Measuring Decision Weights of Ambiguous Events: Adapting de Finetti's Betting-Odds Method to Choquet Expected Utility

1. Measuring Decision Weights of Ambiguous Events by Adapting de Finetti's Betting-Odds Method to Choquet Expected Utility Peter P. Wakker & Enrico Diecidue & Marcel ZeelenbergRUD ‘04Lecture will be on my homepage on July 8. Paper is there too. Don’t forget to make yellow comments invisible RUD: 1. Experimental study. Not very common in this conference. It is, however, an empirical measurement of capacities, the concepts introduced bySchmeidler’89. WE MAKE CAPACITIES, David’s idea, VISIBLE, So this is our birthday present. Many people have talked and written about nonadditive measures, capacities. But few have actually “seen” them (Fox & Tversky, Abdellaoui & Vossmann & Weber, Wu & Gonzalez). We will demonstrate how you can make them visible EASILY. Things such as “the capacity of rain tomorrow is 0.7 for \Mr. Jones. Hej, it is only 0.5 for Ms. Jones,” few if none have faced such information. Today you will see it! We will use our measurements to test properties of those decision weights. Topic: Quantitative measurement of capacities under ambiguity. • Ambiguity will concern Dow Jones & Nikkei indexes today: • U: both go Up () • D: both go Down () • R: Rest event (=; one up other down, or • at least one constant) Question: How do people perceive of these uncertainties? How do they decide w.r.t. these? Concretely: A simple way to directly measure nonadditive beliefs/decision weights for ambiguity quantitatively.

2. Most people here know these things. 2 Some History For gains only today. Then Choquet expected utility (CEU) = prospect theory (= rank-dependent utility). 1950-1980: nonEU desirable, nonlinear probability desirable. 1981 (only then): Quiggin introduced rank-dependence for risk (given probabilities). 1982/1989: Schmeidler did the same independently. Big thing:Schmeidler did it for uncertainty (no probabilities given). Greatest idea in decision theory since 1954!?!? Up to that point, no implementable theory for uncertainty to deviate from SEU. Uncertainty before 1990: prehistorical times! Only after, Tversky & Kahneman (1992) could develop a sound prospect theory, thanks to Schmeidler. Multiple priors had existed long before. Often used in theoretical studies. I am not aware of a study that empirically measured multiple priors, and do not know how to do that in a tractable manner. So, MP is not yet empirically tractable. We did not even have a language to speak about uncertainty!

3. 3 Restrictive Assumption about Utility in Our Analysis We use de Finetti’s betting-odds system, a variation thereof. All its current applications assume linear utility. We do too. Reasonable? Outcomes between Dfl10 (€4.5) and Dfl99 (€45). Are moderate, and not very close to zero. Then utility is approximately linear. References supporting it: de Finetti 1937; Edwards 1955; Fox, Rogers, & Tversky 1996; Lopes & Oden 1999 p. 290; Luce 2000 p. 86; Rabin 2000; Ramsey 1931 p. 176; Samuelson 1959 p. 35; Savage 1954 p. 91. Special dangers of zero-outcome: Birnbaum. Modern view: Risk aversion for such amounts is due to other factors than utility curvature (Rabin 2000). Axiomatizations of CEU with linear utility: Chateauneuf (1991, JME), Diecidue & Wakker (2002, MSS). Yes! Is reasonable(!?)

4. U D R U D R 9 7 5 2 8 6 ( ( ) ) For philosphers: You can claim that probabilities should be nonadditive, but for decision theory that as such doesn’t mean anything. 4 A Reformulation of CEU (= prospect theory = Rank-Dependent Utility) through Rank-Dependence of Decision Weights For specialists, remark that there are two middle weights but for simplicity we ignore difference. • (Subjective) expected utility (linear utility): m w b evaluated through U9 + D7 + R5. w m b evaluated through U2 + D8 + R6. Choquet expected utility generalizes expected utility by adding rank-dependence (“decision-way” of expressing nonadditivity of belief). Properties of rank-dependent decision weights:

5.  “Uncertainty aversion” No rank-dependence U m U U m m > = < U U U U U U U U U U U U U b w w b w b w b b w m m m > > > > = < > p 5 Note that we do unknown probs; figures only suggestive. linear Classical: (rational !?) Economists usually want pessimism for equilibria etc. Pessimism: convex (overweighting of bad outcomes) Optimism: concave (overweighting of good outcomes) (Likelihood) insensitivity: inverse-S (overweighting of extreme outcomes) Empirical findings: inverse-S (lowered) (Primarily insensitivity; also pessimism; Tversky & Fox, 1997; Gonzalez & Wu 1999;Abdellaoui, Vossmann, & Weber 2004 )

6. U m U U w b > > 6 Our empirical predictions: 1. The decision weights depend on the ranking position. 2. The nature of rank-dependence: 3. Violations of CEU … see later. Those violations will come quite later. First I explain things within CEU and explain and test those. Only after those results comes the test of the violations. But one violation will be strong, so, if you don’t like CEU, keep on listening!

7. 7 • Direct test, and real test of (novelty of) rank-dependence, needs at least 3-outcome acts (e.g. for defining m's). • Empirical studies of CEU with 3 outcomes (mostly with known probs): • Many studies in “probability triangle.” Unclear results; triangle is not suited for testing CEU. • Other qualitative studies with three outcomes: • Wakker, Erev, & Weber (‘94, JRU) • Fennema & Wakker (‘96, JRU) • Birnbaum & McIntosh (‘96, OBHDP) • Birnbaum & Navarrete (‘98, JRU) • Gonzalez & Wu (2004) • Lopes et al. on many outcomes, complex results. • Summarizing: no clear results! Most here is for DUR.

8. Which would you choose? ? ) ) ( ( U D R U D R 103 47 12 94 64 8 8 Shows how hard 3-outcome-act choices are. We developed special layout to make such choices transparent. Taking stock: Say firmly, taking public strongly by the hand. • Taking stock. We: • Test the novelty of CEU; • directly measure decision weights of events in varying ranking positions, quantitatively; • through choices between three-outcome acts; • that are transparent to the subjects by appealing to de Finetti’s betting-odds system (through stating “reference acts”): see next slides; • The latter is how we want to make nonadditive measures/decision weights “visible.”

9. + +++ 3 U > : Classical method (de Finetti) to “check” if 20 U ­­ Check if 13 D ¯¯ 46 U D R U D R U D R 3 3 U ? . b this reveals that How check if R > > 20 20 20 0 0 33 46 65 20 0 0 65 ¹ Answer: add a “reference gamble” (side payment). Check if p p Choice = · ( ( ( ) ). ) U D R U D R U D R U  w refer- ence gamble +13 +46 +65 333 333 164968 +13 +46 +65 16 33 ( ( ( ) ) )  46 49 i.e., 68 65  3 U > . Then we can conclude w 20 9 In explanation make clear that “check” means elicit from an individual from his choices. Say that the very idea to verify from prefs, while well-known today, was an impressive step forward. Mention that we are finding out about fair price (CE-equivalent) of U. We: Before Figure-layout: So this is algebra. But, we also want it psychological, I.e., in the minds of our subjects. How can we let this take place in the minds of our subjects? This was the most difficult question in our research. We spent a year or so trying all kinds of stimuli, before we came to choose this figure. Then relate back to difficult choice on p. 8, that now it is clearer. Layout of stimuli Explain in terms of how many utility units more than the reference gamble You can see de Finetti’s intuition “shine” through, embedded in rank-dependence.

10. 10 + + + + +++ +++ +++ +++ U ­­ ­­ D ¯¯ ¯¯ 13 13 13 13 13 13 13 13 13 13 R 46 46 46 46 46 46 46 46 46 46 p p p p p p p p Choice Choice Choice Choice 65 65 65 65 65 65 65 65 65 65 ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ + + + + +++ +++ +++ +++ = = = = = = = = = = · · · · · · · · · · U ­­ ­­ D ¯¯ ¯¯ U w R 40 33 33 28 33 31 33 34 33 33 33 43 33 16 33 19 37 33 22 25 ­­ ­­ ­­ ­­ p p p p p p p p Choice Choice Choice Choice 70 55 73 46 76 46 58 46 46 46 52 46 49 46 64 46 46 67 61 46 + + ¯¯ ¯¯ ¯¯ ¯¯ +++ +++ U ­­ ­­ 65 65 65 86 68 92 65 95 65 65 65 77 80 65 74 65 89 83 71 65 9/20 < D ¯¯ ¯¯ R 12/20. < p p p p Choice Choice x x x x This provides a tractable manner for quantitatively measuring decision weights under ambiguity. Combines de Finetti’s betting odds schemes with rank-dependence. (It is also more transparent to subjects than proper scoring rules.) x x x x Imagine the following choices: 9 more for sure  20 more under U  12 more for sure x x

11. 11 The Experiment • Stimuli: explained before. • N = 186 participants. Tilburg-students,NOT economics or medical. • Classroom sessions, paper-&-pencil questionnaires;one of every 10 students got one random choice for real. • Written instructions • brief verbal comment on likelihood of increases/decreases of Dow Jones & Nikkei. • graph of performance of stocks during last two months.

12. 12 Performance of Dow Jones and Nikkei from March 16, 2001 till May 15, 2001

13. ordercompletely randomized 13 • Order of questions • 2 learning questions • questions about difficulty etc. • 2 experimental questions • 1 filler • 6 experimental questions • 1 filler • 10 experimental questions • questions about emotions, e.g. regret Skip most details of it.

14. 14 Results 1st hypothesis (existence rank-dependence). ANOVA with repeated measures. Event U: F(2,328) = 9.44, p < 0.001; Event D: F(2,322) = 5.77, p = 0.003; Event R: F(2,334) = 2.80, p = 0.06. So, first hypothesis is confirmed: There is rank-dependence.

15. * * * * * 15 Main effect is likelihood and is just fine. Bigger overestimation of unlikely events suggests likelihood insensitivity. The *’s are violations of SEU. 2nd hypothesis (nature of rank-dependence; t-tests) middle worst best Down-event: suggests insensitivity D .34 (.18) .31 (.17) .34 (.17) middle worst best Up-event: suggests pessimism U .44 (.18) .48 (.20) .46 (.18) middle worst best Rest-event: no significant effects (some optimism?) R .52 (.18) .50 (.19) .50 (.18)

16. 16 Degeneracy effects: working name, only for this paper. CEU can explain more of the variance in choice than any other theory. But the total variance explained is still low. Third hypothesis (violations of CEU): CEU accommodates the certainty effect. Are there factors beyond CEU in the certainty effect? We call them degeneracy effects. Example: collapse effects (Loomes & Sugden, Luce, Humphrey, Birnbaum, etc.); these weaken the certainty effect. We had no clear prediction about the direction of degeneracy effects.

17. + + + + + + +++ +++ +++ +++ +++ +++ U ­­ ­­ D ¯¯ ¯¯ 13 16 16 13 16 13 R 46 46 46 46 46 46 p p p p p p U Choice Choice Choice w,n 65 65 46 46 65 46 ¹ ¹ ¹ ¹ ¹ ¹ U w,d = = = = = = · · · · · · U ­­ ­­ D ¯¯ ¯¯ 19 19 33 46 22 33 46 16 46 33 25 22 R ­­ ­­ 49 46 49 46 55 46 46 52 46 46 55 52 p p p p p ¯¯ ¯¯ p Choice Choice Choice 65 71 46 74 65 55 46 65 68 52 46 49 17 Stimuli to test degeneracy effects: … …

18. * * *** * * * * • Results concerning factors beyond CEU 18 middle worst best Down-event: .35 (.20) .35 (.19) degenerate suggests insensitivity D .34 (.18) .31 (.17) .34 (.17) .33 (.19) .33 (.18) nondeg. middle worst best .43 (.17) Up-event: .41 (.18) degenerate U .44 (.18) .48 (.20) .46 (.18) suggests pessimism .51 (.23) .46 (.22) nondeg middle worst best .49 (.20) Rest-event: .51 (.20) degenerate R .52 (.18) .50 (.19) .50 (.18) suggests optimism .50 (.20) .53 (.20) nondeg

19. 19 Conclusions • We adapted de Finetti’s betting odds to CEU/rank-dependence. Gives: easy method for directly measuring nonadditive decision weights quantitatively. • Thus, we empirically investigated properties of rank-dependence. • Rank-dependent violations of classical model were found. • Support for pessimism and likelihood insensitivity. • Some degeneracy effects, violating CEU.