Ambiguity will concern Dow Jones & Nikkei indexes today:

Topic: Quantitative measurement of capacities 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.

Peter P. Wakker & Enrico Diecidue & Marcel ZeelenbergRUD ‘04

Lecture will be on my homepage on July 8. Paper is there too.

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

Expected Utility

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

Don’t forget to make yellow comments invisible

Don’t forget to make yellow comments invisible

RUD: 1. Experimental study. Not very common in this conference. It is, however, an empirical measurement of capacities, the concepts introduced bySchmeidler’89. WE MAKE CAPACITIES, David’s idea, VISIBLE, So this is our birthday present.Many people have talked and written about nonadditive measures, capacities. But few have actually “seen” them (Fox & Tversky, Abdellaoui & Vossmann & Weber, Wu & Gonzalez). We will demonstrate how you can make them visible EASILY.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.

RUD: 1. Experimental study. Not very common in this conference. It is, however, an empirical measurement of capacities, the concepts introduced bySchmeidler’89. WE MAKE CAPACITIES, David’s idea, VISIBLE, So this is our birthday present.Many people have talked and written about nonadditive measures, capacities. But few have actually “seen” them (Fox & Tversky, Abdellaoui & Vossmann & Weber, Wu & Gonzalez). We will demonstrate how you can make them visible EASILY.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.

For gains only today. Then Choquet expected utility (CEU) = prospect theory (= rank-dependent utility).1950-1980: nonEU desirable, nonlinear probability desirable.1981 (only then): Quiggin introduced rank-dependence for risk (given probabilities).1982/1989: Schmeidler did the same independently.Big thing: Schmeidler did it for uncertainty (no probabilities given). Greatest idea in decision theory since 1954!?!?Up to that point, no implementable theory for uncertainty to deviate from SEU.Uncertainty before 1990: prehistorical times!Only after, Tversky & Kahneman (1992) could develop a sound prospect theory, thanks to Schmeidler.

2

Some History

Multiple priors had existed long before. 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. So, MP is not yet empirically tractable.

Multiple priors had existed long before. 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. So, MP is not yet empirically tractable.

Most people here know these

things.

Most people here know these

things.

We did not even have a language to speak about

uncertainty!

We did not even have a language to speak about

uncertainty!

We use de Finetti’s betting-odds system, a variation thereof.All its current applications assume linear utility. We do too.Reasonable?Outcomes between Dfl 10 (€4.5) and Dfl 99 (€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 CEU with linear utility:Chateauneuf (1991, JME), Diecidue & Wakker (2002, MSS).

3

Restrictive Assumption about Utility in Our Analysis

Yes! Is reasonable(!?)

• (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 CEU (= prospect theory = Rank-Dependent Utility)

through Rank-Dependence of Decision Weights

Choquet expected utility generalizes expected utility by adding 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;Abdellaoui, Vossmann, & Weber 2004 )

“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 CEU … see later.Those violations will come quite later. First I explain things within

CEU 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 CEU, keep on listening!

Those violations will come quite later. First I explain things within CEU 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 CEU, keep on listening!

– Many studies in “probability triangle.”

Unclear results; triangle is not suited for testing CEU.– 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

• Direct test, and real test of (novelty of) rank-dependence, needs at least 3-outcome acts (e.g. for defining m's).

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

Most here is for DUR.

Most here is for DUR.

• Taking stock. We:

8

Shows how hard 3-outcome-act choices are.

We developed special layout to make such choices transparent.

– Test the novelty of CEU;– directly measure decision weights of events in varying ranking

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

betting-odds system (through stating “reference acts”): see next slides;

– The latter 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?

Taking stock: Say firmly, taking

public strongly by the hand.

Taking stock: Say firmly, taking

public strongly by the hand.

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. 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. 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. (It is also more transparent to subjects than proper scoring rules.)

This provides a tractable manner for quantitatively measuring decision weights under ambiguity. Combines de Finetti’s betting odds schemes with rank-dependence. (It is also more transparent to subjects than proper scoring rules.)

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.

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

no significant effects (some 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 CEU):

CEU accommodates the certainty effect.

Are there factors beyond CEU in the certainty effect?

We call them degeneracy effects.

Example: collapse effects (Loomes & Sugden, Luce, Humphrey, Birnbaum, etc.); these weaken the certainty effect.

We had no clear prediction about the direction of degeneracy effects.

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

CEU 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 CEU

Conclusions

• We adapted de Finetti’s betting odds to CEU/rank-dependence. Gives: easy method for directly measuring nonadditive decision weights quantitatively.

• Thus, we empirically investigated properties of rank-dependence.

• Rank-dependent violations of classical model were found.• Support for pessimism and likelihood insensitivity.• Some degeneracy effects, violating CEU.

19