Choquet OK?

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Choquet OK?

• Gianna Lotito (Università del Piemonte Orientale,
  Italy
• John D Hey (LUISS, Italy and York, UK)
• Anna Maffioletti (University of Turin )

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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.

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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.

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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.

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The Beautiful Bingo Blower

• Treatment 1
• 2 pink
• 5 blue
• 3 yellow

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The Beautiful Bingo Blower

• Treatment 2
• 4 pink
• 10 blue
• 6 yellow

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The Beautiful Bingo Blower

• Treatment 3
• 8 pink
• 20 blue
• 12 yellow

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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.

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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.

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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.

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The Basic Screen
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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.)

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The Preference Functionals

• Subjective Expected Utility (SEU)
• Prospect Theory (PT)
• Choquet Expected Utility (CEU)
• Maximin
• Maximax
• Minimax Regret

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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.

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Notation

• We denote a lottery by

Here Si is the state (one of
in which the lottery pays out xi.
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Subjective Expected Utility
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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 .

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Choquet Expected Utility
In this model we estimate
Note that there is no necessity that wde wd
we for any d or e.
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Maximin
Here l1, l2 and l3 denote the three outcomes on
one of the two lotteries, L, ordered from the
worst to the best, and m1, m2 and m3 denote the
outcomes on the other lottery, M, also ordered
from the worst to the best
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Maximax
Here l1, l2 and l3 denote the three outcomes on
one of the two lotteries, L, ordered from the
worst to the best, and m1, m2 and m3 denote the
outcomes on the other lottery, M, also ordered
from the worst to the best
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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.

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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.

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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.

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The Bottom Line
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Additivity
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Summary Statistics on Additivity
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Attitude to Ambiguity
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Scatters of S3 against S1
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Scatters of S4 against S1
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Scatters of S5 against S1
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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.

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