Adapting de Finetti's Proper Scoring Rules for Measuring Subjective Beliefs to Modern

Decision Theories of Ambiguity

Gijs van de Kuilen, Theo Offerman, Joep Sonnemans, & Peter P. Wakker

June 23, 2006FUR, Rome

Topic: Our chance estimates of various soccer-teams to become world-champion.E: Brasil will win. not-E: other team.

Imagine following bet:You choose 0 r 1, as you like.We call r your reported probability of Brasil,and 1–r your reported probability of not-Brasil.You receive

E not-E

1 – (1– r)2 1 – r2

What r should be chosen?

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Rational model: Subjective expected utility (SEU). Moderate amounts: U is linear.So: SEV.After some algebra:

.

.

.

Optimal r = your true subjective probability of Brasil winning.

!!! Wow !!!

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"Bayesian truth serum" (Prelec, Science, 2005).Medicine against "frequentism."Superior to elicitations through preferences .Superior to elicitations through indifferences ~ (BDM).

Widely used: Hanson (Nature, 2002), Prelec (Science 2005). In accounting (Wright 1988), Bayesian statistics (Savage 1971), business (Stael von Holstein 1972), education (Echternacht 1972), medicine (Spiegelhalter 1986), psychology (Liberman & Tversky 1993; McClelland & Bolger 1994), experimental economics (Nyarko & Schotter 2002).

We want to introduce these very nice things into the FUR-nonEU world.

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Survey

Part I. Deriving r from theories (SEV, SEU, RDU for probabilistic sophistication, RDU for ambiguity ("CEU").

Part II. Deriving theories from observed r. In particular: Derive beliefs/ambiguity attitudes. Will turn out to be surprisingly easy.

Proper scoring rules <==> Nonexpected utility:Mutual benefits.

Part III. Implementation of our method in an experiment.

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Part I. Deriving r from Theories (SEV, and then 3 deviations).

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Let us assume you very strongly believe in Brasil (Ronaldinho …)

Your "true" subj. prob.(Brasil) = 0.75.

SEV: Then your optimal rE = 0.75.

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0.25 0.50 0.75 10p

8Reported probability R(p) = rE as function of true probability p, under:

nonEU

0.69

R(p)

EU

0.61

rEV

EV

rnonEU

rnonEUA

rnonEUA: nonexpected utility for unknown probabilities ("Ambiguity").

(c) nonexpected utility for known probabilities, with U(x) = x0.5 and with w(p) as common;

(b) expected utility with U(x) = x (EU);

(a) expected value (EV);

rEU

next p.

go to p. 11, Example EU

go to p. 15, Example nonEU

0

0.50

1

0.25

0.75

go to p. 19, Example nonEUA

So far we assumed SEV (as does no-one at FUR, but as does the whole ocean of literature that uses proper scoring rules ...)

Deviation 1 from SEV. What if you want to bet on Brasil with larger stakes [SEU with U nonlinear]?

Now optimizepU(1 – (1– r)2) + (1 – p)U(1 – r2)

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10

U´(1–r2)U´(1 – (1–r)2)

(1–p)p +

pr =

Reversed (and explicit) expression:

U´(1–r2)U´(1 – (1–r)2)

(1–r)r +

rp =

How bet on Brasil? [Expected Utility]. EV: rEV = 0.75.Expected utility, U(x) = x: rEU = 0.69. You now bet less on Brasil. Closer to safety.(Winkler & Murphy 1970.)

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go to p. 8, with figure of R(p)

Deviation 2 from SEV: nonexpected utility for probabilities (Allais 1953, Machina 1982, Kahneman & Tversky 1979, Quiggin 1982, Schmeidler 1989, Gilboa 1987, Gilboa & Schmeidler 1989, Gul 1991, Tversky & Kahneman 1992, etc.)

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For two-gain prospects, virtually all those theories are as follows:

For r 0.5, nonEU(r) = w(p)U(1 – (1–r)2) + (1–w(p))U(1–r2).

r < 0.5, symmetry; soit!Different treatment of highest and lowest outcome: "rank-dependence."

p

w(p)

1

1

0 1/3

Figure. The common weighting function w.w(p) = exp(–(–ln(p))) for = 0.65.

w(1/3) 1/3;

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1/3

w(2/3) .51

2/3

.51

Now

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U´(1–r2)U´(1 – (1–r)2)

(1–w(p))w(p) +

w(p)r =

U´(1–r2)U´(1 – (1–r)2)

(1–r)r +

rp =

Reversed (explicit) expression:

w –1(

)

How bet on Brasil now? [nonEU with probabilities]. EV: rEV = 0.75.EU: rEU = 0.69.Nonexpected utility, U(x) = x, w(p) = exp(–(–ln(p))0.65).rnonEU = 0.61.You bet even less on Brasil. Again closer to safety.

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go to p. 8, with figure of R(p)

Deviations from EV and Bayesianism were at level of behavior so far; were not at level of beliefs. Now for something different; more fundamental.

3rd violation of EV: Ambiguity (unknown probabilities; belief/decision-attitude? Yet to be settled).

No objective data on probabilities.How deal with unknown probabilities?

Have to give up Bayesian beliefs descriptively.According to some even normatively.

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17Instead of additive beliefs p = P(E), nonadditive beliefs B(E) (Dempster&Shafer, Tversky&Koehler, etc.)

All currently existing decision models:For r 0.5, nonEU(r) = w(B(E))U(1 – (1–r)2) + (1–w(B(E)))U(1–r2).Don't recognize? Write W(E) = w(B(E)): is just Schmeidler's Choquet expected utility! Can always write B(E) = w–1(W(E)).

For binary gambles: Pfanzagl 1959; Luce ('00 Chapter 3); Ghirardato & Marinacci ('01, "biseparable").

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U´(1–r2)U´(1 – (1–r)2)

(1–w(B(E)))w(B(E)) +

w(B(E))rE =

U´(1–r2)U´(1 – (1–r)2)

(1–r)r +

rB(E) =

Reversed (explicit) expression:

w –1(

)

How bet on Brasil now? [Ambiguity, nonEUA]. rEV = 0.75.rEU = 0.69.rnonEU = 0.61 (under plausible assumptions).Similarly,

rnonEUA = 0.52.r's are close to always saying fifty-fifty."Belief" component B(E) = w–1(W) = 0.62.

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go to p. 8, with figure of R(p)

B(E): ambiguity attitude /=/ beliefs??Before entering that debate, first:How measure B(E)?Our contribution: through proper scoring rules with "risk correction."

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We reconsider reversed explicit expressions:

U´(1–r2)U´(1 – (1–r)2)

(1–r)r +

rp = w

–1(

)

U´(1–r2)U´(1 – (1–r)2)

(1–r)r +

rB(E) = w

–1(

)

Corollary. p = B(E) if related to the same r!!

Part II. Deriving Theoretical Models from Empirical Observations of r

22If - for event E, subject has rE = r;- for probability p, subject has R(p) = r;then B(E) = p.

Need not measure w, W, U!

We simply measure the R(p) curves, and use their inverses: B(E) = R–1(rE) follows.

Applying R–1 is called risk correction.

Directly implementable empirically. We did so in an experiment, and found plausible results.

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Our proposal takes the best of several worlds!

Need not measure U,W, and w.

Get "canonical probability" without measuring indifferences (BDM …; Holt 2006).

Calibration without needing many repeated observations.

Do all that with no more than simple proper-scoring-rule questions.

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We bring the insights of modern nonEU to proper scoring rules, making them empirically more realistic. (SEV in 2006 is not credible …)

We bring the insights of proper scoring rules to modern nonEU, making B very easy to measure and analyze.

Part III. Experimental Test of Our Correction Method

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Method

Participants. N = 93 students. Procedure. Computarized in lab. Groups of 15/16 each. 4 practice questions.

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27Stimuli 1. First we did proper scoring rule for unknown probabilities. 72 in total.

For each stock two small intervals, and, third, their union. Thus, we test for additivity.

28Stimuli 2. Known probabilities: Two 10-sided dies thrown. Yield random nr. between 01 and 100.Event E: nr. 75 (etc.).Done for all probabilities j/20.

Motivating subjects. Real incentives. Two treatments. 1. All-pay. Points paid for all questions.6 points = €1. Average earning €15.05.2. One-pay (random-lottery system).One question, randomly selected afterwards, played for real. 1 point = €20. Average earning: €15.30.

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Results

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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000

Reported probability

Cor

rect

ed p

roba

bilit

y

ONE (rho = 0.70) ALL (rho = 1.14) 45°

Average correction curves.

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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

ρ

F(ρ )

treatmentone

treatmentall

Individual corrections

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Figure 9.1. Empirical density of additivity bias for the two treatments

Fig. b. Treatment t=ALL

0.60

20

40

60

80

100

120

140

160

0.4 0.2 0 0.2 0.4 0.6

Fig. a. Treatment t=ONE

0

20

40

60

80

100

120

140

160

0.6 0.4 0.2 0 0.2 0.4 0.6

For each interval [, ] of length 0.05 around , we counted the number of additivity biases in the interval, aggregated over 32 stocks and 89 individuals, for both treatments. With risk-correction, there were @ > 60 additivity biases between 0.375 and 0.425 in the treatment t=ONE, and without risk-correction there were @<100 such; etc.

corrected

corrected

uncorrected

uncorrected

Summary and ConclusionModern decision theories: proper scoring rules are heavily biased.We correct for those biases, with benefits for proper-scoring rule community and for nonEU community.Experiment: correction improves quality; reduces deviations from ("rational"?) Bayesian beliefs.Do not remove all deviations from Bayesian beliefs. Beliefs seem to be genuinely nonadditive/nonBayesian/sensitive-to-ambiguity.

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