Measuring the impact of uncertainty resolution

Mohammed AbdellaouiCNRS-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

wit(p)U(x) + (1-wi

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: 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 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 stimuli

The method: 4 stepsSteps Objective

Assessed Quantity

Indifference

Gains

Step 1

Elicitation ofU(.) and w0

+(.)

Gi/6

G13/6

G23/6

G33/6

G43/6

G53/6

Gi/6 ~ (1000, i/6; 0)0, i = 1,…,6

G13/6 ~ (2000, 3/6; 0 )0

G23/6 ~ (2000, 3/6; 1000)0

G33/6 ~ (1000, 3/6; 500 )0

G43/6 ~ (1500, 3/6; 1000)0

G53/6 ~ (2000, 3/6; 1500)0

Step 2

Elicitation ofwT

+(.)

g1

g2

g3

g4

g5

(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)T

Losses

Step 3

Elicitation of U(.) and w0

-(.)

Li/6

L13/6

L23/6

L33/6

L43/6

L53/6

-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)0

Step 4

Elicitation ofwT

-(.)

l1

l2

l3

l4

l5

(-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 2

• U 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 indifferences

wT+(i/6) = w0

+(i/6)[U(1000)/U(gi)],

i = 1, …, 5.

Results

t = 0 t = T = 6

Median Mean Std. Median Mean Std.

Gains

wt+

(1/6) 0.1971 0.1863 0.0588 0.1576 0.1567 0.0534

wt+ (2/6) 0.3223 0.3107 0.0636 0.2867 0.2802 0.0635

wt+ (3/6) 0.4032 0.4012 0.0646 0.3867 0.3841 0.0697

wt+ (4/6) 0.5291 0.5268 0.0822 0.5285 0.5261 0.0876

wt+ (5/6) 0.7133 0.7031 0.0852 0.7229 0.7238 0.0939

Losses

wt-(1/6) 0.1547 0.1395 0.0747 0.1500 0.1381 0.0702

wt-(2/6) 0.2788 0.2585 0.1003 0.2772 0.2556 0.0973

wt-(3/6) 0.3886 0.3601 0.1032 0.3798 0.3620 0.1091

wt-(4/6) 0.5188 0.4913 0.1275 0.5074 0.4929 0.1426

wt-(5/6) 0.7010 0.6945 0.1176 0.7233 0.7034 0.1602

Results

1/6 2/6 3/6 4/6 5/60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Probabilities

Med

ian

Dec

isio

n W

eig

ht

w+0 (.)

w+6 (.)

w-0 (.)

w-6 (.)

NS

ResultsGains Losses

# : > 0 t test Corr. # : > 0 t test Corr.

w0(1/6) vs. w6(1/6) 49 10.04 0.927 24 0.60 0.973

w0(2/6) vs. w6(2/6) 48 10.82 0.945 27 1.20 0.983

w0(3/6) vs. w6(3/6) 33 4.63 0.917 20 -0.64 0.979

w0(4/6) vs. w6(4/6) 21 0.13 0.897 22 -0.28 0.957

w0(5/6) vs. w6(5/6) 9 -2.90 0.827 18 -0.74 0.837

Results

0 1/6 2/6 3/6 4/6 5/6 10

1/6

2/6

3/6

4/6

5/6

1 PWFs for gains: Non-delayed vs. delayed uncertainty resolution

Probability p

w+ 0(p

), w

+ 6(p)

w(p)= p

w+0(.)

w+6(.)

Results

0 1/6 2/6 3/6 4/6 5/6 10

1/6

2/6

3/6

4/6

5/6

1 PWFs for losses: Non-delayed vs. delayed uncertainty resolution

Probability p

w- 0(p

), w

- 6(p)

w(p)= p

w-0(.)

w-6(.)

Results

Gains Losses

t = 0 t = T = 6 t = 0 t = T = 6

LSA

wt(1/6) - wt(0) vs. wt(3/6) - wt(2/6) 9.51 5.60 3.01 2.78

wt(1/6) - wt(0) vs. wt(4/6) - wt(3/6) 5.09 1.31 0.70 0.64

USA

wt(1) - wt(5/6) vs. wt(3/6) - wt(2/6) 15.84 11.66 11.15 7.86

wt(1) - wt(5/6) vs. wt(4/6) - wt(3/6) 10.94 8.02 8.45 6.19

NS

Results

w0(p) - w0(p-(1/6)) vs. w6(p) – w6(p-(1/6))

Gains Losses

# : > 0 t test # : > 0 t test

w0(1/6) - w0( 0 ) vs. w6(1/6) – w6( 0 ) 49 10.043 24 0.609

w0(2/6) - w0(1/6) vs. w6(2/6) – w6(1/6) 37 0.381 32 0.828

w0(3/6) - w0(2/6) vs. w6(3/6) – w6(2/6) 18 -3.682 25 -1.619

w0(4/6) - w0(3/6) vs. w6(4/6) – w6(3/6) 9 -5.194 26 0.070

w0(5/6) - w0(4/6) vs. w6(5/6) – w6(4/6) 7 -6.133 20 -0.912

Results

1/6 2/6 3/6 4/6 5/6-0.03

-0.02

-0.01

0

0.01

0.02

0.03

Probability p

[w

0(p)

- w

0(p-1

/6)]

- [

w6(p

) -

w6(p

-1/6

)]

Gains

Losses

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)

end

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.