game theory optimal strategies formulated in conflict mgmt e-5070
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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
YY11
( use radio )( use radio )
YY22
( use newspaper )( use newspaper )
XX11
( use radio )( use radio )3 5
XX22
( 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 &
StrategiesStrategies
YY11
( use radio )( use radio )
YY22
( use newspaper )( use newspaper )
XX11
( use radio )( use radio )3 5
XX22
( use newspaper )( use newspaper )1 - 2
MAXIMUM 3 5
MIN
IMU
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 mini- mum 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 &
StrategiesStrategies
YY11
( use radio )( use radio )
YY22
( use newspaper )( use newspaper )
XX11
( use radio )( use radio )3 5
XX22
( use newspaper )( use newspaper )1 - 2
Pure StrategyPure Strategy
PlayersPlayers++
StrategiesStrategies
YY11
( use radio )( use radio )YY22
( use paper )( use paper )
XX11
( use radio )( use radio )
XX22
( use paper )( use paper )
3 5
1 - 2
Minimum
Maximum 3 5
3
- 2
3
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 se- lect 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 &
StrategiesStrategies
YY11
( use radio )( use radio )
YY22
( use newspaper )( use newspaper )
XX11
( use radio )( use radio )2 - 4
XX22
( use newspaper )( use newspaper )6 10
Pure StrategyPure Strategy
PlayersPlayers++
StrategiesStrategies
YY11
( use radio )( use radio )YY22
( use paper )( use paper )
XX11
( use radio )( use radio )
XX22
( use paper )( use paper )
2 - 4
10
Minimum
Maximum 6 10
- 4
66
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 se- lect 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 &
StrategiesStrategies
YY11
( use radio )( use radio )
YY22
( use newspaper )( use newspaper )
XX11
( use radio )( use radio )4 2
XX22
( use newspaper )( use newspaper )1 10
Mixed Strategy GameMixed Strategy Game
PlayersPlayers++
StrategiesStrategies
YY11
( use radio )( use radio )YY22
( use paper )( use paper )
XX11
( use radio )( use radio )
XX22
( use paper )( use paper )
4 2
10
Minimum
Maximum 4 10
2
11
EXAMPLEEXAMPLE
11
THERE IS NOSADDLEPOINT
Upper Value
Lower Value
Mixed Strategy GameMixed Strategy Game
PlayersPlayers+ +
StrategiesStrategies
XX11
XX22
YY11 YY22
44 22
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”
Mixed Strategy GameMixed Strategy Game
PlayersPlayers+ +
StrategiesStrategies
( 1-Q )( 1-Q )
PP ( 1-P )( 1-P )
44 22
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%
Mixed Strategy GameMixed Strategy Game
PlayersPlayers+ +
StrategiesStrategies
( 1-Q )( 1-Q )
PP ( 1-P )( 1-P )
44 22
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+ +
StrategiesStrategies
( 1-Q )( 1-Q )
PP ( 1-P )( 1-P )
44 22
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
YesYes
NoNoYesYes
NoNo
NoNo
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
Principle of DominancePrinciple of Dominance
Y1 Y2
XX11 4 4 3 3
XX22 2 2 20 20
X3 1 1
11stst EXAMPLE EXAMPLE
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
3
2
11
MINIMUMPAYOFF
rejected
Principle of DominancePrinciple of Dominance
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
11stst EXAMPLE EXAMPLE
PLAYER YPLAYER Y
PLAYER XPLAYER X
Principle of DominancePrinciple of Dominance
YY11 YY22 YY33 YY44
XX11 - 5- 5 44 66 - 3- 3
XX22 - 2- 2 66 22 - 20- 20
22ndnd EXAMPLE EXAMPLE
PLAYERPLAYERYY
PLAYERPLAYERXX
Principle of DominancePrinciple of Dominance
YY11 Y2 Y3 YY44
XX11 - 5- 5 4 6 - 3- 3
XX22 - 2- 2 6 2 - 20- 20
22ndnd EXAMPLE EXAMPLE
PLAYERPLAYERYY
PLAYERPLAYERXX
PLAYER ‘Y’ CAN ALWAYS DO BETTER PLAYING STRATEGY YPLAYER ‘Y’ CAN ALWAYS DO BETTER PLAYING STRATEGY Y11 OR Y OR Y44
rejected
Principle of DominancePrinciple of Dominance
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
22ndnd EXAMPLE EXAMPLE
PLAYERPLAYERYYPLAYERPLAYER
XX
Solving 3x3 GamesSolving 3x3 Games
YY11
YY22 YY33
XX11 22 33 00
XX22 11 22 33
XX33 44
11 22
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
XX11
SLACKSLACK
XX22
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
VARIABLESVARIABLES YY11 YY22 YY33
SLACKSLACK
XX11
SLACKSLACK
XX22
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
VARIABLESVARIABLES YY11 YY22 YY33
SLACKSLACK
XX11
SLACKSLACK
XX22
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
Linear ProgrammingLinear ProgrammingSOLUTION INTERPRETATIONSOLUTION INTERPRETATION
Y1 = .125 or 12.5%
Y2 = .250 or 25.0%
Y3 = .125 or 12.5%
Linear ProgrammingLinear ProgrammingSOLUTION INTERPRETATIONSOLUTION INTERPRETATION
The Value of the GameThe Value of the Game
11
YY11 + + YY22 + + YY33
11
.125 + .25 + .125.125 + .25 + .125
11
.50.50
== 2.02.0
Linear ProgrammingLinear ProgrammingSOLUTION INTERPRETATIONSOLUTION INTERPRETATION
Player Y ’s Optimal Strategy
2.0 ( .125 , .25 , .125 )
=
.25 , .50 , .25
Y plays Y1 25% of the time
Y plays Y2 50% of the time
Y plays Y3 25% of the time
MEANING:
Y1 Y2 Y3VALUE
OFGAME
Linear ProgrammingLinear ProgrammingSOLUTION INTERPRETATIONSOLUTION INTERPRETATION
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
XX11
SLACKSLACK
XX22
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
Linear ProgrammingLinear ProgrammingSOLUTION INTERPRETATIONSOLUTION INTERPRETATION
Player X’s Optimal Strategy
2.0 ( .125 , .25 , .125 ) = .25 , .50 , .25
XX11 XX22 XX33
MEANING:
X plays X1 25% of the time
X plays X2 50% of the time
X plays X3 25% of the time
VALUE OF
GAME
Linear ProgrammingLinear ProgrammingSOLUTION INTERPRETATIONSOLUTION INTERPRETATION
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
The Game GraphThe Game GraphPURE STRATEGY EXAMPLEPURE STRATEGY EXAMPLE
YY11 YY22
XX11 55
XX22 11 - 2- 2
PLAYER YPLAYER Y
PLAYER XPLAYER X
33
- 2- 2
33 55
3saddle point
The Saddle Point = “3”
X will always play X1
Y will always play Y1
Value of Game = “3”
The Game GraphThe Game GraphPURE STRATEGY EXAMPLEPURE STRATEGY EXAMPLE
ValueValue
55
- 2- 2
Strategy Strategy XX11
33
StrategyStrategyXX22
11
00 11
33
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.
The Game GraphThe Game GraphPURE STRATEGY EXAMPLEPURE STRATEGY EXAMPLE
ValueValue
55
- 2- 2
Strategy Strategy YY22
33StrategyStrategy
YY11
00 11
33
PLAYER YPLAYER Y
11
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
The Game GraphThe Game GraphMIXED STRATEGY EXAMPLEMIXED STRATEGY EXAMPLE
ValueValue
1010
Strategy Strategy YY22
44
StrategyStrategyYY11
00 11
3.45453.4545
PLAYER YPLAYER Y
2211
.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 approx- imately 18% of the time.
The Game GraphThe Game GraphMIXED STRATEGY EXAMPLEMIXED STRATEGY EXAMPLE
ValueValue
1010
Strategy Strategy XX22
44
StrategyStrategyXX11
00 11
3.45453.4545
PLAYER XPLAYER X
1122
.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 approx- imately 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
YY11
YY22 YY33
XX11 22 33 00
XX22 11 22 33
XX33 44
11 22
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”
Menu
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
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