1936 - Burke's Complete Cocktail and Tastybite Recipe by Harman Burney Burke
A Perfect Cocktail Recipe: Mixing Decision Theories and Models of Stochastic Choice
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Transcript of 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
“Bank” offers a price or an exchange (5 boxes remain unopened)
Accept price Open 3 boxes
“Bank” offers a price or an exchange (2 boxes remain unopened)
Accept price Open 2 boxes
Accept price
List of Possible Prizes (DOND UK )
Timing of the gameOpen 5 boxes
“Bank” offers a price (17 unopened boxes left)
“Bank” offers a price (14 unopened boxes left)
Open 3 boxesAccept price
“Bank” offers a price (8 unopened boxes left)
Open 3 boxes
“Bank” offers a price (5 unopened boxes left)
Accept price Open 3 boxes
“Bank” offers a price (2 unopened boxes left)
Accept price Open 2 boxes
Accept price
“Bank” offers a price (11 unopened boxes left)
Open 3 boxesAccept price
Open 3 boxesAccept 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
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
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
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
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