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CRP 834: Decision Analysis

CRP 834: Decision Analysis. Week Two Notes. Review. Statistical decision theory Decision Theory Framework A set of strategies A set of possible futures (state of natures) Umbrella Example. • Generalized Form of a Payoff Matrix. M k =. S i = possible strategy

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CRP 834: Decision Analysis

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  1. CRP 834: Decision Analysis Week Two Notes

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

  3. • Generalized Form of a Payoff Matrix Mk= Si = possible strategy Nj = possible future “States of Nature” (the uncontrolled occurrence of a state of nature Nj after selecting strategy Si ) Pijk = the value of payoff-type k for strategy i and state of nature j, “payoffs”. Mk = payoff matrix

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

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

  6. Analysis of the No-Experiment Alternatives Statement of the problem: 1000 urns parted in 2 categories: (800) q1 : urn contains 4 red balls + 6 black ones (200) q2 : urn contains 9 red balls + 1 black ones Three possible strategies: • A1: guess the urn is of type q1 • A2: guess the urn is of type q2 • A3: refuse to play The payoffs are as follows:

  7. Expected Monetary Value (EMV): • As the probability of q1 is 0.8, and that of q2 0.2, 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. 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. $4 q1 Refuse to play A1 q2 -$20 q1 -$5 L0: A2 $ 0.0 q2 $100 L1: L2: L0 Path (Decision Tree Branch 0)

  10. $4 Refuse to play A1 q2 -$20 -$5 L0: A2 q1 $100 R q2 q1 $4 L1: -$8.0 A1 B q2 L2: q1 A2 L3: q2 $100 L1 Path (Decision Tree Branch 1)

  11. $4 $4 $4 q1 q1 q1 A1 A1 A1 q2 q2 q2 -$20 -$20 -$20 A2 A2 A2 q1 q1 q1 -$5 -$5 -$5 q2 q2 q2 $100 $100 $100 RR L2: RB or BR -$12.0 BB L2 Path (Decision Tree Branch 2)

  12. R $4 $4 q1 q1 Replace A1 A1 -$20 -$20 q2 q2 (L3, R) A2 A2 Continue q1 q1 -$5 -$5 -$4.5 B $100 $100 q2 q2 Stop No replace R R Same as (L2, RR) Same as (L1, R) L3 B Same as (L2, RB) -$9.0 B L3 Path (Decision Tree Branch 3)

  13. $4 $4 q1 q1 A1 A1 -$20 -$20 q2 q2 A2 A2 q1 q1 -$5 -$5 $100 $100 q2 q2 R L3 -$9.0 No replace Same as (L1, B) R Same as (L2, BR) Same as (L2, BB) Stop B B R (L3, B) Continue Replace -$4.5 B L3 Path – continued

  14. Review of Probability Joint probability Bayes Formula:

  15. Review of Probability—example

  16. 0.32 0.32 ? ? R (0.4) q1? q1 (0.64) R R (0.5) q1 (0.8) q1 R q2 ? q2 (0.36) B B (0.6) 0.48 ? 0.18 ? q1 (0.96) q1 ? R R (0.9) ? 0.18 ? 0.48 B q2 (0.2) q2 B (0.5) B (0.1) B q2 (0.04 ) q2 ?) ? 0.02 ? 0.02 Probability Assignment Case 2: Case 1:

  17. $4 q1 Refuse to play A1 q2 -$20 q1 -$5 L0: A2 $ 0.0 q2 $100 L1: L2: Averaging Out-Folding Back – L0 Path

  18. $4 Refuse to play A1 q2 -$20 -$5 L0: A2 q1 $100 R q2 q1 $4 L1: -$8.0 A1 B q2 L2: q1 A2 L3: q2 $100 Averaging out-Folding Back -- L1 Path

  19. $4 $4 $4 q1 q1 q1 A1 A1 A1 q2 q2 q2 -$20 -$20 -$20 A2 A2 A2 q1 q1 q1 -$5 -$5 -$5 q2 q2 q2 $100 $100 $100 RR L2: RB or BR -$12.0 BB Averaging out-Folding Back – L2 Path

  20. R $4 $4 q1 q1 Replace A1 A1 -$20 -$20 q2 q2 (L3, R) A2 A2 Continue q1 q1 -$5 -$5 -$4.5 B $100 $100 q2 q2 Stop No replace R R Same as (L2, RR) Same as (L1, R) L3 B Same as (L2, RB) -$9.0 B Averaging out-Folding Back– L3 Path

  21. $4 $4 q1 q1 A1 A1 -$20 -$20 q2 q2 A2 A2 q1 q1 -$5 -$5 $100 $100 q2 q2 R L3 -$9.0 No replace Same as (L1, B) R Same as (L2, BR) Same as (L2, BB) Stop B B R (L3, B) Continue Replace -$4.5 B Averaging out-Folding Back– L3 Path– continued

  22. 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?

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