Measuring the impact of uncertainty resolution Mohammed Abdellaoui CNRS-GRID, ESTP & ENSAM, France Enrico Diecidue & Ayse Onçüler INSEAD, France ESA conference, Roma, June 2007

Research question & motivations How 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 month End-of-year bonus as a stock option or cash New product development Medical tests

Research question & motivations Intuition: 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 quo Prospects rank-ordered

Background and notation (cont.) Value of the temporal prospect (x, p; y) t w i t (p)U(x) + (1-w i t (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: w i t (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 w i t (p) and w i 0 (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’ [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 questions Task: choice between two temporal prospects Six iterations (i.e. choice questions) to obtain an indifference Iterations 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 method: 4 steps StepsObjective Assessed Quantity Indifference Gains Step 1 Elicitation of U(.) and w 0 + (.) G i/6 G 1 3/6 G 2 3/6 G 3 3/6 G 4 3/6 G 5 3/6 G i/6 ~ (1000, i/6; 0) 0, i = 1,…,6 G 1 3/6 ~ (2000, 3/6; 0 ) 0 G 2 3/6 ~ (2000, 3/6; 1000) 0 G 3 3/6 ~ (1000, 3/6; 500 ) 0 G 4 3/6 ~ (1500, 3/6; 1000) 0 G 5 3/6 ~ (2000, 3/6; 1500) 0 Step 2 Elicitation of w T + (.) g1g2g3g4g5g1g2g3g4g5 (1000, 1/6; 0) 0 ~ (g 1, 1/6; 0) T (1000, 2/6; 0) 0 ~ (g 2, 2/6; 0) T (1000, 3/6; 0) 0 ~ (g 3, 3/6; 0) T (1000, 4/6; 0) 0 ~ (g 4, 4/6; 0) T (1000, 5/6; 0) 0 ~ (g 5, 5/6; 0) T Losses Step 3 Elicitation of U(.) and w 0 - (.) L i/6 L 1 3/6 L 2 3/6 L 3 3/6 L 4 3/6 L 5 3/6 -L i/6 ~ (-1000, i/6; 0) 0, i = 1,…,6 -L 1 3/6 ~ (-2000, 3/6; 0 ) 0 -L 2 3/6 ~ (-2000, 3/6; -1000) 0 -L 3 3/6 ~ (-1000, 3/6; -500 ) 0 -L 4 3/6 ~ (-1500, 3/6; -1000) 0 -L 5 3/6 ~ (-2000, 3/6; -1500) 0 Step 4 Elicitation of w T - (.) l1l2l3l4l5l1l2l3l4l5 (-1000, 1/6; 0) 0 ~ (-l 1, 1/6; 0) T (-1000, 2/6; 0) 0 ~ (-l 2, 2/6; 0) T (-1000, 3/6; 0) 0 ~ (-l 3, 3/6; 0) T (-1000, 4/6; 0) 0 ~ (-l 4, 4/6; 0) T (-1000, 5/6; 0) 0 ~ (-l 5, 5/6; 0) T

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

Results t = 0t = T = 6 MedianMeanStd.MedianMeanStd. Gains w t + (1/6) w t + (2/6) w t + (3/6) w t + (4/6) w t + (5/6) Losses w t - (1/6) w t - (2/6) w t - (3/6) w t - (4/6) w t - (5/6)

Results NS

Results GainsLosses # :  > 0 t testCorr. # :  > 0 t testCorr. w 0 (1/6) vs. w 6 (1/6) w 0 (2/6) vs. w 6 (2/6) w 0 (3/6) vs. w 6 (3/6) w 0 (4/6) vs. w 6 (4/6) w 0 (5/6) vs. w 6 (5/6)

Results

GainsLosses t = 0t = T = 6t = 0t = T = 6 LSA w t (1/6) - w t (0) vs. w t (3/6) - w t (2/6) w t (1/6) - w t (0) vs. w t (4/6) - w t (3/6) USA w t (1) - w t (5/6) vs. w t (3/6) - w t (2/6) w t (1) - w t (5/6) vs. w t (4/6) - w t (3/6) NS

Results w 0 (p) - w 0 (p-(1/6)) vs. w 6 (p) – w 6 (p-(1/6)) GainsLosses # :  > 0 t test # :  > 0 t test w 0 (1/6) - w 0 ( 0 ) vs. w 6 (1/6) – w 6 ( 0 ) w 0 (2/6) - w 0 (1/6) vs. w 6 (2/6) – w 6 (1/6) w 0 (3/6) - w 0 (2/6) vs. w 6 (3/6) – w 6 (2/6) w 0 (4/6) - w 0 (3/6) vs. w 6 (4/6) – w 6 (3/6) w 0 (5/6) - w 0 (4/6) vs. w 6 (5/6) – w 6 (4/6)

Results

Conclusions First individual elicitation of utility and pwf to understand the impact of delayed resolution: measured decision weights for immediate and delayed resolution of uncertainty Observed temporal dimension of the uncertainty; pwf depends on the timing of resolution of uncertainty Gains: detected difference for small probabilities Losses: detected no significant difference Found U for “more convex” (consistent with recent study by Noussair & Wu)

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Roadmap Research question + motivating examples Measurement: method and results

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