Stochastic choice under risk

1. Stochastic choice under risk Pavlo Blavatskyy June 24, 2006

2. Talk Outline • Introduction • Binary choice between a risky and a degenerate lottery • Fourfold pattern of risk attitudes • Discrepancy between certainty equivalent and probability equivalent elicitation methods • Preference reversal phenomenon • Binary choice between two risky lotteries • Generalized common consequence effect (Allais paradox) • Common ratio effect • Violations of the betweenness • Fit to experimental data • Conclusion

3. Introduction • Repeated choice under risk is often inconsistent • In 31.6% of all cases (Camerer, 1989) • In 26.5% of all cases (Starmer and Sugden, 1989) • In ~25% of all cases (Hey and Orme, 1989) • Stochastic nature of choice under risk is persistently documented in experimental data • … but remains largely ignored in the majority of decision theories

4. Conscious randomization? • Machina (1985) and Chew et al. (1991): stochastic choice as a result of deliberate randomization • individuals with quasi-concave preferences (like randomization) • The most preferred lottery is outside the choice set • Hey and Carbone (1995): does not fit the data

5. Models of stochastic choice • Core deterministic decision theory is embedded into a stochastic choice model • e.g. when estimating the parameters of decision theory from experimental data • Three models proposed in the literature

6. 1. Harless and Camerer (1994) • individuals generally choose among lotteries according to some deterministic decision theory • …but there is a constant probability that this deterministic choice pattern reverses (as a result of pure tremble) • Carbone (1997) and Loomes et al. (2002): fails to explain the experimental data

7. 2. Hey and Orme (1994) Random error / Fechner model • Deterministic decision theory → → net advantage of one lottery over another → → distorted by random errors • independently and identically distributed errors, zero mean and constant variance • Hey (1995) and Buschena and Zilberman (2000): heteroscedasticity • Camerer and Ho (1994) and Wu and Gonzalez (1996): choice probability as a logit function of net advantage • Loomes and Sugden (1998): predicts too many violations of first order stochastic dominance

8. 3. Loomes and Sugden (1995) • Individual preferences over lotteries are stochastic • Represented by random utility • Sopher and Narramore (2000): variation in individual decisions is not systematic, which strongly supports random error rather than random utility model

9. So… • Different models of stochastic choice • generate stochastic choice pattern from a deterministic core decision theory • successful in explaining some choice anomalies • not suitable for accommodating all known phenomena

10. New theory • Explain major stylized empirical facts as a consequence of random mistakes • …that individuals commit when evaluating a risky lottery • Make explicit predictions about stochastic choice patterns • …directly accessible for econometric testing on empirical data

11. Binary choice between a risky and a degenerate lottery • An individual has deterministic preferences over lotteries L(x1,p1;…,xn,pn ), x1<…<xn • represented by von Neumann-Morgenstern utility function u:R→R • Observed binary choices of an individual are, however, stochastic • …due to random errors that an individual commits when evaluating a risky lottery

12. … • An individual chooses lottery L over outcome x for certain if U(L) ≥ u(x) • Perceived expected utility of a lottery U(L) is equal to… • “true” expected utility of a lottery μL=Σi pi u(xi )according to individual preferences • plus a random error ξL • An individual always chooses lottery L over outcome x for certain if U(L) > u(x) • An individual behaves as if maximizing the perceived expected utility

13. No transparent errors ! • Assumption 1 (internality axiom) An individual always chooses lottery L over outcome x for certain if outcome x is smaller than x1 • …and an individual always chooses outcome x for certain over lottery L, if outcome x is higher than xn • → no errors in choice under certainty

14. Small errors are non-systematic ! • CEL is an outcome s.t. u(CEL)=μL • Assumption 2 For any ε>0 and a risky lottery L s.t. CEL  ε[x1,xn] the following events are equally likely to occur: • Lottery L is chosen over outcome CEL - ε for certain but not over outcome CEL for certain • Lottery L is chosen over outcome CEL for certain but not over outcome CEL+ ε for certain

15. Results • Assuming that individual maximizes perceived expected utility… • …together with assumptions 1-2 about the distribution of random errors… • we can explain: • Fourfold pattern of risk attitudes • Discrepancy between certainty equivalent and probability equivalent elicitation methods • Preference reversal phenomenon

16. Fourfold pattern of risk attitudes • Empirical observation that individuals often exhibit risk aversion when dealing with probable gains or improbable losses • … and the same individuals often exhibit risk seeking when dealing with improbable gains or probable losses • e.g. a simultaneous purchase of insurance and public lottery tickets

17. How do we explain?

18. Discrepancy between certainty equivalent and probability equivalent elicitation methods • Consider lottery L(x1,0.5;x2,0.5) • Outcome c is a minimum outcome that an individual is willing to accept in exchange for lottery L • Probability p is the highest probability s.t. an individual is willing to accept outcome c for certain in exchange for lottery L’(x1,1-p;x2,p)

19. … • Any deterministic decision theory predicts that p = 0.5 • Hershey and Schoemaker (1985): individuals, who initially reveal high c also declare that p > 0.5 one week later • Robust finding both for gains and losses

20. Explanation (rather logic behind it) • An individual makes random mistakes when evaluating a risky lottery L • → the perceived CE of L is equally likely to be below or above certain outcome ML • For risk-neutral guy, ML is simply (x1+x2)/2 • Accidentally, an individual has too high realization of the perceived CE, c >> ML • Now he or she searches for PE of this high outcome c

21. Explanation, continued • An individual is most likely to associate the sure outcome c with a lottery L’ • …whose perceived certainty equivalent is equally probable to be below or above c • For such lottery p>0.5 • If it were exactly 0.5 lottery L’ coincides with original lottery L • Median of distribution of CE of L is ML • But c >> ML

22. The preference reversal phenomenon • 2 lotteries of similar expected value • R yields a relatively high outcome with low probability (a dollar-bet) • S yields a modest outcome with probability ~1 (a probability-bet) • Individuals often choose S over R in a direct binary choice • … and simultaneously reveal a higher min selling price for R

24. Binary choice between two risky lotteries • Individual chooses lottery L over lottery L’ if μL+ξL ≥ μL’+ξL’ or μL+ξL,L’ ≥ μL’ • The same choice rule as in the Fechner model • But different assumptions about the distribution of an error term ξL,L’ • Large positive errors ξL,L’ ≥u(xn)-u(y1)+μL’ – μL large negative errors ξL,L’ ≤ u(x1)-u(ym)+μL’ – μL are not possible due to A1

25. Small errors are non-systematic (A2) • Error term ξL,L’ is symmetrically distributed on the utility scale • Assumption 2a For any ε>0 and any lotteries L(x1,p1;…xn,pn) & L’(y1,q1;…ym,qm) such that ε≤u(xn)-u(y1)+μL’ – μLand -ε≥u(x1)-u(ym)+μL’ – μL: prob(-ε≤ ξL,L’≤ 0)=prob( 0 ≤ ξL,L’ ≤ ε)

26. No error for “almost sure things” • A1 implies that an individual makes no errors when choosing among degenerate lotteries • When choosing between “almost sure things”, the dispersion of random errors is progressively narrower • … the closer are risky lotteries to the degenerate lotteries

27. Formally…

28. Results • “Fechner” choice rule together with assumptions 1, 2a and 3 explains: • Common consequence effect (Allais paradox) • Common ratio effect • Violations of betweenness (Blavatskyy, EL, 2006)

29. Fit to experimental data • Estimate: • stochastic decision theory (errors drawn from truncated normal distribution) and • RDEU (CPT) + standard Fechner error • on experimental data: • Loomes and Sugden (1998), 92 subjects make 46 binary choices twice • Hey and Orme (1994), 80 subjects make 100 binary choices twice

30. Loomes and Sugden (1998)

31. Hey and Orme (1994)

32. Conclusion • Individuals often make inconsistent decisions in repeated choice under risk • => preferences are stochastic (random utility) => observed randomness is due to random errors • Existing error models: • prob. of error is constant (Harless and Camerer, 1994) • distribution of errors is constant (Hey and Orme, 1994) • Too simple: • No errors are observed in choice between “sure things” • 20% - 30% of inconsistencies in non-trivial choice