1. 1 CRP 834: Decision Analysis Week Two Notes

2. 2 Review Statistical decision theory Decision Theory Framework –A set of strategies –A set of possible futures (state of natures) –Umbrella Example

3. 3 Generalized Form of a Payoff Matrix N1N1 N2N2 ………..NjNj NmNm S1S1 P 11k P 12k ………..P 1jk ………..P 1mk S2S2 P 21k P 22k ………..P 2jk ………..P 2mk :::………..: : ::: : : SiSi P ilk P i2k ………..P ijk ………..P imk :::………..: : ::: : : SnSn P nlk P n2k ………..P njk ………..P nmk Mk=Mk= S i = possible strategy N j = possible future “States of Nature” (the uncontrolled occurrence of a state of nature Nj after selecting strategy Si ) P ijk = the value of payoff-type k for strategy i and state of nature j, “payoffs”. M k = payoff matrix

4. 4 More Decision Rules –The Maximin criterion –The Maximax criterion –The Hurwicz criterion –The Bayes (Laplace) Criterion –The Minimax regret criterion –Mixed strategy

5. 5 Experimentation and Sequential Decision Analysis Analysis of no-experiment alternatives Decision-flow diagram (Decision Tree)

6. 6 Analysis of the No-Experiment Alternatives Statement of the problem: 1000 urns parted in 2 categories: (800)  1 : urn contains 4 red balls + 6 black ones (200)  2 : urn contains 9 red balls + 1 black ones Three possible strategies: A1: guess the urn is of type  1 A2: guess the urn is of type  2 A3: refuse to play The payoffs are as follows:

7. 7 Expected Monetary Value (EMV): As the probability of  1 is 0.8, and that of .2, we have the payoff for: A1 : 0.8 ($40.0) + 0.2 (-$20)= $28 A2 : 0.8 (-$5.0) + 0.2 ($100)= $16 A3 : 0.8 ( $0.0) + 0.2 (-$0.0)= $0

8. 8 Decision-Flow Diagram Allow the following experimental options before making the decision: –no observation at cost $ 0.00 –L1: a single observation at cost $ 8.00 (you can draw a single ball at random from the unidentified urn on the table) –L2: a pair of observation at cost $12.00 –L3: a single observation at cost $ 9.00 with the privilege of another observation at $ 4.50.

9. 9

10. 10 L 0 Path (Decision Tree Branch 0) L:L: L:L: L:L: Refuse to play $ 0.0 AA AA    $4 -$5 -$20 $100

11. 11 L 1 Path (Decision Tree Branch 1) L:L: L:L: L:L: L:L: Refuse to play -$8.0 R B  AA AA   $4 -$5 -$20 $100  AA AA   $4 $100

12. 12 L 2 Path (Decision Tree Branch 2) L:L: -$12.0 RR BB -$20 -$5  AA AA   $4 $100  -$20 -$5  AA AA   $4 $100  -$20 -$5  AA AA   $4 $100  RB or BR

13. 13 L 3 Path (Decision Tree Branch 3) L3L3 -$9.0 R B (L 3, R) Continue Stop Same as (L 1, R) -$20 -$5  AA AA   $4 $100  Replace R B -$20 -$5  AA AA   $4 $100  No replace R B Same as (L 2, RR) Same as (L 2, RB) -$4.5

14. 14 L 3 Path – continued L3L3 -$9.0 R B (L 3, B) Stop Same as (L 1, B) Continue -$4.5 No replace R B Same as (L 2, BR) Same as (L 2, BB) Replace -$20 -$5  AA AA   $4 $100  R B -$20 -$5  AA AA   $4 $100 

15. 15 Review of Probability Joint probability Bayes Formula:

16. 16 Review of Probability—example  R0.320.180.5 B0.480.020.5 0.80.21

17. 17 Probability Assignment Case 1: ? ? B   B R ? ? R Case 2: ? ?    R B     ? ?  0.18 0.48    R (0.5) B (0.5)     0.32 0.02  0.48 0.18 B    (0.8)   (0.2) B  R  0.32 0.02 R 

18. 18 Averaging Out-Folding Back – L 0 Path L:L: L:L: L:L: Refuse to play $ 0.0 AA AA    $4 -$5 -$20 $100

19. 19 Averaging out-Folding Back -- L 1 Path L:L: L:L: L:L: L:L: Refuse to play -$8.0 R B  AA AA   $4 -$5 -$20 $100  AA AA   $4 $100

20. 20 Averaging out-Folding Back – L 2 Path L:L: -$12.0 RR BB -$20 -$5  AA AA   $4 $100  -$20 -$5  AA AA   $4 $100  -$20 -$5  AA AA   $4 $100  RB or BR

21. 21 L3L3 -$9.0 R B (L 3, R) Continue Stop Same as (L 1, R) -$20 -$5  AA AA   $4 $100  Replace R B -$20 -$5  AA AA   $4 $100  No replace R B Same as (L 2, RR) Same as (L 2, RB) -$4.5 Averaging out-Folding Back– L3 Path

22. 22 L3L3 -$9.0 R B (L 3, B) Stop Same as (L 1, B) Continue -$4.5 No replace R B Same as (L 2, BR) Same as (L 2, BB) Replace -$20 -$5  AA AA   $4 $100  R B -$20 -$5  AA AA   $4 $100  Averaging out-Folding Back– L3 Path – continued

23. 23 What is your decision: –Make experiment or not Make experiment? –If make experiment, which option? –Having decided to take an experiment option, what action you will take according to the experiment result? –What is the benefit of information?