On Dow Jones & Nikkei indexes today:

1. Don’t forget to make yellow comments invisible Measuring Decision Weights for Unknown Probabilities by Means of Prospect Theory Peter P. Wakker & Enrico Diecidue & Marcel ZeelenbergThis file will be on my homepage on coming Monday. Domain: Decisions with unknown probabilities (“ambiguity”). • On 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) We will introduce a convenient method for measuring decision weights for ambiguity quantitatively, building on a classical idea of de Finetti, and use it to test properties of those measures.There has been a revival of the interest in such measurements through experimental economics, where people want to quantify beliefs of players. These, however, are still based on the classical Bayesian models. Question: How do people perceive of these uncertainties? How do they decide w.r.t. these? Concretely: A way to measure nonadditive decision weights for ambiguity quantitatively? We will analyze in terms of prospect theory..

2. 2 Some History of Prospect Theory I think that people in this society don’t know very well what I will tell in the 2nd half of this page. Rank-dependence: greatest idea in decision theory since 1980. Who invented rank-dependence? Quiggin (1982). Yaari (1987), Lopes (1984), Allais (1988) are essentially the same as Quiggin. Birnbaum’s configural weighting theory is a bit different. Who else invented rank-dependence? More important? Schmeidler (1989; first version 1982)! (Luce later,knew Quiggin) Big thing:Schmeidler did it for uncertainty, when no probabilities are given. Up to 1990 no serious decision theory for uncertainty (“subjective probability”) existed (except SEU)! Uncertainty before 1990: prehistory! Only after 1990 Tversky & Kahneman (1992) could do serious prospect theory, thanks to Schmeidler. You may find “greatest” overblown; this page will explain why I think so. At end of page remind them of why rank-dependent was called so great.

3. U D R U D R 9 7 5 2 8 6 ( ( ) ) You can claim that probabilities should be nonadditive, but for decision theory that as such doesn’t mean anything. 3 A Reformulation of Prospect Theory (1992) through Rank-Dependence For specialists, remark that there are two middle weights but for simplicity we ignore difference. • (Subjective) expected utility (linear utility): Linear utility: payments were moderate, below Dfl. 99, and remote from zero, above Dfl. 10. Zero-outcome gives trouble (Birnbaum). Specialists who were anticipating something sophisticated may be disappointed here: We are simply assuming linear utility! We think that linear utility is reasonable if outcomes are moderate and remote from zero, and using such outcomes for linear utility is our advice. m w b evaluated through U9 + D7 + R5. w m b 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, so prospect theory = rank-dependent utility.)

4. Economists usually want pessimism for equilibria etc.  Uncertainty aversion U m U m < > U U U U U U U U U U U w b w w b w b b m m m > > > < > > p 4 Note that we do unknown probs; figures only suggestive. Pessimism: (overweighting of bad outcomes) Optimism: (overweighting of good outcomes) (Likelihood) insensitivity: (overweighting of extreme outcomes) Empiricalfindings: (Primarily insensitivity; also pessimism; Tversky & Fox, 1997; Gonzalez & Wu 1999 )

5. U m U U w b > > 5 Our empirical predictions: 1. The decision weights depend on the ranking position. 2. The nature of rank-dependence: 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 the violations will be strong, so, if you don’t like PT, keep on listening!

6. 6 • 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 unsuited 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.

7. ? ) ) ( ( U D R U D R 103 47 12 94 64 8 7 What 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.”

8. + +++ 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 We: 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 33 16 ( ( ( ) ) )  49 46 i.e., 65 68  3 U > . Then we can conclude w 20 Now well-understood. But in his days creative novelty. I consider it the most important idea of decision theory of the past century: for something as intangible as beliefs, quantifications are conceivable, and can be derived from actions. 8 In explanation make clear that “check” means elicit from an individual from choice. Layout of stimuli You can see de Finetti’s intuition “shine” through, embedded in rank-dependence.

9. + + + + +++ +++ +++ +++ 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 34 28 33 37 33 31 33 33 33 33 16 43 33 25 19 33 33 33 22 40 ­­ ­­ ­­ ­­ p p p p p p p p Choice Choice Choice Choice 64 46 73 46 61 76 46 46 46 58 67 49 46 52 70 46 55 46 46 46 + + ¯¯ ¯¯ ¯¯ ¯¯ +++ +++ U ­­ ­­ 65 65 65 86 68 92 65 95 65 65 65 71 83 74 80 65 65 89 65 77 9/20 < D ¯¯ ¯¯ R 12/20. < p p p p Choice Choice 9 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. x x x x Imagine the following choices: 9 more for sure  20 more under U  12 more for sure x x

10. 10 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 • graph of performance of stocks during last two months • brief verbal comment on likelihood of increases/decreases of Dow Jones & Nikkei.

11. ordercompletely randomized 11 • 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

12. * * * * * 12 • 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. 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: suggests optimism R .52 (.18) .50 (.19) .50 (.18)

13. 13 Prospect theory can explain more of the variance in choice than any other theory. But the total variance explained is still way below half. One more thing … Two more effects to test, that can falsify prospect theory … • Collapsing (Loomes & Sugden, Luce, Birnbaum, etc.);only for certainty effect. Prospect theory accommodates the certainty effect. Do factors beyond prospect theory, such as collapse, reinforce or weaken the certainty effect? (2) Can direct assessments of emotions (e.g., regret) predict future choices better than past choices can predict those?

14. + + + + + + +++ +++ +++ +++ +++ +++ 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,c = = = = = = · · · · · · 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 14 Stimuli to test collapsing effects: … …

15. * * *** * * * * Falsifying a decision theory is easy. Getting a theory that does something for you is hard. • Results concerning factors beyond prospect theory 15 middle worst best Down-event: .35 (.20) .35 (.19) collapse suggests insensitivity D .34 (.18) .31 (.17) .34 (.17) .33 (.19) .33 (.18) noncoll. middle worst best .43 (.17) Up-event: .41 (.18) collapse U .44 (.18) .48 (.20) .46 (.18) suggests pessimism .51 (.23) .46 (.22) noncoll. middle worst best .49 (.20) Rest-event: .51 (.20) collapse R .52 (.18) .50 (.19) .50 (.18) suggests optimism .50 (.20) .53 (.20) noncoll.

16. U U D b,c w,c w,c Two of the 3 authors were surprised by this. 16 0.177 p = .019 0.172 p = .023 0.183 p = .015 regret 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!