Non-Probability-Multiple-Prior Models of
Decision Making Under Ambiguity:
new experimental evidence
John Heya and Noemi Paceb
University of York, UK
bLUISS Guido Carli, Italy
a
Thursday Workshop. DERS, York, 16th December 2010
Aim of the Research (1)
• We examine the performance of non-probability-multipleprior 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:
3
SEU   pi u ( xi )
i 1
where
pi is the subjective probability of state i (and p + p + p =1).
1
2
3
• We have to specify the utility function; we assume the CARA form:
x1 r
u ( x) 
1 r
• 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
3
CEU   i u ( xi )
i 1
• where xi is the payoff in outcome i and
• where the weights wi depend upon the ordering of the
outcomes and upon 6 capacities
c(O1 ), c(O2 ), c(O3 ), c(O2  O3 ), c(O1  O3 ) and c(O1  O2 )
• Note that
c()  0 and c(O1  O2  O3 )  1
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:
3
3
AEU   min  pi u ( xi )  (1   ) max  pi u ( xi )
pD
i 1
pD
i 1
• 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
 3


VEU   pi u ( xi )  A    pi jiu ( xi ) 



i 1
i

1


0

j

n


3
• Where
pi baseline subjective probabilities
n finite integer between 0 and i-1
 j satisfies E p ( j )  3 pi ji  0
i 1
adjustment
function
that reflects attitudes toward ambiguity
A
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
3
3
i 1
i 1
V ( f )   min  pi u ( zi )  (1   )  Pu
i ( zi )
pP
• 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
pi  pi for i  1, 2,3

then Pi  pi  (1  p1  p2  p3 ) / 3 for i  1, 2,3




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
CEU
AEU
VEU
NEU
XEU
Subject
EU
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
EU
CEU (Choquet)
AEU (Alpha)
VEU (Vector)
NEU (maxmiN)
XEU (maxmaX)
Treatment 1
5
4
2
1
5
2
Treatment 2
8
1
3
2
4
2
24
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.
25
Thank you
26