Chapter 23
23.1
a1
a2
s1
0
29
s2
0
5
s3
14
0
s4
36
0
23.3 EMV(a 1 ) = .4(55) + .1(43) + .3(29) + .2(15) = 38.0
EMV(a 2 ) = .4(26) + .1(38) + .3(43) + .2(51) = 37.3
23.5
a1
a2
a3
s1
0
15
21
s2
0
3
4
s3
20
5
0
23.7a
Produce
Demand
a0
a1
a2
a3
s0
0
-3.00
-6.00
-9.00
s1
0
5.00
2.00
-1.00
s2
0
5.00
10.00
7.00
s3
0
5.00
10.00
15.00
b
Produce
Demand
a0
a1
a2
a3
s0
0
3.00
6.00
9.00
s1
5.00
0
3.00
6.00
s2
10.00
5.00
0
3.00
s3
15.00
10.00
5.00
0
529
c
23.9
a 1 (flat fee)
a 2 Pay per snowfall
s0
-40,000
0
s1
-40,000
-18,000
s2
-40,000
-36,000
s3
-40,000
-54,000
s4
-40,000
-72,000
23.11a Payoff Table
s 100
s 150
s 200
s 250
a 100
a 200
a 300
12(100)-10(100)
12(100)-9(200)+6(100)
12(100)-8.50(300)+6(200)
= 200
=0
= -150
12(100)- 10(100)
12(150)-9(200)+6(50)
12(150)-8.50(300)+6(150)
= 200
= 300
= 150
12(100)-10(100)
12(200)-9(200)
12(200)-8.50(300)+6(100)
= 200
= 600
= 450
12(100)-10(100)
12(200)–9(200)
12(250)-8.50(300)+6(50)
530
= 200
b
= 600
= 750
Opportunity Loss Table
a 100
a 200
a 300
s 100
0
200
350
s 150
100
0
150
s 200
400
0
150
s 250
550
150
0
c
23.13 P(s 0 ) = .607, P(s 1 ) = .303, P(s 2 ) = .076, P(s 3 ) = .012, P(s 4 ) = .002
Payoff Table
a0
a1
a2
a3
s0
0
-6,000
-12,000
-18,000
s1
0
7,000
1,000
-5,000
s2
0
7,000
14,000
8,000
s3
0
7,000
14,000
21,000
Opportunity Loss Table
531
a0
a1
a2
a3
s0
0
6,000
12,000
18,000
s1
7,000
0
6,000
12,000
s2
14,000
7,000
0
6,000
s3
21,000
14,000
7,000
0
23.14a EMV(Small) = .15(-220) + .55(-330) + .30(-440) = -346.5
EMV(Medium) = .15(-300) + .55(-320) + .30(-390) = -338.0
EMV(Large) = .15(-350) + .55(-350) + .30(-350) =-350.0
EMV decision: build a medium size plant; EMV*= -338.0
b
Opportunity Loss Table
Small
Medium
Large
Low
0
80
130
Moderate
10
0
30
High
90
40
0
c EOL(Small) = .15(0) + .55(10) + .30(90) = 32.5
EOL(Medium) = .15(80) + .55(0) + .30(40) = 24.0
EOL(Large) = .15(130) + .55(30) + .30(0) = 36.0
EOL decision: build a medium size plant
23.15a P(s 10 ) = 9/90 = .10, P(s 11 ) = 18/90 = .20, P(s 12 ) = 36/90 = .40, P(s 13 ) = 27/90 = .30
Payoff Table
s 10
s 11
s 12
s 13
a 10
a 11
a 12
a 13
30
10(5)- 11(2)+2
10(5)-12(2)+3.50
10(5)-13(2)+4.50
= 30
= 29.50
= 28.50
11(5)-11(2)
11(5)-12(2)+2
11(5)-13(2)+3.50
= 33
= 33
= 32.50
11(5)-11(2)
12(5)-12(2)
12(5)- 13(2)+2
= 33
= 36
= 36
11(5)-11(2)
12(5)-12(2)
13(5)-13(2)
= 33
= 36
= 39
30
30
30
b EMV(a 10 ) = 30
EMV(a 11 ) = .10(30) + .20(33) + .40(33) + .30(33) = 32.70
532
EMV(a 12 ) = .10(29.50) + .20(33) + .40(36) + .30(36) = 34.75
EMV(a 13 ) = .10(28.50) + .20(32.50) + .40(36) + .30(39) = 35.45
EMV decision: buy 13 bushels
23.17 EPPI = .10(110) + .25(150) + .50(220) + .15(250) = 196
EMV(a 1 ) = .10(60) + .25(40) + .50(220) + .15(250) = 163.5
EMV(a 2 ) = .10(110) + .25(110) + .50(120) + .15(120) = 116.5
EMV(a 3 ) = .10(75) + .25(150) + .50(85) + .15(130) = 107
EVPI = EPPI – EMV* = 196 – 163.5 = 32.5
23.19 EPPI = .5(65) + .5(110) = 87.5
EMV(a 1 ) = .5(65) + .5(70) = 67.5
EMV(a 2 ) = .5(20) + .5(110) = 65.0
EMV(a 3 ) = .5(45) + .5(80) = 62.5
EMV(a 4 ) = .5(30) + .5(95) = 62.5
EVPI = EPPI – EMV* = 87.5 – 67.5 = 20
23.21 As the difference between the two prior probabilities increases EVPI decreases.
23.23 Posterior Probabilities for I 1
s
j
P(s j )
P(I 1 |s j )
P(s j and I 1 )
P(s j | I 1 )
__________________________________________________________________________
s1
.5
.98
(.5)(.98) = .49
.49/.515 = .951
s2
.5
.05
(.5)(.05) = .025
.025/.515 = .049
P(I 1 ) = .515
Posterior Probabilities for I 2
s
j
P(s j )
P(I 2 |s j )
P(s j and I 2 )
P(s j | I 2 )
__________________________________________________________________________
s1
.5
.02
(.5)(.02) = .01
.01/.485 = .021
s2
.5
.95
(.5)(.95) = .475
P(I 2 ) = .485
533
.475/.485 = .979
23.25 Prior probabilities: EMV(a 1 ) = .333(60) + .333(90) + .333(150) = 100
EMV(a 2 ) = 90
EMV* = 100
Posterior Probabilities for I 1
s
j
P(s j )
P(I 1 |s j )
P(s j and I 1 )
P(s j | I 1 )
__________________________________________________________________________
s1
.333
.7
(.333)(.7) = .233
.233/.467 = .499
s2
.333
.5
(.333)(.5) = .167
.167/.467 = .358
s3
.333
.2
(.333)(.2) = .067
.067/.467 = .143
P(I 1 ) = .467
Posterior Probabilities for I 2
s
j
P(s j )
P(I 2 |s j )
P(s j and I 2 )
P(s j | I 2 )
__________________________________________________________________________
s1
.333
.3
(.333)(.3) = .100
.100/.534 = .187
s2
.333
.5
(.333)(.5) = .167
.167/.534 = .313
s3
.333
.8
(.333)(.8) = .267
.267/.534 = .500
P(I 2 ) = .534
I 1 : EMV(a 1 ) = .499(60) + .358(90) + .143(150) = 83.61
EMV(a 2 ) = 90
I 2 : EMV(a 1 ) = .187(60) + .313(90) + .500(150) = 114.39
EMV(a 2 ) = 90
EMV` = .467(90) + .534(114.39) = 103.11
EVSI = EMV` - EMV* = 103.11 – 100 = 3.11
23.27 Prior probabilities: EMV(a 1 ) = .90(60) + .05(90) + .05(150) = 66
EMV(a 2 ) = 90
EMV* = 90
Posterior Probabilities for I 1
s
j
P(s j )
P(I 1 |s j )
P(s j and I 1 )
P(s j | I 1 )
__________________________________________________________________________
s1
.90
.7
(.90)(.7) = .63
.63/.665 = .947
s2
.05
.5
(.05)(.5) = .025
.025/.665 = .038
s3
.05
.2
(.05)(.2) = .01
.01/.665 = .015
P(I 1 ) = .665
534
Posterior Probabilities for I 2
s
j
P(s j )
P(I 2 |s j )
P(s j and I 2 )
P(s j | I 2 )
__________________________________________________________________________
s1
.90
.3
(.90)(.3) = .27
.27/.335 = .806
s2
.05
.5
(.05)(.5) = .025
.025/.335 = .075
s3
.05
.8
(.05)(.8) = .04
.04/.335 = .119
P(I 1 ) = .335
I 1 : EMV(a 1 ) = .947(60) + .038(90) + .015(150) = 62.49
EMV(a 2 ) = 90
I 2 : EMV(a 1 ) = .806(60) + .075(90) + .119(150) = 72.96
EMV(a 2 ) = 90
EMV` = .665(90) + .335(90) = 90
EVSI = EMV` - EMV* = 90 – 90 = 0
23.29
Payoff Table
Demand
Purchase lot
Don’t purchase lot
10,000
10,000(5)-125,000 = -75,000
0
30,000
30,000(5) – 125,000 = 25,000
0
50,000
50,000(5)-125,000 = 125,000
0
EMV(purchase) = .2(-75,000) + .5(25,000) + .3(125,000) = 35,000
EMV(don’t purchase) = 0
EPPI = .2(0) + .5(25,000) + .3(125,000) = 50,000
EVPI = EPPI – EMV* = 50,000 – 35,000 = 15,000
23.31 Likelihood probabilities (binomial probabilities)
P(I | s 1 ) = P(x = 3, n= 25 | p = .05) = .0930
P(I | s 2 ) = P(x = 3, n= 25 | p = .10) = .2265
P(I | s 3 ) = P(x = 3, n= 25 | p = .15) = .2174
535
Posterior Probabilities
sj
P(s j )
P(I | s j )
P(s j and I)
P(s j | I)
__________________________________________________________________________
s1
.15
.0930
(.15)(.0930) = .0140
.0140/.2029 = .0690
s2
.45
.2265
(.45)(.2265) = .1019
.1019/.2029 = .5022
s3
.40
.2174
(.40)(.2174) = .0870
P(I) = .2029
.0870/.2029 = .4288
EMV(produce) = .0690(-28 million) + .5022(2 million) + .4288(8 million) = 2.503 million
EMV (don’t produce) = 0
EMV decision: produce
23.33
Payoff Table
Don’t proceed
Participating Households
Proceed
50,000
50(500) – 55,000 = -30,000
0
100,000
100(500) – 55,000 = -5,000
0
200,000
200(500) – 55,000 = 45,000
0
300,000
300(500) – 55,000 = 95,000
0
Likelihood probabilities (binomial probabilities)
P(I | s 1 ) = P(x = 3, n= 25 | p = .05) = .0930
P(I | s 2 ) = P(x = 3, n= 25 | p = .10) = .2265
P(I | s 3 ) = P(x = 3, n= 25 | p = .20) = .1358
P(I | s 4 ) = P(x = 3, n= 25 | p = .30) = .0243
Posterior Probabilities
sj
P(s j )
P(I | s j )
P(s j and I)
P(s j | I)
__________________________________________________________________________
s1
.5
.0930
(.5)(.0930) = .0465
.0465/.1305 = .3563
s2
.3
.2265
(.3)(.2265) = .0680
.0680/.1305 = .5211
s3
.1
.1358
(.1)(.1358) = .0136
.0136/.1305 = .1042
s4
.1
.0243
(.1)(.0243) = .0024
P(I) = .1305
.0024/.1305 = .0184
EMV(proceed) = .3563(-30,000) + .5211(-5,000) + .1042(45,000) + .0184(95,000) = -6,858
EMV (don’t proceed = 0
EMV decision: don’t proceed
536
23.35 Posterior Probabilities for I 1
s
j
P(s j )
P(I 1 |s j )
P(s j and I 1 )
P(s j | I 1 )
__________________________________________________________________________
s1
.15
.5
(.15)(.5) = .075
.075/.30 = .25
s2
.55
.3
(.55)(.3) = .165
.165/.30 = .55
s3
.30
.2
(.30)(.2) = .06
.06/.30 = .20
P(I 1 ) = .30
Posterior Probabilities for I 2
s
j
P(s j )
P(I 2 |s j )
P(s j and I 2 )
P(s j | I 2 )
__________________________________________________________________________
s1
.15
.3
(.15)(.3) = .045
.045/.435 = .103
s2
.55
.6
(.55)(.6) = .33
.33/.435 = .759
s3
.30
.2
(.30)(.2) = .06
.06/.435 = .138
P(I 2 ) = .435
Posterior Probabilities for I 3
s
j
P(s j )
P(I 3 |s j )
P(s j and I 3 )
P(s j | I 3 )
__________________________________________________________________________
s1
.15
.2
(.15)(.2) = .03
.03/.265 = .113
s2
.55
.1
(.55)(.1) = .055
.055/.265 = .208
s3
.30
.6
(.30)(.6) = .18
.18/.265 = .679
P(I 3 ) = .265
I 1 : EMV(a 1 ) = .25(-220) + .55(-330) + .20(-440) = -324.5
EMV(a 2 ) = .25(-300) + .55(-320) + .20(-390) = -329.0
EMV(a 3 ) = .251(-350) + .55(-350) + .20(-350) = -350
Optimal act: a 1
I 2 : EMV(a 1 ) = .103(-220) + .759(-330) + .138(-440) = -333.85
EMV(a 2 ) = .103(-300) + .759(-320) + .138(-390) = -327.59
EMV(a 3 ) = .103(-350) + .759(-350) + .138(-350) = -350
Optimal act: a 2
I 3 : EMV(a 1 ) = .113(-220) + .208(-330) + .679(-440) = -392.26
EMV(a 2 ) = .113(-300) + .208(-320) + .679(-390) = -365.28
EMV(a 3 ) = .113(-350) + .208(-350) + .679(-350) = -350
Optimal act: a 3
537
EMV` = .30(-324.5) + .435(-327.59) + .265(-350) = -332.60
EVSI = EMV` - EMV* = -332.60 – (-338) = 5.40
23.37
Likelihood probabilities (binomial probabilities)
P(I | s 1 ) = P(x = 2, n= 25 | p = .05) = .2305
P(I | s 2 ) = P(x = 2, n= 25 | p = .10) = .2659
P(I | s 3 ) = P(x = 2, n= 25 | p = .20) = . 0708
Posterior Probabilities for I
sj
P(s j )
P(I|s j )
P(s j and I)
P(s j | I)
__________________________________________________________________________
s1
.4
.2305
(.4)(.2305) = .0922
.0922/.2127 = .4334
s2
.4
.2659
(.4)(.2659) = .1064
.1064/.2127 = .5000
s3
.2
.0708
(.2)(.0708) = .0142
.0142/.2127 = .0667
P(I) = .2127
EMV(switch) = .4334(-200,000) + .5000(300,000) + .0667(1,300,000) = 149,873
EMV(don’t switch) = 285,000
Optimal act: don’t switch
23.39
Payoff Table
Percentage change
Change ad
Don’t change
-2
-258,000
0
-1
-158,000
0
0
-58,000
0
1
42,000
0
2
142,000
0
EMV(Change ad) = -1(-258,000) + .1(-158,000) + .2(-58,000) + .3(42,000) + .3(142,000) = 2,000
EMV (don’t change) = 0.
Optimal decision: change ad
538
23.41
Likelihood probabilities (binomial probabilities)
P(I | s 1 ) = P(x = 1, n = 5 | p = .30) = .3602
P(I | s 2 ) = P(x = 1, n = 5 | p = .31) = .3513
P(I | s 3 ) = P(x = 1, n = 2 | p = .32) = . 3421
P(I | s 4 ) = P(x = 1, n = 2 | p = .33) = . 3325
P(I | s 5 ) = P(x = 1, n = 2 | p = .34) = . 3226
Posterior Probabilities for I
sj
P(s j )
P(I|s j )
P(s j and I)
P(s j | I)
__________________________________________________________________________
s1
.1
.3602
(.1)(.3602) = .0360
.0360/.3361 = .1072
s2
.1
.3513
(.1)(.3513) = .0351
.0351/.3361 = .1045
s3
.2
.3421
(.2)(.3421) = .0684
.0684/.3361 = .2036
s4
.3
.3325
(.3)(.3325) = .0997
.0997/.3361 = .2968
s5
.3
.3226
(.3)(.3226) = .0968
P(I) = .3361
.0968/.3361 = .2879
EMV(Change ad) = .1072(-258,000) + .1045(-158,000) + .2036(-58,000) + .2968(42,000)
+ .2879(142,000) = -2,620
EMV (don’t change) = 0.
Optimal decision: don’t change ad
23.43a EMV(Model 101) = .2(20 million) + .4(100 million) + .4(210 million) = 128 million
EMV (Model 202) = .1(70 million) + .4(100 million) + .5(150 million) = 122 million
Optimal decision: Model 101
b Likelihood probabilities (binomial distribution) for Model 101
P(X =1, n = 10| p = .05) = .3151
P(X =1, n = 10| p = .10) = .3874
P(X =1, n = 10| p = .15) = .3474
Posterior Probabilities for Model 101
sj
P(s j )
P(I|s j )
P(s j and I)
P(s j | I)
__________________________________________________________________________
s1
.2
.3151
(.2)(.3151) = .0630
.0630/.3570 = .1766
s2
.4
.3874
(.4)(.3874) = .1550
.1550/.3570 = .4341
s3
.4
.3474
(.4)(.3474) = .1390
.1390/.3570 = .3893
P(I) = .3570
539
EMV(Model 101) = .1766(20 million) + .4341(100 million) + .3893(210 million) = 128.7 million
Likelihood probabilities (binomial distribution) for Model 202
P(X =9, n = 20| p = .30) = .0654
P(X =9, n = 20| p = .40) = .1597
P(X =9, n = 20| p = .50) = .1602
Posterior Probabilities for Model 202
sj
P(s j )
P(I|s j )
P(s j and I)
P(s j | I)
__________________________________________________________________________
s1
.1
.0654
(.1)(.0654) = .0065
.0065/.1505 = .0434
s2
.4
.1597
(.4)(.1597) = .0639
.0639/.1505 = .4245
s3
.5
.1602
(.5)(.1602) = .0801
.0801/.1505 = .5321
P(I) = .1505
EMV(Model 101) = .0434(70 million) + .4245(100 million) + .5321(150 million) = 125.3 million
Optimal decision: Model 101
540