Chapter 16 Notes: Chapter 16 Notes: Decision Making under Decision Making under UncertaintyUncertainty• Introduction• Laplace Rule• Maximin Rule• Maximax Rule• Hurwicz Rule• Minimax Regret Rule• Examples

Homework: 2, 4, 8

IntroductionIntroduction• Uncertainty- One step beyond risk on

the unknown scale-Now, we not only don’t know which state of nature may arise, we don’t even know likelihoods, probabilities of each state

• This chapter explores various strategies for dealing with uncertainty

Laplace Method (Rule)Laplace Method (Rule)• Given no information on

probabilities of future states of nature, assume they are all equally likely– Prob (statei), i=1...n=1/n

• Basically, assign probabilities when you aren’t given them– Use expected value methodology of

Ch. 15• Since all states of nature are

equally weighted, we can take a straight average or sum of payoffs

e.g.: Laplace Rulee.g.: Laplace Rule• Ice Cream Vendor--

– Ignoring Stated Possibilities• Assume H,M,C are equally likely: 1/3 chance of each.

E(small) = 9(1/3) + 10(1/3) + 11(1/3)=10E(med) = 1/3(2 + 12 + 15)=92/3E(large) = 1/3(-1)+ 1/3(13)+ 1/3(25)=121/3Still choose large, even though hot is less likely

• Comparing in total payout for each strategy: s=30,n=29, c=37, yields same results since all are equally weighted

Maximin RuleMaximin Rule• Pessimistic Approach• Maximize the Minimum Payoff• Assures the best outcome in the

worst-case scenario• i alternatives; j states of nature, P-

payoff• MAX [ MIN Pij ]i j

Maximin--ContinuedMaximin--Continued• Basically, take minimum payoff

across all states of nature for each alternative

• Choose alternative with the largest (maximum) minimum payoff

• Very Conservative

e.g.: Maximin Rulee.g.: Maximin Rule• Ice Cream Vendor

Min(small) = 9 (9,10,11)Min(med) = 2 (2,12,15)Min(large) = -1 (-1,13,25)

• Max of the mins is small order• Order based entirely on the worst

weather-cold-even though it is least likely

• Pessimistic strategy limits your potential losses, but usually limits potential gains

• Worst case planning rarely takes advantage of best case possibilities

C M H

i j

Maximax RuleMaximax Rule• Optimistic Approach• Maximize the maximum payoff• Assures the best outcome in the

best-case scenario• i-alternatives; j-states of nature; p-

payoff• Max [Max pij]

Maximax--ContinuedMaximax--Continued• Basically, take the maximum

payoff across all states of nature for each alternative

• Choose the alternative with the largest (max) maximum payoff

• Very Aggressive!

e.g.: Maximax Rulee.g.: Maximax RuleIce Cream Vendor

Max(small) = 11 (9,10,11)Max(med) = 15 (2,12,15)Max(large) = 25 (-1,13,25)

• Max of Maxes is large order• Order based entirely on best-case

scenario-hot• Optimizes strategy maximizes

potential return, but exposes to severe loss

• Best Case planning opens up possibility of worst-case catastrophe

Hurwicz RuleHurwicz Rule• Mix of optimism & pessimism• Quantify the extent of optimism &

pessimism explicitly = degree of optimism (1)

= Pessimistic (Maximin) = 1 Optimistic (Maximax)

= degree of pessimism ( =1-)

• MAX [[MAX pij] + [1-][MIN pij]]

maximax minimax

Hurwicz--ContinuedHurwicz--Continued• Basically, take times row max

plus times row min; Choose maximum of the sum.

• Allows middle ground for two extreme strategies

e.g. Hurwicz Rulee.g. Hurwicz Rule

• Ice Cream Vendor– Small: .5(11) + .5(9) = 10– Med: .5(15) + .5(2) = 8– Large: .5(25) + .5(-1) = 12

let let = .5 = .5

= .5= .5

Choose Large

More Pessimistic More Pessimistic Examples:Examples:

– Small= .25(11) + .75(9) = 8.75– Med= .25(15) + .75(2) = 5.25– Large= .25(25) + .75(-1) = 3.5

• All payoffs are lower (more pessimism.), but small order now is best

If If = .25, = .25,

= .75= .75

Note: Although blending good with the bad, Moderate weather still has no bearing(cold)

(hot)

Minimax Regret RuleMinimax Regret Rule• “Second-Guesser’s Rule”• How much better could we have done,

given the state of nature that arises?• Regret= Maximum payoff for a given

state of nature minus payoff of alternative chosen (How much better you could have done)

• Could be interpreted as opportunity cost of alternative--foregone benefit of better option

• Put yourself in a position of no matter what happens (state of nature), makes little difference

e.g. Minimal Regret Rulee.g. Minimal Regret Rule– Ice Cream Vendor

C M HSm 9 10 11M 2 12 15L -1 13 25

Max Reg

payoff payoff

matrixmatrix

Sm 0 3 14M 7 1 10L 10 0 0

(9-9) (13-10) (25-14)

(9-2) (13-12) (25-15)

(9--1) (13-13) 25-25

1410

10

regret regret

matrixmatrix

•Minimize greatest regret with medium or large order•Small order is safe, but full of regret

Ch 16: ProblemsCh 16: ProblemsL1 L2 L3 L4 L5

M1(R) 10N 20 30 40 50X

M2(R) 20N 25 25 30 35X

M3(R) 50X 40 5N 15 20

M4(R) 40X 35 30 25N 25

M5(R) 10N 20 25 30X 20

4040XX 20 20 0 0 00 00

3030XX 1515 5 5 10 10 15 15

00 0 0 25 25 2525 3030XX

1010 5 5 0 0 15 15 25 25XX

4040XX 20 20 5 5 10 10 30 30

• •

•

• •

•

M1 2 3 4 5(10, 20, 5, 25, 10)A) Maximin M4

B) Maximax M1 or M3(50, 35, 50, 40, 30)

“X”“N”

C) Hurwicz M4D) Minimax Regret

M4(40, 30, 30, 25, 40)“•”

.4(50)+.6(10)=26M1

.4(35)+.6(20)=26M2

.4(50)+.6(25)=23M3

.4(40)+.6(25)=31M4

.4(30)+.6(10)=18M5

1 2 3 4 5

66504030 25 20 15 10 5

M4M1 M3

M2

M5

M3M3M1

0 .1 .4 .5 .9 1.0

•M4 is chosen 0.6•M4 is chosen .61•M5 can be tossed for any (?) decision rule--

22S1 S2 S3 S4 R Max Min Lap

A1(R) A2(R)

A3(R)

A4(R)

2 6 0 0 6 0 8 2 2 2 2 2 2 8 0 8 0 0 8 0 8 4 4 0 2 4 0 10

22 2 2 2 2 2 2 2 •2 •

22 6 6 0 0 0 0 66

44 0 0 2 2 2 2 44

00 4 4 2 2 0 4 0 4

1) Laplace: A42) Minimax: A23) Maximax: A14) Hurcwicz: A3 .6(6)+.4(0)=3.6 A1 .6(2)+.4(2)=2.0 A2

.6(8)+.4(0)=4.8 A3

.6(4)+.4(0)=2.4 A45) Minimax Regret: A1

44L1 L2 L3 L4 min max

max maxI1 15 11 12 9 9 15I2 7 9 12 20 7 20I3 8 8 14 17 8 17I4 17 5 5 5 5 17I5 6 14 8 19 6 191) Maximin: I1

2) Maximax: I23) Hurwicz: =.7

I1) .7(9)+.3(15)= 10.8 I2) .7(7)+.3(20)= 10.9 I3) .7(8)+.3(17)= 10.7 I4) .7(5)+.3(17)= 8.6 I5) .7(6)+.3(19)= 9.9

4) Minimax Regret:LI=17 L2=14 L3=14 L4=20 Reg

Max

Regret Table

I1 2 3 2 11 11I2 10 5 2 0 10I3 9 6 0 3 9I4 0 9 9 15 15I5 11 0 6 1 11

I3 wins Minimax Regret

88Cost Matrix

A B C Max Min LaplaceMini Mini

E1 100 90 60 100 60 250E2 70 80 90 90 70 240E3 30 30 140 140 30 200E4 100 20 120 120 20 240

a) Minimax Maximin: E2 Cost Revenue

b) Minimin Maximax: E4 Cost Revenue

c) Hurwicz; = .03 (Almost completely pessimistic): E2d) Minimax Regret: E2A=300 B=20 C=160 Reg

MaxE1 70 70 10 70E2 40 60 30 60E3 10 10 80 80E4 70 10 60 70

e) Laplace: E3 (Min Avg./Total Cost

Regret

Based

on Min.

Cost!!