Ambiguity will concern Dow Jones & Nikkei indexes today:

Peter P. Wakker & Enrico Diecidue & Marcel ZeelenbergLecture + paper are on my homepage

Domain: Quantitative measurement of beliefs under ambiguity.

Question: How do people perceive of these uncertainties? How do they decide w.r.t. these?

Concretely: A simple way to directly measure nonadditive beliefs/decision weights for ambiguity quantitatively.

Measuring Decision Weights of Ambiguous Events by Adapting de Finetti's Betting-Odds Method to Prospect

Theory

U: both go Up ()D: both go Down ()R: Rest event (=; one up other down, or at least one constant)

We will analyze in terms of prospect theory..We will analyze in terms of prospect theory..

Don’t forget to make yellow comments invisible

Don’t forget to make yellow comments invisible

ESA: The measurement of subjective beliefs is popular in game theory and experimental economics today (Andrew Schotter). The measurement methods used are based on classical theories. Subjects, however, deviate from classical models. This was pointed out by prospect theory. We show how subjective beliefs can be measured under prospect theory. We use our method and investigate if and how subjective beliefs deviate from the classical models.

RUD: Many people have worked on nonadditive measures, or read about it. But few have actually “seen” them. Things such as “the capacity of rain tomorrow is 0.7 for \Mr. Jones. Hej, it is only 0.5 for Ms. Jones,” few if none have faced such information. Today you will see it! We will use our measurements to test properties of those decision weights.

Prospect theory does not cause trouble for the classical economic concepts, but instead shows how to defend them against deviating biases, and how to correct for those biases.

ESA: The measurement of subjective beliefs is popular in game theory and experimental economics today (Andrew Schotter). The measurement methods used are based on classical theories. Subjects, however, deviate from classical models. This was pointed out by prospect theory. We show how subjective beliefs can be measured under prospect theory. We use our method and investigate if and how subjective beliefs deviate from the classical models.

RUD: Many people have worked on nonadditive measures, or read about it. But few have actually “seen” them. Things such as “the capacity of rain tomorrow is 0.7 for \Mr. Jones. Hej, it is only 0.5 for Ms. Jones,” few if none have faced such information. Today you will see it! We will use our measurements to test properties of those decision weights.

Prospect theory does not cause trouble for the classical economic concepts, but instead shows how to defend them against deviating biases, and how to correct for those biases.

For gains only today. Then prospect theory = rank-dependent utility = Choquet expected utility.1950-1980: nonEU desirable, nonlinear probability desirable.1981 (only then): Quiggin introduced rank-dependence for risk (given probabilities). Greatest idea in decision theory since 1954!?!?1989: Schmeidler did the same independently.Big thing: Schmeidler did it for uncertainty (no probabilities given).Up to that point, no implementable theory for uncertainty to deviate from SEU. (Multiple priors not yet implementable!?)Uncertainty before 1990: prehistorical times!Only after, Tversky & Kahneman (1992) could develop a sound prospect theory, thanks to Schmeidler.

2

Some History of Prospect Theory

Multiple priors had existed long before. It is often used in theoretical studies. I am not aware of a study that empirically measured multiple priors, and do not know how to do that in a tractable manner.

Multiple priors had existed long before. It is often used in theoretical studies. I am not aware of a study that empirically measured multiple priors, and do not know how to do that in a tractable manner.

ESA’04: skip rest pageESA’04: skip rest page

de Finetti’s betting-odds system assumes linear utility.Our analysis maintains this assumption.Reasonable?Outcomes between Dfl 10 (€4.5) and Dfl 100 (€45). Are moderate, and not very close to zero. Then utility is approximately linear.References supporting it:de Finetti 1937; Edwards 1955; Fox, Rogers, & Tversky 1996; Lopes & Oden 1999 p. 290; Luce 2000 p. 86; Rabin 2000; Ramsey 1931 p. 176; Samuelson 1959 p. 35; Savage 1954 p. 91. Special dangers of zero-outcome: Birnbaum.Modern view: Risk aversion for such amounts is due to other factors than utility curvature (Rabin 2000).Axiomatizations of prospect theory with linear utility:Chateauneuf (1991, JME), Diecidue & Wakker (2002, MSS).

3

Restrictive Assumption about Utility in Our Analysis

Reasonable!?

ESA’04: say rest page in one lineESA’04: say rest page in one line

• (Subjective) expected utility (linear utility):

4

U D R9 7 5

( ) evaluated through U9 + D7 + R5.

U D R2 8 6

( ) evaluated through U2 + D8 + R6.

A Reformulation of Prospect-Theory (=Rank-Dependent-Utility = Choquet-Expected-Utility)

through Rank-Dependence of Decision Weights

(Cumulative) prospect theory generalizes expected utility by rank-dependence(“decision-way” of expressing nonadditivity of belief).

b

b

m

m

w

w

Properties of rank-dependent decision weights:

For specialists, remark that there are two middle weights but for simplicity we ignore difference.

For specialists, remark that there are two middle weights but for simplicity we ignore difference.

For philosphers: You can claim that probabilities should be nonadditive, but for decision theory that as such doesn’t mean anything.

For philosphers: You can claim that probabilities should be nonadditive, but for decision theory that as such doesn’t mean anything.

5

Pessimism: Uw > U

m Ub>

(overweighting of bad outcomes)

Optimism: Uw < m

U Ub<

(overweighting of good outcomes)

(Likelihood)insensitivity:

(overweighting of extreme outcomes)

Empirical findings: (Primarily insensitivity; also pessimism;Tversky & Fox, 1997; Gonzalez & Wu 1999 )

Uncertainty aversion

Uw > m

U

Ub > m

U

Uw > U

b > mU

convex

concave

inverse-S

inverse-S

(lowered)

p

linearClassical:

Uw = m

U Ub=

No rank-dependence

(rational!)

Economists usually want

pessimism for equilibria etc.

Economists usually want

pessimism for equilibria etc.

Note that we do unknown probs;

figures only suggestive.

Note that we do unknown probs;

figures only suggestive.

Our empirical predictions:

1. The decision weights depend on the ranking position.

2. The nature of rank-dependence:

6

Uw >

Ub >

Um

3. Violations of prospect theory … see later.Those violations will come quite later. First I explain things within PT and explain and test those. Only after those results comes the

test of the violations. But one violation will be strong, so, if you don’t like PT, keep on listening!

Those violations will come quite later. First I explain things within PT and explain and test those. Only after those results comes the

test of the violations. But one violation will be strong, so, if you don’t like PT, keep on listening!

– Many studies in “probability triangle.”

Unclear results; triangle is not suited for testing PT.– Other qualitative studies with three outcomes:

• Wakker, Erev, & Weber (‘94, JRU)• Fennema & Wakker (‘96, JRU)• Birnbaum & McIntosh (‘96, OBHDP)• Birnbaum & Navarrete (‘98, JRU)• Gonzalez & Wu (2004)

– Lopes et al. on many outcomes, complex results.– Summarizing: no clear results!

7

• Real test of (novelty of) rank-dependence needs at least 3-outcome prospects (e.g. for defining m's).

• Empirical studies of PT with 3 outcomes (mostly with known probs):

Most here is for DUR.

Most here is for DUR.

ESA’04: rest page in one sentenceESA’04: rest page in one sentence

• Taking stock. We:

8

Shows how hard 3-outcome-prospect choices are.

We developed special layout to make such choices transparent.

– Tests the novelty of PT– by measuring decision weights of events in varying ranking

positions– through choices between three-outcome prospects– that are transparent to the subjects by appealing to de Finetti’s

betting-odds system (through stating “reference prospects”): see next slides.

– This is how we want to make nonadditive measures/decision weights “visible.”

? U D R( )103 47 12 U D R( ) 94 64 8

Which would you choose?ESA: skip this page.ESA: skip this page.

9

+13

+46

+65

+13

+46

+65

i.e.,

U D R33 46 65

( ) U D R16 49 68( )

20

3U > .wThen we can conclude

=

13

46

65

33

46

65

16

49

68

Choice

+

+++

U

D

R

Layout of stimuli

Classical method (de Finetti) to “check” if3

20

U > :

U D R20 0 0

( ) U D R3 3 3( ).Check if

this reveals that bU

3

20

> . U

w

3

20

> ?How check if

U D R20 0 0

( ) U D R3 3 3( )

Answer: add a “reference gamble” (side payment). Check if

refer- ence gamble

We:

You can see de Finetti’s intuition “shine” through, embedded in rank-dependence.

You can see de Finetti’s intuition “shine” through, embedded in rank-dependence.

In explanation make clear that “check” means elicit from an individual from his choices. At

Say that the very idea to verify from prefs, while well-known today, was an impressive step

forward.

Mention that we are finding out about fair price (CE-equivalent) of U.

In explanation make clear that “check” means elicit from an individual from his choices. At

Say that the very idea to verify from prefs, while well-known today, was an impressive step

forward.

Mention that we are finding out about fair price (CE-equivalent) of U.

Before Figure-layout: So this is algebra. But, we also want it psychological, I.e., in the minds of our

subjects. How can we let this take place in the minds of our subjects? This was the most difficult question in our research. We spent a year or so

trying all kinds of stimuli, before we came to choose this figure.

Then relate back to difficult choice on p. 8, that now it is clearer.

Before Figure-layout: So this is algebra. But, we also want it psychological, I.e., in the minds of our

subjects. How can we let this take place in the minds of our subjects? This was the most difficult question in our research. We spent a year or so

trying all kinds of stimuli, before we came to choose this figure.

Then relate back to difficult choice on p. 8, that now it is clearer.

Explain in terms of how many

utility units more than the reference

gamble

Explain in terms of how many

utility units more than the reference

gamble

+

+++

Choice

13

46

65

33

46

65

3770

89

=

=

13

46

65

33

46

65

2861

80

Choice

+

+++

U

D

R

Choice

13

46

65

33

46

65

31

64

83

=

+

+++

Choice

13

46

65

33

46

65

34

67

86

=

+

+++

=

13

46

65

33

46

65

40

73

92

Choice

+

+++

U

D

R

Choice

13

46

65

33

46

65

43

76

95

=

+

+++

10+

+++

Choice

13

46

65

33

46

65

25

58

77

=

Choice

=

13

46

65

33

46

65

16

49

68

+

+++

U

D

R

Choice

13

46

65

33

46

65

19

52

71

=

+

+++

Choice

13

46

65

33

46

65

22

55

74

=

+

+++

x x x x

Imagine the following choices:

x x x x

x x

9 more for sure 20 more under U 12 more for sure

U

w <

9/20 <

12/20.

This provides a tractable manner for quantitatively measuring decision weights under ambiguity. Combines de Finetti’s

betting odds schemes with rank-dependence.

This provides a tractable manner for quantitatively measuring decision weights under ambiguity. Combines de Finetti’s

betting odds schemes with rank-dependence.

11

The Experiment

• Stimuli: explained before.• N = 186 participants. Tilburg-students,

NOT economics or medical.• Classroom sessions, paper-&-pencil questionnaires;

one of every 10 students got one random choice for real. • Written instructions

– brief verbal comment on likelihood of increases/decreases of Dow Jones & Nikkei.

– graph of performance of stocks during last two months.

ESA’04: rest page in one sentenceESA’04: rest page in one sentence

DAILY RESULTS

90

95

100

105

110

115

120

3/16

/200

1

3/18

/200

1

3/20

/200

1

3/22

/200

1

3/24

/200

1

3/26

/200

1

3/28

/200

1

3/30

/200

1

4/1/

2001

4/3/

2001

4/5/

2001

4/7/

2001

4/9/

2001

4/11

/200

1

4/13

/200

1

4/15

/200

1

4/17

/200

1

4/19

/200

1

4/21

/200

1

4/23

/200

1

4/25

/200

1

4/27

/200

1

4/29

/200

1

5/1/

2001

5/3/

2001

5/5/

2001

5/7/

2001

5/9/

2001

5/11

/200

1

5/13

/200

1

5/15

/200

1

DAYS

DJ NK

12Performance of Dow Jones and Nikkei from March 16, 2001 till May 15,

2001

• Order of questions– 2 learning questions– questions about difficulty etc.– 2 experimental questions– 1 filler– 6 experimental questions– 1 filler– 10 experimental questions– questions about emotions, e.g. regret

ordercompletely randomized

13

Skip most details of it.

Skip most details of it.

1st hypothesis (existence rank-dependence).

ANOVA with repeated measures.

Event U: F(2,328) = 9.44, p < 0.001;

Event D: F(2,322) = 5.77, p = 0.003;

Event R: F(2,334) = 2.80, p = 0.06.

So, first hypothesis is confirmed:

There is rank-dependence.

14

Results

.44 (.18)

worst

best

middle

worst

Rest-event:

R.52 (.18)

.50 (.19)

.50 (.18)

2nd hypothesis (nature of rank-dependence; t-tests)15

*

best

middle

worst

suggests insensitivity

D.34 (.18)

.31 (.17)

.34 (.17)

*

*

Up-event:best

middle

U.48 (.20)

.46 (.18)

*

*

Down-event:

suggests pessimism

suggests optimism

Main effect is likelihood and is just fine. Bigger overestimation of unlikely events suggests likelihood insensitivity.

The *’s are violations of SEU.

Main effect is likelihood and is just fine. Bigger overestimation of unlikely events suggests likelihood insensitivity.

The *’s are violations of SEU.

16

Third hypothesis (violations of prospect theory):

Prospect theory accommodates the certainty effect.

Are there factors beyond prospect theory (Loomes & Sugden, Luce, Humphrey, Birnbaum, etc.)?

We call them degeneracy effects.

If existing, do they reinforce or weaken the certainty effect?

Prospect theory can explain more of the variance in choice than any other theory. But the total variance explained is still low.

Prospect theory can explain more of the variance in choice than any other theory. But the total variance explained is still low.

Degeneracy effects: working name, only for this

paper.

Degeneracy effects: working name, only for this

paper.

13

46

65

33

46

65

19

52

71

=

Choice

+

+++

+

+++

Choice

13

46

65

33

46

65

22

55

74

=

Choice

=

13

46

65

33

46

65

16

49

68

+

+++

U

D

R

…

+

+++

16

46

46

46

46

46

22

52

52

=

Choice

+

+++

Choice

16

46

46

46

46

46

25

55

55

=

+

+++

=

16

46

46

46

46

46

19

49

49

Choice

U

D

R

…

17

Uw,n

Uw,d

Stimuli to test degeneracy effects:

.44 (.18)

*

worst

degenerate .41 (.18)

.43 (.17)

best

middle

worst

degenerate .51 (.20)

.49 (.20)

Rest-event: R

.52 (.18)

.50 (.19)

.50 (.18)

nondeg .53 (.20)

.50 (.20)

• Results 18

*

best

middle

worst

.35 (.20)

.35 (.19)

suggests insensitivity

D.34 (.18)

.31 (.17)

.34 (.17)

degenerate

nondeg. .33 (.18)

.33 (.19)

*

*

Up-event:best

middle

U.48 (.20)

.46 (.18)

nondeg .46 (.22)

.51 (.23)

*

*

***

Down-event:

suggests pessimism

suggests optimism

concerning factors beyond prospect theory

Conclusions

• For empirically measuring subjective beliefs, prospect theory has improvements to bring.

• De Finetti’s betting odds can be adapted to prospect theory/rank-dependence: easy way to directly measure nonadditive decision weights quantitatively.

• Using measurements, we investigated properties of rank-dependence.

• Rank-dependent violations of expected utility were found.• Support for pessimism and likelihood insensitivity.• Some degeneracy effects, violating prospect theory.

19