Numbered Figures forProspect Theoryfor Risk and · Web viewNumbered Figures forProspect Theoryfor Risk and Ambiguity by Peter P. Wakker (2010); provided on internet July 2013 (with

Numbered Figures forProspect Theory

for Risk and Ambiguity

by Peter P. Wakker (2010);

provided on internet July 2013 (with permission of CUP)

The figures were made using 2009 software, mainly the drawing

facilities of MS-Word. They were updated March 2017 to a new

version of MS Word. If no elucidation is added to a figure,

then it was made using only facilities of MS Word. Sometimes

there are curves “drawn by hand” which means using the curve-

mouse-drawing facilities of MS-Word.

Sometimes I used graphs of functions. Those graphs I made

using the program Scientific Workplace. I would then turn

them into wmf windows metafiles. Those I introduced as

picture in the MS Word drawing program. (It works better to

first introduce pictures in Powerpoint, and then transfer them

from powerpoint to MS Word, so this is how I did it.) I would

then only take the curve from the wmf file and nothing else,

so I would drop all letters, axes, and so on from the wmf

file. Those I would all make using MS Word.

Apart from 3 exceptions (added where relevant), I never kept

the Sc. Workplace TeX input file, if they are easy to redo.

p. 26:

++++

<

yj;1 … yj;nxj;1 … xj;n

<

(x1;1 , ... , x1;n) (y1;1 ,... , y1;n)

. . .

(xm;1 , ... , xm;n) (ym;1 ,... , ym;n)

FIGURE 1.5.1. Arbitrage (a Dutch book)

2

p. 42:

.

.

.

.

.

.

.

.

.

Decomposing a general prospect

xn

0

0

En

E1

E2

0

x2

0

En

E1

E2

0

0

x1

En

E1

E2

Combining prospects

Amplifyingevent indicators P(En)xn

~P(E2)x2

~

.. . +

xn

x2

x1

En

E1

E2

.

.

.

FIGURE 1.11.1. Deriving expected value

~P(E1)x1 + P(E2)x2 + . . . + P(En)xn

P(E1)x1

~

++=

3

p. 51:

10 106

10 106

50 106

.89

.10

(g)

.01 0

110 106

50 106

(f)

.05

.95

.04

.96

0

0

10 106

50 106

(h)

.11

.89

.10

.90

0

0

10 106

50 106

1

(e)

.8

.2 0

FIGURE 2.4.1.

(a)

.5

.5

.5

.5

70

95

60

80

(d)

.31

.69

.97

.03

0

16

0

4

501

(c)

60

80.3

.7

(b)

.6

.4

.7

.3

20

40

20

40

4

p. 52:

FIGURE 2.4.2

(b)1 ...~

0

4

.03

.97

(a)1 ...~

0

96

.05

.95

5

p. 54:

the expected utility (EU) of the prospect

p1U(x1) + … + pnU(xn).

.

.

xn

x1

pn

p1 FIGURE 2.5.1

6

p. 56:

U

$

FIGURE 2.5.2. Two indifferences and the resulting U curve

0.4

0.81

00 10070300.20

0.80

0100

~700.60

0.40

0100

~30

7

p. 56:

FIGURE 2.5.3. The SG probability p of

m

M

1p

~

p

8

p. 59:

FIGURE 2.6.1.

0

200

100

2/3

1/6

1/6

0

100

2/3

1/3

0

200

2/3

1/3

1/2

1/2

9

p. 60:

(c)

(b)

(a)

10002/3

1/3

20002/3

1/3

FIGURE 2.6.2xy

1

10

p. 60:

1/5

4/5

0

2

8

5

1/5

1/2

1/10

1/52

5

1/2

1/2

0

5

8

1/4

2/4

1/4

;x =

0

5

8

1/4

2/4

1/4FIGURE 2.6.3.

and is equal toThe mixture x4/5ycan be depicted as

( = 4/5);y =2

5

1/2

1/2

11

p. 61:

1-

C

1-

C

~ m

M

1-p

p

FIGURE 2.6.4. The lottery-equivalent method of McCord & de Neufville (1986) (> 0)

12

p. 62:

for all outcomes , M, m, all probabilities p and , and all prospects C.

implies

1-

C

1-

C~ m

M

1-p

pFIGURE 2.6.5. SG consistency holds if

m

M

1-p

~

p

13

p. 65:

FIGURE 2.7.1. The sure-thing principle for risk

implies

p

.

.

.

yn

y2

qn

q2

p

xm

x2

pm

p2

p

.

.

.

yn

y2

qn

q2

p

xm

x2

pm

p2

.

.

.

.

.

.

14

p. 66:

FIGURE 2.8.1

(b)1 ...~

0

16

.69

.31

(a)1 ...~

24

96

.05

.95

15

p. 68:

FIGURE 2.9.1

=m

Mp1U(x1) + ... + pnU(xn)

1 p1U(x1) ... pnU(xn)

~m

MU(xj)

1U(xj)

p1

pj

.

.

.

.

.

.

pn

m

MU(xn)

1U(xn)

m

MU(x1)

1U(x1)

~m

MU(xj)

1U(xj)

x1p1

pj

.

.

.

xn

.

.

.

pn

xj

x1p1

pj

.

.

.

xn

.

.

.

pn

16

p. 70:

: death

surgery

FIGURE 3.1.1. Choice between radio-therapy or surgery for a patient with larynx-cancer (stage T3)

artificial speech

0.3

0.7

normalvoice

artificial speechcure

0.3

0.7

recurrency

cure

0.4

0.6

artificial speech

0.4

0.6

recurrency, surgery

radio-therapy

17

p. 71:

artificial speech ~

normal voice

p

1p

FIGURE 3.1.2. The SG question: For which p is the gamble equivalent to the certain outcome?

18

p. 72:

ELUCIDATION: This Figure was made using only MS Word. I drew

the curves by hand.

FIGURE 3.2.1. Risk aversion

*

p

1p

U

$p + (1p)

*

U()

pU() + (1p)U()

U(p + (1p))

U()

For the prospect , the expected utility, *, is lower than *, the utility of the expected value.

19

p. 72:


the curves by hand.

FIGURE 3.2.2. Concavity, linearity, and convexity

U convex

U

U linear

U

U concave

U

20

p. 75:

qM + (1q)m = xj, so that the means are the same.

m

Mq

.

.

.

x1

xn

.

.

.

pn

pj

p1

1q...

FIGURE 3.3.1. Aversion to elementary mean-preserving spreads

xj

x1

xn

.

.

.

pn

pj

p1

21

p. 79:

ELUCIDATION: This Figure contains a graph of the following function, drawn fat, and indicated in the figure by =0:

, further the function, also drawn fat, and indicated in the

figure by =1:u() = 1

and further the functions (not drawn fat)

for the other values indicated in the figure ( = 20, 5, 2, 1, 0.5, 0.1, 0.1, 0.5, 2, 5, and 30).I made the graphs using Scientific Workplace (did not keep input files) as explained above.

(=ln) =0

=1

1

0

–1

1 2

FIGURE 3.5.1. Power utility curves, normalized at 1 and 2

u() = ln(2) - 1

ln() - 1

u() = 2 - 1 - 1

22

p. 81:

ELUCIDATION: This Figure contains graphs of the function:

u() = (indicated in the figure by =0)

and of the functions

for the other ’s as indicated ( = 2. 0.6, 6, and 2).

I made the graphs using Scientific Workplace (did not keep input files) as explained above.

U

1

2

1 2

2

1

12

$

FIGURE 3.5.2. Exponential utility, normalized at 0 and 1.

= 2

= 0.6

= 0

= 0.6

= 2

u() = 1 - exp()1 - exp()

23

p. 86:

FIGURE 3.7.1. SG invariance

~(H,0)

(H,M)p(H,T)

1p~

(Q,0)

(Q,M)p(Q,T)

1p

24

p. 87:

.

.

. p1U(x1

1,...,x1m) + ... + pnU(xn

1,...,xnm)

(xn1,...,xn

m)

(x21,...,x2

m)

pn

p2

FIGURE 3.7.2. A prospect with multiattribute outcomes and its expected utility

(x11,...,x1

m)p1

25

p. 88:

prospect of Eq. 3.7.3(20 years, blind)

(5 years, healthy)½

½ the marginals

(healthy)(blind)½

½marginal for health

FIGURE 3.7.3. Two prospects with the same marginals

prospect of Eq. 3.7.2(20 years)

(5 years)½

½(20 years, healthy)

(5 years, blind)½

½

marginal for life duration

26

p. 96:

8

3

cand2 wins

cand1 wins

~1

...

cand2 wins

cand1 wins

(d) Your switching value on the dotted line is 4.

8

2

cand2 wins

cand1 wins

~1

...

cand2 wins

cand1 wins

(c) Your switching value on the dotted line is 3.

8

1

cand2 wins

cand1 wins

~1

...

cand2 wins

cand1 wins

(b) Your switching value on the dotted line is 2.

8

10

cand2 wins

cand1 wins

~1

...

cand2 wins

cand1 wins

(a) Your switching value on the dotted line is 1.

FIGURE 4.1.1 [TO Upwards]. Eliciting 1 … 4 forunknown probabilities

Indicate in each Fig. which outcome on the dotted line ... makes the two prospects indifferent (the switching value).

27

p. 97:

...

G

3

cand2 wins

cand1 wins

~g

...

cand2 wins

cand1 wins


G

2

cand2 wins

cand1 wins

~g

...

cand2 wins

cand1 wins


G

1

cand2 wins

cand1 wins

~g

...

cand2 wins

cand1 wins


10

cand2 wins

cand1 wins

~g

1

cand2 wins

cand1 wins

(a) Your switching value on the dotted line is G.

FIGURE 4.1.2 [2nd TO Upwards]. Eliciting 2, 3, 4

Indicate in each fig. which outcome on the dotted line ... makes the two prospects indifferent (the switching value).

28

p. 98:

Indicate in each Fig. which outcome on the dotted line ..., if received with certainty, is indifferent to the prospect.

FIGURE 4.1.3 [CEs]. Eliciting 2,1,3

2

...

4

0.5

0.5

0.5

0.5

0.5

0.5

~

(c) Elicitation of γ3.

0

...

2

~

(b) Elicitation of γ1.

0

...

4

~

(a) Elicitation of γ2.

29

p. 99:

1

1

cand2 wins

cand1 wins

~8

...

cand2 wins

cand1 wins


1

2

cand2 wins

cand1 wins

~8

...

cand2 wins

cand1 wins


1

3

cand2 wins

cand1 wins

~8

...

cand2 wins

cand1 wins


1

4

cand2 wins

cand1 wins

~8

...

cand2 wins

cand1 wins

(a) Your switching value on the dotted line is 3.

FIGURE 4.1.4 [TO Downwards]. Eliciting 3 … 0

Indicate in each fig. which outcome on the dotted line ... makes the two prospects indifferent (the switching value).

30

p. 100:

3

0

... 4

~

(c) Elicitation of PE3.1 ...

2

0

... 4

~

(b) Elicitation of PE2.1 ...

Indicate in each Fig. which probability on the dotted lines ... makes the prospect indifferent to receiving the sure amount to the left.

FIGURE 4.1.5 [PEs]. ElicitingPE1, PE2, PE3

1

0

... 4

~

(a) Elicitation of PE1.1 ...

31

p. 104:


the curves by hand.

this point indicates the prospect (cand1:1, cand2:8)

(= 10)

FIGURE 4.3.1. Your indifferences in Figure 4.1.1

outcomeunder cand2

Curves designate indifference.

outcomeunder cand1

4

8

1

3210

32

p. 104:

FIGURE 4.3.2. Utility graph derived from Figure 4.1.1

U1

¾

½

$

432100

10=

¼

33

p. 109:


the curves by hand.

prospect Ey

xEc

Curves designate indifference. instead of apparently offsets yEc instead of xEc, and so does instead of .

outcome under cand1

yEc

FIGURE 4.5.1. ~t outcomeunder cand2

34

p. 114:

p2, …, pm: outcome probabilities of x beyond p;q2, …, qn: outcome probabilities of y beyond p.p > 0.

and ~

p

.

.

.

ynqn

q2

p

xm

x2

pm

p2

~

p

yn

y2

qn

q2

p

.

.

.

xmpm

p2

.

.

.

.

.

.

FIG. 4.7.1b. ~t for risk

E2, …, Em: outcome events of x beyond E;B2, …, Bn: outcome events of y beyond E.E is nonnull.

and ~.

.

.

yn

y2

Bn

B2

xm

x2

Em

E2

~

E

yn

y2

Bn

B2

E

.

.

.

xm

x2

Em

E2

.

.

.

.

.

.

FIG. 4.7.1a. ~t for uncertainty

x2 y2

E E

35

p. 120:

$0

$1

0.7~$0

$1

not all rain

all rain

FIGURE 4.9.1. Matching proba-bility of all rain (tomorrow) is 0.3.

0.3

36

p. 121:

$0

$1

0.7

0.3

no rain $0

$1

$00.6

0.4

~no rain or all rain

some rain

$0

$1

$0

$1

0.8

0.2

FIGURE 4.9.2. Violation of additivity (Raiffa 1968 §4)

~not all rain

all rain

$0

$1

rain (all or some)

~

For additivity to hold, the bold probability 0.4 should have been 0.3 + 0.2 = 0.5.

$1

37

p. 121:

x3

x2

x10.30.20.5

~x3

x2

x1all rainsome rainno rain

The first three indifferences imply the fourth for all x1, x2, x3, and thus transfer EU from risk to uncertainty.

rain (all or some)

$00.5~

no rain

$0

$1~

no rain or all rain

some rain

$0

$1

$00.8~

not all rain

all rain

$0

$1

$0

$1

0.7

0.3

FIGURE 4.9.3. Probabilistic matching

$10.2 $10.5

38

p. 123:

FIGURE 4.9.4. Different presentations and evaluations of multi-stage prospects

i=1;m qi(j=1;n pju(xj

i)): A rewriting of Eq. 3.7.7.

j=1;n pj(i=1;m qiu(xj

i)): the evaluation by Eq. 3.7.7.

FIG. 4.9.4c. A step in the evalua-tion of prospects in Anscombe & Aumann's model

p1U(x1m) + ... + pnU(xnm)

p1U(x12) + ... + pnU(xn2)

p1U(x11) + ... + pnU(xn1)

hm

h2

h1

.

.

.p1p2

pn

x2mxnm

x1m...

x

12x

22pn

x

11x

21pn xn1...

p1p2

hm

h2

h1

.

.

.

FIG. 4.9.4b. Anscombe & Aumann’s model as mostly used today

h1...

hm

h2xnmx

n2

x

n1

h1...

hm

h2x2mx

22

x

21

h1...

hm

h2x1mx

12

x

11

FIG. 4.9.4a. An analog of the mul-tiattribute utility prospect of Figure 3.7.2

pn

p2

p1

.

.

.

p1

xn2...

p2

39

p. 126:

h1...

hm

h2xn

xn

xn

h1...

hm

h2x2

x2

x2

h1...

hm

h2x1

x1

x1

FIG. 4.9.5. (p1:x1, …, pnxn) in the roulette-horse Example 4.9.6

pn

p2

p1

.

.

.

40

p. 126:

FIG. 4.9.6. (p1:x1, …, pnxn) in the horse-roulette Example 4.9.7

p1p2

pn

x2

xn

x1

...

x1

xn

...p1p2 x2

pn

x1x2

pn xn

...

p1p2

hm

h2

h1

.

.

.

41

p. 134:

0.06 0.06

0.87 0.87(b)

FIGURE 4.12.1. An example of the Allais paradox for risk

0.070.07 025K0.070.07 r

75K

s

25K

0

0

r

75K

0

s 25K

25K

&0.060.06

0.87 0.8725K 25K(a)

42

p. 134:

LL

MM 025K

25K 25K

MM

LL 00

025K

(b)(a)

HHHH

r

75K

s

25K

r

75K

s

25K

&

FIGURE 4.12.2. The certainty effect (Allais paradox) for uncertainty

43

p. 140:


the curves by hand.

prospect (E1:3,E2:1)

4

3

2

Curves designate indifference.

FIGURE 4.15.1. Illustration of standard sequences

outcome under E1

outcomeunder E2

4

1

0

3210

44

p. 146:

FIGURE 5.1.1. Five SG observations

(e)(d)(c)(b)(a) 0.10

0.90

0

10081 ~

0.30

0.70

0

10049 ~

0.50

0.50

0

10025 ~

0.70

0.30

0

1009 ~

0.90

0.10

0

1001 ~

45

p. 146:


the curves by hand. The right curve should be obtained from

the left one by rotating left and flipping horizontally.

Under Eq. 5.1.2, the curve can be interpreted as the probability weighting function w, to be normalized at the extreme amounts(w = 0 at $0 and w = 1 at $100).

Under expected utility, the curve can be interpreted as the utility function, normalized at the extreme amounts.

FIGURE 5.1.2. Two pictures to summarize the data of Figure 5.1.1

FIG. b. An alternative way to display the same data

(e)

(d)

(b)(a)

0.70.3 10

$30

$70

$100

$0

p

$

FIG. a. A display of the data

(e)

(d)

(b)(a)

$

$30 $70

0.7

0.3

1

0$100$0

p

(c)(c)

46

p. 150:

The area shaded by is the expected value p1x1 + p2x2 + ... + pnxn.

p1

0

xn

x3

x2

. . . pnp3

.

. . . .

. . .. . .

. . . .

FIGURE 5.2.1. Expected value

. . . .

x1

. . . .

p2

47

p. 150:

0xnx2x1

The area shaded by is the expected value.

FIG. 5.2.2a. Expected value after rotating left

. .

. .

. .

. .

. .

. ..

. .

.

p1

p2

pn

p3

. . .x3

. .

.

. . .

.

x2 x1

The area shaded by is the expected value.

. .

. .

. .

. .. .

.

FIG. 5.2.2b. Expected value after (rotating left and) flipping horizontally

. .

.

p1

p2

pn

p3

. .

. .

0 xn

.

. . .

. .

. .

Height is G(), the probability of receiving an outcome to the right of , i.e., better than . (G() is therank of .)

x3

48

p. 151:

Expected utilityp1U(x1) + p2U(x2)

+ ... + pnU(xn) is area .

0

U(x2)

p2 pnp3

U(xn) . . . .. . . .

U(x1)

FIGURE 5.2.3. Expected utility

U(x3)

To calculate expected utility, the distance from xj (“all the way”) down to the x-axis has been transformed into the distance U(xj), for all j.

. . .

p1

. . .

.

. . . .

. . . .. . .

49

p. 152:


the curve by hand.

FIGURE 5.2.4. A probability weighting function

0

1w

p

10

50

p. 152:

The height of each single layer, i.e., the distance of each endpoint down to its lower neighbor, has been transformed.

x2 x1

w(p1)x1 + w(p2)x2 + ... + w(pn)xn is the area (value of the prospect).

0 xn . . . x3

FIGURE 5.2.5. Transforming probabilities of fixed outcomes (the “old” model)

w(p3)

. .

. .

.

.

.

. .

. .

. .

. .. .

. .

w(p2)

w(p1)

w(pn)

. . .

. .

. .

51

p. 154:

-

additi-onal area

w(p1+p2)x1w(p1)x1 + w(p2)x2 + ... + w(pn)xn is the area .

FIG. 5.3.1c. x1 hits x2.

0 xn x3

w(p1)

x2 = x1

. . .

w(p3)

. .

. .

.

.

.

. .

. .

. .

. .

. .

. .. .

. .

w(p2)

w(pn)

. . .

x10 xn x3

w(p1)

x2. . .

w(p3)

. .

. .

.

.

.

. .

. .

. .

. .

. .

. .. .

. .

w(p2)

w(pn)

. . .

w(p1)x1 + w(p2)x2 + ... + w(pn)xn is the area .

FIG. 5.3.1b. Reducing x1 further.

FIGURE 5.3.1. Eq. 5.2.1 violates stochastic dominance

area lost

FIG. 5.3.1a. Reducing x1 somewhat.

0 xn x3 x2. . .

w(p1)x1 + w(p2)x2 + ... + w(pn)xn is the area .

w(p3)

. .

. .

.

.

.

. .

. .

. .

. .

. .

. .. .

. .

w(p2)

w(pn)

. . .

ori-ginal x1

redu-ced x1

w(p1)

w(

p1

+

p2)

52

p. 157:


the curves by hand.

Fig. b displays the same prospects as Fig. a, but now in terms of ranks, i.e., the probability of receiving a strictly better outcome, which is 1 minus the usual “distribution function.”

FIG. b. Ranks, being 1 minus the distribution function

xy

outcome

1

0

FIG. a. Probability densities, the continuous analogs of outcome probabilities

x

y

outcome

probab.density

0

FIGURE 5.4.1. The usefulness of ranks

ra

n

53

p. 162:

FIG c.

FIG b.

FIG a.

FIGURE 5.5.1. Combination of preceding figures, with rank dependence as an application of an economic technique to a psychological dimension.

. . .Old (psycholo-gists’) probabi-listic sensitivity

. . .

.

. . .

.

. . .

. .

. .

.. . .

.

x2 x1

w(p3)

w(p2)

w(p1)

w(pn)

. . .

.

. . .

.

. . .

.

x2 x10 . . .0

p3+p2+p1

p2 + p1

p1

pn+...+p1

pn1+...+p1

. . .. . .

.. . . .

. .

.

.

.

.

.

xn x3xn x3

. . .. . .

Rank-dependent probabilisticsensitivity

x2 x1

. . .

.

. . .

.

. . .

.. . . .

.w(p3+p2+p1)

w(pn+...+p1)w(pn1+...+p1)

w(p2+p1)

w(p1)

.

.

.

.

.

.

0

. . .

.

. . .

.

x2 x1

p3+p2+p1

p2 + p1

p1

pn+...+p1

pn1+...+p1

. . .. . .

.. . . .

..

.

.

.

.

.

xn x3. . .00 0

non-risk-neutral value of prospectway to model risk attitude

Expected value (risk neutrality)

p1 pnp3p2 . . .0

x2

x1

xn

x3. . .

p2 pnp3p1 . . .0

. . .U(x2)

U(xn)

U(x1)

U(x3)

(Economists’)outcomesensitivity (EU)

. . . .

. . ..

. . . .

. . .

.

. . . .. . .

. . . .. . . .. . . .

. . . .. . . .

. .

. .

0 . . .

xn x3

54

p. 163:

w(pn+...+p1) w(pn1+...+p1)

1 = w(pn+...+p1)

w(pn1+ ...+p1)

w(p2+p1)

x2 x1

The area shaded by is the value of the prospect. Distances of endpoints of layers (“all the way”) down to the x-axis are transformed, similar to Figure 5.2.3. The endpoint of the last layer now remains at a distance of 1 from the x-axis, reflecting normalization of the bounded probability scale.

FIGURE 5.5.2. Rank-dependent utility with linear utility

.

.

.

0 xn

.

. . .

w(G()): the w-transformed rank

x3

. .

. .

. .

. .

. .

. .. .

.. .

. .

w(p2+p1) w(p1)

w(p1)

55

p. 164:

w(pn + ... + p1) w(pn1 + ... + p1)

w(pn1 + ... + p1)

w(p2+p1)

.

.

.

0

w-transformed probability of receivingutility > .

. .

. .

w(p1)

. .

. .

. .

. .

FIGURE 5.5.3. Rank-dependent utility with general utility

... U(x3)U(xn)

For points on the y-axis (“endpoints of layers”), their distance down to the x-axis are transformed using w. For points on the x-axis (“endpoints of columns”), their distances leftwards to the y-axis are transformed using U.

U(x2)

w(p2+p1) w(p1)

U(x1)

1 = w(pn + ... + p1)

.

. .

.. .

. .

56

p. 164:

Relative to Figure 5.5.3, this figure has been rotated left and flipped horizontally.

. . .

w(pn + ...+p1) w(pn1+...+p1)

w(pn1 +... + p1)w(p2+p1)

0

w-transformed probability of receivingutility > .

FIGURE 5.5.4. Another illustration of general rank-dependent utility

...

U(x3)

U(xn)

U(x1)

w(pn+... + p1) (= 1)

U(x2)

w(p1)

w(p2+p1) w(p1)

57

Documents

Numbered Figures forProspect Theoryfor Risk and · Web viewNumbered Figures forProspect Theoryfor Risk and Ambiguity by Peter P. Wakker (2010); provided on internet July 2013 (with