Making Rank-Dependent Utility Tractable for the Study of Ambiguity

1. Making Rank-Dependent Utility Tractable for the Study of Ambiguity Make yellow comments invisible. ALT-View-O Peter P. Wakker, June 16, 2005MSE, Université de Paris I Aim: Make rank-dependent utility tractable to a general public and specialists alike, in particular for ambiguity. Tool:Ranks! Spinoff: Some changes of minds:

2. 2 Question 1 to audience: From what can we best infer that people deviate from EU for risk (given probabilities)? Allais paradox. Ellsberg paradox. Nash equilibria.

3. 3 Question 2 to audience: From what can we best infer that people deviate from SEU for uncertainty (unknown probabilities)? Allais paradox. Ellsberg paradox. Nash equilibria.

4. 4 Question 3 to audience: Assume rank-dependent utility for unknown probabilities (Choquet Expected utility). From what can we best infer that nonadditive measures are convex (= superadditive)? Allais paradox. Ellsberg paradox. Nash equilibria.

5. 5 After this lecture: Answer to Question 1 ("nonEU for risk") is: Allais paradox. Answer to Question 3 ("capacities convex in RDU = CEU") is: Allais paradox! Answer to Question 2 ("nonEU for uncertainty") is: both Allais and Ellsberg paradox. P.s.: I do think that the Ellsberg paradox has more content than the Allais paradox. Explained later.

6. 6 Other change of mind: The inequality Decision under risk  Decision under uncertainty in the strict sense of [ Decision under risk  Decision under uncertainty =  ] is incorrect! Decision under risk  Decision under uncertainty ! That's how it is!

7. 7 At 1 and 2: They know well. Lines and notation may give new insights Line of DUR  DUU, and of first seeing how to measure the subjective concepts in a theory, and then how to axiomatize the theory. For instance, this can't be done yet for multiple priors. Outline of lecture: Expected Utility for Risk. Expected Utility for Uncertainty. Rank-Dependent Utility for Risk, Defined through Ranks. Where Rank-Dependent Utility Differs from Expected Utility for Risk. Where Rank-Dependent Utility Agrees with Expected Utility for Risk, and some properties. Rank-Dependent Utility for Uncertainty, Defined through Ranks. Where Rank-Dependent Utility Differs from Expected Utility for Uncertainty as it Did for Risk. Where Rank-Dependent Utility Agrees with Expected Utility for Uncertainty. Where Rank-Dependent Utility Differs from Expec-ted Utility for Uncertainty Differently than for Risk. Applications of Ranks.

8. (p1:x1,…,pn:xn) = p1 p1 x1 x1 . . . . . . . . . . . . xn xn pn pn Expectedutility:  p1U(x1 ) + ... + pnU(xn) 8 • Expected Utility for Risk is prospect yielding €xj with probability pj, j=1,…,n.

9. p  ~   1–p 9 U: subjective index of risk attitude (watch out:only under expected utility!!!!!) How measure U from preferences? Set U() = 1, U() = 0. Find, for each   , probability p such that Then U() = p.

10. 0.10 0.30 0.50 0.90 0.70 €100 €100 €100 €100 €100 ~ ~ ~ ~ ~ €25 €81 €49 €9 €1 0 0 0 0 0 0.70 0.50 0.10 0.30 0.90 (b) (c) (d) (a) (e) 1 €100 p € (e) (e) 0.7 €70 (d) (d) (c) (c) (b) 0.3 €30 (b) (a) (a) 0 €0 €0 €100 €30 €70 0 1 0.3 0.7 p € go to p.27,RDU next p. 10 Assume following data deviating from expected value U(100) = 1, U(0) = 0 U(1) = 0.10U(100) + 0.90U(0) = 0.10. EU: EU: U(x) = pU(100) = p. Here is graph of U(x): EU: U(9) = 0.30U(100) = 0.30. Psychology: x = w(p)100 Here is graph of w(p): Psychology: 1 = w(.10)100

11. p   Format ~ has empirical problems:  1–p p  q q   ~ 1–p Q Q 1–q 1–q 11 Certainty effect! Alternative format (McCord & de Neufville '86) Consistency in utility measurement (substition): Upper and lower p should be the same. = substitution for 1&2-outcome prospects.  vNM independence.

12. 12 Theorem. Expected Utility  Continuity in probabilities; monotonicity; weak ordering; consistency in utility measurement ("substitution"). 

13. 13 Proof.

14. q q    P Q 1–q 1–q q q    go to p. 34, where RDU = EU for risk P Q 1–q 1–q next p. 14 moderately- rank- Well-known implication: Independence from common consequence ("sure-thing principle"): rank- r r  r r

15. w w .89 .89 0 0 .10 .10 .01 .01 0 1M  b b .10 .10 5M 1M > < .10 .10 .89 .89 EU 1M 1M w w .01 .01 0 1M  b b .10 .10 5M 1M go to p. 33, where RDU  EU for risk next p. 15 Well-known violation: Allais paradox. M: million € Is the certainty effect. OK for RDU.

16. 16 In preparation for rank-dependence and decision under uncertainty, remember: In (p1:x1,…,pn:xn), we have liberty to choose x1... xn.

17. 17 2. Expected Utility for Uncertainty Wrong start for DUU: Let S = {s1,…,sn} denote a finite state space. x : S  is an act, also denoted as an n-tuple x = (x1,…,xn). Is didactical mistake for rank-dependence! Why wrong? Later, for rank-dependence, ranking of outcomes will be crucial. Should use numbering of xj for that purpose; as under risk! Should not have committed to a numbering of outcomes for other reasons. So, start again:

18. 18 S: state space, or universal event. Act is function x : S  with finite range. x = (E1:x1, …, En:xn): yields xj for all sEj, with: x1,…,xn are outcomes. E1, …, En are events partitioning S. No commitment to a numbering of outcomes! As for risk.Important notational point for rank-dependence (which will come later). If E1,…,En understood, we may write (x1, …,xn).

19. E1 E1 x1 x1 . . . . . . . . . . . . (E1:x1,…,En:xn) = xn xn En En Subjective Expected utility: (E1)U(x1 ) + ... + (E1)U(xn) 19 U: subjective index of utility. : subjective probability. How measure these? Difficult, because two unknown scales. If can measure one, then other is easy (Ramsey).

20. 20 First measure : Savage (1954), Abdellaoui & Wakker (2005). First measure U: Several papers. Is our approach today.

21. next p. next p. 21 Notation:Ex is (x with outcomes on Ereplaced by ): R and ranking position of E is R 10E1x = (E1:10,E2:x2,.., En:xn); Enx = (E1:x1,.., En-1:xn-1, En:); etc. Monotonicity:   Ex  Ex;

22. then ~  * rank-dependent Lemma.Under (subj) expected utility,  ~ U() – U() = U() – U() . * next p. next p. R 22 If there exist x,y, nonnull E with: Ex~E y R R andEx~E y R R r r This is how we measure U under SEU. Need not know !

23. RDU next p. next p. 23 If ~* and'~* for ' > , then, under SEU, U() – U() = U() – U() and U(') – U() = U() – U(): Inconsistency! r r rank- Tradeoff consistency precludes such inconsistencies. That is: improving any outcome in a ~* relationship breaks the relationship. r

24. rank-dependent rank- next p. next p. 24 Theorem. The following two statements are equivalent: (i) (cont. subj) expected utility. (ii) four conditions: (a) weak ordering; (b) monotonicity; (c) continuity; (d) tradeoff consistency. Tradeoff consistency also gives ! Inconsistencies in those generate such in ~*. r

25. rank- go to p.42, RDU=EU next p. 25 Well-known implication: sure-thing principle: ExE y R R  ExE y R R

26. w w L L 0 0 H H M M  0 25K (77%) b b H H 25K 75K > < H H L SEU L 25K 25K w w M M  25K (66%) 0 b b H H 25K 75K go to p.41, RDUEU next p. Almost-unknown implication 26 (Not-so-well-known) violation (MacCrimmon & Larsson '79; here Tversky & Kahneman '92). Within-subjects expt, 156 money managers. d:DJtomorrw–DJtoday.L:d<30 ;M:30d35;H:d>35; K: $1000. OK for RDU: pessimism. Certainty-effect & Allais hold for uncertainty in general, not only for risk!

27. go to p. 10, with ut.curv. 27 3. Rank-Dependent Utility for Risk, Defined through Ranks Empirical findings: nonlinear treatment of probabilities. Hence RDU. Two steps for getting the theory. Step 1. Deviations from expected value in Section 1: nonlinear perception/processing of probability, through w(p). Explain to public that the two steps go together, and in isolation are vacuous. Only jointly they constitute a decision theory. Step 2. Turn this into decision theory through rank-dependence.

28. ? Rank-dependent utility of p1 x1 . . . . . . xn pn 28 • First rank-order x1>…> xn. • Decision weight of xj will depend on: • pj; • pj–1 + … + p1, the probability of receiving something better. The latter will be called a rank.

29. ? So, rank-dependent utility of p1 x1 . . . . . . xn pn 29 First rank-order x1>…> xn. Then rank-dependent utility is 1U(x1 ) + … + nU(xn) where j = w(pj + pj–1 + …+ p1) – w(pj–1 + …+ p1). The decision weight j depends on pj and on pj–1 + …+ p1: pj–1 + …+ p1 is the rank of pj,xj, i.e. the probability of receiving something better.

30. 30 Ranks and ranked probabilities (formalized hereafter) are proposed as central concepts in this lecture. Were introduced by Abdellaoui & Wakker (Theory and Decision, forthcoming in July 2005). With them, rank-dependent life will be much easier than it was ever before!

31. 31 In general, pairs pr, also denoted p\r, with p+r  1 are called ranked probabilities. r is the rank of p. (pr) = w(p+r) – w(r) is the decision weight of pr.

32. Again, rank-dependent utility of p1 x1 . . . . . . xn pn 32 with rank-ordering x1… xn: (p1r1)U(x1) + … + (pnrn)U(xn) with rj = pj–1 + …+ p1 (so r1 = 0). The smaller the rank r in pr, the better the outcome. The best rank, 0, is also denoted b, as in pb=p0. The worst rank for p, 1–p, is also denoted w, as in pw = p1–p.

33. go to p. 15, with Allais 33 4. Where Rank-Dependent Utility Differs from Expected Utility for Risk Allais paradox explained by rank dependence. Now the expression rank dependence can be taken literally!

34. go to p. 14, with risk-s.th.pr 34 5. Where Rank-Dependent Utility Agrees with Expected Utility for Risk, and Properties Common consequence implication of EU goes through completely for RDU if we replace probability by ranked probability. Some properties, suggested by Allais paradox, follow now (more to come later). Now see Fig. of w-shaped.doc

35. 35 w convex (pessimism): r < r'  w(p+r) – w(r)  w(p+r') – w(r') Equivalent to: (pr) increasing in r. Remember: big rank is bad outcome. "Decision weight is increasing in rank." w concave (optimism) is similar. 2 more pessimistic than 1, i.e. w2 more convex than w1: r < r', 1(pr) = 1(qr') 2(pr) 2(qr')

36. 1 w+ 0 g 1 p 0 b bad-outcome region good-outcome region insensitivity-region 36 Inverse-S Good probs weighted morer than in insensitive region: (pb)  (pr) on [0,g]  [g,b] Bad probs weighted more than in insensitive region: (pw)  (pr) on [g,b]  [b,1].

37. 37 6. Rank-Dependent Utility for Uncertainty, Defined through Ranks x = (E1:x1, …, En:xn) was act, with Ej's partitioning S. Rank-dependent utility for uncertainty: (also called Choquet expected utility) W is capacity, i.e. (i) W() = 0;(ii) W(S) = 1 for the universal event S;(iii) If A  B then W(A)  W(B) (monotonicity with respect to set inclusion).

38. 38 ER, with ER = , is ranked event, with R the rank. (ER) = W(ER) – W(R) is decision weight of ranked event. RDU of x = (E1:x1, …, En:xn), with rank-ordering x1… xn, is:jn(EjRj)U(xj) with Rj = Ej–1 … E1 (so R1 = ). Compared to SEU, ranks Rj have now been added, expressing rank-dependence.

39. 39 The smaller the rank R in ER, the better the outcome. The best (smallest) rank, , is also denoted b, as in Eb = E. The worst (biggest) rank for E, Ec, is also denoted w, as in Ew = EEc.

40. 40 Difficult notation in the past: S = {s1,…,sn}. For RDU(x1,…,xn), take a rank-ordering r of s1,...,sn such that xr1 ...  xrn. For each state srj, prj = W(srj,…, sr1) – W(srj-1,…, sr1) RDU = pr1U(xr1) + … + prnU(xrn) Due to -notation, difficult to handle. (2) = 5: Is state s2 fifth-best, or is state s5 second-best? I can never remember!

41. go to p. 26, Allais for uncertainty 41 7. Where Rank-Dependent Utility Differs from Expected Utility for Uncertainty as It Did for Risk Convexity of W follows from Allais paradox! Easily expressable in terms of ranks! Comment on double exclamation marks.

42. go to p. 21, TO etc. 42 8. Where Rank-Dependent Utility Agrees with Expected Utility for Uncertainty The whole measurement of utility, and preference characterization, of RDU for uncertainty is just the same as SEU, if we simply use ranked events instead of events!

43. 43 9. Where Rank-Dependent Utility Differs from Expected Utility for Uncertainty Differently than It Did for Risk Allais: deviations from EU. Pessimism/convexity of w/W, or insensitivity/inverse-S. For risk and uncertainty alike. Deviations from EU in an absolute sense. Ellsberg: more deviations from EU for uncertainty than for risk. More pessimism/etc. for uncertainty than for risk. Deviations from EU in an relative sense. Deviations from EU: byproduct.

44. 44 Historical coincidence: Schmeidler (1989) assumed EU for risk, i.e. linear w. Then: more pessimism/convexity for uncertainty than for risk (based on Ellsberg),  pessimism/convexity for uncertainty. Voilà source of numerous misunderstandings.

45. 45 Big idea to infer of Ellsberg is not, I think, ambiguity aversion. Big idea to infer from Ellsberg is, I think, within-person between-source comparisons. Not possible for risk, because risk is only one source. Typical of uncertainty, where there are many sources. Uncertainty is rich domain, with no patterns to be expected to hold in great generality. In this rich domain, many phenomena are present and are yet to be discovered. Many, even prominent, economists haven't yet caught up on this point.

46. 46 10. Applications of Ranks General technique for revealing orderings (AR) (BR') from preferences: Abdellaoui & Wakker (2005, Theory and Decision). Thus, preference foundations can be given for everything written hereafter.

47. 47 What is null event? Important for updating, equilibria, etc. E is null if W(E) = 0? E is null if W(Ec) = 1? E is null if (H:, E:, L:) ~ (H:, E:, L:)? For some   , H …? Or for all   , H …? ? We: Wrong question! Better refer to ranked events! Plausible condition is null-invariance: independence of nullness from rank. (Eb) = 0. (Ew) = 0. (EH) = 0.

48. 48 W convex:  increases in rank. W concave:  decreases in rank. W symmetric: (Eb) = (Ew). Inverse-S: There are [G,B], event-region of insensitivity; with [,G] good-event region, [B,S] the bad-event region. Good-event inequality (weighing good events better than insensitive): (Eb)  (ER) on event-interval [,G]  [G,B] and bad-event inequality (weighing bad events better than insensitive): (Ew)  (ER) on event-interval [G,B]  [B,S].

49. 49 Theorem. 2 is more ambiguity averse than 1 in sense that W2 is more convex than W1 iff 1(BR') = 1(AR) with R'  R  2(BR') 2(AR).

50. 50 Theorem. Probabilistic sophistication holds  [(AR) (BR)  (AR') (BR')]. In words: ordering of likelihoods is independent of rank.