On Dow Jones & Nikkei indexes today: Peter P. Wakker & Enrico Diecidue & Marcel Zeelenberg This file will be on my homepage on coming Monday. Domain: Decisions with unknown probabilities (“ambiguity”). 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? Measuring Decision Weights for Unknown Probabilities by Means of Prospect Theory U: both go Up (  ) D: both go Down (  ) R: Rest event (  =; one up other down, or at least one constant) We will analyze in terms of prospect theory.. Don’t forget to make yellow comments invisible 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.

Rank-dependence: greatest idea in decision theory since 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. 2 Some History of Prospect Theory I think that people in this society don’t know very well what I will tell in the 2 nd half of this page. 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.

(Subjective) expected utility (linear utility): 3 UDRUDR 9 75 () evaluated through  U 9 +  D 7 +  R 5. UDRUDR 2 86 () evaluated through  U 2 +  D 8 +  R 6. A Reformulation of Prospect Theory (1992) through Rank- Dependence (Cumulative) prospect theory generalizes expected utility by rank-dependence (“decision-way” of expressing nonadditivity of belief). b b m m w w (We consider only gains, so prospect theory = rank-dependent utility.) For specialists, remark that there are two middle weights but for simplicity we ignore difference. 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. 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. You can claim that probabilities should be nonadditive, but for decision theory that as such doesn’t mean anything.

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

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

– 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! 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): Most here is for DUR.

Our experiment: 7 ? UD R ( ) UD R ( ) What would you choose? Shows how hard 3-outcome-prospect choices are. –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.”

i.e., UDR ()  U D R () 20 3  U >. w Then we can conclude ¹· =   p p Choice U D R Layout of stimuli Classical method (de Finetti) to check if 3 20  U > : UDR 2000 () U D R ().).  Check if We: this reveals that b U U 3 20 >. U U w 3 20 > ? How check if UDR 2000 ()  U D R () Answer: add a “reference gamble” (side payment). Check if refer- ence gamble 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. You can see de Finetti’s intuition “shine” through, embedded in rank-dependence. In explanation make clear that “check” means elicit from an individual from choice.

+ +++ Choice   ¹· = p p ¹· =   p p Choice U D R Choice   ¹· = p p Choice   ¹· = p p ¹· =   p p Choice U D R Choice   ¹· = p p Choice   ¹· = p p Choice ¹· =   p p U D R Choice   ¹· = p p Choice   ¹· = p p xx xx Imagine the following choices: xx xx x x 9 more for sure  20 more under U  12 more for sure U U w < 9/20 < 12/20. This provides a tractable manner for quantitatively measuring decision weights under ambiguity. Combines de Finetti’s betting odds schemes with rank- dependence.

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.

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 order completely randomized 11

.44 (.18) worst best middle worst Rest-event: RR.52 (.18).50 (.19).50 (.18) Results under prospect theory 12 * best middle worst suggests insensitivity DD.34 (.18).31 (.17).34 (.17) * * Up-event: best middle UU.48 (.20).46 (.18) * * Down-event: suggests pessimism suggests optimism 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. 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.

13 One more thing … Two more effects to test, that can falsify prospect theory … (1)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? Prospect theory can explain more of the variance in choice than any other theory. But the total variance explained is still way below half.

  ¹· = p p Choice Choice   ¹· = p p Choice ¹· =   p p U D R …   ¹· = p p Choice Choice   ¹· = p p ¹· =   p p Choice U D R … 14 UU w,n UU w,c Stimuli to test collapsing effects:

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

regret correlations between regret and decision weights UU b,c p =.019 UU w,c p =.023 DD w,c p =.015 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! 16 Two of the 3 authors were surprised by this.