Ten “New Paradoxes” Refute Cumulative Prospect Theory of Risky Decision Making Michael H. Birnbaum Decision Research Center California State University, Fullerton

2 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.

3 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.

4 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

5 Allais (1953) “Constant Consequence” Paradox Called “paradox” because preferences contradict Expected Utility. A: $1M for sure  B:.10 to win $2M.89 to win $1M.01 to win $0 C:.11 to win $1M  D:.10 to win $2M.89 to win $0.90 to win $0

6 Expected Utility (EU) Theory A  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  D. So, Allais Paradox refutes EU.

7 Allais and Ellsberg Paradoxes Allais “Constant Ratio” Paradox Ellsberg Paradoxes These violated EU and SEU generalization to uncertain events.

8 Cumulative Prospect Theory/ Rank-Dependent Utility (RDU)

9 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)

10 RAM/TAX Models

11 RAM Model Parameters

12 RAM implies inverse- S

13 Allais “Constant Consequence” Paradox Can be analyzed to compare CPT vs RAM/TAX A: $1M for sure  B:.10 to win $2M.89 to win $1M.01 to win $0 C:.11 to win $1M  D:.10 to win $2M.89 to win $0.90 to win $0

14 Allais Paradox Analysis Transitivity: A  B and B  C  A  C Coalescing: GS = (x, p; x, q; z, r) ~ G = (x, p + q; z, r) Restricted Branch Independence:

15 A: $1M for sure  B:.10 to win $2M.89 to win $1M.01 to win $0 A ’ :.10 to win $1M  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  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  D:.10 to win $2M.89 to win $0.90 to win $0

16 Decision Theories and Allais Paradox Branch Independence CoalescingSatisfiedViolated SatisfiedEU, CPT* OPT* RDU, CPT* ViolatedSWU, OPT*RAM, TAX

17 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. ”

18 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.

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20 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,

21 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.

22 Preferences Satisfy Stochastic Dominance Liberal Standard: If A stochastically dominates B, Reject only if Prob of choosing B is signficantly greater than 1/2.

23 RAM/TAX  Violations of Stochastic Dominance

24 Which gamble would you prefer to play? Gamble AGamble B 90 reds to win $96 05 blues to win $14 05 whites to win $12 85 reds to win $96 05 blues to win $90 10 whites to win $12 70% of undergrads choose B

25 Which of these gambles would you prefer to play? Gamble CGamble D 85 reds to win $96 05 greens to win $96 05 blues to win $14 05 whites to win $12 85 reds to win $96 05 greens to win $90 05 blues to win $12 05 whites to win $12 90% choose C over D

26 RAM/TAX  Violations of Stochastic Dominance

27 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).

28 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?

29 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)

30 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.

31 Analysis: SD in TAX model

32 Formats: Birnbaum & Navarrete (1998) $12 $14 $96$12 $90 $96

33 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)

34 Web Format (1999b)

35 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

36 Pie Charts

37 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

38 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

39 Semi-Split List 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

40 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

41 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)

42 New Formats hot off the Web: Ticket Tables

43 Unaligned Table: Coalesced

44 Unaligned Table: Split

45 Aligned Table: Coalesced

46 Aligned Table: Split Form

47 Violations of SD: hot from Web-Since 4/2/04 New Tickets Format 84% 04% Unaligned Table 81% 12% Aligned Table 72% 08% Violations of SD in coalesced and split forms--Choices 5 and 11. (No. Participants: between-Ss: 141, 141, 151)

48 Coalescing and SD Gamble AGamble B 90 red to win $96 05 white to win $12 05 blue to win $12 85 green to win $96 05 yellow to win $96 10 orange to win $12 Here coalescing  A = B, but 67% of 503 Judges chose B.

49 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?

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

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

52 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

53 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.

54 Effect of Middle Branch

55 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

56 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)

57 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.

58 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)

59 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 $ to win $110 R''': 34% S''': 66%.10 to win $10.20 to win $40.90 to win $98.80 to win $98

60 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

61 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.

62 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

63 5 Other “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 Independence 9. 3-Upper Distribution Independence Distribution Independence.

64 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.

65 Inverse-S and RDU/CPT

66 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

67 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 (n = 1075)

68 3-Lower Distribution Ind. S ’ :.80 to win $2.10 to win $40.10 to win $44 68% S2 ’ :.10 to win $2.45 to win $40.45 to win $44 70% 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 Prior TAX: S &S2; CPT predicts R&R2.

Lower Distribution Ind. S ’ :.04 to win $2.48 to win $40 66%.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’  S’ and S2’  R2’.

70 Distribution Independence: Summary Prior CPT model implies opposite violations from patterns observed. RAM model implies no violations. TAX model can violate 4-DI and 3-UDI. Prior TAX model correctly predicts major trends. Fewer studies of these properties than of others.

71 Summary There are 10 “new paradoxes” of CPT, all of which are compatible with TAX model. Marley and Luce have proposed Gains Decomposition Utility, which, like TAX, violates coalescing.

72 For More Information: Download recent papers from this site. Follow links to “brief vita” and then to “in press” for recent papers.