Measuring the impact of uncertainty resolution Mohammed AbdellaouiCNRS-GRID, ESTP & ENSAM, France

Enrico Diecidue & Ayse Onler INSEAD, FranceESA conference, Roma, June 2007

Research question & motivationsHow does the evaluation of prospects change when they are to be resolved in the future?

Examples: Lottery ticket to be drawn today versus in a monthEnd-of-year bonus as a stock option or cashNew product developmentMedical tests

Research question & motivationsIntuition: sooner rather than later uncertainty resolution is preferred. Motivations: i) value of perfect information cannot be negative (Raiffa 1968) ii) psychological disutility for waiting (Wu 1999) iii) opportunity for planning and budgeting.

Related literature

Markowitz (1959), Mossin (1969), Kreps & Porteus (1978), Machina (1984), Segal (1990), Albrecht & Weber (1997), Smith (1998), Wakker (1999), Klibanoff & Ozdenoren (2007)Wu (1999): model for evaluating lotteries with delayed resolution of uncertainty. Model is rank-dependent utility with time dependent probability weighting functions.

Background and notation

Interested in (x, p; y)t uncertainty resolved at t in [0, T], (temporal prospects)Outcomes received at T, expressed as changes wrt status quoProspects rank-ordered

Background and notation (cont.)

Value of the temporal prospect (x, p; y)t

wit(p)U(x) + (1-wit(p))U(y),

where i = + for gains & i = - for losses.The decision maker selects the temporal prospect that has the highest evaluation.

Background and notation

Interested in 3 functions: wit(p) and U() The utility function U reflects the desirability of outcomes and satisfies U(0) = 0. Outcomes received at the same T, we consider the same utility function U. Probability weighting functions strictly increasing satisfy w+t(0) = w-t(0) = 0 w+t(1) = w-t(1) = 1 for all t in [0, T]. The impact of uncertainty resolution at a resolution date t for an event of probability p can be quantified through the comparison of wit(p) and wi0(p).

Background and notation

Preferences for two temporal prospects (either gain prospects or loss prospects) with common outcomes but different resolution dates depend only on the probabilities and resolution dates, and not the common outcomes. The usefulness of this condition is also emphasized in Wu (1999, p. 172): weak independence and formulated as follows: if a temporal prospect (x, p; y)t is preferred to the temporal prospect (x, q; y)t for x > y > 0 [x < y < 0] then, for all x > y [x < y < 0], the prospect (x, p; y)t should be preferred to the prospect (x, q; y)t.

Measuring the impact of uncertainty resolution

56 individual interviews, instructions, training sessions, random draw mechanism, hypothetical questionsTask: choice between two temporal prospectsSix iterations (i.e. choice questions) to obtain an indifferenceIterations generated by a bisection method.Counterbalance; control for response errors: repeated the third choice question of all indifferences at the end of each step described in Table 1.

The stimuli

The stimuli

The method: 4 steps

StepsObjectiveAssessed QuantityIndifferenceGainsStep 1Elicitation ofU(.) and w0+(.)Gi/6 G13/6G23/6G33/6G43/6G53/6Gi/6 ~ (1000, i/6; 0)0, i = 1,,6G13/6 ~ (2000, 3/6; 0 )0G23/6 ~ (2000, 3/6; 1000)0G33/6 ~ (1000, 3/6; 500 )0G43/6 ~ (1500, 3/6; 1000)0G53/6 ~ (2000, 3/6; 1500)0Step 2Elicitation ofwT+(.)g1g2g3g4g5(1000, 1/6; 0)0 ~ (g1, 1/6; 0)T(1000, 2/6; 0)0 ~ (g2, 2/6; 0)T(1000, 3/6; 0)0 ~ (g3, 3/6; 0)T(1000, 4/6; 0)0 ~ (g4, 4/6; 0)T(1000, 5/6; 0)0 ~ (g5, 5/6; 0)TLossesStep 3Elicitation of U(.) and w0-(.)Li/6L13/6L23/6L33/6L43/6L53/6-Li/6 ~ (-1000, i/6; 0)0, i = 1,,6-L13/6 ~ (-2000, 3/6; 0 )0-L23/6 ~ (-2000, 3/6; -1000)0-L33/6 ~ (-1000, 3/6; -500 )0-L43/6 ~ (-1500, 3/6; -1000)0-L53/6 ~ (-2000, 3/6; -1500)0Step 4Elicitation ofwT-(.)l1l2l3l4l5(-1000, 1/6; 0)0 ~ (-l1, 1/6; 0)T(-1000, 2/6; 0)0 ~ (-l2, 2/6; 0)T(-1000, 3/6; 0)0 ~ (-l3, 3/6; 0)T(-1000, 4/6; 0)0 ~ (-l4, 4/6; 0)T(-1000, 5/6; 0)0 ~ (-l5, 5/6; 0)T

On step 2U and w0+(.) known. wT+(.) can be elicited from the indifferences (1000, i/6; 0)0 ~ (gi, i/6; 0)T, i = 1,,5 From these indifferenceswT+(i/6) = w0+(i/6)[U(1000)/U(gi)], i = 1, , 5.

Results

t = 0t = T = 6MedianMeanStd.MedianMeanStd.Gainswt+ (1/6) 0.19710.18630.05880.15760.15670.0534wt+ (2/6) 0.32230.31070.06360.28670.28020.0635wt+ (3/6) 0.40320.40120.06460.38670.38410.0697wt+ (4/6) 0.52910.52680.08220.52850.52610.0876wt+ (5/6) 0.71330.70310.08520.72290.72380.0939Losseswt-(1/6)0.15470.13950.07470.15000.13810.0702wt-(2/6)0.27880.25850.10030.27720.25560.0973wt-(3/6)0.38860.36010.10320.37980.36200.1091wt-(4/6)0.51880.49130.12750.50740.49290.1426wt-(5/6)0.70100.69450.11760.72330.70340.1602

ResultsNS

Results

GainsLosses#: > 0t testCorr.#: > 0t testCorr.w0(1/6) vs. w6(1/6)4910.040.927240.600.973w0(2/6) vs. w6(2/6)4810.820.945271.200.983w0(3/6) vs. w6(3/6)334.630.91720-0.640.979w0(4/6) vs. w6(4/6)210.130.89722-0.280.957w0(5/6) vs. w6(5/6)9-2.900.82718-0.740.837

Results

Results

ResultsNS

GainsLossest = 0t = T = 6t = 0t = T = 6LSAwt(1/6) - wt(0) vs. wt(3/6) - wt(2/6)9.515.603.012.78wt(1/6) - wt(0) vs. wt(4/6) - wt(3/6)5.091.310.700.64USAwt(1) - wt(5/6) vs. wt(3/6) - wt(2/6)15.8411.6611.157.86wt(1) - wt(5/6) vs. wt(4/6) - wt(3/6)10.948.028.456.19

Results

w0(p) - w0(p-(1/6)) vs. w6(p) w6(p-(1/6))GainsLosses#: > 0t test#: > 0t testw0(1/6) - w0( 0 ) vs. w6(1/6) w6( 0 )4910.043240.609w0(2/6) - w0(1/6) vs. w6(2/6) w6(1/6)370.381320.828w0(3/6) - w0(2/6) vs. w6(3/6) w6(2/6)18-3.68225-1.619w0(4/6) - w0(3/6) vs. w6(4/6) w6(3/6)9-5.194260.070w0(5/6) - w0(4/6) vs. w6(5/6) w6(4/6)7-6.13320-0.912

Results

ConclusionsFirst individual elicitation of utility and pwf to understand the impact of delayed resolution: measured decision weights for immediate and delayed resolution of uncertaintyObserved temporal dimension of the uncertainty; pwf depends on the timing of resolution of uncertainty Gains: detected difference for small probabilitiesLosses: detected no significant differenceFound U for more convex (consistent with recent study by Noussair & Wu)

end

Roadmap

Research question + motivating examples

Measurement: method and results

RemarkTransformation of probabilities is robust phenomenon in decision under riskKahneman & Tversky 1979Empirically: Inverse-S shape for probability weighting function Abdellaoui 2000, Bleichrodt & Pinto 2000, Gonzalez & Wu 1999.

Start with the idea that in real life delayed resolution is the norm. then do example a la Wu, then talk about EU.Start with example on going to rotterdam and realizing that you dont have the wallet. The outcomes are both losses but you dont know now which one is the outcome. You will know once you go home.Use examples on pregnancy: in 9 months you will see (healthy/unhealthy). TThink about adoption: first you need to qualify, then there is an uncertain time for the resolution of uncertainty. Can be today or in 3 years. This gives you disutility, anxiety.What we did so far is CEt=(xT, pt; yT, 1-pt) with t=0,T. uT=u0, but wTw0.What we can do is CEt=(xt, pt; yt, 1-pt) with t=0,T1,T2, T-risky.

I am presenting you some research interestFeel free to interrupt me at any time

RECOMMENDATION: DO A SLIDE ON HOW TO POSITIONATE YOUR RESEARCH.

Need to present an example a la Wu: 50-50 lottery (0, 500), money will be paid in 5 years, uncertainty resolved now or in 5 years. What if outcome is 10 million, or health?Removed Tenure decisionBecause it is about anxiety.KP: They assume expected utility at every single-stage but give up the reduction of compound lotteries assumption and, thus, permit nonindifference to the timing of the resolution of uncertainty. Smith has delayed premium while evaluating income stream. It is a nice case where the timing of the resolution of uncertainty can rationally matter because of intermediate decisions. Preference for delayed resolution cannot be captured by standard models.Segal: gives an EU and RDU axiomatization without the reduction axiom.Wakker presents equivalent dynamic principles that surprisingly imply Bayesianism. foregone event independence, dynamic consistency, and reduction. The only manner to deal with delayed resolution of uncertainty is in the domain of nonEU and as a consequence, one of the above dynamic principles has to be given up.

Weak independence @ decide what to say about wu and where

Dont do the following unless weber is in the audience: Studies on relation between delay and riskAlbrecht & Weber 1997, Keren & Roelofsma 95, Rachlin et al 1991, Weber & Chapman 2005Findings: risky outcomes discounted less heavily than certain ones, less risk aversion for future outcomesLack of studies on probability weighting function. This is our focus.

Machina: Nice sentences on p.199, hard to think of any type of risk which does not involve the delayed resolution of uncertainty. Mossin argues that temporal prospects are the rule rather than the exception.Removed :Caplin & Leahy (2001)

There is an interesting comment from Mohammed. Economists have long thought on how to describe uncertainty resolution. Wu (1999) is the only theoretical attempt. However the natural benchmark for Wu is EU. However EU you don't observe it. our natural benchmark is CPT at time zero. This does not mean that we are biased in favor of CPT: if we dont detect probability transformation we dont detect it. Period. It is not ideological.

We agree with Wu when says that preferences are independent from common outcomes. So, preferences are outcomes independent, however we detect that preferences are not sign independent. We found sign dependence. Wu does not have it.

These objects do not exist under EU. . In an EU/SEU world the time of resolution of uncertainty is immaterial, because of the axioms of reduction of compound lotteries and compound independence. Insert table 1 and Comment verbally: The elicitation procedure consists of four steps and is summarized in Table 1. The second column of the table describes the objective of each elicitation step. The third column gives the quantity that is assessed, the fourth the indifference that is sought with specified outcomes and probabilities.The first step use certainty equivalents of instantaneously resolved prospects (i.e., prospects (x, p; y)0) to elicit U(.) and w0+(.) for gains. Certainty equivalents Gj3/6, j = 1, , 5, are used to estimate the parameter of the power utility function. No parametric form was used for the probability weighting function w0+(.). Once U is known, the decision weights w0+(i/6) can be determined from indifferences Gi/6 ~ (1000, i/6; 0)0, i = 1,,6.Having elicited U and w0+(.), the delayed probability weighting function wT+(.) can be elicited from the indifferences (1000, i/6; 0)0 ~ (gi, i/6; 0)T, i = 1,,5 constructed in the second step. We infer from these indifferences thatwT+(i/6) = w0+(i/6)[U(1000)/U(gi)], i = 1, , 5.The two last steps are exactly similar to steps one and two except that they concern losses instead of gains.

Insert table 1a and picture 1: They go together. One is the first contact with DW. Use colors for medians so that people can focus on this only.For picture 1, use sequential effect where you put the differences (only for the significant ones). The picture is the qualitative translation of table.Put these effects especially for the dark blue.Insert table 1a and picture 1: They go together. One is the first contact with DW. Use colors for medians so that people can focus on this only.For picture 1, use sequential effect where you put the differences (only for the significant ones). The picture is the qualitative translation of table.Put these effects especially for the dark blue.Here, I guess is where anxiety comes into the picture.Use colors to emphasize what you want to stress. First column tells that for losses the number of people is really 50-50, while for gains it decreases from 49 to 9.The correlations show that the data are not dispersed: people may think that the t-tests we have are the result of highly dispersed data. Not it is not the case.The values are not for one-side of two-side tests. Are just tests, then you decide how to use it. if we reject at 2 side, we reject at one side but not vice versa. At one side p-value is 4%, at 2 side is 8%. So which one is more conservative?Picture 2 says that for gains, large probabilities we dont detect any difference. We dont detect difference for losses as well. So, our technology is very refined: it detects when there is something!Table 3 is consistent with pictures 2 and 3 and consistent with past literature. The entries of table 3 are t-tests.Picture 2 says that for gains, large probabilities we dont detect any difference. We dont detect difference for losses as well. So, our technology is very refined: it detects when there is something!Table 3 is consistent with pictures 2 and 3 and consistent with past literature. The entries of table 3 are t-tests.Picture 2 says that for gains, large probabilities we dont detect any difference. We dont detect difference for losses as well. So, our technology is very refined: it detects when there is something!Table 3 is consistent with pictures 2 and 3 and consistent with past literature. The entries of table 3 are t-tests.

LSA: increments from p=0 have larger impact than identical increments in the interior.USA: increments from p=1 have larger impact than identical increments in the interior.

Insert picture 4 and table 4: they go together. Start explanation with losses. Table 4 and picture 4 are related. Number of people tell you that for losses is 50-50 while for gains there is an inversion from 49 to 7.Use nice examples like pregnancy test. What matters for anxiety and hope are small probabilities, not the large one.We also have results on curvature and elevation, but I dont present them.Insert picture 4 and table 4: they go together. Start explanation with losses. Table 4 and picture 4 are related. Number of people tell you that for losses is 50-50 while for gains there is an inversion from 49 to 7.Use nice examples like pregnancy test. What matters for anxiety and hope are small probabilities, not the large one.For a probability weighting function w(.) the difference [w(p) w(p-(1/6))] measures the psychological impact of passing from probability p-(1/6) to probability p; or equivalently, the psychological impact of a probability increase of 1/6 from probability p-(1/6).The figure below shows that for losses, the psychological impact of the probability interval [p-1/6, p] stays relatively constant for non-delayed uncertainty than for delayed for when p increases from 1/6 to 5/6. On the contrary, for gains, the impact of the above mentioned interval seems to be a decreasing function of p. This means that increasing a small probability of an event has more impact when the timing of resolution of uncertainty is shorter. And that increasing a probability in the second half of the unit interval has less impact when the timing of uncertainty resolution is shorter (see also the table below). Here is where you make comments on anxiety and hope.It tells you that when you have a probability of .8 to make a gain you care less to know it now or later. What matters are small probability events: there you want to know now.Overweighting of good outcomesThis means that increasing a small probability of an event has more impact when the timing of resolution of uncertainty is shorter than longer. And that increasing a probability in the second half of the unit interval has less impact when the timing of uncertainty resolution is shorter than longer.

In theory: choice, resolution, and payment are all the same time.Machina: Nice sentences on p.199, hard to think of any type of risk which does not involve the delayed resolution of uncertainty. Mossin argues that temporal prospects are the rule rather than the exception.

I dont plan to use these 20 minutes to talk about these studies