1 crp 834: decision analysis week two notes. 2 review statistical decision theory decision theory...
TRANSCRIPT
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
W1 W2D1 2 5D2 7.6 4.5D3 3.6 3.5D4 2.8 2D5 9.2 3D6 8.4 1.5D7 4.4 0.5D8 10 0
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) 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
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
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
10
L0 Path (Decision Tree Branch 0)
L:
L:
L:
Refuse to play
$ 0.0
A
A
$4
-$5
-$20
$100
11
L1 Path (Decision Tree Branch 1)
L:
L:
L:
L:
Refuse to play
-$8.0
R
B
A
A
$4
-$5
-$20
$100
A
A
$4
$100
12
L2 Path (Decision Tree Branch 2)
L:
-$12.0
RR
BB
-$20
-$5
A
A
$4
$100
-$20
-$5
A
A
$4
$100
-$20
-$5
A
A
$4
$100
RB or BR
13
L3 Path (Decision Tree Branch 3)
L3
-$9.0
R
B
(L3, R)Continue
Stop
Same as (L1, R)
-$20
-$5
A
A
$4
$100
Replace
R
B -$20
-$5
A
A
$4
$100
No replace
R
B
Same as (L2, RR)
Same as (L2, RB)
-$4.5
14
L3 Path – continued
L3
-$9.0
R
B
(L3, B)
Stop
Same as (L1, B)
Continue
-$4.5
No replace
R
B
Same as (L2, BR)
Same as (L2, BB)
Replace
-$20
-$5
A
A
$4
$100
R
B -$20
-$5
A
A
$4
$100
15
Review of Probability
Joint probability
Bayes Formula:
)(*)(
)(*)()(
1kj
mkk
ijiji ABPAP
ABPAPBAP
)(*)()(*)()( BAPBPABPAPBAP
k
kjkk
jkj ABPAPBAPBP )(*)()()(
16
Review of Probability—example
R 0.32 0.18 0.5
B 0.48 0.02 0.5
0.8 0.2 1
2.0)( 2 P
8.0)( 1 P
1.0)(
9.0)(
6.0)(
4.0)(
2
2
1
1
BP
RP
BP
RP
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
Averaging Out-Folding Back – L0 Path
L:
L:
L:
Refuse to play
$ 0.0
A
A
$4
-$5
-$20
$100
19
Averaging out-Folding Back -- L1 Path
L:
L:
L:
L:
Refuse to play
-$8.0
R
B
A
A
$4
-$5
-$20
$100
A
A
$4
$100
20
Averaging out-Folding Back – L2 Path
L:
-$12.0
RR
BB
-$20
-$5
A
A
$4
$100
-$20
-$5
A
A
$4
$100
-$20
-$5
A
A
$4
$100
RB or BR
21
L3
-$9.0
R
B
(L3, R)Continue
Stop
Same as (L1, R)
-$20
-$5
A
A
$4
$100
Replace
R
B -$20
-$5
A
A
$4
$100
No replace
R
B
Same as (L2, RR)
Same as (L2, RB)
-$4.5
Averaging out-Folding Back– L3 Path
22
L3
-$9.0
R
B
(L3, B)
Stop
Same as (L1, B)
Continue
-$4.5
No replace
R
B
Same as (L2, BR)
Same as (L2, BB)
Replace
-$20
-$5
A
A
$4
$100
R
B -$20
-$5
A
A
$4
$100
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?