Mixture Models of Choice under Risk

1. Mixture Models of Choice under Risk Peter G. MOFFATT University of East Anglia, Norwich, United Kingdom. (with Anna CONTE and John D. HEY, both LUISS, Rome)

2. Estimating preference functionals for choice under risk from the choice behaviour of individuals. • Data: Hey (2001, Exp. Ec.). • 53 subjects • a sequence of 500 tasks • spread over 5 days • they choose between risky prospects of the following kind:

3. Lottery q Lottery p €50 €100 €50 €100 €0 €0 Random Lottery Incentive system used (one chosen lottery selected at random and played for real).

4. All 500 problems involved three of the four outcomes £0, £50, £100 and £150. Let us denote them by xi (i= 1, 2, 3, 4) and the corresponding utility values by ui (i = 1, 2, 3, 4). The probabilities of the four outcomes in these two lotteries in pairwise-choice problem t (t = 1, … , 500) are p1t p2t p3t p4t and q1t q2t q3t q4t

5. Subjects compute: (p-lottery) where the P’s and Q’s are not necessarily the correct probabilities, but they are derived from the correct probabilities in the following manner: (q-lottery)

6. Here the function w(.) is a probability weighting function.

7. THE MIXTURE ASSUMPTION • EITHER the subject is EU ( ), • OR the subject is Rank Dependent EU (RDEU). RDEU: Two versions: • Power weighting function: • Quiggin weighting function:

8. Subjects choose the lottery p (q) if and only if where here is a Fechnerian error term, with represents extent of . computational error by subjects.

9. Assume CRRA utility: r represents risk attitude. Observed binary dependent variable:

10. DISTRIBUTIONAL ASSUMPTIONS • Risk attitude (EU maximisers): • Risk attitude and rank-dependent parameter (RDEU maximisers):

11. Probability ω that subject loses concentration and chooses randomly. Tremble parameter Mixing proportion A proportion p of the population is RDEU; a proportion 1-p is EU.

12. Likelihood contribution for a given subject: Sample log-likelihood: LogL maximised using maximum simulated likelihood.

13. Results: Maximised LogL

14. (Marginal) Log-likelihood function (Quiggin)

16. Computation of posterior probabilities:

17. Posterior Probabilities (Quiggin)

18. Distribution of RD parameter

19. 95% bounds for weighting function (Quiggin)

20. Summary (1) • We have taken into account heterogeneity in behaviour between individuals and within individuals:

21. Summary (2) • by ‘heterogeneity between individuals’ we mean that people are different, not only in terms of which type of preference functional that they have, but also in terms of their parameters for these functionals. This means that trying to estimate a ‘representative agent’ preference functional to represent the preference functional of all the individuals may well lead to biased estimates.

22. Summary (3) • by ‘heterogeneity within individuals’ we mean that behaviour can differ for a given individual solving the same choice problem.

23. We have proposed: • solutions to both these problems, concentrating particularly on using a mixture model and introducing unobserved heterogeneity terms to capture the heterogeneity of preference functionals across individuals. • an econometric approach that deals with technical difficulties associated with these issues.

24. Results: • 80% of the population are RDEU with their own (Quiggin) weighting function; 20% are EU. • Mixture model tells us (with high probability) which subjects are of which type.

25. Mixture Models of Choice under Risk Peter MOFFATT University of East Anglia, Norwich (with Anna CONTE and John D. HEY, both LUISS, Rome) Thank you