making decisions against an opponent: an application of mathematical optimization isye 2030 monday,...

Making Decisions Making Decisions Against an Opponent:Against an Opponent:

An application ofAn application of Mathematical OptimizationMathematical Optimization

Making Decisions Making Decisions Against an Opponent:Against an Opponent:

An application ofAn application of Mathematical OptimizationMathematical Optimization

ISYE 2030ISYE 2030

Monday, October 2, 2000Monday, October 2, 2000

Dr. Joel SokolDr. Joel Sokol

[email protected]@isye.gatech.edu

ISYE 2030ISYE 2030

Monday, October 2, 2000Monday, October 2, 2000

Dr. Joel SokolDr. Joel Sokol

[email protected]@isye.gatech.edu

Making Decisions Making Decisions Against an OpponentAgainst an Opponent

Making Decisions Making Decisions Against an OpponentAgainst an Opponent

• Two people/teams in competitionTwo people/teams in competition• Two people/teams in competitionTwo people/teams in competition

• Many examples in business, sports,Many examples in business, sports, economics, and even everyday lifeeconomics, and even everyday life

• Many examples in business, sports,Many examples in business, sports, economics, and even everyday lifeeconomics, and even everyday life

• Outcome depends on both sides’ Outcome depends on both sides’ simultaneoussimultaneous decisions decisions

• Outcome depends on both sides’ Outcome depends on both sides’ simultaneoussimultaneous decisions decisions

Simpsons Decision-MakingSimpsons Decision-MakingSimpsons Decision-MakingSimpsons Decision-Making

Bart LisaBart Lisa

vs.vs.

Rock-Paper-ScissorsRock-Paper-ScissorsRock-Paper-ScissorsRock-Paper-Scissors

RockRock Paper ScissorsPaper Scissors

RockRock 0 0 -1-1 11

PaperPaper 11 0 0 -1-1

ScissorsScissors -1-1 11 0 0

11 = Bart wins = Bart wins -1-1 = Lisa wins = Lisa wins

Poor, predictableBart…always chooses rock. Good old rock…

You can always count on rock. Lisa’s OptimalLisa’s Optimal

ChoiceChoice

What if…?What if…?What if…?What if…?


RockRock 0 0 -1-1 11

PaperPaper 11 0 0 -1-1

ScissorsScissors -1-1 11 0 0


Good old rock… 60% of the time,

you can count on rock.

?

What if…?What if…?What if…?What if…?


60% Rock60% Rock 0 0 -1-1 11

20% Paper20% Paper 11 0 0 -1-1

20% Scissors20% Scissors -1-1 11 0 0


Average = 0 -0.4 +0.4Average = 0 -0.4 +0.4

Good old rock… 60% of the time,

you can count on rock.

Paper!

So, what’s the outcome?So, what’s the outcome?So, what’s the outcome?So, what’s the outcome?

RockRock Paper ScissorsPaper ScissorsAverage = 0 -0.4 +0.4Average = 0 -0.4 +0.4

• Lisa should choose the column with theLisa should choose the column with thebest results (paper), and wins 40% morebest results (paper), and wins 40% more

• Lisa should choose the column with theLisa should choose the column with thebest results (paper), and wins 40% morebest results (paper), and wins 40% more

• In general, Bart’s overall result equals theIn general, Bart’s overall result equals thevalue of Lisa’s best (lowest) column.value of Lisa’s best (lowest) column.

• In general, Bart’s overall result equals theIn general, Bart’s overall result equals thevalue of Lisa’s best (lowest) column.value of Lisa’s best (lowest) column.

• But…But… if Bart knows Lisa will choose paper, if Bart knows Lisa will choose paper,he should choose scissors… and so onhe should choose scissors… and so on

• But…But… if Bart knows Lisa will choose paper, if Bart knows Lisa will choose paper,he should choose scissors… and so onhe should choose scissors… and so on

Bart’s Mathematical ModelBart’s Mathematical ModelBart’s Mathematical ModelBart’s Mathematical Model x = probability that Bart chooses rock, y = x = probability that Bart chooses rock, y =

probability that Bart chooses paper, z = probability that Bart chooses paper, z = probability that Bart chooses scissorsprobability that Bart chooses scissors

v = total value (smallest of column totals)v = total value (smallest of column totals)

x = probability that Bart chooses rock, y = x = probability that Bart chooses rock, y = probability that Bart chooses paper, z = probability that Bart chooses paper, z = probability that Bart chooses scissorsprobability that Bart chooses scissors

v = total value (smallest of column totals)v = total value (smallest of column totals)

RockRock Paper ScissorsPaper Scissors(x) Rock(x) Rock 0 0 -1-1 11(y) Paper(y) Paper 11 0 0 -1-1(z) Scissors(z) Scissors -1-1 11 0 0Average = y - z z - x x - Average = y - z z - x x - yy

• Let’s use a Let’s use a Linear Programming ModelLinear Programming Model• Let’s use a Let’s use a Linear Programming ModelLinear Programming Model

• All linear functionsAll linear functions– No powers (xNo powers (x22))– No square roots, trig functions, logs, etc.No square roots, trig functions, logs, etc.– No variables multiplied by each otherNo variables multiplied by each other

• All linear functionsAll linear functions– No powers (xNo powers (x22))– No square roots, trig functions, logs, etc.No square roots, trig functions, logs, etc.– No variables multiplied by each otherNo variables multiplied by each other

• Find values for the variables thatFind values for the variables that– satisfy all constraintssatisfy all constraints– give the give the bestbest (optimal) objective function value (optimal) objective function value

• Find values for the variables thatFind values for the variables that– satisfy all constraintssatisfy all constraints– give the give the bestbest (optimal) objective function value (optimal) objective function value

Linear Programming ModelsLinear Programming ModelsLinear Programming ModelsLinear Programming Models• ““Objective Function” - minimize or Objective Function” - minimize or

maximizemaximize

• ““Constraints” - mathematical restrictionsConstraints” - mathematical restrictions

• ““Objective Function” - minimize or Objective Function” - minimize or maximizemaximize

• ““Constraints” - mathematical restrictionsConstraints” - mathematical restrictions

Bart’s Linear ProgramBart’s Linear ProgramBart’s Linear ProgramBart’s Linear Program

MaximizeMaximize vv

MaximizeMaximize vv

RockRock Paper ScissorsPaper Scissors(x) Rock(x) Rock 0 0 -1-1 11(y) Paper(y) Paper 11 0 0 -1-1(z) Scissors(z) Scissors -1-1 11 0 0Average = y - z z - x x - yAverage = y - z z - x x - y

Subject toSubject to v v y - z y - z (Rock)(Rock)

Subject toSubject to v v y - z y - z (Rock)(Rock)

v v z - x z - x (Paper)(Paper)

v v z - x z - x (Paper)(Paper)

v v x - y x - y (Scissors)(Scissors)

v v x - y x - y (Scissors)(Scissors)

x + y + z = 1x + y + z = 1

x x 0, y 0, y 0, z 0, z 0 0

x + y + z = 1x + y + z = 1

x x 0, y 0, y 0, z 0, z 0 0

Lisa’s Mathematical ModelLisa’s Mathematical ModelLisa’s Mathematical ModelLisa’s Mathematical Model a = probability that Lisa chooses rock, b = probability a = probability that Lisa chooses rock, b = probability

that Lisa chooses paper, c = probability that Lisa that Lisa chooses paper, c = probability that Lisa chooses scissorschooses scissors

d = total value (largest of row totals)d = total value (largest of row totals)

• Another Another Linear Programming ModelLinear Programming Model

a = probability that Lisa chooses rock, b = probability a = probability that Lisa chooses rock, b = probability that Lisa chooses paper, c = probability that Lisa that Lisa chooses paper, c = probability that Lisa chooses scissorschooses scissors

d = total value (largest of row totals)d = total value (largest of row totals)

• Another Another Linear Programming ModelLinear Programming Model

Rock(a) Paper(b) Scissors(c) Rock(a) Paper(b) Scissors(c) Avg.Avg.RockRock 0 0 -1-1 1 1 c - bc - bPaperPaper 11 0 0 -1 -1 a - ca - cScissorsScissors -1-1 11 0 0 b - ab - a

Lisa’s Linear ProgramLisa’s Linear ProgramLisa’s Linear ProgramLisa’s Linear Program

MinimizeMinimize dd

Subject toSubject to d d b - c b - c (Rock)(Rock)

d d c - a c - a (Paper)(Paper)

d d a - b a - b (Scissors)(Scissors)

a + b + c = 1a + b + c = 1

a a 0, b 0, b 0, c 0, c 0 0

MinimizeMinimize dd

Subject toSubject to d d b - c b - c (Rock)(Rock)

d d c - a c - a (Paper)(Paper)

d d a - b a - b (Scissors)(Scissors)

a + b + c = 1a + b + c = 1

a a 0, b 0, b 0, c 0, c 0 0

Rock(a) Paper(b) Scissors(c) Rock(a) Paper(b) Scissors(c) Avg.Avg.RockRock 0 0 -1-1 1 1 c - bc - bPaperPaper 11 0 0 -1 -1 a - ca - cScissorsScissors -1-1 11 0 0 b - ab - a

– Solution: x = y = z = 1/3Solution: x = y = z = 1/3– Value = 0Value = 0

– Solution: x = y = z = 1/3Solution: x = y = z = 1/3– Value = 0Value = 0

– Maximize outcomeMaximize outcome

– Outcome = Outcome = lowest column lowest column

averageaverage

– Maximize outcomeMaximize outcome

– Outcome = Outcome = lowest column lowest column

averageaverage

Two Optimization ModelsTwo Optimization ModelsTwo Optimization ModelsTwo Optimization Models

• Lisa’s ModelLisa’s Model• Lisa’s ModelLisa’s Model

Note: The two values will Note: The two values will alwaysalways be equal be equalNote: The two values will Note: The two values will alwaysalways be equal be equal

• Bart’s ModelBart’s Model• Bart’s ModelBart’s Model

– Minimize outcomeMinimize outcome

– Outcome = Outcome = highest row averagehighest row average

– Minimize outcomeMinimize outcome

– Outcome = Outcome = highest row averagehighest row average

– Solution: a = b = c = 1/3Solution: a = b = c = 1/3– Value = 0Value = 0

– Solution: a = b = c = 1/3Solution: a = b = c = 1/3– Value = 0Value = 0

A More Interesting Example:A More Interesting Example:2-point conversions in football2-point conversions in footballA More Interesting Example:A More Interesting Example:

2-point conversions in football2-point conversions in football

• Offense gets the ball at 3 yard line, Offense gets the ball at 3 yard line, has one play to scorehas one play to score

• Many offensive play choicesMany offensive play choices

• Many defensive strategiesMany defensive strategies

• Offense gets the ball at 3 yard line, Offense gets the ball at 3 yard line, has one play to scorehas one play to score

• Many offensive play choicesMany offensive play choices

• Many defensive strategiesMany defensive strategies

So, let’s simplify a little bit… So, let’s simplify a little bit… So, let’s simplify a little bit… So, let’s simplify a little bit…

Simplified Model: Run or PassSimplified Model: Run or PassSimplified Model: Run or PassSimplified Model: Run or Pass

• Offense can choose to run or to passOffense can choose to run or to pass

• Defense can choose run defense or Defense can choose run defense or pass defensepass defense

• Offense can choose to run or to passOffense can choose to run or to pass

• Defense can choose run defense or Defense can choose run defense or pass defensepass defense

RunRun Pass PassDefenseDefense DefenseDefense

RunRun 30% 30% 50% 50%PassPass 80% 80% 40% 40%

Success RatesSuccess Rates

What do you expect?What do you expect?

Two Linear ProgramsTwo Linear ProgramsTwo Linear ProgramsTwo Linear Programs

OffenseOffense

MaximizeMaximize v v

Subject toSubject to v v .3x + .8y .3x + .8y

v v .5x + .4y .5x + .4y

x + y = 1x + y = 1

x x 0, y 0, y 0 0

OffenseOffense



v v .5x + .4y .5x + .4y

x + y = 1x + y = 1

x x 0, y 0, y 0 0

DefenseDefense

MinimizeMinimize d d

Subject toSubject to d d .3a + .5b .3a + .5b

d d .8a + .4b .8a + .4b

a + b = 1a + b = 1

a a 0, b 0, b 0 0

DefenseDefense



d d .8a + .4b .8a + .4b

a + b = 1a + b = 1

a a 0, b 0, b 0 0

Run (a)Run (a) Pass (b)Pass (b)DefenseDefense DefenseDefense

Run (x)Run (x) 30% 30% 50% 50%PassPass (y)(y) 80% 80% 40% 40%



d d .8a + .4b .8a + .4b

a + b = 1a + b = 1

a a 0, b 0, b 0 0



d d .8a + .4b .8a + .4b

a + b = 1a + b = 1

a a 0, b 0, b 0 0Minimize Minimize dd

Subject toSubject to d d .5 – .2a .5 – .2a

d d .4 + .4a .4 + .4a

0 0 a a 1 1

Minimize Minimize dd

Subject toSubject to d d .5 – .2a .5 – .2a

d d .4 + .4a .4 + .4a

0 0 a a 1 1• Substitute b = 1 – aSubstitute b = 1 – a• Substitute b = 1 – aSubstitute b = 1 – a

Defense’s OptimizationDefense’s OptimizationDefense’s OptimizationDefense’s Optimization

aa0 .5 10 .5 1

11

.5.5

00

dd

d d .5 – .2a .5 – .2a

d d .4 + .4a .4 + .4a

a=1/6, d= .47a=1/6, d= .47



v v .5x + .4y .5x + .4y

x + y = 1x + y = 1

x x 0, y 0, y 0 0



v v .5x + .4y .5x + .4y

x + y = 1x + y = 1

x x 0, y 0, y 0 0• Substitute y = 1 – xSubstitute y = 1 – x• Substitute y = 1 – xSubstitute y = 1 – x

Maximize Maximize vv

Subject toSubject to v v .8 – .5x .8 – .5x

v v .4 + .1x .4 + .1x

0 0 x x 1 1

Maximize Maximize vv

Subject toSubject to v v .8 – .5x .8 – .5x

v v .4 + .1x .4 + .1x

0 0 x x 1 1

Offense’s OptimizationOffense’s OptimizationOffense’s OptimizationOffense’s Optimization

xx0 .5 10 .5 1

11

.5.5

00

vv

v v .8 – .5x .8 – .5x

v v .4 + .1x .4 + .1x

x=2/3, v= .47x=2/3, v= .47

SummarySummarySummarySummary

• Start with a complex problem Start with a complex problem (making decisions against an opponent)(making decisions against an opponent)

• Start with a complex problem Start with a complex problem (making decisions against an opponent)(making decisions against an opponent)

Slides at www.isye.gatech.edu/~jsokol/2030.pptSlides at www.isye.gatech.edu/~jsokol/2030.pptSlides at www.isye.gatech.edu/~jsokol/2030.pptSlides at www.isye.gatech.edu/~jsokol/2030.ppt

• Create a simplified mathematical model Create a simplified mathematical model (Optimization/linear programming)(Optimization/linear programming)

• Create a simplified mathematical model Create a simplified mathematical model (Optimization/linear programming)(Optimization/linear programming)

• Use the model to suggest a strategyUse the model to suggest a strategy (in this case, a surprising strategy!)(in this case, a surprising strategy!)

• Use the model to suggest a strategyUse the model to suggest a strategy (in this case, a surprising strategy!)(in this case, a surprising strategy!)

making decisions against an opponent: an application of mathematical optimization isye 2030 monday,...

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