CRP 834: Decision Analysis

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
N1 N2 .. Nj .. Nm
S1 P11k P12k .. P1jk .. P1mk
S2 P21k P22k .. P2jk .. P2mk
.. ..
.. ..
Si Pilk Pi2k .. Pijk .. Pimk
.. ..
.. ..
Sn Pnlk Pn2k .. Pnjk .. Pnmk
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
(No Transcript)
10
L0 Path (Decision Tree Branch 0)
11
L1 Path (Decision Tree Branch 1)
12
L2 Path (Decision Tree Branch 2)
13
L3 Path (Decision Tree Branch 3)
14
L3 Path continued
15
Review of Probability

• Joint probability
• Bayes Formula

16
Review of Probabilityexample
  q1 q2  
R 0.32 0.18 0.5
B 0.48 0.02 0.5
  0.8 0.2 1
17
Probability Assignment
Case 2

• Case 1

18
Averaging Out-Folding Back L0 Path
19
Averaging out-Folding Back -- L1 Path
20
Averaging out-Folding Back L2 Path
21
Averaging out-Folding Back L3 Path
22
Averaging out-Folding Back L3 Path continued
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?