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					Non-Probability-Multiple-Prior Models of
  Decision Making Under Ambiguity:
      new experimental evidence


             John Heya and Noemi Paceb
                  a University of York, UK
                  b
                   LUISS Guido Carli, Italy



   Thursday Workshop. DERS, York, 16th December 2010
          Aim of the Research (1)
• We examine the performance of non-probability-multiple-
  prior models of decision making under ambiguity from the
  perspective of their descriptive and predictive power.

• We try to answer the question as to whether some new
  theories of behaviour under ambiguity are significantly
  better than Subjective Expected Utility (SEU).

• We reproduce ambiguity in the laboratory in a transparent
  and non-probabilistic way, using a Bingo Blower.

                                                              2
          Aim of the Paper (2)
• In contrast with previous experiments, rather
  than carrying out statistical tests comparing
  the various theories, we estimate a number of
  different models using just part of our
  experimental data and then use the estimates
  to predict behaviour on the rest of the data,
  and see which models produce better
  predictions.
• We might call this the Wilcox method.
                                                  3
    Theories under Investigations (1)
• Focus on the class of non-probability-multiple prior models that
  proceed through the use of a preference functional:
• Subjective Expected Utility (SEU).
• Prospect Theory (as SEU but with probabilities that do not sum to 1).
• Choquet Expected Utility (CEU).
• (Cumulative Prospect Theory - is the same as CEU.)
• Alpha Expected Utility (AEU), with subcases - Maxmax, Maxmin.
• Vector Expected Utility (VEU).
• *Variational Representation of Preferences (Maccheroni, Marinacci and
  Rustichini 2006).
• *Confidence Function (Chateneuf and Faro, 2009).
• *Contraction Model (Gajdos, Hayashi, Tallon and Vergnaud, 2008).

                                                                          4
   Theories under Investigations (2)
• SEU: agents attach subjective probabilities (which satisfy the usual
  probability laws) to the various possible events and choose the
  lottery which yields the highest expected utility:



where
    is the subjective probability of state i (and p1+ p2+ p3=1).
• We have to specify the utility function; we assume the CARA form:



• Prospect theory – same except that the probabilities do not add to 1.

                                                                          5
  Theories under Investigations (3)
• CEU (Schmeidler, 1989): an uncertain prospect with three
  possible mutually exclusive outcomes (O1, O2 and O3) has
  Choquet Expected Utility given by



• where xi is the payoff in outcome i and
• where the weights wi depend upon the ordering of the
  outcomes and upon 6 capacities

• Note that


                                                             6
  Theories under Investigations (4)
• AEU (Ghirardato et al. 2004): the decisions are made on the
  basis of a weighted average of the minimum expected utility
  over the nonempty, weak compact and convex set D of
  probabilities on and the maximum expected utility over
  this set:

• where
   is the index of the ambiguity aversion of the decision maker
  =1 pessimistic evaluation: Maxmin Expected Utility Theory
  =0 optimistic evaluation: Maxmax Expected Utility Theory
• Here the D is a set of possible probabilities.
                                                                  7
   Theories under Investigations (5)
• VEU (Siniscalchi, 2009): an uncertain prospect is assessed according to a
  baseline expected utility evaluation and an adjustment that reflects the
  individual’s perception of ambiguity and her attitude toward it. This
  adjustment is itself a function of the exposure to distinct sources of
  ambiguity, and its variability



• Where
   baseline subjective probabilities
   finite integer between 0 and i-1
   satisfies
  adjustment function that reflects attitudes toward ambiguity


                                                                              8
Other models under consideration
• Variational Representation of Preferences
  (Maccheroni, Marinacci and Rustichini 2006).
• Confidence Function (Chateneuf and Faro,
  2009).
• Contraction Model (Gajdos, Hayashi, Tallon
  and Vergnaud, 2008).
• Cumulative Prospect Theory is the same as
  Choquet Expected Utility in our context.

                                                 9
Variational Model
    with just 2 outcomes




                           10
           Contraction Model


• Where α measures imprecision aversion and Pi
  (i=1,…,3) is the Steiner Point of the set P.
• If the set P is the set (p1, p2, p3) such that
  p1+p2+p3=1 and



                                               11
         Previous Contributions
• There have been a number of attempts to test a
  number of theories, but few to estimate
  preference functionals. Amongst these latter:
• Hey, Lotito, Maffioletti (2010), Journal of Risk and
  Uncertainty.
• Andersen et al. (2009).
• Most people test between various theories.
• We prefer our methodology (fit and predict) to
  testing.
                                                     12
      The Beautiful Bingo Blower
• These are videos of the York Bingo Blower.
• A pilot experiment was carried out in the CESARE
  lab at LUISS and a full-scale study at York (data
  not yet analysed) We had two treatments:
• Treatment 1: 2 pink, 5 blue, 3 yellow.
• Here is a video showing the first treatment.
• Treatment 2: 8 pink, 20 blue, 12 yellow.
• Here is a video showing the second treatment.

                                                  13
             The Experiment
• We asked the subjects a total of 76 questions.
• There were two types of question:
  1. The first type of question was to allocate a
  given quantity of tokens between two of the
  three colours in the Bingo Blower.
  2. The second type of question was to allocate
  a given quantity of tokens between one of the
  three colours in the Bingo Blower and the
  other two.
                                                14
      The First Type of Question
• In this type we gave the subject a given quantity
  of tokens and asked him or her to allocate the
  tokens between two of the three colours in the
  Bingo Blower.
• That is, between pink and blue, or between blue
  and yellow, or between yellow and pink.
• We also told the exchange rate between tokens
  and money for each colour.
• An allocation of tokens between the two colours
  implies an amount of money for each of the two
  colours.
                                                      15
     The Second Type of Question
• In this type we gave the subject a quantity of tokens and
  asked him or her to allocate the tokens between one of the
  three colours in the Bingo Blower and the other two.
• That is, between pink and not-pink (that is, blue and yellow),
  or between blue and not-blue (that is, yellow and pink), or
  between yellow and not-yellow (that is, pink and blue).
• We also told the exchange rate between tokens and money
  for each colour.
• An allocation of tokens between the one colour and the other
  two implies an amount of money for the one colour and the
  other two.

                                                               17
                  Payment
• At the end of the experiment, for each
  subject, one of the 76 questions was picked at
  random.
• The subject then ejected one ball from the
  Bingo Blower (he or she could not manipulate
  the ejection).
• Its colour determined their payment, as we
  show in the following slides.

                                               19
   Payment if this problem selected at
                 random




• If the ball ejected was yellow you would get paid £13.50, if the
  ball ejected was blue you would get paid £23.00 and if the ball
  ejected was pink you would get paid nothing.
   Payment if this problem selected at
                 random




• If the ball ejected was pink you would get paid £20.02, if the ball
  ejected was blue you would get paid £9.99 and if the ball ejected
  was yellow you would get paid £9.99.
              Some Preliminary Results
             2 subjects from treatment 1
Subject         EU             CEU         AEU         VEU         NEU         XEU
number 1
fitted lls      -190.2071   -183.0185   -181.5304   -188.1796   -181.7416   -190.2071
predicted lls   -48.8033    -60.5265    -58.7214    -63.0753    -59.1122    -48.8033



Subject         EU            CEU         AEU         VEU         NEU         XEU
number 11
fitted lls      -205.1974   -199.6514   -201.5773   -198.3144   -199.5326   -205.1972
predicted lls   -50.7617    -57.6046    -58.5032    -59.6637    -58.0319    -50.7617




                                                                                       22
              Some Preliminary Results
             2 subjects from treatment 2
Subject         EU             CEU         AEU         VEU         NEU         XEU
number 21
fitted lls      -204.0909   -196.9011   -194.3246   -204.0388   -200.8982   -197.8221
predicted lls    -54.2818    -70.2031    -62.0990    -54.3734    -68.4758    -54.6955




Subject         EU            CEU         AEU         VEU         NEU         XEU
number 31
fitted lls      -222.5434   -211.0259   -214.0669   -222.4006   -214.1413   -220.6171
predicted lls   -61.9205    -60.7264    -62.4350    -61.5515    -60.5216    -63.2526


                                                                                        23
       Summary of ‘Winners’ on
       predicted log-likelihoods
Model           Treatment 1   Treatment 2
EU                   5             8
CEU (Choquet)        4             1
AEU (Alpha)          2             3
VEU (Vector)         1             2
NEU (maxmiN)         5             4
XEU (maxmaX)         2             2


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                 Next Steps
• We have now carried out at York an experiment
  with 89 subjects and the same 76 questions.
• We are now about to analyse the data.
• We are planning to fit Subjective EU, Prospect
  Theory, Choquet EU, Alpha EU, Vector EU, the
  Variational model, the Contraction Model and
  perhaps others.
• We await suggestions and specific forms.

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Thank you


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