Choquet OK? Gianna Lotito (Università del Piemonte Orientale, Italy John D Hey (LUISS, Italy and York, UK) Anna Maffioletti (University of Turin )
Ambiguity • The theoretical literature is vast… • ..with many stories told and many preference functionals proposed. • The experimental literature too is vast… • …with many different ways of producing ambiguity in the laboratory. • But there remains a fundamental gap between the two.
Ambiguity • Here we propose a novel way of creating ambiguity in the laboratory, and varying the amount of ambiguity… • … and use our data to estimate preference functionals (rather than test between them)… • …so we can see which fits best and how varying ambiguity changes things.
The Beautiful Bingo Blower • Our method for creating ambiguity in the lab is through the Beautiful Bingo Blower… • …like the Dodo, once common in British seaside resorts… • …found after a 4-year search on e-bay.
The Beautiful Bingo Blower • Treatment 1 • 2 pink • 5 blue • 3 yellow
The Beautiful Bingo Blower • Treatment 2 • 4 pink • 10 blue • 6 yellow
The Beautiful Bingo Blower • Treatment 3 • 8 pink • 20 blue • 12 yellow
The BBB • The experimenter cannot manipulate the implementation of the ambiguity device. • The device is transparent. • Probabilities cannot be calculated on an objective basis (and there are not second-order probabilities). • The existence and amount of ambiguity is not subject-specific.
The Design • Participation fee of £10. • 3 colours (pink, blue and yellow) . • 3 amounts of money (-£10, £10 and £100). • Pairwise choice questions. • 33 = 27 different lotteries and hence 27x26/2 = 351 possible pairwise choices. • Omitting those with dominance leaves us with 162 pairwise choice questions. • Order and left-right juxtaposition randomised.
The Treatments • Treatment 1: 2 pink, 5 blue, 3 yellow • Treatment 2: 4 pink, 10 blue, 6 yellow • Treatment 1: 8 pink, 20 blue, 12 yellow • In Treatment 1, the balls can be counted. • In Treatment 2, the pink can be counted and possibly the yellow but not the blue. • In Treatment 3, no colour can be counted.
The Data • 48 subjects -15 on Treatment 1, 17 on Treatment 2 and 16 on Treatment 3. • £44.37 average payment (note: answering at random gives expected payment = £33.33; if risk-neutral and knew the true probabilities,expected payment = £46.63). • For each subject we have 162 responses. • (Note: minimum 30 seconds for each.)
The Preference Functionals • Subjective Expected Utility (SEU) • ‘Prospect Theory’ (PT) • Choquet Expected Utility (CEU) • Maximin • Maximax • Minimax Regret
Normalisation • In SEU, ‘PT’ and CEU there is a utility function. • We normalise so that • u(-£10) = 0 • u(£10) = u • u(£100) = 1 • We estimate u along with other parameters.
Notation • We denote a lottery by Here Siis the state (one of in which the lottery pays out xi.
‘Prospect Theory’ • Exactly the same as SEU except that we do not impose the condition that • pa + pb + pc = 1 • In SEU we estimate u, pa, pb, and pc subject to pa + pb + pc = 1. • In PT we estimate u, pa, pb, and pc .
Choquet Expected Utility In this model we estimate Note that there is no necessity that wde = wd + we for any d or e.
Maximin Here l1, l2and l3denote the three outcomes on one of the two lotteries, L, ordered from the worst to the best, and m1, m2and m3denote the outcomes on the other lottery, M, also ordered from the worst to the best:
Maximax Here l1, l2and l3denote the three outcomes on one of the two lotteries, L, ordered from the worst to the best, and m1, m2and m3denote the outcomes on the other lottery, M, also ordered from the worst to the best:
Minimax Regret With this preference functional, the decision maker is envisaged as imagining each possible ball drawn, calculating the regret associated with choosing each of the two lotteries, and choosing the lottery for which the maximum regret is minimized. Again there are no parameters to estimate, though it is assumed that there is a larger regret associated with a larger difference between the outcome on the chosen lottery and the outcome on the non-chosen lottery.
Estimation • We proceed using Maximum Likelihood implemented with GAUSS. • Stochastic Specification… • …we assume a Fechnerian error story – preferences are measured with error such that the differences in the evaluations of the two lotteries is N(0, s2). • We estimate s along with the other parameters.
Tests • All the analysis done subject by subject. • We have tested the hypothesis that subjects choose at random – rejected. • We have carried out likelihood ratio tests among the nested models (SEU, ‘PT’ and CEU). • We have carried out non-nested Vuong tests between the non-nested models. • We arrive at the Bottom Line….
Conclusions • CEU fits well. • Effect of increasing ambiguity seems to be minor – particularly in terms of additivity. • Rules of thumb identified and may be more important as additivity increases. • Additivity measures interesting. • Choquet OK?... • … yes.