1. Chapter 16 Notes: Decision Making under Uncertainty Introduction Laplace Rule Maximin Rule Maximax Rule Hurwicz Rule Minimax Regret Rule Examples Homework: 2, 4, 8

2. Introduction 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

3. Laplace Method (Rule) Given no information on probabilities of future states of nature, assume they are all equally likely –Prob (state i ), 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

4. e.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 )=10 E(med) = 1 / 3 (2 + 12 + 15)=9 2 / 3 E(large) = 1 / 3 (-1)+ 1 / 3 (13)+ 1 / 3 (25)=12 1 / 3  Still 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

5. Maximin 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 ] ij

6. Maximin--Continued Basically, take minimum payoff across all states of nature for each alternative Choose alternative with the largest (maximum) minimum payoff Very Conservative

7. e.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

8. i j Maximax 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]

9. Maximax--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!

10. e.g.: Maximax Rule Ice 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

11. Hurwicz 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

12. Hurwicz--Continued Basically, take  times row max plus  times row min; Choose maximum of the sum. Allows middle ground for two extreme strategies

13. e.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  =.5  =.5 Choose Large

14. More Pessimistic 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  =.25,  =.75  Note: Although blending good with the bad, Moderate weather still has no bearing (cold) (hot)

15. Minimax 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

16. e.g. Minimal Regret Rule –Ice Cream Vendor CMHCMH Sm 9 10 11 M 2 12 15 L -1 13 25 Max Reg payoff matrix Sm 0 3 14 M 7 1 10 L 10 0 0 (9-9) (13-10) (25-14) (9-2) (13-12) (25-15) (9--1) (13-13) 25-25 14 10  regret matrix Minimize greatest regret with medium or large order Small order is safe, but full of regret

17. Ch 16: Problems L1 L2 L3 L4 L5 M1 (R) 10 N 20 30 40 50 X M2 (R) 20 N 25 25 30 35 X M3 (R) 50 X 40 5 N 15 20 M4 (R) 40 X 35 30 25 N 25 M5 (R) 10 N 20 25 30 X 20 40 X 20 000 30 X 15 5 10 15 0 0 25 25 30 X 0 0 25 25 30 X 10 5 0 15 25 X 40 X 20 5 10 30 M12345 (10, 20, 5, 25, 10)A) Maximin  M4 B) Maximax  M1 or M3 (50, 35, 50, 40, 30) “X” “N” C) Hurwicz  M4 D) 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

18.  6 6 6 6 50 40 30 25 20 15 10 5 M4 M1 M3 M2 M5 M3 M1 0.1.4.5.9 1.0 M4 is chosen 0   .6 M4 is chosen.6    1 M5 can be tossed for any (?) decision rule--  

19.  2 2 2 2 S1S2S3S4R MaxMinLap 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 2 2 2 22 2 2 2 22 2 6 0 06 2 6 0 06 4 0 2 24 4 0 2 24 0 4 2 0 4 0 4 2 0 4      1) Laplace: A4 2) Minimax: A2 3) Maximax: A1 4) 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 A4 5) Minimax Regret: A1

20.  4 4 4 4 L1L2L3L4min max max I1151112 9 9 15 I2 7 91220 7 20 I3 8 81417 8 17 I417 5 5 5 5 17 I5 614 819 6 19 1) Maximin: I1 2) Maximax: I2 3) 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=14L3=14 L4=20Reg Max Regret Table I123 2 11 11 I2 105 2 0 10 I396 0 3 9 I409 9 15 15 I5 110 6 1 11 I3 wins Minimax Regret

21.  8 8 8 8 Cost Matrix ABCMaxMinLaplace Mini E1100 90 60 100 60 250 E2 70 80 90 9070 240 E3 30 30 140 14030 200 E4 100 20 120 120 20 240 a) Minimax  Maximin: E2 Cost Revenue b) Minimin  Maximax: E4 Cost Revenue c) Hurwicz;  =.03 (Almost completely pessimistic): E2 d) Minimax Regret: E2 A=300 B=20C=160 Reg Max E170 70 10 70 E240 6030 60 E310 1080 80 E470 1060 70 e) Laplace: E3 (Min Avg./Total Cost Regret Based on Min. Cost!!