain.

The maximum minimum payoff for player ‘X’ is “+3”, therefore ‘X’ will play strategy X1 ( use radio ) .

“+3” is the lower value of the game. The lower value equals the maxi-min strategy for player ‘X’.

Mini-Max Mini-Max Criterion for Player ‘Y’Criterion for Player ‘Y’

A player using the mini-max criterion will select the

strategy that minimizes the maximum possible loss.

The minimum maximum loss for player ‘Y’ is “+3”,

( actually “-3” ) , therefore ‘Y’ will play strategy Y1

( use radio ) .

“3” ( actually “-3” ) is the upper value of the game.

The upper value equals the mini-max strategy for

player ‘Y’.

Mini-Max =Mini-Max = Maxi-Min Maxi-Min !!

Players ‘X’ and ‘Y’ are simultaneously employing both criteria when choosing their strategies.

Minimizing one’s maximum losses is tantamount to maximizing one’s minimum gains !

Game TheoryGame TheoryADDITIONAL TERMINOLOGYADDITIONAL TERMINOLOGY

Pure StrategyPure Strategy : A game in which both players will always play just one strategy each.

Saddle Point GameSaddle Point Game : A game that has a pure strategy.

Value of the GameValue of the Game : The expected winnings of the game if the game is played a large number of times.

Pure Strategy GamePure Strategy Game

Occurs when the upper value of the game and the lower value of the game are identical, that is, UV = LV.

The above value is also the value of the game.

“ UV = LV ” is described as an equilibrium or saddlepoint condition.

Pure StrategyPure StrategyEXAMPLEEXAMPLE

Players &Players &

Pure StrategyPure Strategy

PlayersPlayers++

( use radio )( use radio )YY22

( use paper )( use paper )

Minimum

Maximum 3 5

EXAMPLEEXAMPLE

Saddlepoint

UpperValue

LowerValue

Pure StrategyPure Strategy Game GameEXAMPLEEXAMPLE

UV = LV = 3 , meaning that every time the advertising game is played, player ‘X’ will select strategy ‘X1’ and player ‘Y’ will select strategy ‘Y1’ .

Moreover, each time the advertising game is played, player ‘X’ will gain a 3% market share and player ‘Y’ will lose a 3% market share.

Pure StrategyPure Strategy22ndnd EXAMPLE EXAMPLE

Players &Players &

( use radio )( use radio )2 - 4

( use newspaper )( use newspaper )6 10

Pure StrategyPure Strategy

PlayersPlayers++

Minimum

Maximum 6 10

22ndnd EXAMPLE EXAMPLE

Saddlepoint

Upper Value Lower Value

Pure Strategy GamePure Strategy Game22ndnd EXAMPLE EXAMPLE

UV = LV = 6 , meaning that every time the advertising game is played, player ‘X’ will select strategy ‘X2’ and player ‘Y’ will select strategy ‘Y1’ .

Moreover, each time the advertising game is played, player ‘X’ will gain a 6% market share and player ‘Y’ will lose a 6% market share.

Largest Share Search Engine

Mixed Strategy GameMixed Strategy Game

INVOLVES USE OF THE ALGEBRAIC APPROACH.INVOLVES USE OF THE ALGEBRAIC APPROACH.

Occurs when there is no saddlepoint, that is,no pure strategy

The overall objective of each player is todetermine what percentage of the time he or she should play each strategy, inorder to maximize winnings, regardless

of what the other player does

Mixed Strategy GameMixed Strategy GameEXAMPLEEXAMPLE

Players &Players &

( use newspaper )( use newspaper )1 10

PlayersPlayers++

Minimum

Maximum 4 10

EXAMPLEEXAMPLE

THERE IS NOSADDLEPOINT

Upper Value

Lower Value

PlayersPlayers+ +

YY11 YY22

11 1010

EXAMPLEEXAMPLE

The Algebraic AThe Algebraic Approachpproach

• “Q” = percentage of time player X plays strategy “X1”

• “1-Q” = percentage of time player X plays strategy “X2”

• “P” = percentage of time player Y plays strategy “Y1”

• “1-P” = percentage of time player Y plays strategy “Y2”

PlayersPlayers+ +

( 1-Q )( 1-Q )

PP ( 1-P )( 1-P )

11 1010

EXAMPLEEXAMPLE

Player X StrategyPlayer X Strategy

SET THE COLUMNS EQUAL TO ONE ANOTHER AND SOLVE FOR ‘ Q ‘

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

4Q + 1 - 1Q = 2Q + 10 - 10Q

4Q - 1Q - 2Q + 10Q = - 1 + 10

11Q = 9

THEREFORE Q = 9/11 or 82% and (1-Q) = 2/11 or 18%

PlayersPlayers+ +

( 1-Q )( 1-Q )

PP ( 1-P )( 1-P )

11 1010

EXAMPLEEXAMPLE

Player Y StrategyPlayer Y Strategy

SET THE ROWS EQUAL TO ONE ANOTHER ANDSOLVE FOR ‘ P ‘

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

4P + 2 – 2P = 1P + 10 – 10P

4P – 2P – 1P + 10P = - 2 + 10

11P = 8

THEREFORE P = 8/11 or 73% and (1-P) = 3/11 or 27%

Value of the GameValue of the Game

PlayersPlayers+ +

( 1-Q )( 1-Q )

PP ( 1-P )( 1-P )

11 1010

CALCULATIONSCALCULATIONS

.82.82

.18.18

.73.73 .27.27

Value of the GameValue of the GameCALCULATIONSCALCULATIONS

• 1st way: .73(4) + .27( 2) = 3.46

• 2nd way: .73(1) + .27(10) ≈ 3.46

• 3rd way: .82(4) + .18( 1) = 3.46

• 4th way: .82(2) + .18(10) ≈ 3.46

Procedure for Solving Two-Person, Zero Procedure for Solving Two-Person, Zero Sum GamesSum Games

Develop Strategiesand Payoff Matrix

Is ThereA Pure

StrategySolution?

IsGame2x2?

Solve Problem forSaddle Point Solution

Solve with LinearProgramming

Can DominanceBe Used To

Reduce Matrix?

Solve for MixedStrategy Probabilities

YesYes

NoNoYesYes

The Principle of DominanceThe Principle of Dominance

Used to reduce the size of games byeliminating strategies that would

never be played

A strategy for a player can be eliminated if that player can always do as well or

better by playing another strategy

Principle of DominancePrinciple of Dominance

YY11 Y Y22

XX11 4 4 33

XX22 2 2 20 20

XX33 1 1 11

11stst EXAMPLE EXAMPLE

PLAYER YPLAYER Y

PLAYER XPLAYER X

XX11 4 4 3 3

XX22 2 2 20 20

X3 1 1

PLAYER YPLAYER Y

PLAYER XPLAYER X

PLAYER ‘X’ CAN ALWAYS DO BETTER PLAYING STRATEGY XPLAYER ‘X’ CAN ALWAYS DO BETTER PLAYING STRATEGY X11 OR X OR X22

MINIMUMPAYOFF

rejected

YY11 YY22

XX11 44 33

XX22 22 2020

THE NEW GAME AFTER ELIMINATION OF ONE “X” STRATEGYTHE NEW GAME AFTER ELIMINATION OF ONE “X” STRATEGY

PLAYER YPLAYER Y

PLAYER XPLAYER X

YY11 YY22 YY33 YY44

XX11 - 5- 5 44 66 - 3- 3

XX22 - 2- 2 66 22 - 20- 20

PLAYERPLAYERYY

PLAYERPLAYERXX

YY11 Y2 Y3 YY44

XX11 - 5- 5 4 6 - 3- 3

XX22 - 2- 2 6 2 - 20- 20

PLAYERPLAYERYY

PLAYERPLAYERXX

PLAYER ‘Y’ CAN ALWAYS DO BETTER PLAYING STRATEGY YPLAYER ‘Y’ CAN ALWAYS DO BETTER PLAYING STRATEGY Y11 OR Y OR Y44

rejected

YY11 YY22

XX11 - 5 - 3

XX22 - 2 - 20

THE NEW GAME AFTER ELIMINATION OF TWO “Y” STRATEGIESTHE NEW GAME AFTER ELIMINATION OF TWO “Y” STRATEGIES

PLAYERPLAYERYYPLAYERPLAYER

Solving 3x3 GamesSolving 3x3 Games

YY22 YY33

XX11 22 33 00

XX22 11 22 33

XX33 44

VIA LINEAR PROGRAMMINGVIA LINEAR PROGRAMMING

PLAYERPLAYERYY

PLAYERPLAYERXX

Linear ProgrammiLinear Programming Formulationng FormulationEXAMPLEEXAMPLE

Objective Function:

Subject to:

Non-negativityConstraint:

Maximize Y1 + Y2 + Y3

2Y1 + 3Y2 + 0Y3 <= 1

1Y1 + 2Y2 + 3Y3 <= 1

4Y1 + 1Y2 + 2Y3 <= 1

Y1 , Y2 , Y3 => 0

Linear ProgramminLinear Programming Formulationg FormulationEXAMPLEEXAMPLE

2Y1 + 3Y2 + 0Y3 + 1X1 + 0X2 + 0X3 = 1

1Y1 + 2Y2 + 3Y3 + 0X1 + 1X2 + 0X3 = 1

4Y1 + 1Y2 + 2Y3 + 0X1 + 0X2 + 1X3 = 1

CONVERT TO LINEAR EQUALITIES( ADD SLACK VARIABLES )

LinearLinear Programming ProgrammingCOMPUTER-GENERATED 1COMPUTER-GENERATED 1stst FEASIBLE SOLUTION FEASIBLE SOLUTION

BASISBASIS

VARIABLESVARIABLES YY11 YY22 YY33

SLACKSLACK

XX33QUANTITY

XX11 22 33 00 11 00 00 1

XX22 11 22 33 00 11 00 1

XX33 44 11 22 00 00 11 1

Z jZ j 00 00 00 00 00 00 00

C j - Z jC j - Z j 11 11 11 00 00 00

LinearLinear Programming ProgrammingCOMPUTER-GENERATED OPTIMAL SOLUTIONCOMPUTER-GENERATED OPTIMAL SOLUTION

BASICBASIC

SLACKSLACK

XX33QUANTITY

YY22 00 11 00 .3125.3125 .125.125 -.1875-.1875 0.25

YY33 00 00 11 -.2188-.2188 .3125.3125 .0312.0312 0.125

YY11 11 00 00 .0313.0313 -.1875-.1875 .2812.2812 0.125

Z jZ j 11 11 11 .125.125 .25.25 .125.125 0.500.50

C j - Z jC j - Z j 00 00 00 -.125-.125 -.25-.25 -.125-.125

Linear ProgrammingLinear ProgrammingSOLUTION INTERPRETATIONSOLUTION INTERPRETATION

BASICBASIC

SLACKSLACK

XX33QUANTITYQUANTITY

YY22 00 11 00 .3125.3125 .125.125 -.1875-.1875 0.250.25

YY33 00 00 11 -.2188-.2188 .3125.3125 .0312.0312 0.1250.125

YY11 11 00 00 .0313.0313 -.1875-.1875 .2812.2812 0.1250.125

Z jZ j 11 11 11 .125.125 .25.25 .125.125 0.500.50

C j - Z jC j - Z j 00 00 00 -.125-.125 -.25-.25 -.125-.125

Y1 = .125 or 12.5%

Y2 = .250 or 25.0%

Y3 = .125 or 12.5%

The Value of the GameThe Value of the Game

YY11 + + YY22 + + YY33

.125 + .25 + .125.125 + .25 + .125

.50.50

== 2.02.0

Player Y ’s Optimal Strategy

2.0 ( .125 , .25 , .125 )

.25 , .50 , .25

Y plays Y1 25% of the time

MEANING:

Y1 Y2 Y3VALUE

OFGAME

X1 = - .125

X2 = - .250

X3 = - .125

THE VALUES ARE THE SHADOW PRICES OF THE SLACK

VARIABLES IN THE Cj - Zj ROW

Linear ProgrammingLinear ProgrammingSOLUTION INTERPRETATION SOLUTION INTERPRETATION

BASICBASIC

VARIABLESVARIABLES Y1Y1 Y2Y2 Y3Y3

SLACKSLACK

XX33QUANTITYQUANTITY

YY22 00 11 00 .3125.3125 .125.125 -.1875-.1875 0.250.25

YY33 00 00 11 -.2188-.2188 .3125.3125 .0312.0312 0.1250.125

YY11 11 00 00 .0313.0313 -.1875-.1875 .2812.2812 0.1250.125

Z jZ j 11 11 11 .125.125 .25.25 .125.125 0.500.50

C j - Z jC j - Z j 00 00 00 -.125-.125 -.250-.250 -.125-.125

Player X’s Optimal Strategy

2.0 ( .125 , .25 , .125 ) = .25 , .50 , .25

XX11 XX22 XX33

MEANING:

X plays X1 25% of the time

VALUE OF

X1 = .125The dual solution : X2 = .250 X3 = .125

Y1 = .125The primal solution : Y2 = .250 Y3 = .125

THE DUAL SOLUTION IS THE INVERSE OF THE PRIMAL SOLUTION

Game Theory with QM for WindowsGame Theory with QM for Windows

Click on

“MODULE”

to access all menus

Select and Click

“GAME THEORY”

Module

Click “File”,Scroll,

Click “New File”

The DATA CREATION TABLEasks for the

“Number of Row Strategies”( X1, X2, X3, etc. )

and“Number of Column Strategies”

( Y1, Y2, Y3, etc. )

Then click the “OK” box

The Data Input Tableappears with a 2x2

matrix

The column and rowheadings may be

changed at this point

The payoffs need to beentered

The Game GraphThe Game GraphPURE STRATEGY EXAMPLEPURE STRATEGY EXAMPLE

YY11 YY22

XX11 33 55

XX22 11 - 2- 2

PLAYER YPLAYER Y

PLAYER XPLAYER X

YY11 YY22

XX11 55

XX22 11 - 2- 2

PLAYER YPLAYER Y

PLAYER XPLAYER X

- 2- 2

3saddle point

The Saddle Point = “3”

X will always play X1

Y will always play Y1

Value of Game = “3”

ValueValue

- 2- 2

Strategy Strategy XX11

StrategyStrategyXX22

PLAYER XPLAYER X

The Game GraphThe Game GraphPURE STRATEGY EXAMPLE INTERPRETATIONPURE STRATEGY EXAMPLE INTERPRETATION

Player X can win payoffs of “3”“3” or “5”“5” via strategy Xstrategy X11

Player X can win payoffs of “1”“1” or “-2”“-2” via strategy Xstrategy X22

The dashed horizontal red linedashed horizontal red line labeled “3”“3” shows the expected value of the game to be “3”

The dashed vertical red linedashed vertical red line labeled “1”“1” shows that player X will play strategy X1 100% (11) of the time.

ValueValue

- 2- 2

Strategy Strategy YY22

33StrategyStrategy

PLAYER YPLAYER Y

The Game GraphThe Game GraphPURE STRATEGY EXAMPLE INTERPRETATIONPURE STRATEGY EXAMPLE INTERPRETATION

Player Y can win payoffs of “-1”“-1” or “-3”“-3” via strategy Ystrategy Y11

Player Y can win payoffs of “+2”“+2” or “-5”“-5” via strategy Ystrategy Y22

The dashed horizontal red linedashed horizontal red line labeled “3”“3” shows the expected value of the game to be “3”

The dashed vertical red linedashed vertical red line labeled “1”“1” shows that player Y will play strategy Y1 100% ( (11) ) of the time.

Game Theory Game Theory UsingUsing

The Game GraphThe Game GraphMIXED STRATEGY EXAMPLEMIXED STRATEGY EXAMPLE

YY11 YY22

XX11 44 22

XX22 11 1010

PLAYER YPLAYER Y

PLAYER XPLAYER X

ValueValue

Strategy Strategy YY22

StrategyStrategyYY11

3.45453.4545

PLAYER YPLAYER Y

.8181.8181

The Game GraphThe Game GraphMIXED STRATEGY EXAMPLE INTERPRETATIONMIXED STRATEGY EXAMPLE INTERPRETATION

Player Y can win payoffs of “-1”“-1” or or “-4”“-4” via strategy Ystrategy Y11

Player X can win payoffs of “-2”“-2” or “-10”“-10” via strategy Ystrategy Y22

The dashed horizontal red linedashed horizontal red line labeled “3.4545”“3.4545” shows the expected value of the game to be “3.4545” The dashed vertical red linedashed vertical red line labeled “.8181”“.8181” shows that player X will play strategy X1 approximately 82% of the time, inferring that player X will play strategy X2 approximately 18% of the time.

ValueValue

Strategy Strategy XX22

StrategyStrategyXX11

3.45453.4545

PLAYER XPLAYER X

.7272.7272

The Game GraphThe Game GraphMIXED STRATEGY EXAMPLE INTERPRETATIONMIXED STRATEGY EXAMPLE INTERPRETATION

Player X can win payoffs of “2”“2” or “4”“4” via strategy Xstrategy X11

Player X can win payoffs of “1”“1” or or “10”“10” via strategy Xstrategy X22

The dashed horizontal red linedashed horizontal red line labeled “3.4545”“3.4545” shows the expected value of the game to be “3.4545” The dashed vertical red linedashed vertical red line labeled “.7272”“.7272” shows that player Y will play strategy Y1 approximately 73% of the time, inferring that player Y will play strategy Y2 approximately 27% of the time.

Game Theory via Linear Game Theory via Linear Programming with QM for WindowsProgramming with QM for Windows

Solving 3x3 GamesSolving 3x3 Games

YY22 YY33

XX11 22 33 00

XX22 11 22 33

XX33 44

VIA LINEAR PROGRAMMINGVIA LINEAR PROGRAMMING

PLAYERPLAYERYY

PLAYERPLAYERXX

Linear ProgrammiLinear Programming Formulationng FormulationEXAMPLEEXAMPLE

Objective Function:

Subject to:

Non-negativityConstraint:

Maximize Y1 + Y2 + Y3

2Y1 + 3Y2 + 0Y3 <= 1

1Y1 + 2Y2 + 3Y3 <= 1

4Y1 + 1Y2 + 2Y3 <= 1

Y1 , Y2 , Y3 => 0

Scroll To The

“LINEAR PROGRAMMING”

The three (3) variablesin the problem are:

Y1 , Y2 , Y3

The Objective Functionis always

maximized

Y1 = .125Y2 = .250Y3 = .125

X1 = .125X2 = .250X3 = .125

Y1 = .125 or 12.5%Y2 = .250 or 25.0%Y3 = .125 or 12.5%

The Value of the Game:

1 / ( .125 + .250 + .125 ) = 1 / .50 = 2.0

2.0 ( .125 , .250 , .125 )=

( 25% , 50% , 25% )

Y plays Y1 25% of the timeY plays Y2 50% of the timeY plays Y3 25% of the time

X1 = - .125X2 = - .250X3 = - .125

2.0 ( .125 , .250 , .125 ) = .25 , .50 , .25

Therefore:

X plays X1 25% of the timeX plays X2 50% of the timeX plays X3 25% of the time

VALUEOF THEGAME

Linear ProgrammingApproach

Game Theory

OptimalStrategies

Formulatedin

Conflict

MGMT E-5070

Game Theory Optimal Strategies Formulated in Conflict MGMT E-5070.

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