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

2. 2 Classical Paradoxes: Contradictions with Expected Value Risk Aversion: People prefer small sum to gamble with higher EV. St. Petersburg Paradox: People prefer small sum of cash to a chance to play the gamble with infinite EV.

3. 3 Expected Utility Theory Could explain why people would buy and sell gambles Explain sales and purchase of insurance Explain the (original) St. Petersburg Paradox Explain risk aversion

4. 4 Allais and Ellsberg Paradoxes Allais “Constant Ratio” Paradox Allais “Constant Consequence” Ellsberg Paradoxes These violated EU and SEU generalization to uncertain events.

5. 5 Cumulative Prospect Theory/ RDU Tversky & Kahneman (1992) CPT, 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. Implies violations of restricted branch independence. Shared Nobel Prize in Econ. (2002)

6. 6 Case against CPT/RDU 1. Violations of Stochastic Dominance 2. Violations of Coalescing (Event-Splitting Effects) 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)

7. 7 Case against CPT (violations of inverse- S ) 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 10. 3-2 Distribution Independence.

8. 8 “ Configural Weight” Models Birnbaum’s RAM and TAX models predicts the first six of the new paradoxes. (3 in advance of expts) TAX model accounts for the last four, which violate RAM. Predictions preceded the experiments. Basic ideas date to early-1970s; descriptive equations. Marley & Luce: new GDU model similar to TAX.

9. 9 RAM & TAX models These are configural weight models in which branches carry weight depending on probability and rank of consequence. Splitting the upper branch makes gamble better. Splitting the lower branch makes it worse.

10. 10 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

11. 11 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:

12. 12 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

13. 13 Decision Theories and Allais Paradox Branch Independence CoalescingSatisfiedViolated SatisfiedEU, CPT* OPT* RDU, CPT* ViolatedSWU, OPT*RAM, TAX

14. 14 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. ”

15. 15 Web-Based Research Series of studies testing 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.

16. 16

17. 17 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 violations of coalescing (violations of BI run in opposition). See JMP 2004, 48, 87-106.

18. 18 Stochastic Dominance If the probability to win t or more given A is greater than or equal to the corresponding probability given gamble B, and is strictly higher for at least one t, we say A Dominates B by First Order Stochastic Dominance.

19. 19 Preferences Satisfy Stochastic Dominance If A stochastically dominates B, Reject only if prob of choosing B is signficantly greater than 1/2.

20. 20 RAM/TAX  Violations of SD

21. 21 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

22. 22 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

23. 23 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.

24. 24 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?

25. 25 Do RAM/TAX models imply that people “often” violate stochastic dominance? No. 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)

26. 26 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.

27. 27 Analysis: SD in TAX model

28. 28 Formats: Birnbaum & Navarrete (1998).05.05.90.10.05.85 $12 $14 $96$12 $90 $96

29. 29 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)

30. 30 Web Format (1999b)

31. 31 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

32. 32 Pie Charts

33. 33 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

34. 34 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

35. 35 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

36. 36 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

37. 37 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)

38. 38 New Formats hot off the Web: Ticket Tables

39. 39 Unaligned Table: Coalesced

40. 40 Unaligned Table: Split

41. 41 Aligned Table: Coalesced

42. 42 Aligned Table: Split Form

43. 43 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: 433-- between-Ss: 141, 141, 151)

44. 44 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.

45. 45 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?

46. 46 G– = ($96,.85 – r; $90,.05; $12,.1 + r)

47. 47 G– = ($96,.85 – r; $90,.05 + r; $12,.1)

48. 48 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

49. 49 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.

50. 50 Effect of Middle Branch

51. 51 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

52. 52 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)

53. 53 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.

54. 54 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.

55. 55 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

56. 56 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.

57. 57 Summary There are 10 “new paradoxes” of CPT. All are compatible with TAX model. Marley and Luce have proposed Gains Decomposition Utility, which, like TAX, violates coalescing. These models remain candidates descriptive models of decision making.

58. 58 For More Information: http://psych.fullerton.edu/mbirnbaum/ Download recent papers from this site. Follow links to “brief vita” and then to “in press” for recent papers. mbirnbaum@fullerton.edu