Risk Aversion and Expected Utility Theory: A Field Experiment with Large and Small Stakes

1. Risk Aversion and Expected Utility Theory: A Field Experiment with Large and Small Stakes Matilde Bombardini (UBC) Francesco Trebbi (University of Chicago GSB) Expected Utility: A Field Experiment with Large Stakes

2. Motivation How well does expected utility theory explain the behavior of an average person? • Several criticisms of expected utility theory. Here we address: • Rabin’s critique – unreasonable rate of decline of marginal utility of wealth • Consistency within the same individual • Bayesian updating • Why this game • High-stakes: 500,000 euros, 250,000 euros,… • Non-infinitesimal probabilities • Simple structure • No ability involved in answering questions • Ability only involved in Bayesian updating Expected Utility: A Field Experiment with Large Stakes

3. Timing of the game 20 Boxes opening and offers alternate: • 6 boxes opened • First offer (always a change) • 3 boxes opened • Second offer (always monetary) • 3 boxes opened • Third offer • 3 boxes opened • Fourth offer • 3 boxes opened • Fifth offer • Uncover content of contestant’s box Expected Utility: A Field Experiment with Large Stakes

4. The contestants • Selection • Based on photogenic looks and personality • No questions asked about gambling and/or other games • Characteristics • Gender and age range • Marital status and children • Region/City • Urban/Rural • Occupation (to approximate income) • Self-selection • Risk lovingness does not seem to play a role • The producers: “People want to be on TV” • Data allows to reject self-selection Expected Utility: A Field Experiment with Large Stakes

5. Information structure • What the “infamous” (man on the phone) knows: • The content of all the boxes • The past behavior of contestants • What the contestant knows: • The distribution of prizes (constant for all episodes) • The past history of the episode • The strategy of the “infamous” (empirical distribution of offers) for E(lottery)>20,000 Euros • Stage 4: Corr (Relative offer, Relative prize in hand) = 0.32 • Stage 5: Corr (Relative offer, Relative prize in hand) = 0.36 • Both correlations significant at 1 percent Expected Utility: A Field Experiment with Large Stakes

6. What the data can tell • Risk preferences • Acceptance of monetary offers reveals a lower bound on degree of risk aversion • Rejection of monetary offers reveals an upper bound on degree of risk aversion • Box changes place upper bounds and lower bounds if offers are informative • Consistency of the choice set • Sequence of acceptances and rejections may reveal inconsistencies • Bayesian updating • If consider informativeness of the signal “offer” then changes can reveal information on Bayesian updating Expected Utility: A Field Experiment with Large Stakes

7. The contestant problem • Consider only monetary offers • Stage • Monetary offer at stage s: • Lottery at stage s: with • is the set of prizes not discovered at stage s • Let if the offer is accepted at stage s • The problem of the contestant if offers are not informative is: • The problem of the contestant if offers are informative Expected Utility: A Field Experiment with Large Stakes

8. Prize probabilities • Non-informative offers • Priors are not affected by offers • Where is the number of boxes in set • Informative offers • Offers affect probabilities according to empirical likelihood: • The posterior, conditional on the offer is calculated by Bayes rule: Expected Utility: A Field Experiment with Large Stakes

9. Solution of the optimization problem • Finite horizon dynamic optimization problem • Recursive solution: • At stage 5 given two prizes left • Given offer m5 • Compute expected utility and compare it to utility from offer • Decision to accept the offer: • Compute for all possible offers at s=5 and all possible paths • Obtain continuation value at s=4 and compare to utility from offer m4 • Solve recursively Expected Utility: A Field Experiment with Large Stakes

10. Empirical Model Non informative offers (NIO) • Take CRRA utility function: • Calculate continuation value at each stage s as function of g and: • Possible future box openings • Future empirical distribution of offers • Optimal behavior in future offers • Empirical probability of change vs. monetary offer • Empirical distribution of offers as a function of expected value of the lottery • Compare continuation value with utility from monetary offer and find threshold g: • Acceptance: • Rejection: Expected Utility: A Field Experiment with Large Stakes

11. Empirical Model Informative offers (IO) • Same basic structure as for NIO • Obtain empirical signal (offer) likelihood from the data: • For each stage s = 3, 4, 5 • For each box held by participant at stage s (relative to remaining boxes) • Segment offers in terms of percentage of expected value (5 segments: 0-20, 20-40, 40-60, 60-80, 80-100) • Obtain frequency of offers at stage s as function of box held by contestant (three likelihood matrices L3, L4, L5) • Bayesian updating of priors conditional on offer observed at stage s • Calculate continuation value at each stage s as function of g and: • Updated probabilities • Future Bayesian updating and optimal behavior Expected Utility: A Field Experiment with Large Stakes

12. Empirical Model Maximum Likelihood: • Assume unobserved heterogeneity in risk preferences: for individual j: • Partition the set of individuals in 3 subsets: • NA : individuals for which only observed • NR : individuals for which only observed • NAR : observations for which both and observed Expected Utility: A Field Experiment with Large Stakes

13. Empirical Model • Multiple rejections: • Log Likelihood function Expected Utility: A Field Experiment with Large Stakes

14. Expected Utility: A Field Experiment with Large Stakes

18. Expected Utility: A Field Experiment with Large Stakes

19. Expected Utility: A Field Experiment with Large Stakes

23. Expected Utility: A Field Experiment with Large Stakes

25. Conclusions • People are approximately risk neutral when stakes are small  Reasonable rate of decline of marginal utility of wealth • Rabin’s critique: • “Data sets dominated by modest-risk investment opportunities are likely to yield much higher estimates of risk aversion than data sets dominated by larger-scale investment opportunities” -- (Rabin, Econometrica 2000) We obtain the opposite, consistently with expected utility theory • We find that it is reasonable to use the same underlying expected utility function for small and large stakes • People make fewer mistakes when stakes are high Expected Utility: A Field Experiment with Large Stakes