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

new experimental evidence

John Heya and Noemi Paceb

aUniversity of York, UKbLUISS 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.

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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).

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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.

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3

1

( )i ii

SEU p u x

ip

1

( )1

rxu x

r

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

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3

1

( )i ii

CEU u x

1 2 3 2 3 1 3 1 2( ), ( ), ( ), ( ), ( ) and ( )c O c O c O c O O c O O c O O

1 2 3( ) 0 and ( ) 1c c O O O

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.

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3 3

1 1

min ( ) (1 ) max ( )i i i ip D p D

i i

AEU p u x p u x

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

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3 3

1 1 0

( ) ( )i i i ji ii i j n

VEU p u x A p u x

ip

n

j 3

1

( ) 0p j i jii

E p

A

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.

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Variational Modelwith just 2 outcomes

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Contraction Model

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• 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

3 3

1 1

( ) min ( ) (1 ) ( )i i i ip Pi i

V f p u z Pu z

1 2 3

for 1, 2,3

then (1 ) / 3 for 1,2,3

i i

i i

p p i

P p p p p i

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.

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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.

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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.

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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.

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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.

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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

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Subject

number 1

EU CEU AEU VEU NEU XEU

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

number 11

EU CEU AEU VEU NEU XEU

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

Some Preliminary Results 2 subjects from treatment 2

Subject

number 21

EU CEU AEU VEU NEU XEU

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

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Subject

number 31

EU CEU AEU VEU NEU XEU

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

Summary of ‘Winners’ on predicted log-likelihoods

Model Treatment 1 Treatment 2EU 5 8CEU (Choquet) 4 1AEU (Alpha) 2 3VEU (Vector) 1 2NEU (maxmiN) 5 4XEU (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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