1

Ten “New Paradoxes” of Risky Decision Making

Michael H. BirnbaumDecision Research CenterCalifornia State University,

Fullerton

Classical Paradoxes:Contradictions with Expected

Value

St. Petersburg Paradox: People prefer small sum of cash to a chance to play the gamble with infinite EV.

Risk Aversion: People prefer small sum to gambles with higher EV.

Bernoulli (1738)If a poor man had a lottery ticket that would pay 20,000 ducats or nothing with equal probability, he would NOT be ill-advised to sell it for 9,000 ducats. A rich man would be ill-advised to refuse to buy it for that price.

Expected Utility Theory

• Could explain why people would buy and sell gambles

• Explain sales and purchase of insurance

• Explain the St. Petersburg Paradox• Explain risk aversion

Allais (1953) “Constant Consequence” Paradox

Called “paradox” because preferences contradict Expected Utility.

A: $1M for sure f B: .10 to win $2M.89 to win $1M

.01 to win $0

C: .11 to win $1M p D: .10 to win $2M.89 to win $0 .90 to win $0

Expected Utility (EU) Theory

€

EU(G ) = pii =1

n

∑ u(xi )

A f B u($1M) > .10u($2M) + .89u($1M) +.01u($0) Subtr. .89u($1M): .11u($1M) > .10u($2M)+.01u($0)

Add .89u($0): .11u($1M)+.89u($0) > .10u($2M)+.90u($0)

C f D. So, Allais Paradox refutes EU.

Allais and Ellsberg Paradoxes

• Allais “Constant Ratio” Paradox• Ellsberg Paradoxes• These violated EU and SEU

generalization to uncertain events.

Cumulative Prospect Theory/ Rank-Dependent

Utility (RDU)

€

CPU(G ) = [W ( pj )− W ( pj )j =1

i −1

∑j =1

i

∑i =1

n

∑ ]u(xi )

Probability Weighting Function, W(P)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Decumulative Probability

Decumulative Weight

CPT Value (Utility) Function

0

20

40

60

80

100

120

140

0 20 40 60 80 100 120 140

Objective Cash Value

Subjective Value

Cumulative Prospect Theory/ RDU

• Tversky & Kahneman (1992) CPT is more general than EU or (1979) PT, accounts for risk-seeking, risk aversion, sales and purchase of gambles & insurance.

• Accounts for Allais Paradoxes, chief evidence against EU theory.

• Imply certain violations of restricted branch independence.

• Shared Nobel Prize in Econ. (2002)

RAM/TAX Models

€

x1 > x2 > K > xi > K > xn > 0

€

RAMU(G ) =

a( i,n)t( pi )u(xi )i =1

n

∑

a( i,n)t( pi )i =1

n

∑

RAM Model Parameters

Probability Weighting Function, t(p)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1Objective Probability, p

€

a(1,n) = 1; a(2,n) = 2;K ; a( i,n) = i;K ; a(n ,n) = n

RAM implies inverse-SCertainty Equivalents of

($100, p; $0)

0

20

40

60

80

100

0 0.2 0.4 0.6 0.8 1

Probability to Win $100

Certainty Equivalent

Allais “Constant Consequence” Paradox

Can be analyzed to compare CPT vs RAM/TAX

A: $1M for sure f B: .10 to win $2M.89 to win $1M

.01 to win $0

C: .11 to win $1M p D: .10 to win $2M.89 to win $0 .90 to win $0

Allais Paradox Analysis

• Transitivity: A f B and B f C A f C

• Coalescing: GS = (x, p; x, q; z, r) ~ G = (x, p + q; z, r)• Restricted Branch Independence:

€

S = (x, p;y,q;z,r) f R = ( ′ x , p; ′ y ,q;z,r)

€

′ S = (x, p;y,q; ′ z ,r) f ′ R = ( ′ x , p; ′ y ,q; ′ z ,r)

€

⇔

A: $1M for sure f B: .10 to win $2M.89 to win $1M

.01 to win $0

A’: .10 to win $1M f B: .10 to win $2M .89 to win $1M .89 to win $1M .01 to win $1M .01 to win $0

A”: .10 to win $1M f B’: .10 to win $2M .89 to win $0 .89 to win $0 .01 to win $1M .01 to win $0

C: .11 to win $1M f D: .10 to win $2M .89 to win $0 .90 to win $0

Decision Theories and Allais Paradox

Branch Independence

Coalescing Satisfied Violated

Satisfied EU, CPT*OPT*

RDU, CPT*

Violated SWU, OPT* RAM, TAX

Kahneman (2003)

“…Our model implied that ($100, .01; $100, .01) — two mutually exclusive .01 chances to gain $100—is

more valuable than the prospect ($100, .02)…The prediction is wrong…of course, because most decision makers will spontaneously transform the former prospect into the latter and treat them as equivalent in subsequent operations of evaluation and choice. To eliminate the problem, we proposed that decision makers, prior to evaluating the prospects, perform an editing operation that collects similar outcomes and adds their probabilities. ”

Web-Based Research

• Series of Studies tests: classical and new paradoxes in decision making.

• People come on-line via WWW (some in lab).

• Choose between gambles; 1 person per month (about 1% of participants) wins the prize of one of their chosen gambles.

• Data arrive 24-7; sample sizes are large; results are clear.

Choice data TAX CE

S R % R S R

15 red $50

85 black $7

10 blue $100

90 white $7

80* 14 < 18

10 red $50

05 blue $50

85 white $7

10 black $100

05 purple $7

85 green $7

49 16 > 15

85 red $100

10 white $50

05 blue $50

85 black $100

10 yellow $100

05 purple $7

63* 68 < 70

85 black $100

15 yellow $50

95 red $100

05 white $7

20* 76 > 62

Allais Paradoxes• Do not require large, hypothetical prizes.• Do not depend on consequence of $0.• Do not require choice between “sure

thing” and 3-branch gamble.• Largely independent of event-framing• Best explained as violation of coalescing

(violations of BI run in opposition).• See JMP 2004, 48, 87-106.

Stochastic Dominance

If the probability to win x or more given A is greater than or equal to the corresponding probability given gamble B, and is strictly Higher for at least one x, we say that A Dominates B by First Order Stochastic Dominance.

€

P(x ≥ t | A) ≥ P(x ≥ t | B)∀ t ⇒ A f B

Preferences Satisfy Stochastic Dominance

Liberal Standard: If A stochastically dominates B,

€

P(A f B) ≥ 12

Reject only if Prob of choosing B is signficantly greater than 1/2.

RAM/TAX Violations of Stochastic Dominance

Which gamble would you prefer to play?

Gamble A Gamble B

90 reds to win $9605 blues to win $1405 whites to win $12

85 reds to win $9605 blues to win $9010 whites to win $12

70% of undergrads choose B

Which of these gambles would you prefer to play?

Gamble C Gamble D

85 reds to win $9605 greens to win $9605 blues to win $1405 whites to win $12

85 reds to win $9605 greens to win $9005 blues to win $1205 whites to win $12

90% choose C over D

RAM/TAX Violations of Stochastic Dominance

Violations of Stochastic Dominance Refute CPT/RDU, predicted by RAM/TAX

Both RAM and TAX models predicted this violation of stochastic dominance before the experiment, using parameters fit to other data. These models do not violate Consequence monotonicity).

Questions

• How “often” do RAM/TAX models predict violations of Stochastic Dominance?

• Are these models able to predict anything?

• Is there some format in which CPT works?

Do RAM/TAX models imply that people should violate stochastic dominance?Rarely. Only in special cases. Consider “random” 3-branch gambles: *Probabilities ~ uniform from 0 to 1. *Consequences ~ uniform from $1 to $100.

Consider pairs of random gambles. 1/3 of choices involve Stochastic Dominance, but only 1.8 per 10,000 are predicted violations by TAX. Random study of 1,000 trials would unlikely have found such violations by chance. (Odds: 7:1 against)

Can RAM/TAX account for anything?

• No. These models are forced to predict violations of stochastic dominance in the special recipe, given these properties:

• (a) risk-seeking for small p and • (b) risk-averse for medium to large

p in two-branch gambles.

Analysis: SD in TAX model

TAX Model

-20

0

20

-1 0 1

Value of δ

γ = 2

γ = 1

γ = .85

γ = .7

γ = .6

γ = .5

Formats:Birnbaum & Navarrete

(1998)

.05 .05 .90 .10 .05 .85$12 $14 $96 $12 $90 $96

I: .05 to win $12 J: .10 to win $12 .05 to win $14 .05 to win $90 .90 to win $96 .85 to win $96

Birnbaum & Martin (2003)

Web Format (1999b)

5. Which do you choose?

I: .05 probability to win $12 .05 probability to win $14 .90 probability to win $96 OR J: .10 probability to win $12 .05 probability to win $90 .85 probability to win $96

Reversed Order

5. Which do you choose?

I: .90 probability to win $96 .05 probability to win $14 .05 probability to win $12 OR

J: .85 probability to win $96 .05 probability to win $90 .10 probability to win $12

Pie Charts

Tickets Format

I: 90 tickets to win $96 05 tickets to win $14 05 tickets to win $12

OR

J: 85 tickets to win $96 05 tickets to win $90 10 tickets to win $12

List Format

I: $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96 $14 $12

OR

J: $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96 $90 $12, $12

Semi-Split List

I: $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96 $14 $12 ORJ: $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96 $90 $12, $12

Marbles: Event-Framing

5. Which do you choose?

I: 90 red marbles to win $96 05 blue marbles to win $14 05 white marbles to win $12 OR

J: 85 red marbles to win $96 05 blue marbles to win $90 10 white marbles to win $12

Decumulative Probability Format

5. Which do you choose?

I: .90 probability to win $96 or more .95 probability to win $14 or more 1.00 probability to win $12 or more OR

J: .85 probability to win $96 or more .90 probability to win $90 or more 1.00 probability to win $12 or more

(This had a significantly higher rate of violation)

New Formats now being tested: Ticket Tables

Unaligned Table: Coalesced

Unaligned Table: Split

Aligned Table: Coalesced

Aligned Table: Split Form

Fresh Data: hot from Web-Since 4/2/04

• New Tickets Format 83% 04%• Unaligned Table 80% 12%• Aligned Table 78% 08%• Violations of SD in coalesced and

split forms--Choices 5 and 11.(No. Participants so far: 209-- between-Ss: 58, 74, 77)

Coalescing and SD

Gamble A Gamble B

90 red to win $9605 white to win $1205 blue to win $12

85 green to win $9605 yellow to win $9610 orange to win $12

Here coalescing A = B, but 67% of 503 Judges chose B.

Studies of SD: models vs. heuristics

• Do people violate SD by simply averaging the consequences and ignoring probabilities?

• RAM or TAX more accurate in predicting when violations ARE or ARE NOT observed?

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Probability to win $96 in G–

Proportion of Violations of SD

Observed

Pred-TAX

Pred_RAM

G– = ($96,.85 – r; $90,.05; $12,.1 + r)

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Probability to win $96 in G–

Proportion of Violations of SD

Observed

Pred_TAX

Pred_RAM

G– = ($96, .85 – r; $90, .05 + r; $12, .1)

SD Study 4: Consequences

• 90 black win $97• 05 yellow win $15• 05 purple win $13

• 85 red to win $95• 05 blue to win $91• 10 white to win $11

Predictions of TAX: 70% for ($97, $13) 68% for ($95, $11). Observed: 72% and 68% n = 315

SD Study 4: middle Branch

• 90 red win $96• 05 blue win $14

• 05 white win $12

• 85 red win $96• 05 blue win $70• 10 white win $12

Predictions of RAM and TAX

Are 64% and 60%, respectively.

Observed is 70%, n = 315.

Effect of Middle BranchG– = ($96, .85; x , .05; $12, .10)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50 60 70 80 90 100

Value of Middle Consequence in G–

Observed Data

TAX prior preds

RAM prior preds

SD Study 5: All 3 conseqs.

• 90 black win $97• 05 yellow win $15• 05 purple win $13

• 85 red win $90• 05 blue win $80• 10 white win $10

Predictions of TAX and RAM

are 63% and 50%, respectively.

Observed is 57%*, n = 394

Summary: 23 Studies of SD, 8653 participants

• Huge effects of splitting vs. coalescing of branches-70% vs 10%

• Small effects of gender, education, in decision-making

• Very small effects of probability format, displays

• Miniscule effects of event framing (framed vs unframed)

Summary SD (continued)

• People respond to changes in probability, contrary to counting heuristic.

• Both RAM and TAX can violate probability monotonicity, data closer to TAX than RAM. (* more)

• People respond to changes in consequences, but not extremely.

Case against CPT/RDU

• 1. Violations of Stochastic Dominance• 2. Violations of Coalescing (Event-

Splitting)• 3. Violations of Lower Cumulative

Independence• 4. Violations of Upper Cumulative

Independence• 5. Violations of Wu’s 3-Upper Tail

Independence ( OI, but tests RDU)

Upper Cumulative Independence

R': 72% S': 28% .10 to win $10 .10 to win $40 .10 to win $98 .10 to win $44 .80 to win $110 .80 to win $110

R''': 34% S''': 66% .10 to win $10 .20 to win $40 .90 to win $98 .80 to win $98

€

′ R f ′ S ⇒ ′ ′ ′ R ff ′ ′ ′ S

Lower Cumulative Independence

R: 39% S: 61% .90 to win $3 .90 to win $3 .05 to win $12 .05 to win $48 .05 to win $96 .05 to win $52

R'': 69% S'': 31%.95 to win $12 .90 to win $12.05 to win $96 .10 to win $52

€

R p S ⇒ ′ ′ R pp ′ ′ S

Summary: UCI & LCI

22 studies with 33 Variations of the Choices, 6543 Participants, & a variety of display formats and procedures. Significant Violations found in all studies.

Wu’s (94) Upper Tail Test

• 50 to win $0• 07 to win $68• 43 to win $92

• 50 to win $0• 07 to win $68• 43 to win $97

34%• 52 to win $0• 48 to win $92

62%• 52 to win $0• 05 to win $92• 43 to win $97

5 “Paradoxes” that contradict CPT model

6. Violations of Restricted Branch Independence opposite predictions of inverse-S weighting function. Allais.

7. Violations of 4-distribution independence, favor TAX over RAM

8. 3-Lower Distribution Independence9. 3-Upper Distribution Independence10. 3-2 Distribution Independence.

Restricted Branch Indep.

S’: .1 to win $40

.1 to win $44 .8 to win $100

S: .8 to win $2 .1 to win $40 .1 to win $44

R’: .1 to win $10

.1 to win $98 .8 to win $100

R: .8 to win $2 .1 to win $10 .1 to win $98

CPT R’S, but SR’ is sig. more freq.

Restricted Branch Independence Summary

• 28 studies of RBI with 7341 participants

• Most find SR’ reversals are more

frequent than RS’• This pattern is opposite the

implications of CPT with inverse-S weighting function

3-Upper Distribution Ind.

S’: .10 to win $40

.10 to win $44 .80 to win $100

S2’: .45 to win $40

.45 to win $44 .10 to win $100 67%*

R’: .10 to win $4

.10 to win $96 .80 to win $100 56%*

R2’: .45 to win $4

.45 to win $96 .10 to win $100

3-Lower Distribution Ind.

S’: .80 to win $2 .10 to win $40 .10 to win $44

S2’: .10 to win $2 .45 to win $40 .45 to win $44

R’: .80 to win $2 .10 to win $4 .10 to win $96

R2’: .10 to win $2 .45 to win $4 .45 to win $96

3-2 Lower Distribution Ind.

S’: .04 to win $2

.48 to win $4066% .48 to win $44

S2’: .50 to win $40

69% .50 to win $44

R’: .04 to win $2 .48 to win $4 .48 to win $96

R2’: .50 to win $4 .50 to win $96

CPT R’ f S’ and S2’ f R2’.

For More Information:

http://psych.fullerton.edu/mbirnbaum/

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mbirnbaum@fullerton.edu