Adapting de Finetti's proper scoring rules for Measuring Subjective Beliefs to Modern Decision Theories ofAmbiguity Peter P. Wakker (& Gijs van de Kuilen, Theo Offerman, Joep Sonnemans) December 22, 2005 Tel Aviv Aim:See title. Modern theories: rank-dependent utility of Giboa ('87) & Schmeidler ('89) and prospect theory (Kahneman & Tversky '79, '92)

Part I (Didactical). de Finetti's Proper Scoring Rules (under Subjective Expected Value) 2 For clarity, I give example of grading, which is not primary application.

Say, you grade a multiple choice exam in geography to test students' knowledge about Statement E: Capital of Zeeland = Assen. 3 Reward: if E true if not E true  E  not-E Assume: Two kinds of students. 1.Those who know. They answer correctly. 2.Those who don't know. Answer Problem: Correct answer does not completely identify student's knowledge. Some correct answers, and high grades, are due to luck. There is noise in the data.

One way out: oral exam, or ask details. Too time consuming. I now promise a perfect way out: Exactly identify state of knowledge of each student. Still multiple choice, taking no more time! Best of all worlds! 4

5 For those of you who do not know answer to E. What would you do? Best to say "don't know." System perfectly well discriminates between students! Reward: if E true if not E true  E  not-E  don't know

New Assumption: Students have all kinds of degrees of knowledge. Some are almost sure E is true but not completely; others have a hunch; etc. Above system does not discriminate perfectly well between such various states of knowledge. One solution (too time consuming): Oral exams etc. Second solution is to let them make many binary choices: 6 (way too time consuming; given here only because binary choice widely used in decision theory).

7  don't know 0.20  E Reward: if E true if not E true 1 0 choice  don't know 0.90  E Reward: if E true if not E true 1 0 choice 9  don't know 0.10  E Reward: if E true if not E true 1 0 choice 1  don't know 0.30  E Reward: if E true if not E true 1 0 choice 3

8 Etc. Can get approximation of true subjective probability p, i.e. degree of belief (!?). Binary-choice ("preference") solution is popular in decision theory. This method works under subjective expected value. Justifiable by de Finetti's (1931) famous book making argument. Too time consuming for us. Rewards for students are noisy this way. Spinoff: This lecture will remind decision theorists that multiple choice can be more informative than binary choice.

9  partly know E, to degree r r r  E Reward: if E true if not E true 1 0 Just ask students for "indifference r," i.e. ask: "Which r makes lower row equally good as upper row?" Problems: Why would student give true answer r = p? What at all is the reward (grade?) for the student? Does reward give an incentive to give true answer? BDM … complex and noisy payment … Third solution [introspection] Student chooses number r ("reported probability").

10 I now promise a perfect way out: de Finetti's dream-incentives. Exactly identifies state of knowledge of each student, no matter what it is. Still multiple choice, taking no more time. Need no BDM. Rewards students fairly, with little noise. Best of all worlds. For the full variety of degrees of knowledge. Student can choose reported probability r for E from the [0,1] continuum, as follows.

11 Reward: if E true if not E true Claim: Under "subjective expected value," optimal reported probability r = true subjective probability p. 1 0 r=1 0 1r=0 1 – (1–r) 2 1–r 2 r don't know r=0.5: : E : not-E degree of belief in E :

Proof. p true probability; r reported probability. EV = p ( 1 – (1–r) 2 ) + (1–p)(1–r 2 ). 1 st order optimality: 2p(1–r) – 2r ( 1–p ) = 0. r = p! Figure!?  One corollary (a test of EV): r is additive. Incentive compatible... Many implications Reward: if E true if not E true 1 – (1–r) 2 1–r 2  r: degree of belief in E To help memory:

Theoretical Example [Expected Value, EV], alternative to "Assen-Zeeland." An urn K ("known") contains 100 balls, 25 G(reen), 25 R(ed), 25 S(ilver), 25 Y(ellow). One ball is drawn randomly. E: ball is not red, so E = {G,S,Y}. p of E = Under expected value, optimal r E is r G + r S + r Y = = 0.75 = r E : r satisfies additivity; as probabilities should! 13 Continued later.

p 14 Reported probability R(p) = r E as function of true probability p, under: nonEU 0.69 R(p) EU 0.61 r EV EV r nonEU r nonEUA r nonEUA refers to nonexpected utility for unknown probabilities ("Ambiguity"). (c) nonexpected utility for known probabilities, with U(x) = x 0.5 and with w(p) as common; (b) expected utility with U(x) = x 0.5 (EU); (a) expected value (EV); reward: if E true if not E true EV  r EV =  r EU = r EU  r nonEU =  r nonEUA = next p. go to p. 19, Example EU go to p. 24, Example nonEU go to p. 31, Example nonEUA

Part II. Adapting de Finetti's Proper Scoring Rules to Deviations from Expected Value and also from Expected Utility 15

However, EV (= risk neutrality) is empirically violated. We will, one by one, incorporate three deviations: 1. Nonlinear utility, with expected utility instead of expected value; 2. Nonexpected utility for given probabilities (Allais paradox etc.); 3. No subjective probabilities for unknown probabilities (no probabilistic sophistication; Ellsberg paradox, ambiguity aversion, etc.). 16

1 st deviation of EV: nonlinear utility Bernoulli (1738) proposed expected utility (EU). 17 EU(r) = pU ( 1 – (1–r) 2 ) + ( 1–p ) U(1–r 2 ). Then r = p need no more be optimal. Reward: if E true if not E true 1 – (1–r) 2 1–r 2 degree of belief in E  r:

Theorem. Under expected utility with true probability p, 18 U´(1–r 2 ) U´(1 – (1–r) 2 ) (1–p) p + p r = U´(1–r 2 ) U´(1 – (1–r) 2 ) (1–r) r + r p = Explicit expression: A corollary to distinguish from EV: r is nonadditive as soon as U is nonlinear.

Example continued [Expected Utility, EU]. (urn K, 25 G, 25 R, 25 S, 25 Y). E: {G,S,Y}; p = EV: r EV = Expected utility, U(x) =  x: r EU = r G + r S + r Y = = 0.93 > 0.69 = r E : additivity violated! Such data prove that expected value cannot hold. 19 Further continued later. go to p. 14, with figure of R(p)

2 nd violation of EV: nonexpected utility Allais (1953), Machina (1982) proposed nonexpected utility (nonEU). Many such models: - original prospect theory (Kahneman & Tversky 1979), - rank-dependent utility (Quiggin 1982), - disappointment aversion (Gul 1991), - new prospect theory (Tversky & Kahneman 1992), etc. 20 For two-gain prospects, all theories are as follows: Reward: if E true if not E true 1 – (1–r) 2 1–r 2 degree of belief in E  r:

21 Reward: if E true if not E true 1 – (1–r) 2 1–r 2  r: degree of belief in E For r  0.5, nonEU(r) = w(p)U ( 1 – (1–r) 2 ) + ( 1–w(p) ) U(1–r 2 ). For r  0.5, nonEU(r) = w(1–p) U(1–r 2 ) + ( 1–w(1–p) ) U ( 1 – (1–r) 2 ). Different treatment of highest and lowest outcome. "Rank-dependence." Quiggin (1982), Gilboa (1987), Schmeidler (1989; first version 1982).

p w(p) /3 Figure. The common weighting fuction. w(p) = exp(–(–ln(p))  ) for  = w(1/3)  1/3; 22 1/3 w(2/3) .51 2/3.51

Theorem. 23 U´(1–r 2 ) U´(1 – (1–r) 2 ) ( 1–w(p) ) w(p) + w(p) r = U´(1–r 2 ) U´(1 – (1–r) 2 ) (1–r) r + r p = Explicit expression: w –1 ( )

Example continued [nonEU]. (urn K, 25 G, 25 R, 25 S, 25 Y). E: {G,S,Y}; p = EV: r EV = EU: r EU = Nonexpected utility, U(x) =  x, w(p) = exp(–(–ln(p)) 0.65 ). r nonEU = r G + r S + r Y = = 1.17 > 0.61 = r E : additivity is strongly violated! 24 go to p. 14, with figure of R(p)

3 rd violation of EV: Ambiguity (unknown probabilities) Preparation for continued example. Random draw from additional urn A. 100 balls, ? G a, ? R a, ? S a, ? Y a. Unknown composition ("Ambiguous"). E a : {G a,S a,Y a }; p = ?. How to deal with unknown probabilities? 25

Proposal: probabilistic sophistication (Machina & Schmeidler 1992): assign "subjective" probabilities to events. Then behave as if known probs (may be nonEU). 26

With E a = {G a, S a, Y a }, choose P(E a ). Then 27 For r  0.5, nonEU(r) = w(P(E a ))U ( 1 – (1–r) 2 ) + ( 1–w(P(E a )) ) U(1–r 2 ). However, also this is often violated empirically: Reward: if E a true if not E a true 1 – (1–r) 2 1–r 2 degree of belief in E  r:

For urn A (100 balls, ? G a, ? R a, ? S a, ? Y a ), By symmetry, P(G a ) = P(R a ) = P(S a ) = P(Y a ). Must all be Then P(E a ) = 0.75 must be. So, 28 must be evaluated the same as (E: known urn) Reward: if E true if not E true 1 – (1–r) 2 1–r 2 degree of belief in E  r: Reward: if E a true if not E a true 1 – (1–r) 2 1–r 2 degree of belief in E a  r: Value of both is w(0.75)U ( 1 – (1–r) 2 ) + ( 1–w(0.75) ) U(1–r 2 ). r E = r E a must hold. Empirical finding: People prefer known probs. > < Probabilistic sophistication!

29 Instead of modeling beliefs through additive P(E a ), beliefs may be nonadditive, B(E a ), with B(E a )  B(G a ) + B(S a ) + B(Y a ). (Dempster&Shafer, Tversky&Koehler, etc.) Reward: if E a true if not E a true 1 – (1–r) 2 1–r 2 degree of belief in E  r: For r  0.5, nonEU(r) = w(B(E a ))U ( 1 – (1–r) 2 ) + ( 1–w(B(E a )) ) U(1–r 2 ). Writing W(E a ) = w(B(E a )): W(E a )U ( 1 – (1–r) 2 ) + ( 1–W(E a ) ) U(1–r 2 ). P.s.: From this, can always write B(E a ) = w –1 (W(E a )).

Theorem. 30 U´(1–r 2 ) U´(1 – (1–r) 2 ) ( 1–w(B(E)) ) w(B(E)) + w(B(E)) r E = U´(1–r 2 ) U´(1 – (1–r) 2 ) (1–r) r + r B(E) = Explicit expression: w –1 ( )

Example continued [Ambiguity, nonEUA]. (urn A, 100 balls, ? G, ? R, ? S, ? Y). E a : {G a,S a,Y a }; p = ? r EV = r EU = r nonEU = Typically, w(B(E a )) << w(B(E)) (E from known urn). r nonEUA is, say, r G + r S + r Y = = 1.44 > 0.52 = r E : additivity is very extremely violated! r's are close to always saying fifty-fifty. Belief component B(E) = go to p. 14, with figure of R(p)

Many debates about whether or not B(E) can be interpreted as belief, or comprises other things about ambiguity attitude. We do not enter such debates. Before entering such debates, question is first how to measure B(E). This is what we do. We show how proper scoring rules can, after some corrections, measure B(E). Earlier proposals for measuring B: 32

Proposal 1 (common in decision theory): Measure U,W, and w from behavior, and derive B(E) = w –1 (W(E)) from it. Problem: Much and difficult work!!! Proposal 2 (common in decision analysis of the 1960s, and in modern experimental economics): measure canonical probabilities, that is, for E a, find event E with objective probability p such that (E a :100) ~ (p:100). Then B(E) = p. Problem: measuring indifferences is difficult. Proposal 3 (common in proper scoring rules): Calibration … Problem: Need many repeated observations. 33

34 Our proposal: Take the best of all worlds! Get B(E) = w –1 (W(E)) without measuring U,W, and w from behavior. Get canonical probability without measuring indifferences, or BDM. Calibrate without needing long repeated observations. Do all that with no more than simple proper- scoring-rule questions.

35 We reconsider explicit expressions: U´(1–r 2 ) U´(1 – (1–r) 2 ) (1–r) r + r p = w –1 ( ) U´(1–r 2 ) U´(1 – (1–r) 2 ) (1–r) r + r B(E) = w –1 ( ) Same right-hand sides. If p and B(E) belong to the same r, then we can immediately substitute B(E) = p. Need not calculate the rhs!! We simply measure directly the R(p) curves, and use their inverses.

Part III. Experimental Test of Our Correction Method 36

Method Participants. N = 93 students. Procedure. Computarized in lab. Groups of 15/16 each. 4 practice questions. 37

38 Stimuli 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.

39 Stimuli 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 € One-pay (random-lottery system). One question, randomly selected afterwards, played for real. 1 point = €20. Average earning: €15.30.

40 Results Preliminary …

41 Average correction curves.

ρ F(ρ) treatment one treatment all Individual corrections.

43 Reported probability Corrected probability Treat- ment one Treat- ment all

Summary and Conclusion. Modern decision theories show that proper scoring rules are heavily biased. We present ways to correct for these biases (get right beliefs even if EV violated). Experiment shows that correction improves quality and reduces deviations from ("rational"?) Bayesian beliefs. Correction does not remove all deviations from Bayesian beliefs. Beliefs seem to be genuinly nonadditive/nonBayesian/sensitive- to-ambiguity. 44