A Perfect Cocktail Recipe: Mixing Decision Theories and Models of Stochastic Choice

A Perfect Cocktail Recipe: Mixing Decision

Theories and Models of Stochastic Choice

Ganna PogrebnaJune 29, 2007

Blavatskyy, Pavlo and Ganna Pogrebna (2007) “Models of Stochastic Choice and Decision Theories: Why Both are Important for Analyzing Decisions” IEW Working Paper 319

Talk Outline

IntroductionTelevision shows

Affari Tuoi Deal or No Deal UK

DataEstimated Models of Stochastic

Choice and Decision TheoriesResults

IntroductionNon-expected utility theories

Response to violations of EUTTested in laboratory experiments

Natural experiment in TV showsMore representative subject pool Significantly higher incentives

Deal or No Dealrisky lottery vs. amount for certainhigh stakesdynamic problem

Affari Tuoi Italian prototype of Deal or No Deal Aired six days a week on RAI Uno All contestants self-select into the show 20 contestants participate in each episode Contestants are randomly assigned sealed

boxes, numbered from 1 to 20 Each box contains one of twenty monetary

prizes ranging from 0.01 to 500,000 EUR Independent notary company allocates

prizes across boxes and seals the boxes

GameContestants receive one multiple-

choice general knowledge question Contestant, who is the first to answer

this question correctly, plays the game: Contestant keeps her own box and

opens the remaining boxes one by one Once a box is opened, the prize sealed

inside is publicly revealed and deleted from the list of possible prizes

List of Possible Prizes (Affari Tuoi )

* Prize 5,000 Euro was replaced with prize 30,000 Euro starting from January 30, 2006

Timing of the gameOpen 6 boxes

Exchange own box for any of 13 remaining unopened boxes?

Open 3 boxes

“Bank” offers a price for contestant’s box (11 boxes remain unopened)

Open 3 boxesAccept price

“Bank” offers a price or an exchange (8 boxes remain unopened)

Open 3 boxes


Accept price Open 3 boxes



Accept price

List of Possible Prizes (DOND UK )

Timing of the gameOpen 5 boxes

“Bank” offers a price (17 unopened boxes left)




Open 3 boxes





Accept price




Data114 Affari Tuoi episodes

September 20, 2005 to March 4, 2006 234 Deal or No Deal UK episodes

October 31, 2005 to July 22, 2006 Distribution of possible prizes,

“bank” offers, prize in own boxGender, age, marital status, region

“Bank” Offers, remarks “Bank” monetary offers are fairly

predictable across episodes In early stages of the game, they are

smaller than EV of possible prizes As the game progresses, the gap between

EV and the monetary offer decreases and often disappears when there are two unopened boxes left.

Offers do not depend on prize in contestants box

OLS Regression Results for Affari Tuoi and DOND UK

Models of Stochastic Choice

Trembles (Harless and Camerer, 1994) Fechner Model of Homoscedastic Random Errors

(Hey and Orme, 1994) Fechner Model of Heteroscedastic Random Errors

(Hey, 1995 and Buschena and Zilberman, 2000) Fechner Model of Heteroscedastic and Truncated

Random Errors (Blavatskyy, 2007) Random Utility Model (Loomes and Sugden, 1995)

Trembles (Harless and Camerer, 1994) Individuals generally choose among

lotteries according to a deterministic decision theory

But there is a constant probability that this deterministic choice pattern reverses (as a result of pure tremble).

- vector of parameters that characterize the parametric form of a decision theory

- utility of a lottery L according to this theory.

θ

θ,Lu

Trembles, continued LL of observing N decisions of contestants

to reject an offer for a risky lottery , can be written as

where is an indicator function i.e. if x is true and if x is false, and is probability of a tremble.

iO iL Ni ,...,1

,,,21log

,,log,,1log

1

11

θθ

θθθθ

ii

N

i

ii

N

iii

N

iR

OuLuI

OuLuIpOuLuIpLL

xI 1xI 0xI 1,0p

Trembles, continuedLL of observing M decisions of

contestants to accept offer for a risky lottery , can be written as

Parameters and p are estimated to maximize log-likelihood .

iO iL Mi ,...,1

,,,21log

,,1log,,log

1

11

θθ

θθθθ

ii

N

i

ii

N

iii

M

iA

OuLuI

OuLuIpOuLuIpLL

AR LLLL θ

Fechner Model of Homoscedastic Random Errors (Hey and Orme, 1994) H&O (1994) estimate a Fechner model

of random errorsWhere a random error distorts the net

advantage of one lottery over another (in terms of utility)

Net advantage is calculated according to underlying deterministic decision theory

The error term is a normally distributed random variable with zero mean and constant standard deviation

Fechner Model of Homoscedastic Random Errors, continued LL of observing N decisions of contestants

to reject an offer for a risky lottery , can be written as

where is the cumulative distribution function (cdf) of a normal distribution with zero mean and standard deviation .

iO iL Ni ,...,1

N

iiiR OuLuLL

1,0 ,,log θθ

.,0

Fechner Model of Homoscedastic Random Errors, continuedLL of observing M decisions of

contestants to accept offer for a risky lottery ,

can be written as

Parameters and are estimated to maximize log-likelihood

iO iL Mi ,...,1

M

iiiA OuLuLL

1,0 ,,1log θθ

θ AR LLLL

Fechner Model of Heteroscedastic Random Errors (Hey, 1995 and Buschena and Zilberman, 2000) Assume that the error term is heteroscedastic

STDEV of errors is higher in certain decision problems, e.g. when lotteries have many possible outcomes

In DOND a natural assumption is that contestants, who face risky lotteries with a smaller range of possible outcomes, have a lower volatility of random errors than contestants, who face risky lotteries with a wider range of possible outcomes.

We estimate a Fechner model of random errors when the standard deviation of random errors is proportionate to the difference between the utility of the highest outcome and the utility of the lowest outcome of a risky lottery L. x

x

Fechner Model of Heteroscedastic Random Errors, continued LL of observing N decisions of contestants

to reject an offer for a risky lottery , can be written asiO iL Ni ,...,1

N

iiixuxuR OuLuLL

ii1

,,,0 ,,log θθθθ

Fechner Model of Heteroscedastic Random Errors, continuedLL of observing M decisions of


can be written as

Parameters and are estimated to maximize log-likelihood .

iO iL Mi ,...,1

M

iiixuxuA OuLuLL

ii1

,,,0 ,,1log θθθθ

θ AR LLLL

Fechner Model of Heteroscedastic and Truncated Random Errors (Blavatskyy, 2007) Truncate the distribution of random errors so that an

individual does not commit transparent errors. E.g. transparent error - an individual values a risky

lottery > than its highest possible outcome for certain or when an individual values a risky lottery < than its lowest possible outcome for certain (known as a violation of the internality axiom).

In DOND a rational contestant would always reject an offer, which is < than the lowest possible prize remaining and accept an offer, which > the highest of the remaining prizes

But in Fechner model - a strictly positive probability that a contestant commits such transparent error.

To disregard such transparent errors, the distribution of heteroscedastic Fechner errors is truncated from above and from below.

Fechner Model of Heteroscedastic and Truncated Random Errors, continued LL of observing N decisions of contestants

to reject an offer for a risky lottery , can be written asiO iL Ni ,...,1

N

i iixuxuiixuxu

iixuxuiixuxuR LuxuLuxu

LuOuLuxuLL

iiii

iiii

1 ,,,0,,,0

,,,0,,,0

,,,,

,,,,log

θθθθθθθθ

θθθθ

θθθθ

Fechner Model of Heteroscedastic and Truncated Random Errors, continuedLL of observing M decisions of


can be written as

Parameters and are estimated to maximize log-likelihood .

iO iL Mi ,...,1

M

i iixuxuiixuxu

iixuxuiixuxuA LuxuLuxu

LuxuLuOuLL

iiii

iiii

1 ,,,0,,,0

,,,0,,,0

,,,,

,,,,log

θθθθθθθθ

θθθθ

θθθθ

θ AR LLLL

Random Utility Model (Loomes and Sugden, 1995) Individual preferences over lotteries are

stochastic and can be represented by a random utility model.

Individual preferences over lotteries are captured by a decision theory with a parametric form that is characterized by a vector of parameters .

We will assume that one of the parameters is normally distributed with mean and standard deviation and the remaining parameters are non-stochastic.

θθR

Rθ

Random Utility Model, continued Let denote a value of parameter Such that given other parameters ,

a contestant is exactly indifferent between accepting and rejecting an offer O for a risky lottery L

i.e.

and for all an individual prefers to accept an offer.

RR θ RRθ

RRRRRR OuLu θθθθ ,,,,

RRR θ

Random Utility Model, continued

LL of observing N decisions of contestants to reject an offer for a risky lottery , can be written asiO iL Ni ,...,1

N

iRRRLL

1,log θ

Random Utility Model, continued

LL of observing M decisions of contestants to accept offer for a risky lottery ,

can be written as

Parameters , and are estimated to maximize log-likelihood .

iO iL Mi ,...,1

M

iRRALL

1,1log θ

Rθ AR LLLL

7 Decision Theories Embedded in Models of Stochastic Choice

Decision theory

Investigated in experimental study?

Camerer (1989)

Starmer (1992)

Harless & Camerer (1994)

Hey and Orme (1994)

Hey (2001)

Risk Neutrality

Expected Utility Theory

Skew-Symmetric Bilinear Utility

Regret Theory

Rank-Dependent Expected Utility

Yaari’s Dual Model

Disappointment Aversion Theory

Risk Neutrality (RN)

Maximize expected value (EV)utility of a risky lottery

that delivers outcome xi with probability pi is

There are no free parameters to be estimated for this decision theory, i.e. vector θ is the empty set

nn pxpxL ,;...;, 11

n

i ii xp1

Expected Utility Theory (EUT)

utility of lottery is u is a (Bernoulli) utility function over money We will estimate expected utility theory with

two utility functions: constant relative risk aversion (CRRA) and expo-power (EP)

CRRA utility function is (vector θ is just r ) EP utility function is (vector θ is )

nn pxpxL ,;...;, 11

n

iii xup

1

1,log1,11

rxrrx

xur

rx r

exu

11

1

,rθ

Regret Theory (RT) and Skew-Symmetric Bilinear Utility Theory (SSB)

SSB: an individual chooses a risky lottery over a sure amount O if

where ψ is a skew-symmetric function

SSB coincides with regret theory if ψ is convex (assumption of regret aversion)

We will estimate RT (SSB) with function

nn pxpxL ,;...;, 11

0,,1

n

iii OxpOL

OxrxrOOxrOrxOx

rr

rr

,11,11,

11

11

RT and SSB, continued

This function satisfies assumption of regret aversion when δ>1

When δ=1, RT(SBB)=EUT+CRRAWhen r=0, RT(SBB)=CPT with

current offer as a reference point, no loss aversion and linear prob. weighting

When δ=1 and r=0 RT(SBB)=RNVector θ is ,rθ

Yaari’s Dual Model (YDM)

the utility of a risky lottery is

probability weighting function

vector consists only of one element—the coefficient of the probability weighting function .

nn pxpxL ,;...;, 11

nxxx ...21

n

ii

i

jj

i

jj xpwpw

1

1

11

11 ppppw

θ

Rank-Dependent Expected Utility Theory

the utility of a risky lottery is

probability weighting function

and CRRA utility function vector consists of two elements— CRRA

coefficient and the coefficient of the probability weighting function:

nn pxpxL ,;...;, 11 nxxx ...21

11 ppppw

θ

n

ii

i

jj

i

jj xupwpw

1

1

11

,rθ

Disappointment Aversion Theory (DAT)

the utility of a risky lottery

is

is a number of disappointing outcomes in lottery L

is a subjective parameter that captures disappointment preferences

vector consists of two elements— CRRA coefficient and the disappointment aversion parameter

nn pxpxL ,;...;, 11

nxxx ...21

θ

n

mniiin

mnii

mn

iiin

mnii

xupp

xupp 1

1

1

1

1

1

1

1

1,...,1 nm

1

,rθ

Results

We estimate static and dynamic decision problem

In a static problem, an individual treats all remaining prizes as equiprobable

In a dynamic problem, an individual anticipates future bank offers

In both problems:Result 1: Estimates of parameters of decision

theories differ substantially, depending on which model of stochastic choice the theories are embedded in

Results (goodness of fit)We use Vuong’s likelihood ratio test (and

Clarke test) for non-nested models Result 2: For every decision theory, the

best fit to the data is obtained when this theory is embedded into a Fechner model with heteroscedastic truncated errors

Result 3: For every model of stochastic choice, the worst fit to the data is obtained when this model is combined with RN.

Results, dynamic vs. static

Tremble model yields the same LL in static and dynamic problems

Random utility model provides a better fit to the data in a dynamic rather than a static decision problem

No statistically significant difference for Fechner model

Estimated parameters are similar in a dynamic and a static problem but differ significantly across different models of stochastic choice

Results, UK static

Decision theories embedded in a tremble model perform significantly worse compared to other models

Decision theories embedded in a Fechner model with homoscedastic errors yield similar goodness of fit as in a random utility model

Best fit to the data:RDEU embedded into a Fechner model with

heteroscedastic errors RDEU and EUT+EP in a truncated Fechner model

Results, AT dynamic In a tremble model, for all theories that

have RN as a special case, the estimates are the same as for RN Mass point in „bank“ offers (2.2%=EV)

In a dynamic decision problem: standard deviation of stochastic parameters in

a random utility model tends to be higher the variance of random errors in a Fechner

model tends to be lower Best fit to the data:

EUT+EP or RT (SSB) in a truncated Fechner model

EUT+EP or RT (SSB) in a random utility model

Results, UK dynamic Tremble model—estimates are the same

as in a static case (except for RT(SSB) and DAT) even w/o mass point

In a dynamic decision problem: standard deviation of stochastic parameters in

a random utility model tends to be higher the variance of random errors in a Fechner

model tends to be higher too (expect for YDM) Best fit to the data:

RDEU or EUT+EP in a truncated Fechner modelRT (SSB) or RDEU in a random utility model

Conclusion Correctly selected model of stochastic

choice matters just as much as a correctly selected decision theory Estimated parameters differ a lot

Best model of stochastic choice is a truncated Fechner model (and a random utility model in dynamic problems)

In this model, EUT performs not significantly worse than non-EUT

CRRA gives (nearly always) worse fit than EP

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A Perfect Cocktail Recipe: Mixing Decision Theories and Models of Stochastic Choice