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Peter P. Wakker (& Gijs van de Kuilen, Theo Offerman, Joep Sonnemans) December 22, 2005 Tel Aviv

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.

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Peter P. Wakker (& Gijs van de Kuilen, Theo Offerman, Joep Sonnemans) December 22, 2005 Tel Aviv

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  1. 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, 2005Tel Aviv Aim: See title. Modern theories: rank-dependent utility of Giboa ('87) & Schmeidler ('89) and prospect theory (Kahneman & Tversky '79, '92)

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

  3. 3  E 1 0 not-E 0 1 Say, you grade a multiple choice exam in geography to test students' knowledge about Statement E: Capital of Zeeland = Assen. Reward: if E true if not E true Assume: Two kinds of students. Those who know. They answer correctly. Those who don't know. Answer 50-50. 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.

  4. 4 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!

  5. Reward: if E true if not E true  E 1 0 not-E 0 1 5 don't know 0.75 0.75 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!

  6. 6 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: (way too time consuming; given here only because binary choice widely used in decision theory).

  7. 7 Reward: if E true if not E true Reward: if E true if not E true Reward: if E true if not E true Reward: if E true if not E true  E  E  E  E choice 3 choice 2 choice 1 choice 9 1 0 1 0 1 0 1 0 don't know don't know don't know don't know 0.90 0.90 0.10 0.10 0.30 0.30 0.20 0.20 . . .

  8. 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. 9 Reward: if E true if not E true  E 1 0  partly know E, to degree r r r Third solution[introspection] Student chooses number r ("reported probability"). 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 …

  10. 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. r=1 degree of belief in E r : r=0.5: 0.75 0.75 r=0 11 Reward: if E true if not E true : E 1 0 1 – (1–r)2 1–r2 don't know : not-E 0 1 Claim: Under "subjective expected value," optimal reported probability r = true subjective probability p.

  12. Reward: if E true if not E true degree of belief in E 1 – (1–r)2 1–r2  r: 12 To help memory: Proof. p true probability; r reported probability. EV = p(1 – (1–r)2) + (1–p)(1–r2). 1st 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 ...

  13. 13 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 = 0.75. Under expected value, optimal rE is 0.75. rG + rS + rY = 0.25 + 0.25 + 0.25 = 0.75 = rE: r satisfies additivity; as probabilities should! Continued later.

  14. 14 1 R(p) rEV rEU 0.69 0.61 rnonEU 0.75 0.50 rnonEUA nonEU EU 0.25 EV 0 1 0 0.25 0.50 0.75 p reward: if E true if not E true EV  rEV=0.75 0.94 0.44 0.8125  rEU=0.69 go to p. 19, Example EU 0.91 0.52 0.8094 go to p. 31, Example nonEUA go to p. 24, Example nonEU  rnonEU=0.61 0.85 0.63 0.7920 next p.  rnonEUA=0.52 0.77 0.73 0.7596 Reported probability R(p) = rE as function of true probability p, under: (a) expected value (EV); (b) expected utility with U(x) = x0.5 (EU); (c) nonexpected utility for known probabilities, with U(x) = x0.5 and with w(p) as common; rnonEUA refers to nonexpected utility for unknown probabilities ("Ambiguity").

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

  16. 16 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.).

  17. Reward: if E true if not E true degree of belief in E 1 – (1–r)2 1–r2  r: 17 1st deviation of EV: nonlinear utility Bernoulli (1738) proposed expected utility (EU). EU(r) = pU(1 – (1–r)2) + (1–p)U(1–r2). Then r = p need no more be optimal.

  18. (1–p) p + U´(1–r2) U´(1–r2) U´(1–(1–r)2) U´(1–(1–r)2) r p = (1–r) r + 18 Theorem. Under expected utility with true probability p, p r = Explicit expression: A corollary to distinguish from EV: r is nonadditive as soon as U is nonlinear.

  19. go to p. 14, with figure of R(p) 19 Example continued [Expected Utility, EU]. (urn K, 25 G, 25 R, 25 S, 25 Y). E: {G,S,Y}; p = 0.75. EV: rEV = 0.75. Expected utility, U(x) = x: rEU = 0.69. rG+rS+rY = 0.31+0.31+0.31 = 0.93 > 0.69 = rE: additivity violated! Such data prove that expected value cannot hold. Further continued later.

  20. Reward: if E true if not E true degree of belief in E 1 – (1–r)2 1–r2  r: 20 2nd 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. Fortwo-gainprospects,alltheoriesareas follows:

  21. Reward: if E true if not E true degree of belief in E 1 – (1–r)2 1–r2  r: 21 For r  0.5, nonEU(r) = w(p)U(1 – (1–r)2) + (1–w(p))U(1–r2). For r  0.5, nonEU(r) = w(1–p) U(1–r2) + (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).

  22. .51 1/3 2/3 22 1 w(p) 1 0 1/3 p Figure.The common weighting fuction. w(p) = exp(–(–ln(p))) for  = 0.65. w(1/3)  1/3; w(2/3)  .51

  23. w(p) r = (1–w(p)) w(p) + U´(1–r2) U´(1–r2) U´(1–(1–r)2) U´(1–(1–r)2) Explicit expression: ) r ( p = w–1 (1–r) r + 23 Theorem.

  24. go to p. 14, with figure of R(p) 24 Examplecontinued[nonEU]. (urn K, 25 G, 25 R, 25 S, 25 Y). E: {G,S,Y}; p = 0.75. EV: rEV = 0.75. EU: rEU = 0.69. Nonexpected utility, U(x) = x, w(p) = exp(–(–ln(p))0.65). rnonEU = 0.61. rG+rS+rY = 0.39+0.39+0.39 = 1.17 > 0.61 = rE: additivity is strongly violated!

  25. 25 3rd violation of EV: Ambiguity (unknown probabilities) Preparation for continued example. Random draw from additional urn A. 100 balls, ? Ga, ? Ra, ? Sa, ? Ya. Unknown composition ("Ambiguous"). Ea: {Ga,Sa,Ya}; p = ?. How to deal with unknown probabilities?

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

  27. Reward: if Ea true if not Ea true degree of belief in E 1 – (1–r)2 1–r2  r: 27 With Ea = {Ga, Sa, Ya}, choose P(Ea). Then For r  0.5, nonEU(r) = w(P(Ea))U(1–(1–r)2) + (1–w(P(Ea)))U(1–r2). However, also this is often violated empirically:

  28. Reward: if E true if not E true Reward: if Ea true if not Ea true degree of belief in E degree of belief in Ea 1 – (1–r)2 1–r2 1 – (1–r)2 1–r2  r:  r: > < Probabilistic sophistication! 28 For urn A (100 balls, ? Ga, ? Ra, ? Sa, ? Ya), By symmetry, P(Ga) = P(Ra) = P(Sa) = P(Ya). Must all be 0.25. Then P(Ea) = 0.75 must be. So, must be evaluated the same as (E: known urn) Value of both is w(0.75)U(1–(1–r)2) + (1–w(0.75))U(1–r2). rE = rEa must hold. Empirical finding: People prefer known probs.

  29. Reward: if Ea true if not Ea true degree of belief in E 1 – (1–r)2 1–r2  r: 29 Instead of modeling beliefs through additive P(Ea), beliefs may be nonadditive, B(Ea), with B(Ea)  B(Ga) + B(Sa) + B(Ya). (Dempster&Shafer, Tversky&Koehler, etc.) For r  0.5, nonEU(r) = w(B(Ea))U(1–(1–r)2) + (1–w(B(Ea)))U(1–r2). Writing W(Ea) = w(B(Ea)): W(Ea)U(1–(1–r)2) + (1–W(Ea))U(1–r2). P.s.: From this, can always write B(Ea) = w–1(W(Ea)).

  30. w(B(E)) rE = (1–w(B(E))) w(B(E)) + U´(1–r2) U´(1–r2) U´(1–(1–r)2) U´(1–(1–r)2) Explicit expression: r ) ( w–1 B(E) = (1–r) r + 30 Theorem.

  31. go to p. 14, with figure of R(p) 31 Examplecontinued[Ambiguity, nonEUA]. (urn A, 100 balls, ? G, ? R, ? S, ? Y). Ea: {Ga,Sa,Ya}; p = ? rEV = 0.75. rEU = 0.69. rnonEU = 0.61. Typically,w(B(Ea))<<w(B(E)) (E from known urn). rnonEUA is, say, 0.52. rG+rS+rY = 0.48+0.48+0.48 = 1.44 > 0.52 = rE: additivity is very extremely violated! r's are close to always saying fifty-fifty. Belief component B(E) = 0.62.

  32. 32 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:

  33. 33 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 Ea, find event E with objective probability p such that (Ea: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.

  34. 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. ) ( w–1 p = ) ( U´(1–r2) U´(1–r2) w–1 B(E) = U´(1–(1–r)2) U´(1–(1–r)2) r r (1–r) (1–r) r + r + 35 We reconsider explicit expressions: 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.

  36. 36 Part III. Experimental Test of Our Correction Method

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

  38. 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. 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 €15.05. 2. One-pay (random-lottery system). One question, randomly selected afterwards, played for real. 1 point = €20. Average earning: €15.30.

  40. 40 Results Preliminary …

  41. Average correction curves. 41

  42. 0.7 0.6 0.5 treatment one 0.4 0.3 0.2 treatment all 0.1 0 42 F( ρ ) 1 0.9 Individual corrections. 0.8 ρ -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

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

  44. 44 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.

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