Game Theory Optimal Strategies Formulated in Conflict MGMT E-5070

Game Theory INTRODUCTION Strategies taken by other firms or individuals can dramatically affect the outcome of our decisions Consequently, business cannot make important decisions today without considering what other firms or individuals are doing or might do

Game Theory INTRODUCTORY TERMINOLOGY  Game Theory  Game Theory is the study of how optimal strategies are formulated in conflict.  game  A game is a contest involving two or more decision makers, each of whom wants to win.  Game Theory  Game Theory considers the impact of the strategies of others on our strategies and outcomes.

Janos (John) von Neumann ( 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 20 th century

John Nash ( )  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 CROWE

Game Theory INTRODUCTORY TERMINOLOGY  Two – Person Game  Two – Person Game : A game in which only two parties ( ‘X’ and ‘Y’ ) can play.  Zero – Sum Game  Zero – 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 Game EXAMPLE ALL PAYOFFS SHOWN IN TERMS OF PLAYER ‘X’ Players & Strategies Y 1 ( use radio ) Y 2 ( use newspaper ) X 1 ( use radio ) 35 X 2 ( use newspaper ) 1 - 2

Two Important Values The Lower Value ( LV )  smallest  Find the smallest number in each row. largest  Select the largest of these numbers.  This is the LV. The Upper Value ( UV )  largest  Find the largest number in each column. smallest  Select the smallest of these numbers.  This is the UV.

Two-Person Zero-Sum Game EXAMPLE Players & Strategies Y 1 ( use radio ) Y 2 ( use newspaper ) X 1 ( use radio ) 35 X 2 ( use newspaper ) MAXIMUM 3 5 MINIMUM THE UPPER VALUE THE LOWER VALUE

Maxi-Min Criterion 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 X 1 ( use radio ).  “+3” is the lower value of the game. The lower value equals the maxi-min strategy for player ‘X’.

Mini-Max 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 Y 1 ( 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 = 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 mini- mum gains !

Game Theory ADDITIONAL TERMINOLOGY  Pure Strategy  Pure Strategy : A game in which both players will always play just one strategy each. Saddle Point Game  Saddle Point Game : A game that has a pure strategy. Value of the Game  Value of the Game : The expected winnings of the game if the game is played a large number of times.

Pure 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 Strategy EXAMPLE Players & Strategies Y 1 ( use radio ) Y 2 ( use newspaper ) X 1 ( use radio ) 35 X 2 ( use newspaper ) 1 - 2

Pure Strategy Players+Strategies Y 1 ( use radio ) Y 2 ( use paper ) X 1 ( use radio ) X 2 ( use paper ) Minimum Maximum EXAMPLE EXAMPLE Saddlepoint Upper Value Lower Value

Pure Strategy Game EXAMPLE  UV = LV = 3, meaning that every time the advertising game is played, player ‘X’ will select strategy ‘X 1 ’ and player ‘Y’ will se- lect strategy ‘Y 1 ’.  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 Strategy 2 nd EXAMPLE Players & Strategies Y 1 ( use radio ) Y 2 ( use newspaper ) X 1 ( use radio ) 2- 4 X 2 ( use newspaper ) 6 10

Pure Strategy Players+Strategies Y 1 ( use radio ) Y 2 ( use paper ) X 1 ( use radio ) X 2 ( use paper ) Minimum Maximum nd EXAMPLE 2 nd EXAMPLE Saddlepoint Upper ValueLower Value

Pure Strategy Game 2 nd EXAMPLE  UV = LV = 6, meaning that every time the advertising game is played, player ‘X’ will select strategy ‘X 2 ’ and player ‘Y’ will se- lect strategy ‘Y 1 ’.  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 Game INVOLVES USE OF THE ALGEBRAIC APPROACH. Occurs when there is no saddlepoint, that is, no pure strategy The overall objective of each player is to determine what percentage of the time he or she should play each strategy, in order to maximize winnings, regardless of what the other player does

Mixed Strategy Game EXAMPLE Players & Strategies Y 1 ( use radio ) Y 2 ( use newspaper ) X 1 ( use radio ) 42 X 2 ( use newspaper ) 1 10

Mixed Strategy Game Players+Strategies Y 1 ( use radio ) Y 2 ( use paper ) X 1 ( use radio ) X 2 ( use paper ) Minimum Maximum EXAMPLE EXAMPLE 1 THERE IS NO SADDLEPOINT Upper Value Lower Value

Mixed Strategy Game Players+Strategies X1X1X1X1 X2X2X2X2 Y1Y1Y1Y1 Y2Y2Y2Y EXAMPLE

The Algebraic Approach “Q” = percentage of time player X plays strategy “X 1 ” “1-Q” = percentage of time player X plays strategy “X 2 ” “P” = percentage of time player Y plays strategy “Y 1 ” “1-P” = percentage of time player Y plays strategy “Y 2 ”

Mixed Strategy Game Players+Strategies Q ( 1-Q ) P ( 1-P ) EXAMPLE

Player X Strategy SET THE COLUMNS EQUAL TO ONE ANOTHER AND SOLVE FOR ‘ Q ‘ 4Q + 1(1-Q) = 2Q + 10(1-Q) 4Q Q = 2Q Q 4Q - 1Q - 2Q + 10Q = Q = 9 THEREFORE Q = 9/11 or 82% and (1-Q) = 2/11 or 18%

Mixed Strategy Game Players+Strategies Q ( 1-Q ) P ( 1-P ) EXAMPLE

Player Y Strategy SET THE ROWS EQUAL TO ONE ANOTHER AND SOLVE FOR ‘ P ‘ 4P + 2(1-P) = 1P + 10(1-P) 4P + 2 – 2P = 1P + 10 – 10P 4P – 2P – 1P + 10P = P = 8 THEREFORE P = 8/11 or 73% and (1-P) = 3/11 or 27%

Value of the Game Players+Strategies Q ( 1-Q ) P ( 1-P ) CALCULATIONS

Value of the Game CALCULATIONS 1 st way:.73(4) +.27( 2) = nd way:.73(1) +.27(10) ≈ rd way:.82(4) +.18( 1) = th way:.82(2) +.18(10) ≈ 3.46

Procedure for Solving Two-Person, Zero Sum Games Develop Strategies and Payoff Matrix Is There A Pure Strategy Solution? Is Game 2x2? Solve Problem for Saddle Point Solution Solve with Linear Programming Can Dominance Be Used To Reduce Matrix? Solve for Mixed Strategy Probabilities YesYesNo Yes NoNo

The Principle of Dominance Used to reduce the size of games by eliminating 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 Dominance Y 1 Y 2 X X X st EXAMPLE PLAYER Y PLAYER X

Principle of Dominance Y 1 Y 2 X X X st EXAMPLE PLAYER Y PLAYER X PLAYER ‘X’ CAN ALWAYS DO BETTER PLAYING STRATEGY X 1 OR X MINIMUM PAYOFF rejected

Principle of Dominance Y 1 Y 2 X X THE NEW GAME AFTER ELIMINATION OF ONE “X” STRATEGY 1 st EXAMPLE PLAYER Y PLAYER X

Principle of Dominance Y 1 Y 2 Y 3 Y 4 X X nd EXAMPLE PLAYERY PLAYERX

Principle of Dominance Y 1 Y 2 Y 3 Y 4 X X nd EXAMPLE PLAYERY PLAYERX PLAYER ‘Y’ CAN ALWAYS DO BETTER PLAYING STRATEGY Y 1 OR Y 4 rejected

Principle of Dominance Y 1 Y 2 X X THE NEW GAME AFTER ELIMINATION OF TWO “Y” STRATEGIES 2 nd EXAMPLE PLAYERY PLAYERX

Solving 3x3 Games Y1Y1Y1Y1 Y2Y2Y2Y2 Y3Y3Y3Y3 X1X1X1X1230 X2X2X2X2123 X3X3X3X34 12 VIA LINEAR PROGRAMMING PLAYERY PLAYERX

Linear Programming Formulation EXAMPLE Objective Function: Subject to: Non-negativity Constraint: Maximize Y 1 + Y 2 + Y 3 2Y 1 + 3Y 2 + 0Y 3 <= 1 1Y 1 + 2Y 2 + 3Y 3 <= 1 4Y 1 + 1Y 2 + 2Y 3 <= 1 Y 1, Y 2, Y 3 => 0

Linear Programming Formulation EXAMPLE 2Y 1 + 3Y 2 + 0Y 3 + 1X 1 + 0X 2 + 0X 3 = 1 1Y 1 + 2Y 2 + 3Y 3 + 0X 1 + 1X 2 + 0X 3 = 1 4Y 1 + 1Y 2 + 2Y 3 + 0X 1 + 0X 2 + 1X 3 = 1 CONVERT TO LINEAR EQUALITIES ( ADD SLACK VARIABLES )

Linear Programming COMPUTER-GENERATED 1 st FEASIBLE SOLUTION BASISVARIABLES Y1Y1Y1Y1 Y2Y2Y2Y2 Y3Y3Y3Y3SLACK X 1 SLACK X 2 SLACK X 3 QUANTITY X1X1X1X X2X2X2X X3X3X3X Z j C j - Z j

Linear Programming COMPUTER-GENERATED OPTIMAL SOLUTION BASICVARIABLES Y1Y1Y1Y1 Y2Y2Y2Y2 Y3Y3Y3Y3SLACK X 1 SLACK X 2 SLACK X 3 QUANTITY Y2Y2Y2Y Y3Y3Y3Y Y1Y1Y1Y Z j C j - Z j

Linear Programming SOLUTION INTERPRETATION BASICVARIABLES Y1Y1Y1Y1 Y2Y2Y2Y2 Y3Y3Y3Y3SLACK X 1 SLACK X 2 SLACK X 3 QUANTITY Y2Y2Y2Y Y3Y3Y3Y Y1Y1Y1Y Z j C j - Z j

Linear Programming SOLUTION INTERPRETATION Y 1 =.125 or 12.5% Y 2 =.250 or 25.0% Y 3 =.125 or 12.5%

Linear Programming SOLUTION INTERPRETATION The Value of the Game 1 Y 1 + Y 2 + Y =2.0

Linear Programming SOLUTION INTERPRETATION Player Y ’s Optimal Strategy 2.0 (.125,.25,.125 ) =.25,.50,.25 Y plays Y 1 25% of the time Y plays Y 2 50% of the time Y plays Y 3 25% of the time MEANING: Y1Y1 Y2Y2 Y3Y3 VALUE OF GAME

Linear Programming SOLUTION INTERPRETATION X 1 = X 2 = X 3 = THE VALUES ARE THE SHADOW PRICES OF THE SLACK VARIABLES IN THE Cj - Zj ROW

Linear Programming SOLUTION INTERPRETATION BASICVARIABLESY1Y2Y3SLACK X 1 SLACK X 2 SLACK X 3 QUANTITY Y2Y2Y2Y Y3Y3Y3Y Y1Y1Y1Y Z j C j - Z j

Linear Programming SOLUTION INTERPRETATION Player X’s Optimal Strategy 2.0 (.125,.25,.125 ) =.25,.50,.25 X1X1X1X1 X2X2X2X2 X3X3X3X3 MEANING: X plays X 1 25% of the time X plays X 2 50% of the time X plays X 3 25% of the time VALUE OF GAME

Linear Programming SOLUTION INTERPRETATION X 1 =.125 The dual solution : X 2 =.250 X 3 =.125 Y 1 =.125 The primal solution : Y 2 =.250 Y 3 =.125 THE DUAL SOLUTION IS THE INVERSE OF THE PRIMAL SOLUTION

Game 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 TABLE asks 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 Table appears with a 2x2 matrix The column and row headings may be changed at this point The payoffs need to be entered

The Game Graph PURE STRATEGY EXAMPLE Y1Y1Y1Y1 Y2Y2Y2Y2 X1X1X1X135 X2X2X2X PLAYER Y PLAYER X

The Game Graph PURE STRATEGY EXAMPLE Y1Y1Y1Y1 Y2Y2Y2Y2 X1X1X1X15 X2X2X2X PLAYER Y PLAYER X saddle point

The Saddle Point = “3” X will always play X1 Y will always play Y1 Value of Game = “3”

The Game Graph PURE STRATEGY EXAMPLE Value Strategy X 1 3 Strategy X PLAYER X

The Game Graph PURE STRATEGY EXAMPLE INTERPRETATION  “3” “5” strategy X 1  Player X can win payoffs of “3” or “5” via strategy X 1  “1” “-2” strategy X 2  Player X can win payoffs of “1” or “-2” via strategy X 2  dashed horizontal red line “3”  The dashed horizontal red line labeled “3” shows the expected value of the game to be “3”  dashed vertical red line “1”  The dashed vertical red line labeled “1” shows that 1 player X will play strategy X 1 100% (1) of the time.

The Game Graph PURE STRATEGY EXAMPLE Value Strategy Y 2 3 Strategy Y PLAYER Y 1

The Game Graph PURE STRATEGY EXAMPLE INTERPRETATION  “-1” “-3” strategy Y 1  Player Y can win payoffs of “-1” or “-3” via strategy Y 1  “+2” “-5” strategy Y 2  Player Y can win payoffs of “+2” or “-5” via strategy Y 2  dashed horizontal red line “3”  The dashed horizontal red line labeled “3” shows the expected value of the game to be “3”  dashed vertical red line “1”  The dashed vertical red line labeled “1” shows that (1) player Y will play strategy Y 1 100% (1) of the time.

Game Theory Using

The Game Graph MIXED STRATEGY EXAMPLE Y1Y1Y1Y1 Y2Y2Y2Y2 X1X1X1X142 X2X2X2X2110 PLAYER Y PLAYER X

The Game Graph MIXED STRATEGY EXAMPLE Value 10 Strategy Y 2 4 Strategy Y PLAYER Y

The Game Graph MIXED STRATEGY EXAMPLE INTERPRETATION  “-1” or “-4” strategy Y 1  Player Y can win payoffs of “-1” or “-4” via strategy Y 1  “-2” “-10” strategy Y 2  Player X can win payoffs of “-2” or “-10” via strategy Y 2  dashed horizontal red line “3.4545”  The dashed horizontal red line labeled “3.4545” shows the expected value of the game to be “3.4545”  dashed vertical red line “.8181”  The dashed vertical red line labeled “.8181” shows that player X will play strategy X 1 approximately 82% of the time, inferring that player X will play strategy X 2 approx- imately 18% of the time.

The Game Graph MIXED STRATEGY EXAMPLE Value 10 Strategy X 2 4 Strategy X PLAYER X

The Game Graph MIXED STRATEGY EXAMPLE INTERPRETATION  “2” “4” strategy X 1  Player X can win payoffs of “2” or “4” via strategy X 1  “1” or “10” strategy X 2  Player X can win payoffs of “1” or “10” via strategy X 2  dashed horizontal red line “3.4545”  The dashed horizontal red line labeled “3.4545” shows the expected value of the game to be “3.4545”  dashed vertical red line “.7272”  The dashed vertical red line labeled “.7272” shows that player Y will play strategy Y 1 approximately 73% of the time, inferring that player Y will play strategy Y 2 approx- imately 27% of the time.

Game Theory via Linear Programming with QM for Windows

Solving 3x3 Games Y1Y1Y1Y1 Y2Y2Y2Y2 Y3Y3Y3Y3 X1X1X1X1230 X2X2X2X2123 X3X3X3X34 12 VIA LINEAR PROGRAMMING PLAYERY PLAYERX

Linear Programming Formulation EXAMPLE Objective Function: Subject to: Non-negativity Constraint: Maximize Y 1 + Y 2 + Y 3 2Y 1 + 3Y 2 + 0Y 3 <= 1 1Y 1 + 2Y 2 + 3Y 3 <= 1 4Y 1 + 1Y 2 + 2Y 3 <= 1 Y 1, Y 2, Y 3 => 0

Scroll To The “LINEAR PROGRAMMING” Menu

The three (3) variables in the problem are: Y1, Y2, Y3 The Objective Function is always maximized

Y 1 =.125 Y 2 =.250 Y 3 =.125 X 1 =.125 X 2 =.250 X 3 =.125

Y 1 =.125 or 12.5% Y 2 =.250 or 25.0% Y 3 =.125 or 12.5% The Value of the Game: 1 / ( ) = 1 /.50 = (.125,.250,.125 ) = ( 25%, 50%, 25% ) Y plays Y 1 25% of the time Y plays Y 2 50% of the time Y plays Y 3 25% of the time

X 1 = X 2 = X 3 = (.125,.250,.125 ) =.25,.50,.25 Therefore: X plays X 1 25% of the time X plays X 2 50% of the time X plays X 3 25% of the time VALUE OF THE GAME

Linear Programming Approach

Game Theory Optimal Strategies Formulated in Conflict MGMT E-5070