RevealedIndifference:UsingResponseTimestoInferPreferences*
ArkadyKonovalov1andIanKrajbich2
October18,2016
Abstract
Revealedpreferenceisthedominantapproachforinferringpreferences,butitrelieson
discrete,stochasticchoices.Thechoiceprocessalsoproducesresponsetimes(RTs)which
arecontinuousandcanoftenbeobservedintheabsenceofinformativechoiceoutcomes.
Moreover, there is a consistent relationship between RTs and strength-of-preference,
namelythatpeoplemakeslowerdecisionsastheyapproachindifference.Thisrelationship
arises from optimal solutions to sequential information sampling problems. Here, we
investigateseveralwaysinwhichthisrelationshipcanbeusedtoinferpreferenceswhen
choiceoutcomesareuninformativeorunavailable.WeshowthatRTsfromasinglebinarychoice problem are enough to usefully rank people according to their degree of loss
aversion. Using a large number of choice problems, we are further able to recover
individual utility-function parameters from RTs alone (no choice outcomes) in three
differentchoicedomains.Finally,weareabletouselongRTstopredictwhichchoicesare
inconsistentwithasubject’sutilityfunctionandlikelytolaterbereversed.Theseresults
provideaproofofconceptforanovel“methodofrevealedindifference”.
JELCodes:C91;D01;D03;D87;D81;D90.
*
TheauthorsthankPaulHealy,LucasCoffman,RyanWebb,DanLevin,JohnKagel,KirbyNielsen,
JeevantRampal,andPujaBhattacharyafortheirhelpfulcommentsandconversationsandYosuke
Morishima,ErnstFehr,ToddHare,ShabnamHakimi,andAntonioRangelforsharingtheirdata.
1
DepartmentofEconomics,TheOhioStateUniversity;[email protected].
2DepartmentofPsychology,DepartmentofEconomics,TheOhioStateUniversity
[email protected].
1
1Introduction
Economics is built around the idea that a person’s preferences can be inferred from
theirchoices.Thisisthestandardrevealedpreferenceapproach.However,choiceitselfis
nottheonlyoutputofthechoiceprocess.Wearealsooftenabletoobserveotherfeatures
such as response times (RT). Moreover, RTs are continuous and so may carry more
information than discrete stochastic choice outcomes (Loomes, 2005; Webb, 2013)3. The
trick lies in discovering how the interaction between preferences and choice options
producesRTs.Ifwecouldcharacterizeandtheninvertthisfunction,thenwecouldinfer
preferencesfromRTs.
Considerthefollowingexample.Supposeweareattemptingtodeterminewhichoftwo
agentsismoreimpatient.Weaskeachofthemthesamequestion:wouldyouratherhave
$25todayor$40intwoweeks?Supposethatbothagentschoosetotakethemoneynow.
With just this information there is no way to distinguish between the agents without
further questioning. Now suppose Agent A made his choice in 5 seconds, while Agent B
madehersin10seconds.Whomightwesuspectismoreimpatient?
OuransweristhatAgentAislikelymoreimpatient.Whilestandardeconomictheoryis
silent in this situation, there is a relevant theoretical framework for answering this
question, namely sequential sampling models. That framework views decisions like this
oneasamentaltug-of-warbetweentheoptions.Foroptionsthataresimilarinstrength
(utility)itwilltakemoretimetodeterminethewinner,andinmanycasestheweakerside
may prevail. Thus there emerges a relationship between strength-of-preference, choice
3
ForareviewontheuseofRTsineconomics,seeClithero(2016a)andSpiliopoulosandOrtmann
(2014).
2
probability,andRT.Returningtoourexample,weknowthatsinceAgentAwasrelatively
faster than Agent B, it is more likely that Agent A faced an easier decision. Since it was
easier for Agent A to choose the sooner option, we can infer that he is likely more
impatient.
If indeed agents reliably make slower decisions when they are closer to indifference,
thiscouldbeusedtoeconomists’andpolicymakers’advantageinanumberofways.First,
onecoulddesignsimplerandshorterpreference-elicitationmechanismssuchasthesingletrial binary-choice example above. Second, one could recover utility functions in cases
whereagents’preferencesfalloutsideoftherangeofagivenelicitationprocedure,whereit
is currently only possible to put a lower or upper bound on extreme preferences. Third,
onecouldestimateutility-functionparametersinthecompleteabsenceofchoiceoutcomes.
Byaskingavarietyofquestions,onecouldseewhichoneselicittheslowestresponsesand
infer that those questions made the person roughly indifferent. This last point is to
highlight that it is still possible to do preference analysis in the absence of informative
choice outcomes. This suggests, for example, that in order to protect their private
information,agentsmustconsiderobservabilityofboththeirchoicesandtheirRTs.
Here,wedemonstratethattheseareindeedpromisingapproaches,usingexperimental
datafromthreeprominentchoicedomains:risk,time,andsocialpreferences.InSection
5.3weshowthatsingle-trialRTscanbeusedtoranksubjectsaccordingtotheirdegreeof
loss aversion. In Section 5.4 we show that RTs on “extreme” trials (where most subjects
choosethesameoption)canalsobeusedtoranksubjectsaccordingtotheirloss,time,and
social preferences. Finally in Sections 5.5 and 5.6 we examine RTs from the full datasets
anduseeachsubject’sRTdatatoestimatetheirutility-functionparameters.Ineverycase
3
theserankings/parametersalignwellwiththoseestimatedfromsubjects’choicesoverthe
full datasets. Taken together, these results serve as a proof of concept that individual
preferencescanbeinferredfromRTsalone.
2Model
2.1.Background
In cognitive psychology it has long been known that the speed and accuracy of
discrimination between two items along some dimension increases with the distance
betweenthoseitemsonthatdimension.Inotherwords,easiercomparisonsarefasterand
moreaccurate.Forinstance,decidingwhethera5lbweightisheavierthana4lbweight
generallytakesmoretimeandislessconsistentthanthesamecomparisonbetweena10lb
anda4lbweight.Extendingthisideatoeconomicchoice,wewouldexpectthatdeciding
betweentwoitemssubjectivelyvaluedat$5and$4wouldgenerallytakemoretimeandbe
lessconsistentthanachoicebetweentwoitemsvaluedat$10and$4.
This kind of comparison process is typically modeled using a class of “sequential
sampling models” (SSM), often also referred to as drift-diffusion (DDM) or evidenceaccumulation models (Bogacz et al. 2009; Gold and Shadlen 2001; Ratcliff and McKoon
2008; Usher and McClelland 2001). For over 50 years, cognitive psychologists have been
usingSSMstoexplainindividualbehaviorinsimpleperceptionandmemorytasks(Brown
and Heathcote 2008; Laming 1979; Link 1975; Ratcliff 1978; Ratcliff and McKoon 2008;
Stone1960;UsherandMcClelland2001)andtoalesserextenteconomicchoice(Agranov
&Ortoleva,2015;Busemeyer&Townsend,1993;Cavanagh,Wiecki,Kochar,&Frank,2014;
Krajbich, Armel, & Rangel, 2010; Krajbich, Lu, Camerer, & Rangel, 2012; Milosavljevic,
4
Malmaud, Huth, Koch, & Rangel, 2010; Philiastides & Ratcliff, 2013; Polania, Krajbich,
Grueschow,&Ruff,2014;Rodriguez,Turner,&McClure,2014a).
Typically, a subject’s task is to discriminate between two alternatives along some
dimension(e.g.brightness,motion,ornumberofitems).Inordertomakeherdecision,the
subjectmustrelysolelyon(typically)noisysignals,whicharesampledincontinuoustime,
graduallyformingaweightofevidenceforeachalternativeonthatdimension.However,
thissamplingprocesstakestime,andtimeiscostly.Thustheagent’srealdecisioninsuch
situationsishowlongtosample.Thisiswhatisknownasthe“speed-accuracytradeoff”.
Theoptimalsolutiontothisproblemdependsonvariousassumptions,butinthesimple
caseofaone-shotdecisionwithafixedtimecostandfixedrewardforacorrectresponse,
theoptimalstoppingruleistosamplesignalsuntilthenetdifferenceinevidencereaches
some critical threshold (Wald, 1945). This decision rule is optimal in the sense that it
minimizestheexpectedRTforanydesiredaccuracyrate.Thisconstantthresholdruleis
the most commonly used model, though some have explored optimality in other settings
(Busemeyer&Rapoport,1988;Frazier&Yu,2007;Fudenberg,Strack,&Strzalecki,2015;
Webb,2013b;Woodford,2014).
The key thing to notice in this class of models is that the agent chooses her stopping
rule prior to the sampling process and so what determines her actual choice and RT,
conditional on that stopping rule, is the sequence of signals that she receives. That
sequence is stochastic, but does depend on the true underlying values. For economic
choice, the speed and accuracy of the decision are monotonically increasing in the
differencebetweentheutilitiesofthetwooptions.Thisquantityisoftenreferredtoasthe
“drift rate”, and in our economic setting it represents strength-of-preference, i.e. the
5
difference in cardinal utility. Thus, when a person is close to indifference, the drift rate
approaches zero, causing a delay in the choice (Chabris, Morris, Taubinsky, Laibson, &
Schuldt, 2009; Dickhaut, Rustichini, & Smith, 2009; Moffatt, 2005; Mosteller & Nogee,
1951).
2.2.Model
Hereweuseastandarddrift-diffusionmodel(DDM)(Ratcliff,1978)toestablishthelink
betweenRTsandtheunderlyingutilitiesinsimplebinarychoicetasks.
Letusassumethatanagentobservesasetofalternatives𝑗 ∈ {1,2}.Thechoiceprocess
involves two components: a constant boundary threshold 𝑏 and a decision variable 𝑦(𝑡)
thatevolvesovertimeaccordingtothefollowingdifferentialequation:
𝑑𝑦 𝑡 = 𝑣 ∙ 𝑑𝑡 + 𝜎 ∙ 𝑑𝑊
(1)
where𝑦(𝑡)isaccumulatedevidencetowardsoption1(with𝑦(0) = 0),𝑣isthedriftrate,
whichisassumedtobealinearfunctionoftheutilitydifference:
𝑣 ≡ 𝑧 ⋅ (𝑢9 𝜃 − 𝑢< 𝜃 ),
(2)
where𝑧 ∈ 𝑅> isascalingparameter,𝑢? (⋅)istheutilityofthegivenalternative,and𝜃isthe
agent-specificutilityfunctionparameter.Finally,𝜎 ∙ 𝑑𝑊isaWienerprocess(i.e.Brownian
motion)thatrepresentsGaussianwhitenoisewithvariance𝜎 < .Withoutlossofgenerality,
wenormalize𝜎 = 1asitcanonlybeidentifieduptoscale(duetothearbitraryunitsony).
We define the response time 𝑅𝑇 as the first time that the absolute value of the
decision variable reaches a boundary 𝑏 ∈ 𝑅> , plus a non-stochastic component known as
non-decisiontime(𝜏 ∈ 𝑅> ,typicallyinterpretedasthetimethatasubjectneedstoprocess
theinformationonthescreen):
6
𝑅𝑇 = min 𝑡: 𝑦 𝑡
≥ 𝑏 + 𝜏.
(3)
(4)
Thechoiceoutcome𝑎 ∈ {1,2}isdefinedasfollows:
𝑎=
1𝑖𝑓𝑦 𝑅𝑇 = 𝑏
2𝑖𝑓𝑦 𝑅𝑇 = −𝑏
Now, assuming without loss of generality 𝑣 ≥ 0, we can calculate the choice
probabilities 𝑝(𝑎 = 𝑗), the expected RT, and an approximate bivariate probability density
function (PDF) for the RTs (minus 𝜏) as follows (Blurton, Kesselmeier, & Gondan, 2012;
Navarro&Fuss,2009;Ratcliff,1978;Srivastava,Feng,Cohen,Leonard,&Shenhav,2015;
Wabersich&Vandekerckhove,2014):
M NOP Q9
𝑝(𝑎 = 1) =
9
<
U
𝑓(𝑡) =
W
XU
𝑒Q
N
𝐸 𝑅𝑇 =
ON Z
N
`
[a9
−1
[Q9
V
<
M NOP QM RNOP
𝑖𝑓𝑣 > 0
𝑖𝑓𝑣 = 0
1−
<
M NOP QM RNOP
+ 𝜏𝑖𝑓𝑣 > 0
𝑏 + 𝜏𝑖𝑓𝑣 = 0
𝑘⋅𝑒
Q
]N ^ N Z
_PN
sin
[W
<
(5)
(𝑒 VU + 𝑒 QVU )
(6)
(7)
Typically,ifchoicedataisavailable,onecanuseequation(5)toestimate
parameterspurelyfromchoices,maximizinglog-likelihood:
𝐿𝐿 =
d(1
𝑎d = 1 ⋅ log 𝑝 𝑎d = 1 𝑏, 𝑣d
+ (1 𝑎d = 2 ⋅ log 1 − 𝑝 𝑎d = 1 𝑏, 𝑣d ,(8)
where𝑛denotestrialnumber,and1(⋅)istheindicatorfunction.
If only RT data is available, we can use the RT probability densities to estimate the
utility parameter for each subject 𝑖 (𝜃i ) given the empirical distribution of RTs by
maximizingthefollowinglikelihoodfunction:
𝐿𝐿 =
d(log(𝑓
𝑅𝑇d , 𝑎d = 1 𝑏, 𝜏, 𝑣d ) + log(𝑓 𝑅𝑇d , 𝑎d = 2 𝑏, 𝜏, 𝑣d )). 7
(9)
2.3.Modelpredictions
There are several important qualitative predictions of the model that allow us make
inferencesbeyondstructuralestimation.
First, the expected RTs decrease as the utility difference between the two options
jk(lm)
becomes larger (
jV
< 0, see Figure 1 for a simulation example). If the drift scaling
parameter and non-decision time are the same (or similar) across subjects, we should
observe a group-level correlation between the RT and the utility function parameter of
interest, conditional on the subjects making the same choice. Thus we should be able to
rank subjects according to their individual preferences using RTs from a single decision
problem.WeexplorethispredictioninSection5.3.
More generally, this relationship can be used to rank subjects using sets of choice
problemswheretheyallmadethesamechoice,andsochoicesareuninformative.Consider
anintertemporalchoiceexamplewithtwoindividuals:apatientpersonwhoisindifferent
between$10todayand$12tomorrow,andanimpatientpersonwhoisindifferentbetween
$10todayand$19tomorrow.Thepatientindividualwilllikelytakesometimetodecide
between$10todayand$11tomorrow,butchooseveryquicklybetween$10todayand$20
tomorrow.Ontheotherhand,theimpatientindividualwillstruggletochoosebetween$10
todayand$20tomorrow,butchooseveryquicklybetween$10todayand$11tomorrow.
Notice that in this example both individuals would most likely choose $10 today over
$11tomorrowand$20tomorrowover$10today,basedontheirindifferencepoints.Thus
arevealedpreferenceapproachwouldbeunlikelytomakeadistinctionbetweenthesetwo
individuals based only on these choices. In contrast, the revealed indifference approach
would tell us that a slow choice between $10 today and $11 tomorrow is indicative of
8
patience, while a slow choice between $10 today and $20 tomorrow is indicative of
impatience.WeconsiderthecaseofuninformativechoiceoutcomesinSection5.4.
Second, long RTs are considerably more informative than short RTs. Sequential
samplingmodelscorrectlypredictthatshortRTscanoccuratanylevelofchoicedifficulty
(i.e.strength-of-preference),butlongRTsalmostexclusivelyoccurfordifficultchoices(i.e.
nearindifference;again,seeFigure1foranexamplesimulation).Theintuitionhereisthat
there is a lot of noise in the decision process and so the quickest choices are those with
noise pointed in the same direction (Ratcliff, Philiastides, & Sajda, 2009; Ratcliff &
Tuerlinckx, 2002). Because these sequences of aligned noise occur independently of the
driftrate,thereistypicallyno(oraveryweak)correlationbetweentheshortestRTsand
difficulty (Ratcliff & McKoon 2008). However, the slowest choices are ones where the
combinationofdriftplusnoiseleadstoroughlyzeronetevidenceaccumulation.Giventhat
thesamplednoiseismean-zero,net-zeroevidenceaccumulationisfarmorelikelytooccur
withsmalldrift(difficultproblems)thanwithlargedrift(easyproblems).Thisproducesa
correlationbetweenthelongestRTsanddifficulty.
By taking advantage of this knowledge, we can improve the RT-based estimation of
utility-function parameters (relative to structural estimation, Section 5.5), either by
focusingonthelongestRTsorbyestimatingthepeakinRTsusingthefullsubjectdataset.
ThesemethodsareexploredinSection5.6.
Finally, the model predicts that most “errors”, or choices inconsistent with the utility
function,shouldhappenwhenthesubjectisclosetoindifferenceandthustendstomake
slowchoices.WetestthishypothesisinSection5.7.
9
2.4.Modellimitations
WithRTs,“slow”isarelativeconceptwhereaschoiceisabsolute.Inordertousethe
revealed indifference approach, we must therefore develop methods to identify slow
decisions for a given individual. Another potential issue is that even though RTs are
continuous,theycanbeinfluencedbyotherfactorsasidefromstrength-of-preferenceand
thusmayappeartobequitenoisy(Krajbichetal.2015a).Forinstance,othershaveargued
thatlongRTsreflectmoreeffortordeliberativethinkingonthepartofthesubject(Gabaix,
Laibson, Moloche, & Weinberg, 2006; Hey, 1995; Rubinstein, 2007; Wilcox, 1993). This
couldinterferewithourabilitytouseRTstoinferpreferences.
There are certain characteristics of the choice environment that we believe may
facilitateestimatingpreferencesfromRTs.First,RTsshouldberecordedwithmillisecond
precisionsincemanysimplebinarychoicestakeonlyafewseconds.Second,choicesshould
bemadeusingakeyboardbuttonpressratherthanusingamousetoselectanoption.This
is simply to minimize noise in RTs due to hand movements. Third, we limit ourselves to
binary decisions since indifference is not straightforward with multiple alternatives.
Fourth,forthelastexercise,whereweusetherevealedindifferenceapproachandallthe
trials to estimate utility functions, the range of indifference points implied by the choice
problemsintheexperimentshouldcovertherangeofparametervaluesinthepopulation.
It should be mentioned that SSMs have been successfully applied to datasets that violate
someoftheserequirements(e.g.Krajbichetal.2015b;KrajbichandRangel2011),butsuch
datasetsarenotidealforourcurrentgoals.
Inadditiontotheserequirements,thereareotherfactorsthatmayaffectourabilityto
successfullyinferindifferencepointsfromRTs.Onequestioniswhetherthereshouldbe
10
constraints on RTs. RT restrictions are common in binary choice tasks in order to keep
subjects focused, but overly restrictive cutoffs may attenuate the effect of strength-ofpreference on RTs (Fudenberg et al., 2015). Another question is whether the method of
revealedindifferencecanbeusedinsituationswheretheutilityfunctioncontainsmultiple
parameters.Thiswillobviouslycomplicatematters,butifoneparameterisrelativelymore
importantthantheothers,themethodmaystillwork.Hereweexaminedatasetsthatvary
on these two dimensions, in order to explore the range of situations in which we can
successfullyemploythemethodofrevealedindifference.
3Relevantliterature
TheideaofapplyingSSMstoeconomicchoicewasfirstintroducedbyBusemeyerand
colleagues in the 1980s (Busemeyer, 1985; Busemeyer & Diederich, 2002; Busemeyer &
Rapoport, 1988; Busemeyer & Townsend, 1993; Johnson & Busemeyer, 2005; Rieskamp,
2008;Roe,Busemeyer,&Townsend,2001).Recentyearshaveseenrenewedinterestin
thisworkduetotheabilityofthesemodelstosimultaneouslyaccountforchoices,RTs,eye
movements, and brain activity in many individual preference domains such as risk and
uncertainty (Busemeyer, 1985; Busemeyer & Townsend, 1993; Fiedler & Glöckner, 2012;
Hunt et al., 2012; Stewart, Hermens, & Matthews, 2015), intertemporal choice (Dai &
Busemeyer, 2014; Rodriguez et al., 2014a), social preferences (Hutcherson et al. 2015;
Krajbich et al. 2015a; Krajbich et al. 2015b), as well as food and consumer choice (De
Martino, Fleming, Garret, & Dolan, 2013; Krajbich et al., 2010, 2012; Krajbich & Rangel,
2011; Milosavljevic et al., 2010; Philiastides & Ratcliff, 2013; Towal, Mormann, & Koch,
11
2013).Finally,Clithero(2016b)usedaDDMtoaugmentchoicedatawithRTinformation,
producingbetterout-of-samplepredictionsinfoodchoice.
ItisimportanttoacknowledgethatwearenotthefirsteconomiststouseonlyRTdata
topredictbehavior.Inadditiontotheworkmentionedabove,SchotterandTrevino(2014)
and Chabris et al. (2009) have used RTs to predict strategic behavior and aggregate
intertemporalpreferences,respectively.
Schotter and Trevino (2014) use the longest (and second longest) RTs to estimate
threshold strategies in a global game; the RT-based estimates are able to explain out-ofsamplechoicebetterthantheequilibriumprediction.Thispaperissimilarinspirittoours,
but studies strategic situations while we focus on individual preference elicitation. The
authorsarguethatlongRTsintheirsettingreflecteitherdiscoveringtheoptimalthreshold
strategyorexplicitcalculation.Theseconsiderationsandexplanationsarequiteseparate
fromtheideasandmodelsexploredinourwork.
In the preference domain, Chabris et al. (2009) and an earlier unpublished working
paperbythesameauthors(Chabris,Laibson,Morris,Schuldt,&Taubinsky,2008)useda
methodinfluencedbythedirectedcognitionmodel(Gabaix&Laibson,2005;Gabaixetal.,
2006)andfullRT-distributiondataatthegroupleveltoestimateaggregatedintertemporal
preferences using a structural estimation approach. This method aims to identify utility
parameters using the fact that RTs increase with choice difficulty. Their study finds a
correlation between the RT-based estimates and choice-based predictions for three
separatelargegroupsofsubjects,butdoesnotattemptanyanalysisofindividualsubjects’
preferences.
12
Our work also contrasts with the work of Rubinstein, Wilcox, and others (Chen &
Fischbacher,2015;Hey,1995;Recalde,Riedl,&Vesterlund,2014;Rubinstein,2007,2013,
2014; Wilcox, 1993), where long RTs are associated with deliberative thought and short
RTsareassociatedwithintuition.Thesepapershaveprimarilystudiedstrategicsettings
where effort is likely to vary more substantially across subjects and so dominate the
strength-of-preferenceeffectsthatdriveourresults.Whilethiscanbeaccountedforwith
varyinglevelsof𝑏intheDDMthatwehavediscussed,itmayalsobethatotherdecision
processesareinvolvedinstrategicdecisions.
4ExperimentalDesign
We analyze four separate datasets: the last two were collected with other research
goalsinmind,butincludedprecisemeasurementsofRTs,whilethefirsttwowerecollected
specificallyforthisstudy.
4.1.Loss-aversionexperiments
These experiments were conducted at The Ohio State University. In each round,
subjects chose between a sure amount of money and a 50/50 lottery that included a
positive amount and a loss (which in some rounds was equal to $0). The set of decision
problemswasadaptedfromSokol-Hessneretal.(2009).Subjects’RTswerenotrestricted.
Intheadaptiveexperiment,eachsubject’schoicedefinedthenexttrial’soptionsusinga
Bayesian procedure (DOSE, see Wang et al. 2010) to ensure an accurate estimate of the
subject’sriskandlossaversionwithinalimitednumberofrounds.Eachsubjectcompleted
the same three unpaid practice trials followed by 30 paid trials. Importantly, every
13
subject’sfirstpaidtrialwasidentical.Eachsubjectreceivedtheoutcomeofonerandomly
selectedtrial.61subjectsparticipatedinthisversionoftheexperiment,earning$17-20on
average4.
In the non-adaptive experiment, each subject first completed a three-trial practice
followed by 276 paid trials. These trials were presented in two blocks of the same 138
choice problems, each presented in random order without any pause between the two
blocks. Subjects were endowed with $17 and additionally earned the outcome of one
randomly selected trial (in case of a loss it was subtracted from the endowment). 39
subjectsparticipatedinthisexperiment,earning$18onaverage.Allsubjectsgavewritten
consent,andthestudieswereapprovedbytheOSUInstitutionalReviewBoard.
For both experiments we assumed a standard prospect theory value function
(Kahneman&Tversky,1979):
𝑈 𝑊, 𝐿, 𝑆 =
0.5𝑊r − 0.5𝜆 −𝐿 r
,
𝑆r
where 𝑊 is the gain in the lottery, L is the loss, S is the sure amount, 𝜌 reflects risk
aversion,and𝜆captureslossaversion.Inthenon-adaptiveexperiment,theutilityfunctions
wereestimatedusingastandardMLEapproachwithalogitchoicefunction.
Similartopriorworkusingthistask,wefoundthatriskaversionplaysaminimalrolein
this task relative to loss aversion, with 𝜌estimates typically close to 1. Therefore,
acknowledging that varying levels of risk aversion could add noise to the RTs, for the
analyses below (both choice- and RT-based) we assumed risk neutrality (𝜌 = 1). For the
non-adaptiveexperiment,weusedonlytrialswithnon-zerolosses(specifically,112outof
4
Inordertocoveranypotentiallossesincurredduringthistask,subjectsfirstcompletedan
unrelatedtaskthatendowedthemwithenoughmoneytocoverforanypotentiallosses.
14
138 decision problems) for both estimation and out-of-sample prediction. Two subjects
withoutlyingestimatesof𝜆(beyondthreestandarddeviationsofthemean)wereremoved
fromtheanalysis.Thesameexclusioncriterionwasusedfortheotherdatasets.
4.2.Temporaldiscountingexperiment
This experiment was conducted while subjects underwent functional magnetic
resonanceimaging(fMRI)attheCaliforniaInstituteofTechnology(Hare,Hakimi,&Rangel,
2014).In each round, subjectschosebetweengetting$25rightaftertheexperimentora
largeramount(upto$54)atalaterdate(7to200days).Therewere108uniquedecision
problemsandsubjectsencounteredeachproblemtwice.All216trialswerepresentedin
random order. Each trial, the amount was first presented on the screen, followed by the
delay,andsubjectswereaskedtopressoneoftwobuttonstoacceptorrejecttheoffer.The
decision was followed by a feedback screen showing “Yes” (if the offer was accepted) or
“No”(otherwise).Thedecisiontimewaslimitedto3seconds,andifasubjectfailedtogive
a response, the feedback screen contained the text “No decision received”. These trials
(2.6%acrosssubjects)wereexcludedfromtheanalysis.Trialswereseparatedbyrandom
intervals(2-6seconds).
41subjectsparticipatedinthisexperiment,earninga$50show-upfeeandtheamount
from one randomly selected choice. The payments were made using prepaid debit cards
thatwereactivatedatthechosendelayeddate.Allsubjectsgavewrittenconsent,andthe
experimentwasapprovedbyCaltech’sInternalReviewBoard.
For this experiment, we used a hyperbolic discounting utility function (Herrnstein,
1981;Laibson,1997):
15
𝑈 𝑥, 𝑡 =
𝑥
,
1 + 𝑘𝑡
where𝑥isthedelayedamount,𝑘isthediscountfactor(higherismoreimpatient),and𝑡
is the delay period in days. One subject that chose $25 now on every trial was removed
from the analysis. The utility functions were estimated using a standard MLE approach
withalogitchoicefunction.Twosubjectswithoutlyingestimatesof𝑘wereremovedfrom
theanalysis.
4.3.Binarydictatorgameexperiment
This dataset was collected while subjects underwent fMRI at the Social and Neural
Systems laboratory, University of Zurich (Krajbich et al. 2015a). Subjects made choices
between two allocations, X and Y, which specified their own payoff and an anonymous
receiver’s payoff. The payoffs were displayed in experimental currency units, and 120
predeterminedallocationswerepresentedinrandomorder.Eachallocationhadatradeoff
betweenafairoption(moreequaldivision)andaselfishoption(withhigherpayofftothe
dictator).72outof120decisionproblemspersubjecthadhigherpayofftothedictatorin
bothoptionsXandY(toidentifyadvantageousinequalityaversion),whiletherestofthe
problems (48/120) had higher payoffs to the receiver in both options (to identify
disadvantageous inequality aversion). In each round, subjects observed a decision screen
that included the two options, and had to make a choice with a two-button box. Subjects
were required to make their decisions within 10 seconds; if a subject failed to respond
under this time limit, that trial was excluded from the analysis (4 trials were excluded).
Intertrialintervalswererandomizeduniformlyfrom3to7seconds.
16
Subjects read written instructions before the experiment, and were tested for
comprehension with a control questionnaire. All subjects passed the questionnaire and
understoodtheanonymousnatureofthegame.Intotal,30subjectswererecruitedforthe
experiment. They received a show-up fee of 25 CHF and a payment from 6 randomly
chosenrounds,averagingatabout65CHF.Allsubjectsprovidedwrittenconsent,andthe
ethicscommitteeoftheCantonofZurichapprovedthestudy.
To fit choices in this experiment, we used a standard Fehr-Schmidt other-regarding
preferencemodel(Fehr&Schmidt,1999):
𝑈 𝑥i , 𝑥? = 𝑥i − 𝛼 ⋅ max 𝑥? − 𝑥i , 0 − 𝛽 ⋅ max 𝑥i − 𝑥? , 0 ,
where𝑥i isthedictator’spayoff,𝑥? isthereceiver’spayoff,𝛼reflectsdisadvantageous
inequality aversion, and 𝛽 reflects advantageous inequality aversion. As described above,
wewereabletoseparatetrialsthatidentified𝛼and𝛽,sothesechoicesweretreatedastwo
separate datasets. The utility functions were estimated using a standard MLE approach
withalogitchoicefunction.Onesubjectwithanoutlyingestimateof𝛼wasremovedfrom
theanalysis.
5Results
5.1.Choice-basedestimations
Thethreeutilityfunctionsweselectedtomodelsubjects’choicesperformedwellabove
chance. To examine the number of choices that were consistent with the estimated
parameter values, we used standard MLE estimates of logit choice functions (for the
estimation procedure details see Appendix A) to identify the “preferred” alternatives in
every trial and compared those to the actual choice outcomes. More specifically, we
17
calculated utilities using parameters estimated purely from choices, and in every trial
predicted that the alternative with the higher utility would be chosen with certainty. All
subjectswereveryconsistentintheirchoiceseveninthedatasetswithalargenumberof
trials:socialchoice𝛼:94%,socialchoice𝛽:93%,intertemporalchoice:83%,non-adaptive
riskychoice:89%(seeAppendixBforthesubject-levelchoicepredictions).
5.2.Responsetimesreflectchoicedifficulty
ThemethodofrevealedindifferenceispredicatedontheDDM-basedideathatasubject
will take more time to decide as the decision becomes more difficult. For the following
analysis,ourmeasureofdifficultywasthedifferencebetweenthesubject’sutilityfunction
parameter (estimated purely from the subject’s choices) and the parameter value that
wouldmakeapersonindifferentbetweenthetwoalternativesinthattrial(werefertothis
asthe“indifferencepoint”).Whenasubject’sparametervalueisequaltotheindifference
point of a trial, we say that the subject is indifferent on that trial and so the choice is
maximallydifficult.
Letusillustratethisconceptwithasimpleexample.Supposethatintheintertemporalchoicetaskasubjecthastochoosebetween$25todayand$40in30days.Ifthesubjecthas
ahyperbolicdiscountingutilityfunction(asweassume),theywouldbeindifferentwithan
individual discount rate𝑘 that is the solution to the equation 25 = 40/(1 − 30𝑘), or k =
0.0125.Thiswouldbetheindifferencepointofthisparticulartrial.Asubjectwiththisk
value would be indifferent on this trial, a subject with a lowerk would favor the delayed
option,andasubjectwithahigherkwouldfavortheimmediateoption.
18
The bigger the absolute difference between the subject’s parameter and the trial’s
indifference point, the easier the decision, and the shorter the average RT. Indeed, this
effectisobservedinallofourdatasets,withRTspeakingatindifference(Figure2).Mixedeffects regression models (treating subjects as random effects) show strong, statistically
significant effects of the absolute distance between the indifference point and subjects’
individual utility parameters on log(RTs)5 for all the datasets (fixed effect of distance on
RT:dictatorgame𝛼:t=-7.5,p<0.001;dictatorgame𝛽:t=-9.1,p<0.001;intertemporal
choice𝑘:t=-9.9,p<0.001;non-adaptiveriskychoice𝜆:t=-9.6,p<0.001,adaptiverisky
choice𝜆:t=-4.4,p<0.001).
5.3.One-trialpreferenceranking
Intheadaptiveriskychoiceexperiment(Section4.1),allsubjectsfacedthesamechoice
problem in the first trial. They had to choose between a 50/50 lottery with a positive
amount($12)andaloss($7.5),andasureamount($0).Assumingriskneutrality,asubject
withalossaversioncoefficientof𝜆 = 1.6shouldbeindifferentbetweenthesetwooptions,
with more loss-averse subjects picking the safe option, and the rest picking the risky
option.
Because the mean loss aversion in our sample was 2.5 (median = 2.46), most of the
subjects(44outof61)pickedthesafeoptioninthisfirsttrial.Now,ifwehadtorestrictour
experimenttojustthisonetrial,theonlywaywecouldclassifysubjects’preferenceswould
5
A key feature of RTs generated by sequential sampling models is that they are roughly lognormally distributed. Thus it is typical to transform RTs with a natural logarithm before
performingstatisticalanalyses.
19
betodividethemintotwogroups:thosewith𝜆 ≥ 1.6andthosewith𝜆 ≤ 1.6.Withineach
groupwewouldnotbeabletosayanythingabouteachindividual’slossaversion.
ByobservingRTswecanestablisharankingofthesubjectsineachgroup.Specifically,
the DDM predicts that subjects with longer RTs would exhibit loss aversion closer to 1.6
(assuming that the threshold, drift, and non-decision time parameters are similar across
subjects). We ranked subjects in each group according to their RTs and then compared
thoserankingstothe“true”lossaversionparametersestimatedfromthe30choicesinthe
fulldataset(seeFigure3).
In line with the results of the previous section, RTs peaked around the indifference
point(𝜆 = 1.6).Therewasasignificantrank-based(Spearman)correlationbetweenRTs
andlossseekinginthe“safeoption”group(r=0.43,p=0.004)andbetweenRTsandloss
aversion in the “risky option” group (r = 0.41, p = 0.1). Thus the single-trial RT-based
rankingsalignedquitewellwiththe30-trialchoice-basedrankings.
As mentioned in the introduction, this approach could have great appeal since
economists and practitioners are often time constrained when collecting data, hence the
appeal of adaptive experiments (Blais & Weber, 2006; Cavagnaro, Myung, Pitt, & Kujala,
2010;Kim,Pitt,Lu,Steyvers,&Myung,2014;Rodriguez,Turner,&McClure,2014b;Wang
etal.,2010,andothers).Ourresultssuggestthatinfactveryfewtrialsmaybeneededto
getareliableestimateofsubjects’preferences.
20
5.4.Uninformativechoices
Another possible use of RT-based inference is the case where an experiment (or
questionnaire) is flawed in a such way that most subjects give the same answer to the
choice problems (or one might consider a situation where people feel social pressure to
give a certain answer, even if it contradicts their true preference). This is similar to the
situationfromthelastsection,exceptthatsubjects’choicesmaybeevenlessinformative,
andwithmultipletrials,theremaybeevenmoretogainfromexaminingtheRTs.
To model this situation, for each dataset (non-adaptive risk choice, intertemporal
choice, and social choice) we isolated trials with the most extreme indifference point,
wheremostsubjectschosethesameoption(e.g.,thelotteries),andlimitedouranalysisto
those subjects who picked this most popular option. In some instances, this involved
several trials since some of the choice problems were repeated or shared the same
indifferencepoint.
We found that the RTs on these trials were strongly correlated with subjects’
preferenceparametersinallfourdomains(riskychoice:r=0.55,p<0.001;intertemporal
choice:r=0.43,p=0.007;socialchoice𝛼:r=0.4,p=0.03;socialchoice𝛽:r=0.68,p<
0.001,Spearmancorrelations).Againweseethatitispossibletoranksubjectsaccording
to their preferences in the absence of distinguishing choice data. Similar to the previous
section, this method could be used to bolster datasets that are limited in scope and so
unabletorecoverallsubjects’preferences.
21
5.5.DDM-basedutilityestimationfromRTs
TheresultsdescribedintheprevioussectionsdemonstratethatwecanuseRTstorank
subjectsaccordingtotheirpreferenceswithoutinformativechoicedata.Inthissection,we
explore ways to estimate individual subjects’ utility-function parameters from their RTs
acrossmultiplechoiceproblems.
The DDM predicts more than just a simple linear relationship between strength-ofpreferenceandmeanRT;itpredictsentireRTdistributions.AsdescribedinSection2,the
DDMapproachassumesthatineachtrial,subjectsmakedecisionsusingaWienerrandom
process that produces a distribution of RTs given three free parameters, which can be
estimatedforeachindividualsubject(WabersichandVandekerckhove2014,seeAppendix
Afortheestimationdetails).Inparticular,thedriftrateinthemodelisalinearfunctionof
the utility difference and so by estimating drift rates we can identify the latent utilityfunctionparameters.
In each dataset6, we found that individual utility-function parameters estimated from
RTsalone(usingthestructuralDDMmodel,withoutchoicedata)werecorrelatedwiththe
sameparametersestimatedfromthechoicedata(socialchoice𝛼:r=0.39,p=0.04;social
choice𝛽:r=0.52,p=0.003;intertemporalchoice𝑘:r=0.57,p<0.001;riskychoice𝜆:r=
0.36,p=0.03;Pearsoncorrelations;Figure4,seeSection2.2andAppendixAforthemodel
andestimationdetails).
Anotherwaytoassessthegoodness-of-fitistoexaminethenumberofchoicesthatare
consistent with the estimated parameter values, as in Section 5.1. To do so, we used the
RT-estimated parameters to identify the “preferred” alternatives in every trial and
6
Wedonotusetheadaptiveriskdatasetinthissectionsincetheadaptivenatureofthetaskcreates
autocorrelationsinthedatathatwilllikelyinterferewithourestimationprocedure.
22
comparedthosetotheactualchoiceoutcomes.RT-basedestimatedparameterswereable
toexplainahighproportionofchoicesinthedatasets(socialchoice𝛼:79%respectively(p
< 0.001); social choice 𝛽: 80% (p < 0.001); intertemporal choice: 76% (p < 0.001); risky
choice:77%(p<0.001);p-valuesdenotetwo-sidedWilcoxonsignedranktestsignificance
at the subject level, comparing these proportions to chance). For a stricter test, we
calculated an average of all indifference points for each experiment, which roughly
corresponds to the mean of the experimenter’s prior parameter distribution, and made
choice predictions for each subject using this single value. The DDM accuracy rates were
abletobeatthisbaselineintwooutoffourcases(fortheintertemporalchoiceandsocial
choice𝛽,p<0.05,two-sidedWilcoxonsignedranktest).
5.6.AlternativeapproachestoutilityestimationfromRTs
The DDM may seem optimal for parameter recovery if that is indeed the data
generatingprocess.However,severalfactorslikelylimititsusefulnessinthissetting.The
DDM has several free parameters that are identified using features of choice-conditioned
RT distributions. Identification thus typically relies on many trials and observing choice
outcomes. Without meeting these two requirements, the DDM approach may struggle to
identifyparametersaccurately.Belowweexplorealternativeapproachestoanalyzingthe
RTs.
5.6.1.TopRTdecilemethod
One alternative approach is to focus on the longest RTs: for instance, Schotter and
Trevino (2014) successfully use the longest RT over a number of trials to identify a
23
decisionthresholdinasimpleglobalgame.AslongRTsindicateindifference,theycouldbe
usedtoidentifytrialswherethesubjectiveutilitiesofthetwooptionswereequal,andthus
obtain an estimate of the utility function parameter from the indifference points in those
trials.Hereweexplorewhetherthismethodworksinourdatasets.
Figure 5 plots RTs as a function of strength-of-preference for every trial in the
experiments. It is easy to see that even though for many subjects the single slowest trial
providesagoodsignalof“true”preference(asdefinedbythechoice-basedestimation),for
others the longest RT is far from indifference. This is especially true for the risky choice
data (possibly due to the two-parameter utility function or the unrestricted RTs). This
suggestedtousthatitmaybebettertousemorethanthesingleslowesttrial.
With these factsin mind, we setaboutconstructing a method for using RTs toinfera
subject’sindifferencepoint.Clearly,focusingontheslowesttrialswouldyieldlessbiased
estimates of subjects’ indifference points. However, using too few slow trials would
increase the variance of those estimates. We settled on a simple method that uses the
slowest10%ofasubject’schoices,thoughwealsoexploredothercutoffs(AppendixD).
Inshort,ourestimationalgorithmforanindividualsubjectincludesthefollowingsteps:
(1) identify trials with RTs in the upper 10% (the slowest decile); (2) for each of these
trials, calculate the value of the utility-function parameter that would make the subject
indifferentbetweenthetwoalternatives;(3)averagethesevaluestogettheestimateofthe
subject’s utility-function parameter (see Figure 6 for an example and Appendix A for
estimation details). It is important to note that this method puts bounds on possible
parameterestimates:theaverageofthehighest10%ofallpossibleindifferencevaluesis
the upper bound, while the average of the lowest 10% of the indifference values is the
24
lower bound. Thus it is crucial to choose choice problems with enough range in their
indifferencevaluestomorethanspantherangeofvaluesinthepopulation.
Again, the parameters estimated using this method were correlated with the same
parameters estimated purely from the choice data, providing a better fit than the DDM
approach (social choice 𝛼: r = 0.44, p = 0.02; social choice 𝛽: r = 0.56, p = 0.001;
intertemporal choice 𝑘: r = 0.71, p < 0.001; risky choice 𝜆: r = 0.64, p < 0.001; Pearson
correlations; Figure 7, see Appendix A for estimation details for both methods).
Furthermore, these parameters provided prediction accuracy that was better than an
informedbaseline(seeSection5.5)inthreeoutoffourcases(excludingsocialchoice𝛼).In
all cases, a random 10% sample of trials produced estimates that were not a meaningful
predictor of the choice-based parameter values (since these estimates are just a mean of
10% random indifference points).7 As noted previously, the RT-based estimations have
upperandlowerboundsduetoaveragingovera10-percentsampleoftrialsandthusare
notabletocapturesomeoutliers(e.g.seeFigure7,bottomleftpanel).Furthermore,the
numberof“extreme”indifferencepointsinthedatasetsthatweconsideredislow,biasing
theRT-basedestimatestowardsthemiddle.
5.6.2.Localregressionmethod
Asthetop10%approachonlyutilizespartofthedata,wedevelopedanotherapproach
based on the peaks in RTs, this time using all the available RT data. The local regression
7
We drew a random 10% sample 1000 times for each individual in each parameter dataset and
estimated the correlation between the average indifference point and the true parameter value
(socialchoice𝛼:meanPearsonr=-0.01;socialchoice𝛽:r=0.01;intertemporalchoice𝑘:r=-0.02;
riskychoice𝜆:r=-0.003).
25
(LOWESS) method also uses the “revealed indifference” approach: for each individual
subject,werunalocalpolynomialregressionofRTsontheindifferenceparametervalues
and use that regression to identify the indifference value that produces the highest
predictedRT(seeFigure6foranexampleandAppendixAfordetails).Asthepeakofthis
line is typically close to the choice-based estimate and the observations with the highest
RTs, this method is quite similar in its predictions to the top-RT-decile method. This
methodrequirestheresearchertochoosethelocalregressionsmoothingparameter.Here
weuseavalueof0.5asitproducesthebestresultsacrossallofthedatasets(thoughother
valuescanworkbetterforspecificdatasets;seeAppendixC).
Althoughthisapproachgenerallyproducesresultssimilartothetopdecileapproach,it
can be affected by outliers (e.g. sparse data and unusually high RTs around extreme
indifference points) and thus sometimes misestimates individual subject parameters,
producing correlations that are in some cases worse than those produced by the top RT
decile approach: social choice 𝛼: r = 0.12, p = 0.52; social choice 𝛽: r = 0.6, p = 0.001;
intertemporal choice 𝑘: r = 0.44, p = 0.01; risky choice 𝜆: r = 0.88, p < 0.001; Pearson
correlations.Thiscanbemitigatedbyusingaspecificsmoothingparameterforeachdata
set,butourgoalwastoidentifyamethodthatworkswellacrossallthedatasets.
5.7.Choicereversals
Finally, we wish to explore one additional set of predictions from the revealed
indifference approach. We know that when subjects are closer to indifference, their
choicesbecomelesspredictable.Wealsoknowthatsubjectsgenerallychoosemoreslowly
26
theclosertheyaretoindifference.Therefore,wehypothesizedthatslowerchoicesareless
predictableandthereforealsolesslikelytoberepeated.
Inallthreedatasets,thechoice-estimatedutilitymodelwassignificantlylessconsistent
withlong-RTchoicesthanwithshort-RTchoices(basedonamediansplitwithinsubject):
80% vs 89% (p <0.001) in the risky choice experiment, 71% vs 79% (p < 0.001) in the
intertemporalchoiceexperiment,88%vs94%(p=0.008)and90%vs96%(p<0.001)in
the dictator game experiment; p-values denote Wilcoxon signed rank test significance on
thesubjectlevel.
Asecond,morenuancedfeatureofSSMsisthatthey(mostly)predictslowerrors,even
conditioningondifficulty.Whiletherearesomeexceptions(seeRatcliff&McKoon2008),
the“slowerror”phenomenoniscommonlyobservedinexperimentsandaccountingforit
wasabreakthroughintheliterature(Ratcliff1978).Totestforslowerrorsinourdata,we
ran mixed-effects regressions of choice consistency on the RTs and the absolute utility
difference between the two options. In all cases we found a strong negative relationship
betweentheRTsandthechoiceconsistency(slowerchoice=lessconsistent)(fixedeffects
of RTs: social choice 𝛼: z = -2.62, p = 0.009; social choice 𝛽: z = -3.3, p < 0.001,
intertemporalchoice:z=-5.28,p<0.001,riskchoice:z=-5.35,p<0.001).Thusweindeed
observeevidenceforslowerrorsinallofourtasks.
In two of the datasets (intertemporal choice and non-adaptive risk choice) subjects
faced the same set of decision problems twice. This allowed us to perform a more direct
test of the slow error hypothesis by seeingwhether slow decisions in the first encounter
weremorelikelytobereversedonthesecondencounter.
27
In the intertemporal choice experiment, the median RT for a later-reversed decision
was 1.36 s, compared to 1.17 s for a later-repeated decision. A mixed-effects regression
effect of first-choice RT on choice reversal, controlling for the absolute utility difference,
was highly significant (z = 4.04, p < 0.001). The difference was even stronger in the risk
choice experiment: subsequently reversed choices took 2.36 s versus only 1.4 s for
subsequently repeated choices. Again, a mixed-effects regression revealed that RT was a
significant predictor of subsequent choice reversals (controlling for absolute utility
difference,z=5.2,p<0.001).
6Discussion
Herewehavedemonstratedaproof-of-conceptforthemethodofrevealedindifference.
The method of revealed indifference contrasts with the standard method of revealed
preference, by using response times (RTs) rather than choices to infer preferences. This
newmethodreliesonthefactthatpeoplegenerallytakelongertodecideastheyapproach
indifference. Using datasets from three different choice domains (risk, temporal, and
social)weestablishedthatpreferencesarehighlypredictablefromRTsalone.Finally,we
also found that long RTs are predictive of choice mistakes, as captured by inconsistency
withtheestimatedutilityfunctionandlaterpreferencereversals.
Throughout the paper we have highlighted ways in which we think RTs may be
important for economists. First, using RTs may allow us to estimate agents’ preferences
using very short and simple decision tasks, even a single binary-choice problem (Section
5.3). For example, if you want to know whether people will buy your product for $50,
knowingthattheywouldbuyitfor$30isnotveryusefulinformation.However,ifyoualso
28
know that they were very quick to say yes at $30, you might reasonably infer that many
wouldstillpurchasetheproductfor$50.
Second,usingRTscanhelpustorecoverpreferenceswhen,evenwithmultiplechoice
problems, some agents always give the same response, meaning that we could otherwise
onlyputboundsontheirpreferences(Section5.4).
Third,thefactthatRTscanbeusedtoinferpreferenceswhenchoicesareunobservable
or uninformative (Sections 5.5 & 5.6) is an important point for those who are concerned
about private information, mechanism design, etc. For instance, while voters are very
concernedabouttheconfidentialityoftheirchoices,theymaynotbethinkingaboutwhat
their time in the voting booth might convey about them. In an election where most of a
community’s voters strongly favor one candidate, a long stop in the voting booth may
signal dissent. Another famous example from outside of economics is the implicit
associationtest(IAT),wheresubjects’RTsareusedtoinferpersonalitytraits(e.g.racism)
thatthesubjectswouldotherwiseneveradmittoorevenbeawareof(Greenwald,McGhee,
&Schwartz,1998).Thusprotectingprivacymayinvolvemorethansimplymaskingchoice
outcomes.
Fourth,ourworkhighlightsamethodfordetectingchoiceerrors.Whilethestandard
revealed preference approach must equate preferences and choices, the revealed
indifferenceapproachallowsustoidentifychoicesthataremorelikelytohavebeenmade
bymistake,orattheveryleast,withverylowconfidence.ThusRTsmayplayanimportant
normativeroleinestablishinghowconfidentlywecansaythatagivenchoicetrulyreveals
thatagent’sunderlyingpreference.
29
InthispaperwehavedescribedaframeworkthatmathematicallylinksRTsandchoices
tounderlyingpreferences.Thisframeworkisnotaheuristic;infact,itarisesastheoptimal
solutiontoasequentialsamplingproblem,wherewhatissampledarestochasticsignalsof
the underlying true preference. In addition to their normative appeal, these SSMs have
enjoyed much empirical success in capturing choice probabilities and RT distributions in
many domains, including economic choice (Fudenberg et al. 2015; Krajbich et al. 2010,
2014; Ratcliff and McKoon 2008; Webb 2013; Woodford 2014). Moreover, they are
appealing from a neuroscience perspective, as SSMs are biologically plausible and align
wellwithneuralrecordingsinbothhumansandotheranimals(Bogaczetal.2009;Polania
etal.2014).However,itisimportanttonotethatSSMsarenotuniqueintheirprediction
ofslowdecisionscorrespondingwithindifference.Forexample,thisisalsoafeatureofthe
directedcognitionmodel(Gabaix&Laibson,2005).
OnequestionthatarisesfromourresultsiswhatistheoptimalwaytomakeuseofRTs
inordertoinfersubjects’preferences?Ofthemethodswetested,thetop10%ruleseems
toworkverywellacrossourdatasetsandisaneasymethodtouseinpractice.However,
this cutoff will generally depend on the number of trials in a particular experiment: the
fewerthetrials,thelargerthetoppercentileneedstobe.Foreachofthethreedatasets,we
calculatedtheoptimalpercentilecutoffandfoundthatthetop10-20%RTsgeneratedthe
bestpredictions.Astheindividual-trialRTinformationisnoisy(fasttrialscanbebotheasy
anddifficult,butslowtrialsarealmostalwaysdifficult),usingfewertrialsproduceshigher
variancepredictions,whilemoretrialsmayintroducebias.
ThereareofcourselimitationstousingRTstoinferindifference.Itimportanttokeep
inmindthatotherfactorsmayinfluenceRTsinadditiontostrength-of-preference,suchas
30
complexity,stakesize,andtrialnumber(Moffatt2005;Krajbichetal.2015b).Aswithany
analysis, it is important to control or account for these factors in order to maximize the
chanceofsuccess.
A second potential criticism of these findings is that we have focused on repeated
decisions which are made quite quickly (1-3 seconds on average) and so may not be
representativeof“realworld”decisionsormaybebeingmadeusingsimpleheuristics.We
haveseveralresponsestothiscriticism.First,theseareallmulti-attributechoiceproblems
and so it is unclear what simple heuristics subjects could be using. Second, the use of
simpleheuristicswouldonlyimpairourabilitytoestimatepreferencesfromRTs,sincein
thosecasesthereshouldbenorelationshipbetweenstrength-of-preferenceandRT.The
less people can rely on heuristics and instead have to evaluate the alternatives to
determinewhichisthebest,themoreeffectivethemethodofrevealedindifferenceshould
be.Third,wewouldarguethatmany,ifnotmost,realworlddecisionsareminorvariants
of other decisions that we make repeatedly over the course of our lives. So while these
tasksmaynot,forexample,fullycapturetheprocessofbuyingahouse,theymaybevery
representativeofroutineeconomicdecisions.Finally,whatresearchhasbeendoneonRTs
inone-shot,slowdecisions,issofarconsistentwiththeSSMpredictions.
For example, Krajbich et al. 2015b study a voluntary contribution public goods game
experimentwheresubjectsmadeonlythreedecisionsandtookonaverage43.5stodecide
eachtime.Inthatexperiment,slowdecisionstendedtofavorthelessattractivealternative,
consistentwithbeingclosertoindifference.Inparticular,withalow-benefitpublicgood,
slow contributions tended to be higher, while with a high-benefit public good, slow
contributionstendedtobelower.
31
More careful analysis is required to distinguish between the SSM and alternative
interpretationsbyRubinsteinandothers(Chen&Fischbacher,2015;Hey,1995;Recaldeet
al., 2014; Rubinstein, 2007, 2013, 2014, 2016),wherelong RTsareassociatedwith more
carefulordeliberativethoughtandshortRTsareassociatedwithintuition.Itmayinfact
bethecasethatinsomeinstancespeopledousealogic-basedapproach,inwhichcasea
longRTmaybemoreindicativeofcarefulthought,whileinotherinstancestheyrelyona
SSM approach, in which case a long RT likely indicates indifference. This could lead to
contradictory conclusions from the same RT data; for example one researcher may see a
long RT and assume the subject is very well informed, while another researcher may see
thatsameRTandassumethesubjecthasnoevidenceonewayortheother.Moreresearch
isrequiredtotestwhetherSSMs,whicharedesignedtoteaseapartsuchexplanations,can
besuccessfullyappliedincomplexeconomicdecisions.
32
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20
utility difference
Figure 1. Example simulations of the drift-diffusion model (DDM). Response times
(RTs)asafunctionofthedifferenceinutilitiesbetweentwooptionsin900simulatedtrials.
Thegraydotsshowindividualtrials,theblackcirclesdenoteaverageswithbinsofwidth
10.Theparametersusedforthesimulationcorrespondtotheparametersestimatedatthe
group level in the time discounting experiment (𝑏 = 1.33, 𝑧 = 0.09, 𝜏 = 0.11). Utility
differencesaresampledfromauniformdistributionbetween-20and20.
42
Social choice (β)
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4
Figure2.RTspeakatindifference.RTinsecondsasafunctionofthedistancebetween
theindividualsubject’sutilityfunctionparameterandtheindifferencepointonaparticular
trial;dataareaggregatedintobinsofwidth0.02(toprow),0.01(bottomleftpanel),and1
(bottomrightpanel),whicharetruncatedandcenteredforillustrationpurposes.Binswith
fewerthan10subjectsarenotshown.Barsdenotestandarderrors,clusteredatthesubject
level.
43
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Figure3.Preferencerankcanbeinferredfromasingledecisionproblem.RTsinthe
first round of the adaptive risk experiment as a function of the individual subject’s lossaversion coefficient from the whole experiment; Spearman correlations displayed. In this
round,eachsubjectwaspresentedwithabinarychoicebetweenalotterythatincludeda
50%chanceofwinning$12andlosing$7.5,andasureoptionof$0.Theleftpaneldisplays
subjects who chose the safe option, and the right panel shows those who chose the risky
option.ThesolidblacklinesareOLSfits.
44
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β estimated from choices
Intertemporal choice
Risk choice
0.05
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r = 0.52
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p = 0.04
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−0.5
0.5
r = 0.39
β estimated from RTs
1.0
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α estimated from RTs
1.5
Social choice (α)
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0.04
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r = 0.36
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0.05
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λ estimated from choices
k estimated from choices
8
Figure 4. The DDM estimates of subjects’ utility function parameters. Subject-level
correlation(Pearson)betweenparametersestimatedfromchoicedataandRTdatausing
thedrift-diffusionmodel(DDM).Thesolidlinesare45degreelines.
45
Social choice (β)
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2
Intertemporal choice
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6
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4
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RT [s]
8
●
−6
−4
−2
0
2
4
subject λ − indifference λ
subject k − indifference k
6
Figure5.SinglelongestRTsareanoisypredictorofindifference.RTinsecondsasa
functionofthedistancebetweentheindividualsubjectutilityfunctionparameterandthe
indifference point on a particular trial; gray dots denote individual trials. Red triangles
denotetrialswiththehighestRTforeachindividualsubject.
46
●
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indifference β
0.5
1.0
Figure6.Exampleofanindividualsubject’sRT-basedparameterestimation.Theplot
shows RTs in all trials as a function of the indifference parameter value on that trial.
ObservationsinthetopRTdecileareshowninred.TheredtriangleshowsthelongestRT
for the subject. The solid vertical red line shows the subject’s choice-based parameter
estimate.ThedottedverticalredlineshowstheaverageindifferencevalueforthetopRT
decile approach. The dotted grey line shows the local regression fit (LOWESS, smoothing
parameter=0.5).
47
Social choice (β)
1.0
1.0
Social choice (α)
0.0
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p = 0.001
−0.5
β estimated from RTs
0.5
r = 0.56
p = 0.02
−0.5
α estimated from RTs
r = 0.44
−0.5
0.0
0.5
1.0
−1.0
0.5
Intertemporal choice
Risk choice
1.0
6
5
r = 0.64
●
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2
λ estimated from RTs
0.03
p < 0.000
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0.00
0.01
0.0
β estimated from choices
r = 0.71
k estimated from RTs
−0.5
α estimated from choices
0.04
−1.0
0.00
0.01
0.02
0.03
0.04
0
1
2
3
4
5
λ estimated from choices
k estimated from choices
6
Figure 7. The top RT decile estimates of subjects’ utility function parameters..
Subject-level correlation (Pearson) between parameters estimated from choice data and
response time (RT) data using trials with RTs in the top decile. The solid lines are 45
degrees.ThedottedredlinesshowtheboundsonRTparameterestimations.
48
ONLINEAPPENDIX
A. Parameterestimationmethodology
Choice-basedmethod
Weestimateeachindividualutilityfunction𝑢(⋅ |𝜃),where𝜃isasubject-specificparameter
described in Section 3, in the standard way as follows. We assume that for each pair of
options and choice 𝑎 = 1,2 the error terms in utilities follow the type I extreme value
distribution,sotheprobabilityofchoosingoption1isalogisticfunction
𝑝(𝑎i = 1) =
1
1 + 𝑒 Q•
‚ƒ (⋅|„)Q‚N (⋅|„ )
,
where 𝜇 and 𝜃 are free parameters that can be estimated for each subject individually
maximizingalikelihoodfunction
𝐿𝐿 =
(log 𝑝(𝑎d = 1) ⋅ 1 𝑎† = 1 + log 1 − 𝑝 𝑎d = 2
⋅ 1(𝑎† = 2)),
d
where𝑛isthetrialnumber,𝑎d isthechoicemadebythesubjectonthattrial,and1(⋅)is
theindicatorfunction.
Appendix B shows the subject level correlations between the predicted and the actual
choices.
Top10%responsetimemethod
For each decision problem on each trial 𝑛, we calculate the indifference parameter value
𝜃did‡ asasolutiontotheequation
𝑢9 ⋅ 𝜃d = 𝑢< ⋅ 𝜃d .
49
ThenweaveragetheindifferencevaluesonthetrialsinthetopRTdeciletoobtainthefinal
parameterestimate:
𝜃=
id‡
d(𝜃d
⋅ 1(𝐹(𝑅𝑇d ) ≥ 0.9))
,
d 1(𝐹(𝑅𝑇d ) ≥ 0.9)
where 𝑅𝑇d is the response time on trial 𝑛, 𝐹(⋅) is the empirical RT distribution for the
specificsubject,and1 ⋅ istheindicatorfunction.
Localregression(LOWESS)method
As in the previous method, for each decision problem on each trial 𝑛, we estimate the
indifferenceparametervalue𝜃did‡ solvingtheequation
𝑢9 ⋅ 𝜃d = 𝑢< ⋅ 𝜃d .
For each individual subject, we regress response time log(𝑅𝑇) in every trial 𝑛 on the
corresponding indifference parameter value 𝜃did‡ using a local polynomial regression
(LOWESS,Cleveland1979)intheRpackagestats:
𝑅𝑇 = 𝑓 𝜃did‡ + 𝜀d .
We set the smoothing parameter to 0.5 as it has provided the best prediction accuracy
acrossallfourdatasets(seeAppendixC).
Then we obtain the parameter estimate 𝜃 by inverting the fitted regression line at the
maximumpredictedresponsetime𝑅𝑇:
𝜃 = 𝑓 Q9 max 𝑅𝑇 .
50
Drift-diffusionmodel(DDM)method
In the DDM (see Section 2) a latent decision variable evolves over time with an average
drift rate plus Gaussian noise (the Wiener diffusion) until it reaches one of two predeterminedboundaries,whichcorrespondtothetwochoiceoptions..Giventheboundary
separation parameter, the drift rate, the non-decision time (the component of RT not
attributable to the decision process itself, e.g. moving one’s hand to indicate the choice),
and the variance of the Gaussian noise, it is possible to calculate choice probabilities and
choice-contingentRTdistributions.
Inourparticularcaseweassumethatchoicesareunknown,andsowecanonlyuse
thecombined(summed)RTdistributiontoestimatethefreeparametersofthemodel.We
assumethatallsubjectssharethesameconstantboundaryparameter𝑏,non-decisiontime
𝜏,anddriftrateparameter𝑧,whichmultipliestheutilitydifferenceoneverysingletrial:
𝑣 ≡ 𝑧 ⋅ 𝑢9 ⋅ 𝜃 − 𝑢< ⋅ 𝜃 .
We use a density function of the Wiener distribution from the RWiener R package
(WabersichandVandekerckhove2014)toestimatethelikelihood(9)fortheobservedRT
on every given trial assuming a set of parameters (𝑏, 𝜏, 𝑧, 𝜽), where 𝜽 is a vector of
individualsubjects’parameters.Essentially,theidentificationoftheindividualparameters
ispossibleduetothefactthatRTsarepredictedtovaryastheutilitydifference𝑣varies
acrosstrialsandsubjects.
51
B. Choice-basedfits
Social choice (β)
0.1
●
●●●●
●●●● ●
r = 0.91
●
0.6
●●●●
0.2
0.3
0.4
0.5
r = 0.99
●
p < 0.000
●●●● ●
0.6
●
0.0
0.2
0.4
0.6
0.8
P(fair) in data
P(fair) in data
Intertemporal choice
Risk choice
1.0
0.1
0.4
●
0.0
0.2
0.4
0.6
0.8
●
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●
r = 0.99
p < 0.000
0.0
0.0
0.2
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r = 0.97
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0.6
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1.0
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0.4
0.8
P(safe) estimated from choices
●
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●
0.2
1.0
0.0
P(later) estimated from choices
●●●●
●
0.0
p < 0.001
0.8
1.0
0.2
●●
●
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0.4
0.3
●
●
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0.2
P(fair) estimated from choices
0.5
0.4
●
0.0
P(fair) estimated from choices
0.6
Social choice (α)
0.8
1.0
0.0
P(later) in data
0.2
0.4
0.6
P(safe) in data
0.8
1.0
FigureB1.Subject-levelcorrelation(Pearson)betweenchoiceproportionsinthedataand
as predicted by choice-estimated utility functions (see Appendix A for details). The solid
linesare45degrees.
52
C. Local regression (LOWESS) method choice prediction accuracy for various
levelsofthesmoothingparameter
Social choice (β)
0.85
0.80
prediction accuracy
0.90
0.85
0.80
prediction accuracy
0.90
0.95
Social choice (α)
0.4
0.6
0.8
1.0
0.0
0.4
0.6
smoothing parameter
Intertemporal choice
Risk choice
0.8
1.0
0.8
1.0
0.85
0.75
0.80
prediction accuracy
0.80
0.75
0.70
prediction accuracy
0.2
smoothing parameter
0.90
0.2
0.85
0.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
smoothing parameter
0.2
0.4
0.6
smoothing parameter
Figure C1. Choice prediction accuracy as a function of the smoothing parameter of the
LOWESS regression model. The solid black lines denote mean prediction accuracy across
subjects,theshadedareasshowstandarderrorsatthesubjectlevel.
53
D. ExploringtheoptimalcutoffforthetopRTestimationmethod
Social choice (β)
0.90
0.85
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
0.75
0.80
prediction accuracy
0.90
0.85
0.80
prediction accuracy
0.95
Social choice (α)
0.80
0
20
40
60
80
100
0
20
40
60
top percentile
top percentile
Intertemporal choice
Risk choice
100
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●
0.82
0.80
0.65
0.76
0.78
prediction accuracy
0.75
0.70
prediction accuracy
0.84
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●
80
0
20
40
60
80
100
0
top percentile
20
40
60
top percentile
80
100
Figure D1. Averaging the longest 10-20% RT trials provides the best choice
predictionaccuracy.Choicepredictionaccuracyasafunctionofthepercentageofslowest
trialsusedintheparameterestimationfrom1to100%.Thesolidblacklinesdenotemean
predictionaccuracyacrosssubjects,theshadedareasshowstandarderrorsatthesubject
level,theredlinesabovethegraphsindicatesignificantdifferencefromthebaselineatthe
p = 0.05 level (Wilcoxon signed rank test). The baseline is the average of all indifference
54
pointsacrosstrials.Itisimportanttoemphasizethatthebaselinetowhichwecomparethe
predictive power is not chance (50%) as almost any experimenter uses some prior
knowledgeoftheparameterdistributioninthepopulationtoselecttheirchoiceproblems.
For example, an experimenter studying intertemporal choice might select a set of choice
problemssothattheaveragesubjectwouldchoosetheimmediateoptionhalfofthetime
andthedelayedoptiontheotherhalfofthetime.Soifyouweretoaveragetheindifference
pointsfromthetrialsinsuchanexperiment,youwouldbeabletopredictbehaviorquite
accurately,onaverage.Insuchanexperiment,behaviorintrialswithextremeindifference
pointswillbeverypredictable.Thatis,onatrialdesignedtomakeaverypatientsubject
indifferent, most subjects will have a strong preference for the immediate option.
Similarly, on a trial designed to make an impatient person indifferent, most subjects will
haveastrongpreferenceforthedelayedoption.Thusbehaviorinmanyofanexperiment’s
trialsisquiteeasytopredictbecausethosetrialsareonlyincludedtoidentifyparameter
values for extreme subjects. For instance, a single loss-aversion coefficient of 𝜆 = 2 can
predictabout75%ofchoicesinourrisky-choicedataset.
55
E. Instructionsfortheriskexperiment
Instructions
Thankyouforparticipatingintoday’sstudy.
Pleasecarefullyreadthematerialonthefollowingpagestounderstand
• Therules
• Thedecisionsyouwillbemakingtoday
Ifyouhaveanyquestionsafterreadingtheseinstructionsorduringtheexperiment,please
askthembeforetheexperimentorduringthedesignatedbreaks.
Therules
• Pleasechecknowtoensurethatyourmobilephoneisonsilentmodeandputitin
yourbagorpocket.
• Pleasedonottalkduringtheexperiment.
Thestudy
Todayyouwillbemakingaseriesofchoices,andyourfinalpaymentwilldependonlyon
yourownchoicesandchance.
Payment
Yourpaymentwillconsistoftwoamounts:
• Afixedendowmentof34experimentalcurrencyunits(ECUs)thatyouaregivenat
thebeginningofthestudy.
• Your earnings from one randomly selected choice round. You may earn additional
moneybeyondthe34ECUs,oryoumaylosesomeofthat34ECUs,dependingon
yourchoicesandonchance.Theminimumamountofmoneyyoucanearntodayis
10ECUs=$5.
• All the amounts in today’s study will be shown in ECUs and will be converted to
dollarsattheendofthestudyatarateof2ECUs=$1.
56
Yourchoices
In each round of the experiment you will be asked to make a choice between one of two
options. Option one consists of two possible amounts, each one with a probability of 50%.
Optiontwoconsistsofoneamount,withaprobabilityof100%.
Belowisanexampledecisionscreen.InthisroundOptionone(ontheleft)consistsofagain(in
green)of30ECUsandaloss(inred)of10ECUs.Ifyoupickthisoption,thecomputerflipsafair
digital coin (chances are 50-50). In case of heads, you would earn 30 ECUs on top of your
endowment. In case of tails, you would lose 10 ECUs, which would then be subtracted from
yourendowment.Tochoosethisoption,youwouldpress1.
Optiontwo(ontheright)isasuregainof15ECUs.Ifyoupickthisoption,youwouldearn15
ECUsontopofyourendowment.Tochoosethisoption,youwouldpress2.
30
vs
15
-10
1
2
Yourfinalearningswillonlydependononeofyourchoices:attheendofthestudy,onlyoneof
theroundswillberandomlyselectedforpayment.Theoutcomefromthisroundwillbeadded
orsubtractedfromyourinitialendowment.
Ifyouhaveanyquestions,pleaseraiseyourhandnow.
57