Making Rank-Dependent Utility Tractable for the Study of Ambiguity Peter P. Wakker, June 16, 2005 MSE, 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: Make yellow comments invisible. ALT-View-O Make yellow comments invisible. ALT-View-O

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

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

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)? a. Allais paradox. b. Ellsberg paradox. c. Nash equilibria. 4

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. 5

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! 6 Decision under risk  Decision under uncertainty ! That's how it is!

Outline of lecture: 1.Expected Utility for Risk. 2.Expected Utility for Uncertainty. 3.Rank-Dependent Utility for Risk, Defined through Ranks. 4.Where Rank-Dependent Utility Differs from Expected Utility for Risk. 5.Where Rank-Dependent Utility Agrees with Expected Utility for Risk, and some properties. 6.Rank-Dependent Utility for Uncertainty, Defined through Ranks. 7.Where Rank-Dependent Utility Differs from Expected Utility for Uncertainty as it Did for Risk. 8.Where Rank-Dependent Utility Agrees with Expected Utility for Uncertainty. 9.Where Rank-Dependent Utility Differs from Expec- ted Utility for Uncertainty Differently than for Risk. 10. Applications of Ranks. 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.

8 Expected utility: x1x1 xnxn p1p1 pnpn  p 1 U(x 1 ) p n U(x n ) (p 1 :x 1,…,p n :x n ) = x1x1 xnxn p1p1 pnpn is prospect yielding €x j with probability p j, j=1,…,n. 1.Expected Utility for Risk

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  1–p p ~   Then U(  ) = p.

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

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. Format p 1–p  ~   has empirical problems: ~ q  1–q Q q Q  1–p p 

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

13 Proof.

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

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

16 In preparation for rank-dependence and decision under uncertainty, remember: In (p 1 :x 1,…,p n :x n ), we have liberty to choose x 1 ...  x n.

Wrong start for DUU: Let S = {s 1,…,s n } denote a finite state space. x : S   is an act, also denoted as an n- tuple x = (x 1,…,x n ). Is didactical mistake for rank-dependence! Why wrong? Later, for rank-dependence, ranking of outcomes will be crucial. Should use numbering of x j for that purpose; as under risk! Should not have committed to a numbering of outcomes for other reasons. So, start again: Expected Utility for Uncertainty

S: state space, or universal event. Act is function x : S   with finite range. x = (E 1 :x 1, …, E n :x n ): yields x j for all s  E j, with: x 1,…,x n   are outcomes. E 1, …, E n 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 E 1,…,E n understood, we may write (x 1, …,x n ). 18

19 (E 1 :x 1,…,E n :x n ) = x1x1 xnxn E1E1 EnEn Subjective Expected utility:   (E 1 )U(x 1 )  (E 1 )U(x n ) x1x1 xnxn E1E1 EnEn 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 First measure  : Savage (1954), Abdellaoui & Wakker (2005). First measure U: Several papers. Is our approach today.

Monotonicity:      E x   E x; Notation:  E x is (x with outcomes on E replaced by  ): E 1 x = (E 1 :10,E 2 :x 2,.., E n :x n );  E n x = (E 1 :x 1,.., E n - 1 :x n - 1, E n :  ); etc. R and ranking position of E is R next p.

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

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

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 ~*. 24 rank-dependent rank- r next p.

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

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

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). 27 Step 2. Turn this into decision theory through rank-dependence. go to p. 10, with ut.curv. Explain to public that the two steps go together, and in isolation are vacuous. Only jointly they constitute a decision theory.

28 First rank-order x 1 > … > x n. Decision weight of x j will depend on: 1.p j ; 2.p j–1 + … + p 1, the probability of receiving something better. The latter will be called a rank. Rank-dependent utility of x1x1 xnxn p1p1 pnpn ?

29 First rank-order x 1 > … > x n. Then rank-dependent utility is  1 U(x 1 ) + … +  n U(x n ) where  j = w(p j + p j–1 + … + p 1 ) – w(p j–1 + … + p 1 ). The decision weight  j depends on p j and on p j–1 + … + p 1 : p j–1 + … + p 1 is the rank of p j,x j, i.e. the probability of receiving something better. So, rank-dependent utility of x1x1 xnxn p1p1 pnpn ?

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! 30

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

32 Again, rank-dependent utility of x1x1 xnxn p1p1 pnpn with rank-ordering x 1  …  x n :  (p 1 r 1 )U(x 1 ) + … +  (p n r n )U(x n ) with r j = p j–1 + … + p 1 (so r 1 = 0). The smaller the rank r in p r, the better the outcome. The best rank, 0, is also denoted b, as in p b = p 0. The worst rank for p, 1–p, is also denoted w, as in p w = p 1–p.

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

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

w convex (pessimism): r < r'  w(p+r) – w(r)  w(p+r') – w(r') Equivalent to:  (p r ) 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. w 2 more convex than w 1 : r < r',  1 (p r ) =  1 (q r' )   2 (p r )   2 (q r' ) 35

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

x = (E 1 :x 1, …, E n :x n ) was act, with E j '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) Rank-Dependent Utility for Uncertainty, Defined through Ranks

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

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

Difficult notation in the past: S = {s 1,…,s n }. For RDU(x 1,…,x n ), take a rank-ordering  of s 1,...,s n such that x  1 ...  x  n. For each state s  j,   j = W(s  j,…, s  1 ) – W(s  j-1,…, s  1 ) RDU =   1 U(x  1 ) + … +   n U(x  n ) Due to  -notation, difficult to handle.  (2) = 5: Is state s 2 fifth-best, or is state s 5 second-best? I can never remember! 40

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

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! Where Rank-Dependent Utility Agrees with Expected Utility for Uncertainty go to p. 21, TO etc.

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 Where Rank-Dependent Utility Differs from Expected Utility for Uncertainty Differently than It Did for Risk

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. 44

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. 45 Many, even prominent, economists haven't yet caught up on this point.

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

What is null event? Important for updating, equilibria, etc. E is null if W(E) = 0? E is null if W(E c ) = 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. 47  (E b ) = 0.  (E w ) = 0.  (E H ) = 0.

W convex:  increases in rank. W concave:  decreases in rank. W symmetric:  (E b ) =  (E w ). 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):  (E b )   (E R ) on event-interval [ ,G]  [G,B] and bad-event inequality (weighing bad events better than insensitive):  (E w )   (E R ) on event-interval [G,B]  [B,S]. 48

Theorem.  2 is more ambiguity averse than  1 in sense that W 2 is more convex than W 1 iff  1 (B R' ) =  1 (A R ) with R'  R   2 (B R' )   2 (A R ). 49

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

Updating on A given B, with A  B. What is W(A) if B is observed? Gilboa (1989a,b): 51 W(A) W(B). Dempster & Shafer: 1 – W(B c ) W(A  B c ) – W(B c ). W(A) + 1 – W ( (B\A) c ) W(A). Jaffray, Denneberg:  (B b )  (A b )  (B w )  (A B c )  (A b ) +  ( (B\A) w )  (A b ) Gilboa & Schmeidler (1993): depends on optimism / pessimism. Ranks formalize this. Cohen,Gilboa,Jaffray,&Schmeidler (2000): lowest one did best.

Conclusion: With ranks and ranked probabilities or events, rank-dependent utility becomes considerably more tractable. 52

Le fin! 53