Peter P. Wakker & Enrico Diecidue & Marcel Zeelenberg Don’t forget to make yellow comments invisible Measuring Decision Weights of Ambiguous Events by Adapting de Finetti's Betting-Odds Method to Prospect Theory Peter P. Wakker & Enrico Diecidue & Marcel Zeelenberg Domain: Decisions with unknown probabilities (“ambiguity”). Ambiguity will concern Dow Jones & Nikkei indexes today: U: both go Up () D: both go Down () We will analyze in terms of prospect theory.. R: Rest event (=; one up other down, or at least one constant) Building on a classical idea of de Finetti. We use it to test properties of those decision weights. Many people here have worked on nonadditive measures, or read about it. But few have actually “seen” them. 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! Question: How do people perceive of these uncertainties? How do they decide w.r.t. these? Concretely: A simple way to directly measure nonadditive decision weights for ambiguity quantitatively.

Some History of Prospect Theory 2 Some History of Prospect Theory For gains only today. Then prospect theory = rank-dependent utility = Choquet expected utility. 1950-1980: nonEU desirable, nonlinear probability desirable. 1981 (only then): Quiggin introduced rank-dependence for risk (given probabilities). Greatest idea in decision theory since 1954!?!? 1989: Schmeidler did the same independently. Big thing: Schmeidler did it for uncertainty (no probabilities given). Up to that point, no implementable theory for uncertainty to deviate from SEU. (Multiple priors not yet implementable!?) 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. It is 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.

Restrictive Assumption about Utility in Our Analysis 3 Restrictive Assumption about Utility in Our Analysis de Finetti’s betting-odds system assumes linear utility. Our analysis maintains this assumption. Reasonable? Outcomes between Dfl 10 (€4.5) and Dfl 100 (€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 prospect theory with linear utility: Chateauneuf (1991, JME), Diecidue & Wakker (2002, MSS). Reasonable!?

( ) ( ) A Reformulation of Prospect-Theory (= You can claim that probabilities should be nonadditive, but for decision theory that as such doesn’t mean anything. 4 A Reformulation of Prospect-Theory (= Rank-Dependent-Utility = Choquet-Expected-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): U D R 9 7 5 ( ) b m w evaluated through U9 + D7 + R5. U D R 2 8 6 ( ) w b m evaluated through U2 + D8 + R6. (Cumulative) prospect theory generalizes expected utility by rank-dependence (“decision-way” of expressing nonadditivity of belief). We consider only gains. (This is why prospect theory = rank-dependent-utility/Choquet-expected-utility.) Properties of rank-dependent decision weights:

(overweighting of bad outcomes) Economists usually want pessimism for equilibria etc. 5 p  Uncertainty aversion Note that we do unknown probs; figures only suggestive. U w > m b convex Pessimism: (overweighting of bad outcomes) U w < m b Optimism: concave (overweighting of good outcomes) U w > m b (Likelihood) insensitivity: inverse-S (overweighting of extreme outcomes) U w > b m Empirical findings: inverse -S (lowered) (Primarily insensitivity; also pessimism; Tversky & Fox, 1997; Gonzalez & Wu 1999 )

Our empirical predictions: 6 Our empirical predictions: 1. The decision weights depend on the ranking position. 2. The nature of rank-dependence: U w > b m 3. Violations of prospect theory … see later. Those violations will come quite later. First I explain things of PT 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 PT, keep on listening!

Empirical studies of PT with 3 outcomes (mostly with known probs): 7 Real test of (novelty of) rank-dependence needs at least 3-outcome prospects (e.g. for defining m's). Empirical studies of PT with 3 outcomes (mostly with known probs): Many studies in “probability triangle.” Unclear results; triangle is not suited for testing PT. 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 (in preparation or done?) Lopes et al. on many outcomes, complex results. Summarizing: no clear results! Most here is for DUR.

( ) ? Which would you choose? U D R 103 47 12 94 64 8 103 47 12 94 64 8 Which would you choose? Shows how hard 3-outcome-prospect choices are. Our experiment: Critically tests the novelty of PT by measuring decision weights of events in varying ranking positions through choices between three-outcome prospects that are transparent to the subjects by appealing to de Finetti’s betting-odds system (through stating “reference prospects”): see next slides. This is how we want to make nonadditive measures/decision weights “visible.”

( ) ). ( ) ( ) p Classical method (de Finetti) to “check” if 3 20 9 Classical method (de Finetti) to “check” if 3 20 U > : In explanation make clear that “check” means elicit from an individual from his choices. At 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. U D R 20 0 0 ( ) U D R 3 3 3 ).  Check if this reveals that b U 3 20 > . U w 3 20 > ? How check if We: U D R 20 0 0 ( )  U D R 3 3 3 Answer: add a “reference gamble” (side payment). Check if Layout of stimuli + 13 + 46 + 65 refer- ence gamble ¹ · = 13 46 65 33 16 49 68 ­­ ¯¯ p Choice + +++ U D R Explain in terms of how many utility units more than the reference gamble i.e., U D R 33 46 65 ( )  U D R 16 49 68 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. You can see de Finetti’s intuition “shine” through, embedded in rank-dependence. 20 3 U > . w Then we can conclude

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

The Experiment Stimuli: explained before. 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 Performance of Dow Jones and Nikkei from March 16, 2001 till May 15, 2001

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

suggests insensitivity 14 Results under prospect theory Main effect is likelihood and is just fine. Bigger overestimation of unlikely events suggests likelihood insensitivity. Don’t forget to mention that we do find significant rank-dependence. The *’s are violations of SEU. best middle worst Down-event: suggests insensitivity D .34 (.18) .31 (.17) .34 (.17) * * best middle worst Up-event: suggests pessimism U .44 (.18) .48 (.20) .46 (.18) * * best middle worst Rest-event: suggests optimism R .52 (.18) .50 (.19) .50 (.18) *

1. We also test a possible violation of prospect theory: 15 Two more things … 1. We also test a possible violation of prospect theory: collapsing (Loomes & Sugden, Luce, Birnbaum, etc.), for the certainty effect. Prospect theory accommodates the certainty effect. Do factors beyond prospect theory, such as collapse, affect the certainty effect? If so, do they reinforce it or weaken it? 2. We also test whether direct assessments of emotions (e.g., regret) can predict future choices better than past choices can predict future choices. Prospect theory can explain more of the variance in choice than any other theory. But the total variance explained is still low.

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

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

Two of the 3 authors were surprised by this. 18 U b,c U w,c D w,c regret 0.177 p = .019 0.172 p = .023 0.183 p = .015 correlations between regret and decision weights Regret correlates positively with almost all decision weights: The more regret, the more risk seeking. It correlates especially strongly in presence of collapsing. Strange finding for economists’ revealed preference approach!

Using measurements, we investigated properties of rank-dependence. 19 Conclusions De Finetti’s betting odds can be adapted to prospect theory/rank-dependence: easy way to directly measure nonadditive decision weights quantitatively. Using measurements, we investigated properties of rank-dependence. Rank-dependent violations of expected utility were found. Support for pessimism and likelihood insensitivity. Collapsing effects, violating prospect theory. Direct introspective judgments predict future choices.