Attitudes toward choice with incomplete preferences: an experimental study

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ContentslistsavailableatScienceDirect

Journal of Economic Behavior and Organization

journalhomepage:www.elsevier.com/locate/jebo

Attitudes toward choice with incomplete preferences:

An experimental study ^R

Ritxar Arlegi

^a^,^∗

, Sacha Bourgeois-Gironde

^b

, Mikel Hualde

^a

a Department of economics, Public University of Navarre and Institute for Advanced Research in Business and Economics (INARBE), Campus Arrosadia s/n- Pamplona 31006,Navarra,Spain

b Centre de Recherches en Economie et Droit - Université Paris 2 - Panthéon-Assas. Institut Jean-Nicod, Ecole Normale Supérieure - PSL, France

a rt i c l e i nf o

Article history:

Received 16 December 2021 Revised 24 August 2022 Accepted 30 September 2022 Available online 12 November 2022 JEL classiﬁcation:

D01 Keywords:

Incomplete preferences Choice aversion Preference for ﬂexibility

a b s t ra c t

Wepresentanexperimenttotestdifferentindividualattitudestowardchoice,suchaspref- erenceforflexibility,choiceaversion,betweennessandchoiceneutrality.Unlikeotherrelated experimentalpapers,wewanttoanalyzewhetherdifferentchoiceattitudescancoexistfor thesamesubject,dependingonthecharacteristicsofthechoicesetsheisfacing.Inpar- ticular,ourmainhypothesisisthatthepresenceofincomparabilityamongthealternatives inthechoiceset,andthetimeatwhichsuchincomparabilityissolved,affectcruciallythe kindofattitudetowardschoicethatthesubjectwillexhibit.Wefindthat,indeed,choice attitudesarenothomogeneousacrosschoicesets,yettheyareconditionalontheprefer- encesoverthealternatives.Wealsofindsomeevidencesupportingthatsubjectstendto valueheuristicallysetsasawhole.

ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/)

1. Introduction

Accordingtothestandardmicroeconomictheorythevalueofabudgetsetisthatofitsbestalternative.Translatedtothe moregeneralsettingofevaluatingopportunitysets(or“menus”)thisevaluationruleisknownastheindirectutilitycriterion (Arrow,1995).Adirectconsequenceofthiscriterion isthat addingtoorremovingfroma menuasuboptimalalternative doesnotaffectitsvalue.

Anincreasing numberoftheoreticalworkschallengethisview claimingthatindividualsmaydisplay differentattitudes towardthe actofchoice,sothatavailingmoreorlessalternativesmayindeedaffectthevalueofthemenu,evenifthese alternatives are suboptimal (or“dominated”). This distortstheindirect utilityparadigmof rationality,wherethe decision

R Financial support from the National Agency for Research (ANR-17-EURE-0017 FrontCog), from the Universidad Pública de Navarra (PID2020- 119703RB_I00 and PJUPNA19-2022), from the Spanish Ministry of Science and Innovation (PID2021-127799NB-I00) and from the Government of Navarre is acknowledged. We are grateful to Jon Benito, Pablo Brañas, Filippos Exadaktylos and Stéphane Luchini for their comments and to the administrative staff of the Inarbe Institute for their help before and during the experimental sessions. We also thank two anonymous reviewers and the associate editor for their valuable suggestions and comments. Open access funding provided by Universidad Pública de Navarra.

∗Corresponding author.

E-mail address: [email protected] (R. Arlegi) .

https://doi.org/10.1016/j.jebo.2022.09.027

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makerisneutraltothemereactofchoice.Wehighlightthreeprominentexamplesofsuchalternativeviews.Anindividual displayschoice-preferencewhenshestrictlyprefersenlarginganopportunitysetevenifsuchenlargementismadebymeans ofdominatedalternatives.Forexample,theindividualmaypreferoptionxtoy,butshemayalsopreferhavingthepossi- bilitytochoosefrom

{

^x,y

}

^than^being^forced^to^choose

{

^x

}

^.^Choiceâversionîsôpposite ^tochoice-preference.Underchoice aversion itcanbe thecasethatanindividual preferseither

{

^x

}

^or

{

^y

}

^rather^than^having^to^make^a^choice^between

{

^x,y

}

^.

Unlikeforthechoice aversionmodels,under“betweenness”,aversion tochoicearisesasaconsequenceofcertaininterre- lations ofthealternatives inthemenu.Betweennessisusually relatedtoa preferenceforcommitmentinthepresenceof regretandtemptationaversion(GulandPesendorfer,2001;Dekeletal.,2009;Sarver,2008).Thisentailsthattheindividual maypreferbeingobligedtochoose

{

^x

}

^rather^than^having^to^choose^from

{

^x,y

}

^,^but^she^may^prefer

{

^x,y

}

^rather^than^being

obligedtochoose

{

^y

}

^.

Althoughtheseviewsactuallypointtodifferentattitudesandhavesometimes beendeemed asexclusiveone fromthe other(forinstance,obviously,choicepreferenceandchoiceaversion),ourpresentpurposeistoshowthatthereexistformal andobservable conditionsfortheir compatibilityoratleastjuxtaposition,within asameindividual, ascan bemanifested by the interaction betweenher valuation of an optionset andher actual choice behavior. Attitudestoward choice ﬁnely dependuponthewayacompletepreferenceandaconsistentvaluationprocesscanbeindividuallyconstructedornotover anoptionset.

Weareinterestedintheintra-personalcompatibilityofdifferentattitudestowardchoice.Weclaimthattheyarenotonly psychologicalcharacteristicsofthedecisionmakerbutthattheydependtoalargeextentonenvironmentalfactors,inparticular,tothefeaturesofthechoiceproblemandofthealternativestobeselected,whichallowsornotfortheformationofa completepreferencerelationandofaconsistentvaluationprocessoftheconsideredsetsofoptions.Arlegietal.(2021)pro- poseatheoreticalmodelwherechoice-preference,choiceaversion andtheindirectutilitycriterion(or“choice-neutrality”) can be madecompatibleunderaproperspecificationofthe model.Inparticular,three typesofbinaryrelationsover the alternatives are defined: a binaryrelationP interpreted asaundoubted strictpreference; abinary relation1 oftentative incomparability,which isexpectedto be solved beforethe final alternative hasto be selectedfromthe set,anda binary relation2 ofassertiveincomparability,whichisnotexpectedtobesolvedbeforethefinalalternativefromasethastobe selected. Eachofthethreebinaryrelations connectswithadifferentattitudetowardchoice: An alternativethat isdomi- natedaccordingtoPdoesnotaddanyvaluewhenaddedtoaset,alternativesrelatedby1involveflexibilityandtherefore a preferencefora greaterchoice,andalternativesrelatedby2 involveapsychological costofincomparabilitythat leads to aversiontochoice. Thethreebinaryrelationscan coexistinan individual decisionproblem, andthereforethedecision makermayexhibitthethreedifferentchoiceattitudes.

Close to our purpose, Le Lec and Tarroux (2020) designed an experiment consisting of a first stage during which participants had to evaluate all the possible choice sets formed by combinations of four alternatives, including singletons. In thesecond stage participants choose one item fromone of the menus randomly selected.The alternatives consisted of 30-minutes access to websites of well-known media of different types: a popular general TV channel;a more youth oriented TV channel;a culture-orientedTV channel anda leading newspaper.In that way, it could not be a pri- ori presumed that the participants preferred any particular alternative over any other. Such a preference is deduced from their evaluation of the corresponding singletons. The results suggest that participants are choice-averse and that adding suboptimal options reduces the value of a set. The value of the set is sensitive to the value of all its alternatives, even when they are suboptimal. The authors provide two possible explanations fortheir findings. One is that individuals fear making bad decisions in the second stage. The other is that subjects value the menus heuristically as a whole, andnot onthe basisof theirfinal consequences. The paradigmusedinthat experimentis instrumentalfor usto investigate possible correlations betweenthe degree of determinacy of a preference (whether it can be presumed complete or not)and various possible attitudes towardchoice (aversion,neutrality orpreference for more available options inaset).

In this paper we present an experiment that essentially combines and develops both the theoretical work by Arlegi etal.(2021) and theexperimental work by Le Lec and Tarroux (2020) (henceforthLT20), in a view to systemati- cally addressthe questionof howattitudes towardchoices are drivenby set-valuation and preferencecompleteness fac- tors,themselvesdrivenbydiversecompositionsofoptionsets.Inourownexperimentalsetting,alternativeswithin option sets are designedin sucha waythat we can presumethat the three typesofpreferenceor incomparabilityrelations, as recalledabove,willoccur. Inthiswaywetrytocheck whichcorrelationbetweenthetypesofbinaryrelationsamongal- ternativesandthedifferenttypesofattitudestowardchoiceariseinthelab.UnlikeinLT20,wherealternatives,consisting of enjoyingreading awebpage during 30min, areﬁnal consumption goods,in ourdesign alternatives are means to ob- tain monetary earnings.Some ofthem give clearlybetterchancesto obtainearnings thanothers (theyarerelatedby P);

amongsomeotherpairsofalternativesthecomparisonintermsofthechancestheygivetoobtainearningsisdeliberately unclear anddifficult to be made(they are relatedby 2) and, finally,some alternatives involve an uncertaintythat will be solved before an optionhasto beselected fromthemenu inthe finalstage of thedecisionprocess (theyare related by1).

Onthebasisofthisdesign,weareinapositiontoaddressourtwomainquestions:i)Howthemoreorlessdeterminate (P,1,2) binary relationsholding over thealternatives we display affectthe valuation ofoptions sets andii)To which extent we observe a correlation betweenthose binary relations andattitudes towardchoice as modelledin Arlegiet al.

(2021).

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Literaturereview

Apart of the mentioned works by Arlegi etal. (2021) andLT20, there are several works that study, by separate, the differentattitudestowardchoiceaddressedatthebeginningoftheintroduction.

Regardingthechoice-preferenceattitude,thereareatleasttwopossibleexplanations.Oneisthatfreedomofchoicehas anintrinsicvalue(Suppes,1987;Sen,1988;PattanaikandXu,1990;2000;Bossertetal.,1994).Anotherpossibleexplanation is a preference for ﬂexibility, according to which the preference for

{

^x,y

}

^over

{

^x

}

^may ârise ^when ^the preferences over the singlealternatives are not totallycertain at themoment ofevaluating the menus.Such uncertaintyis then expected to be solved beforethe final alternativehas tobe selected fromtheset atthe second stageof thedecisionprocess (see Kreps,1979;Nehring,1999;Ahn andSarver,2013;GornoandNatenzon,2018).Intrinsicvalueoffreedomofchoiceenters incontradictionwiththestandardprinciplesofrationality,butpreferenceforflexibilityiscompatiblewiththemunderan appropriatemodellingbymeansofcontingentutilityfunctions.¹

Sonsino andMandelbaum(2001) explain choice aversion asan aversion tothe complexityofthe decisionproblemat hand, which can be directly related withthe cardinality of the options set. Ortoleva (2013) proposes an explanation in termsof“thinkingaversion”,accordingtowhichthedecision-makeranticipatesthecostofthinkingthatisneededinorder toidentifythebestalternativewhentheﬁnalchoicemustbemade.GuerdjikovaandZimper(2008),Dananetal.(2012)and Arlegietal.(2021)linkchoiceaversionwithapsychologicalaversiontoindecisivenessorincompletenessofthepreferences overthesinglealternativesinasituationwheresuchincompletenessisnotexpectedtobesolvedwhentheﬁnalalternative hastobeselectedfromtheopportunityset.

Asforbetweeness,temptationmodels,likeGulandPesendorfer(2001)andDekeletal.(2009),explainitbyconsidering y asatemptingalternative,sothattheindividual anticipatesaself-controlefforttochoosexfrom

{

^x,y

}

^,^which^makes^her

prefer

{

^x

}

^to

{

^x,y

}

^.Ât ^the^same ^time,^sheâlso ^knows^that^she ^prefers^to^choose^x ôver^y^, ^which^should^make ^her ^prefer

{

^x,y

}

^to

{

^y

}

^.În^regret^models⁽^Sarver,²⁰⁰⁸⁾^xîs^superior^to^yⁱⁿêxpectedûtility^terms,^which^makes

{

^x,y

}

^be^preferred^to

{

^y

}

^,^but^there^arecontingencieswhereymaybe better,sothat

{

^x

}

is preferredto

{

^x,y

}

âsâ^way^toâvoid^the^regretôf^not

havingchosenyifsuchcontingenciesﬁnallyhold.

SeveralstudieshavefocusedonexperimentallytestingthedifferentattitudestowardchoicereferredabovepriortoLT20.

Sonsino andMandelbaum (2001) analyse preference forflexibility and complexityaversion in an experiment where different menus consisting ofinvestment planswere offered tothe participants. The plansdiffered in size (either three or eight options). Each optioncontained three possible states ofnature andthere were two treatments for each plan. Un- der their "probabilistictreatment” subjectswere told thatthey would knowtheprobability associated toeach state. Un- der their “complete information” (CI) treatment they were told that they would know the exact state of nature. Thus, complexity is designed at two levels: the size of the menu and the information conditions. The size parameter acts in two opposite directions: preferencefor flexibility andcomplexity aversion. Expected payoffs are designed to isolate the complexity effects. The results show that the size aversion effect has more impact than the preference for flexibility effect under the CI treatment, but the opposite holds under the probabilistic treatment. Moreover, there is a negative cross interactionofcomplexityeffects:complexityaversion withrespecttothe informationconditionsshowsuponly for small sets, andsize aversion showsup only forinformationallysimplertreatments. This finding motivatesfurther inves- tigations aboutthe precise conditions that would make the three attitudes defined above as jointly exemplified among participants.

Toussaert(2018)examinestemptationasasourceofbetweenness.Thetemptingalternativeisthepossibilitytoreadan interestingstoryaboutoneoftheparticipantsintheexperiment.Thereisalsoatasktobeperformed,consistinginlooking to an updating numberandtakingnote of itin a promptingbox that appeared 5times.Subjects received2 poundsper correctanswer.Ifthesubjectchosereadingthestory,thenonly4outofthe5timeswouldberandomlyvalid.Preferences over three menus are examined: {not beingable to read the story}; {beingforced to read the story},or {being able to choose between readingthe story ornot}. 40% of subjects displayedbetweenness; 6.7% showedaversion; 34.1% showed choice-preference,and9.2%wereconsistentwiththeindirectutilitycriterion(theremaining 10%showedother orderings).

Herethereisnoapparentcompatibilitybetweentypesofattitudeswithinparticipants,butthisresultshowstheprevalence oftwomainattitudesacrossthepopulation,choicepreferenceandbetweenness,whicharealsocentraltargetsinourown experimentalsetting.

CettolinandRiedl(2019)conducteddifferentexperimentswhereparticipantsfacedaseriesofdecisionswheretheywere presentedwithariskyandanambiguousprospect.Bothprospectsweresimilarwithrespecttothepossibleoutcomes. In each ofthesituations subjectscould eitherchooseoneofthetwoprospects,orselectanoptionthat delegatesthechoice between the two prospects to a fair random device. The results obtainedshow that there is a significant percentage of subjectsthatchoosethedelegationoptionandsuchachoiceisattributabletoincompletenessintheir preferencesandthe wishtoresorttoadevicethatwoulddefinitelyfixtheircognitiveuncertainty.

In Section2 we detailtheexperimental design,andspell out ourmainhypothesis. Section 3presentsthe resultsand conclusionsareproposedinSection4.

1Arlegi and Nieto (2001a,b) , defend that apparent intrinsic value of freedom of choice may be masking, in fact, a preference for ﬂexibility in Kreps (1979) terms.

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2. Experimentaldesign

The experimentwasconductedintheInarbeLaboratory ofExperimentalEconomicsatthePublicUniversityofNavarra indifferentsessionsinNovember2020andinMay2021.107peopleenrolledintheexperiment,butﬁnally99participants showedupamongthe6sessionswhichtookplace.Allsubjectswereeitherundergraduateorgraduatestudents.Therewere 39menand60women.Participantsspentaround90minandearnedanaverageofe14.63.Theexperimentwasrunusing thesoftwarez-Tree(Fischbacher,2007).

There were foureligible alternatives inthe experiment. Each alternative consistedin guessing a particularvalue ofa magnituderelatedwithaspeciﬁcsportcompetition.Guessinghadtobemadeonthebasisofpastdataofthecorresponding magnitudethat subjectscould observefora limitedtime. Prizesdependeddirectlyontheaccuracyoftheguessing.Some alternativeswereclearlyeasiertoguessthanothersbecause,giventhedata,thereasonablerangesofpossibleanswerswere evidentlysmaller.Otheralternativeswere,inprinciple,incomparable,becausetheywereofsimilarnature;theirrangeswere alikeandbecausethelimitedtimethatsubjectshadtoobservethedatadidnotallowthemtomakeprecisecalculations.In nocaseanyprocessednumericalinformationofthealternatives,suchasthevaluesofthemeanandthestandarddeviation, wereprovidedtotheparticipants.

Oncesubjectshadobservedthefouralternativesatstakeandmadeanideaoftheirdesirability(easetoguess),theyhad toevaluatecertainmenusmadeofthem.Menusdifferedontheinternalrelationshipsofdominanceand/orincomparability oftheirelements.Thevaluationsofthemenuswereusedtocheckourhypothesisabouttheeffectofsuchrelationshipson thedifferentattitudestowardschoice.

Inparticular,thefouralternativeswerethefollowing:

• Handball_1_7:Consistedinguessingthedifferenceinpointsbetweenthefirstandthe7thclassifiedteamsinahandball competition in2011. Forthispurpose subjectsobserved the classification andscores ofthe sixprevious seasons. The rangeofdatais12,withaminimumvalueof12andamaximumvalueof24.

Thiswastheinformationseenbytheparticipants:

20 04/20 05 Wins Losses Points 20 05/20 06 Wins Losses Points 20 06/20 07 Wins Losses Points

First 21 4 43 First 23 2 47 First 23 3 46

Second 18 5 39 Second 19 5 40 Second 23 3 46

Third 17 6 37 Third 18 8 36 Third 20 5 41

Fourth 16 7 35 Fourth 17 7 36 Fourth 19 6 39

Fifth 15 9 32 Fifth 17 9 34 Fifth 13 9 30

Sixth 13 10 29 Sixth 11 11 26 Sixth 12 11 27

Seventh 13 12 27 Seventh 8 10 24 Seventh 12 11 27

2007/2008 Wins Losses Points 2008/2009 Wins Losses Points 2009/2010 Wins Losses Points

First 21 3 44 First 19 4 41 First 23 1 46

Second 17 7 36 Second 20 6 40 Second 19 4 39

Third 17 7 36 Third 19 6 39 Third 17 5 36

Fourth 16 7 35 Fourth 17 5 38 Fourth 16 6 34

Fifth 16 7 35 Fifth 14 7 33 Fifth 16 8 32

Sixth 15 7 34 Sixth 15 11 30 Sixth 13 10 27

Seventh 13 9 30 Seventh 13 10 29 Seventh 11 13 22

• Hockey_Wins_Losses: Consisted inguessingthe difference betweenthewon andthelost matches bythe winnerofa hockey competition in2010. Subjects observed the corresponding information of the sixprevious seasons. The range of thedata of thistaskwas 16,with a minimumvalue of 18 anda maximum value of34. The informationseen by participantswas:

20 04/20 05 Wins Losses 20 05/20 06 Wins Losses 20 06/20 07 Wins Losses

WINNER 41 19 WINNER 52 21 WINNER 53 22

Second 45 24 Second 52 24 Second 48 25

Third 43 23 Third 42 31 Third 40 31

Fourth 41 30 Fourth 41 33 Fourth 42 34

Fifth 37 34 Fifth 29 37 Fifth 35 41

2007/2008 Wins Losses 2008/2009 Wins Losses 2009/2010 Wins Losses

WINNER 47 25 WINNER 53 19 WINNER 45 27

Second 43 31 Second 41 30 Second 44 32

Third 41 29 Third 41 32 Third 39 30

Fourth 39 31 Fourth 36 35 Fourth 39 33

Fifth 36 35 Fifth 34 35 Fifth 30 38

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• Budget_1_5:Consistedinguessingthedifferencebetweenthebudgetsinyear2020oftheteamwiththehighestbudget andtheteamwiththe5thhighestbudgetofacompetition.Subjectsobservedthebudgetsofthesixpreviousseasons.

Therangeofthistaskwas377,500,000,withaminimumvalueof514,000,000andamaximumvalueof891,500,000.

2014 Budget 2015 Budget 2016 Budget

First 647,650,000 First 700,500,000 First 787,200,000 Second 598,450,000 Second 618,500,000 Second 743,100,000 Third 289,000,000 Third 336,000,000 Third 456,900,000 Fourth 137,300,000 Fourth 254,500,000 Fourth 287,900,000 Fifth 122,850,000 Fifth 186,500,000 Fifth 236,400,000

2017 Budget 2018 Budget 2019 Budget

First 772,500,000 First 1,160,000,000 First 1,200,000,000 Second 716,200,000 Second 1,090,000,000 Second 1,190,000,000 Third 585,000,000 Third 872,250,000 Third 844,150,000 Fourth 269,450,000 Fourth 442,650,000 Fourth 493,350,000 Fifth 230,900,000 Fifth 268,500,000 Fifth 337,150,000

• Soccer_Licenses_Or_%Licenses:

Thisalternative couldconsist inguessingeitherthe numberofsoccerlicenses registeredatasoccerfederationorthe percentageoftheselicenseswithrespecttothetotalsportlicensesissuedinacountry.Thetiming oftheresolutionof thisuncertaintyis describedlateron.The rangeofthe task“licenses” was 207,103witha minimumvalue of855,987 andamaximumof1,063,090.Therangeofthetask“percentageoflicenses” was2.3.

2013 Licenses % Licenses 2014 Licenses % Licenses 2015 Licenses % Licenses

Soccer 855,987 25.2 Soccer 874,093 25.8 Soccer 909,761 26

Rugby 4,463 0.1 Rugby 4,833 0.1 Rugby 5,127 0.1

Greyhounds 10,180 0.3 Greyhounds 10,151 0.3 Greyhounds 10,312 0.3

Gymnastics 30,822 0.9 Gymnastics 24,032 0.7 Gymnastics 38,842 1.1

Golf 294,844 8.7 Golf 283,849 8.4 Golf 276,150 7.9

2016 Licenses % Licenses 2017 Licenses % Licenses 2018 Licenses %Licenses

Soccer 942,674 26.3 Soccer 1,027,907 27.3 Soccer 1,063,090 27.5

Rugby 5,907 0.2 Rugby 6,408 0.2 Rugby 7,080 0.2

Greyhounds 11,201 0.3 Greyhounds 11,600 0.3 Greyhounds 121,901 0.3

Gymnastics 40,066 1.1 Gymnastics 45,863 1.2 Gymnastics 49,719 1.3

Golf 271,865 7.6 Golf 270,463 7.2 Golf 270,996 7

Henceforth,wecallthosealternatives,respectively,h,k,b andl.

Theprizeforchoosinganalternativewas15eurominusthedifferencebetweenthecorrectanswerandthegivenanswer (inabsolutevalue).Thus,thepayoffsarecappedat0andat15.

Theexperimentwasdividedintothreestages.

FirstStage:Preferenceformation

Theparticipantsobservedtheinformationofthefoureligiblealternatives,oneaftertheother,during15seach.Theaim ofthisstagewasforthesubjectstoformapreferencerelationoverthealternatives,dependingonthediﬃcultytoguessthe correspondingrequiredvalueandtheassociatedexpectedearnings.Theprovidedtimewassupposedtobeenoughtogetan ideaofthediﬃcultyofthetasks,butnottomakeaprecisecalculationthatallowedtoformaclearpreferenceamongsome alternatives. Inparticular, weassumedthat withinthe 15s intervalsduringwhich theparticipantswere acquaintedwith theoptions inpresence,we supposethatthey hadenoughtimetorealizetherelativerateoferror,duetoapproximation ofthecorrectanswer,thatisliabletoaffectheranswerorguess.Inparticular,theycouldrealizethattheexpectederrorif guessingthebudget,orifguessingthenumberoflicenses,wasclearlygreaterthantheoneforalternativeshandk,andfor theseinturntheexpectederrorwasclearlygreaterthanifguessingthepercentageoflicenses.Similarly,weassumedthat 15swere,ontheotherhand,notenoughtomakealternativeshandkclearlycomparableintermsoftheexpectederror.

Secondstage:Choicesetvaluation.

Subjectshadtovalue,bymeansoftheiterativeMultiplePriceList(iMPL)method(Andersenetal.,2006),differentchoice setsconsistingofcombinationsofthefoureligiblealternatives.Inparticular,thesubjectsrepeatedlystatediftheypreferred eitheranamountofmoneyorchoosinganalternativefromthecorrespondingchoiceset.Theamountofmoneyvariedfrom 1to12inincrementsof€1.Bythistablewegetanestimationofthevaluethatthesubjectattachestothealternative.At each moment of thequestionnaire, if theparticipant didnot choose “indifferent” thenshe hadto ﬁllanother table that containedthe intermediateprices betweenthepriceofthelast time shechose “task” andthepriceofthe ﬁrsttime she chose“money” instepsof€0.1.

Withthisinformationwestatethatthevalueofachoicesetisequaltothe“indifference” answerandincasethereisno suchindifferenceanswerwestatethatitisequaltotheaverageofthepricesatwhichthesubjectswitchesherpreference (e.g. ifasubjectdoesnot answerindifferenceandstatesthat sheprefersdoing theestimation ratherthanreceiving €6.4 butsheprefers€6.5ratherthandoingtheestimation,thenwestatethatshevaluesthesetby€6.45).

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Participantswere announcedthat,inthethirdstage,achoicesetandpricewouldbechosenrandomlyforeachpartici- pant.Ifthesubjectchosemoneyforthatsetshewouldreceivethecorrespondingamountofmoney,ifshechosemakinga choicefromthesetshewouldhavealimitedtimeof40stodoit.

Participantswerealsosaidthat,inthecasethattheselectedsetcontainedthealternativel,thecomputerwoulddecide randomly whichofthetwo estimationsshouldbedone beforehavingtoselectanalternativefromtheset(eitherl orthe accompanyingalternative).

Participantshadtovaluethefollowingsets:

{

^h

}

,

{

^l

}

,

{

^b

}

,

{

^k

}

,

{

^k,b

}

,

{

^h,k

}

,

{

^h,l

}

,

{

^h,k,b

}

ThesetofallvaluablesetsisdenotedbyX,thatis,X=

{{

^h

}

,

{

^l

}

,

{

^b

}

,

{

^k

}

,

{

^k,b

}

,

{

^h,k

}

,

{

^h,l

}

,

{

^h,k,b

}}

Wedidnot includeallthepossiblecombinationsoftheeligiblealternatives inordertominimize randomanswersdue totediousrepetition.

HenceforthwedenotethevaluegivenbythesubjectinthisstagetoasetX∈X byV(^X)^. Thirdstage:Chooseanalternative

Atthebeginningofthethirdstage,achoicesetandapricewerechosenrandomlyforeachparticipant.Ifshehadchosen

“money” at theprevious stageshereceived thechosen amountofmoney,ifshehadchosen “choice set” shehad40sto selectanalternativetomakethecorrespondingcalculationandgiveananswer,asannouncedattheendofstage2.During the40sshehadaccesstotheinformationofallthealternativesintheset(i.eshewasseeingthedataofthesixprevious yearsofallthosealternatives).Iftheparticipanthadchosenindifferencethecomputer selectedeither“money” or“choice set” randomly.

Asalreadysaid,alternativel iscontingent,sothat,iftherandomlyselectedsetbythecomputercontaineditandthepar- ticipant hadchosen“choiceset” insteadofmoneythenthecomputerclariﬁedwhetherl consistedinguessingthenumber oflicenses(thediﬃcultone)orthepercentageoflicenses(theeasyone).

Prior to the experiment theinstructions were read aloud.For the sake ofexplanation some examples of thetype of taskthat subjectswererequiredtodowere presented.Alternativesinthisintroductorystage weredesignedsothat their structure weresimilar totheonesthatsubjectswouldfaceinthetrueexperimentbutdifferentenoughasnottolead to anybiasorlearningeffect.Thenamesofthealternativesreflectedthetaskthathadtobedone.Thereweretwoalternatives withsimilarmeanvalueandstandarddeviation,onealternativewhichwasmoredifficultthantheprevioustwoalternatives anda contingent alternative.Thiscontingent alternative wasnamed Basket_Points_OR_Blocks becausetheparticipants had toguesseitherthepointsscoredinaseasonby thetop scorerofateam(verydifficulttask)ortheblockexecutedby the lowest scorer(veryeasytask)ofabasketballleague.Wecarefullyexplainedthat,incaseofbeingpartofa choicesetthe contingency (i.e ifit consists inguessing the difficulttask orin guessingthe easy one) wasgoing to be solved previous to havingto pickan alternativefromtheset.We alsoemphasized that theyhadto chooseone andonlyone alternative.

Finally they had to answer two control questions related to these two aspects (i.e the contingency and the concept of

“choice” set).

After completingtheexperimentallparticipantswere askedtocompleteashortenedRavenTestandalsoan iMPLcon- sistingofalotterywhichyielded€40withprobability0.5and€0withprobability0.5.Onlyoneparticipantpersessionwas selected randomlyforthe lottery.Thisdecisionwastakenrandomly afterthey hadcompletedthe iMPL.² The Raven Test wasadministeredtomakeanestimationofthecognitiveabilitiesofthesubjectsandthelotteryiMPLinordertoestimate theirriskaversion.

2.1. Hypotheses

Asalreadypointedout,thefouralternativesweredesignedinsuchawaythatweshouldexpectaclearstrictpreference, P,ofthesubjectsbetweensomealternativesbutnotall.Inparticular,forset

{

^h,k,b,l

}

^,^we^should^expect^P=

{

(^h,b),(^k,b)

}

^.

Thereasonisthat,giventherangesofthevaluesofthedifferentalternativesandthewayhowprizesaredesigned(subjects earn€15minus theerrorintheir answer),weassume thatb shouldbe clearlyperceived asinferior tobothhandk.As a directconsequence,weshouldexpectV(

{

^h

}

)>V(

{

^b

}

)^and^V(

{

^k

}

)>V(

{

^b

}

)^.

We have taken the ranges ofthe data series asindicative of the difficulty of the tasks. The values ofthe mean and standard deviationofeach alternativesuggeststhesamekindofdominancerelations: handk clearlydominateb insuch terms and,giventhelimitedtime toobservethedata,ourassumptionisthat itwasnotenough todiscernwhetherhor kwaseasiertoundertake.However,giventhat only6valuesforeachalternativewereavailable,itisdifficulttoguarantee that thedistribution isnormal,sowe haveconsideredtherangeofdata asa betterindicator ofthedifficulty ofthetask thanthemeanandstandarddeviation.

In particular, for h the mean is 18 and the median is 17.5. If the time series continued within the historical range, statingthat thecorrectansweristhemean,itwouldleadtoa minimalearningof€9andamaximal earningof€15.The skewness coeﬃcientis0.10319.Foralternativek themeanis26.33whilethemedianis26.5. Ifthetime seriescontinued withinthehistoricalrange,statingthatthecorrectansweristhemean,thiswouldleadtoaminimalearningof€6.67anda maximalearningof€15.Theskewnesscoeﬁcientis-0.070811.Rememberthatsubjectshadonly15stomakethenecessary

2 The previous instructions as well as the experiment are detailed in the online supplemental material.

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calculations toobtain thehistorial data ofthetask(6 substractions before makingwhatever other calculation). Basedon this,ourassumptionisthathandkaresimilar,independentlyoftheshapeoftheutilityfunctionforrisk.

However, whenlookingattherangesoftasksb andthatofcalculatingthenumberoflicenseswe clearlyobservethat theyaremorediﬃcultthanhandk,whichmakesveryunlikelyobtaininganyearningbyperforminganyofthetwotasks.

Finally,therangeofthetask“percentageoflicenses” is2.3.Ifthetimeseriescontinuedinthehistoricalrange,statingthat thecorrectansweristhemeanwouldleadtoaminimalearningof€13.85andamaximalearningof€15.

Other criteria could have been considered to presume the diﬃculty to do the different guessings. In particular, one mightconsider theeffectoffamiliaritywiththetask.Withthisrespect,people areprobablylessusedto predictbudgets ornumbersoflicensesthanthetypeofscoresrequiredfor“handball” or“hockey”,whicharepreciselyavailable atbetting houses.Nevertheless,this,inprincipleshouldnotaffecttherelationof“incomparability” betweenhandk,sincebothconsist ofguessingscores,anditwouldreinforceourideathatboth,handk,dominateb.

Inanycase,theabsenceofaclearstrictpreferencebetweenhandkshouldleadtoasimilarvaluationofthem(V(

{

^h

}

)≈ V(

{

^k

}

)^). ^Regarding ^the ^special^case^of ^the probabilistic alternative, l,its valuation (as a singleton) should depend on the attitudetowardriskofthesubject,sonoparticularpreferencewaspresumed.

For anyset X∈X we denote by maxX the alternative containedin X to which the subjectgives the maximal value whenitisvaluedasasingleton,thatis,

∀

^X∈X,maxX=x∈X s.tV(

{

^max^X

}

)=max_x_∈_XV(

{

^x

}

)^.^Similarly,^let^min^X=x∈X s.tV(

{

^min^X

}

)=min_x_∈_XV(

{

^x

}

)^.

Wedistinguishthefollowingﬁveattitudestowardchoice:

Attitudes toward choice Comparison between V (X) , V ({ max X}) and V ({ min X}) Choice-Preference V (^X) > V ({ max X})

Choice-Aversion V (X) < V ({ min X})

Betweenness V ({ min X}) ≤ V (X) < V ({ max X}) Neutrality V ({^{min X}}) < V (X) = V ({^{max X}}) Indifference-Neutrality V ({^{min X}}) = V (X) = V ({^{max X}})

When the participantvaluesa set by morethan themostvalued item containedinit we saythat she showschoice- preference.Whenshevaluesthesetlessthantheleastvaluedelementwesaythat sheshowschoice-aversion.Ifshevalues asetbetweenthebestandtheworstelementofthesetwesaythatsheshowsbetweenness.Ifshevaluesa setby means ofitspreferredelementandall theelementsarenotequallyvaluedwecalltothisneutrality.Finally,whentheparticipant valuesequallyalltheitemsaswellasthesetthatcontainsthemwesaythatsheshowsindifference-neutrality.

As pointedout in theIntroduction, Choice-preferencemayarise for,at least,two reasons.Oneis the intrinsicvalue of freedom ofchoice(Sen,1988;PattanaikandXu,1990) accordingtowhich,whateverthecharacteristicsofthealternatives, theindividualprefershavingmoreoptions.Anotherpossiblereasonisthepreferenceforﬂexibility(Kreps,1979;Arlegiand Nieto,2001a;2001b;PejsachowiczandToussaert,2017).Thisariseswhen,inanenvironmentofuncertaintyaboutthefuture preferences,theavailabilityofmorealternativesallowstomakeabetterchoiceoncesuchuncertaintyisresolved.

Inourdesign,whenvaluingasetthatcontainedalternativel,theuncertaintyaboutwhetheritwasbetterorworsethan theothersinthesetwasannouncedtobesolved beforetheﬁnal choicehadtobemade.Infactl wasdesignedinsuch a waythatitcouldbeeitherclearlysuperiortoanyotheralternativeifitresultedinguessingthepercentageoflicenses(the standarddeviationwasverysmallinthatcase),orclearlyinferiortoanyotheralternativeifitresultedinguessingthetotal number oflicenses(because ofthe hugestandard deviationinthiscase).Thus, singleton

{

^l

}

^is^a ^risky^set^but, ^when ^the

subjecthadtovalue anyothersetthatcontainsl asanoption,sheshould knowthatitprovidedan opportunitytopicka veryattractivealternativeifitresultedtobethepercentageguessing.Also,sheshouldrealizethatthisalternativecouldbe rejectedatnocost ifitresultedtobethetotalnumberoflicensesguessing.Asaconsequence,weshouldexpectthat the additionofl toanysetimprovesitsvalue.

Inaddition,weshouldexpectabroaderpreferenceforchoiceinset

{

^h,l

}

^thanⁱⁿ^other^twofold^sets^that^do^not^contain

l because thetwo possiblereasons forthechoice-preferenceattitudeconvergein

{

^h,l

}

^: ^theîntrinsic^value ôf^freedom ôf

choiceandthepreferenceforﬂexibility.

Choice-aversionisbasedontheideaofaversiontoincompletepreferences(Arlegietal.,2021) oraversiontoindecisive- ness (Dananetal., 2012). Thatis,not havinga well-deﬁnedpreferenceover thealternatives could lead tosome decision coststhatthedecisionmakerwouldliketoavoid.Forthisreasontheindividualwouldchoosearestrictedset.

Inourexperimentthediﬃcultytodeﬁneaclearpreferenceforonealternativeoveranothermightcomefromthelimited timetomakeallthenecessarycalculationsorapproximateestimations.Set

{

^h,k

}

^was^the^clear^candidate^for^this^to^happen.

The timeduringwhichsubjectshadtovisualizetheinformationofbothalternatives was,presumably,enoughtoperceive thattheirmeansandstandarddeviationswererathersimilar,butalsoenoughtorealizethattheavailabletimetopickone ofthe twowasinsuﬃcienttomakethe necessarycalculationsthatallowed toelucidatethisambiguity, thatis,toforma clearpreferenceforonealternativeovertheother.Inordertoavoidthepsychologicaldiscomfortofnotbeingabletoform such a preference, thesubjectmightprefer not beingobligedto choose,that is,havingonly either

{

^h

}

^or

{

^k

}

^.^Obviously,

thiskindofaversionislesslikelytoappearwhenthepreferencerelationiswell-deﬁned.Forexample,thelimitedtimeto visualize the informationofalternatives k andb shouldbe enough forthesubjectto perceivethat the latterwasclearly inferior than the former.Thus, there should be no reasonforher to experience choice-aversion inset

{

^k,b

}

^, ^even ^if^she

knewthatshehadalimitedtimetomakethechoice,becauseshecouldneglecttheinformationofb atnocost.

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Table 1

Average value of the singletons.

Alternative Average value

Budget ( b) 3.82

Handball ( h ) 5.54

Hockey ( k ) 5.33

Licenses ( l) 4.82

Thisaversioneffectshouldnotbeblendedwiththeinformationoverloadphenomenon(LeLecetal.,2016)orthechoice overloadone(IyengarandLepper,2000).Inourdesignallthealternativescontainedthesameamountofinformation(six calculationshadtobedone)andthenumberofalternativescontainedineachsetwasverylow(atmostthree).Giventhis, ifeither informationoverloadorchoice overloadwere present, theseattitudes shouldbe equally distributedamongsets, whilstaversionshouldbemorepronouncedinthesetswheretheincomparabilitiescanarise.

Betweennessisusuallyrelatedtotheideaoftemptation(GulandPesendorfer,2001;Toussaert,2018).Underthisinterpre- tationanalternativethat isdominatedandundesiredwhenevaluatingtheopportunitysetmightbetemptingandchosen whentheﬁnal choicehastobemade, andtheindividualisawareofthis.Bythedesignofourexperimentno alternative had, inprinciple,thecharacteristicsofbeingtempting.However, wehaveincludedbetweennessasapossibleattitudefor thesake ofexhaustivenessandalsotoadapttoLT20’sﬁndings.LT20givetwo possibleexplanationstothisphenomenon.³ Oneisthatsubjectsvaluethesetheuristicallymaking somekindofweightedaverageofthevalue oftheitemsintheset.

Theotheroneisthefeartomakeabaddecisionwhensubjectsareabletoanticipatetheirownboundedrationality.

In ourcontext, ifbetweenness arose it could be interpretedas ifthe subjectvaluedthe setas awhole object rather than asan instrumenttopickup one alternative.LT20describe thefear tomakea baddecisionas“

{

...

}

^fear ^of^making

decisionsinthefuturethatarenotthosethattheywouldprefertoday,duetotheirownboundedrationalityorthepossibilityof an undesirablechangein theirpreferences”.Thatis,undersuchan interpretationthere isachance ofchoosingwhatisnot thebestoption.Thisreasoningisclearlyrelatedwithtemptation,wherethereisalsoadiscordancebetweenpreferencesat differentstages.

In ourexperiment,in mostsets thereissupposedly a strongpreferencerelationoversome alternatives,which iscon- stantalong time because their objectivevaluesare ﬁxed.

{

^h,k

}

îs^the ônly^set ^where,îfôbserved,betweennesscould be provoked by the fear of making a bad decision. In the remaining cases (where a dominated alternative is present) the heuristicvaluationrulewouldbethereasonableexplanation.Inthisway,ourdesignshedslightonthepossiblesourcesof thebetweennessattitudeobservedinLT20.

Bothbetweennessandchoice-aversioninvolvevaluingasetlessthanitsbestalternativebut,andasalreadyexplained, choice-aversion wouldbeinterpreted asapsychologicaldiscomfortforhavingtochoosewithouthavingaclearpreference overtheoptions.

According to the neutrality attitudethe subjectvalues the alternatives in the setin a different way,but the value of thesetisexactlythatofitsbestalternative.Thisisconsistentwiththestandardindirectutilitycriterioninmicroeconomic theory. Indifference-neutralityisalso consistentwiththis criterion,butapplies to a situationwhereall the alternatives of thesetareequallyvaluedseparately.Infact,bothcanbeinterpretedasaconsequenceofvaluingV(^X)=V(

{

^max^X

}

)^.^If,ⁱⁿ addition,V(^max^X)>V(

{

^min^X

}

)^then^we^haveneutrality.IfV(

{

^max^X

}

)=V(

{

^min^X

}

)^then^we^haveindifference-neutrality.

Bothrepresentthesameattitudetowardchoiceinthesensethattheavailabilityofmorealternativesisnotvaluableeither positively or negatively asfar asthe maximal alternative isgranted. However, we have distinguishedbetweenneutrality andindifference-neutralitybecausewe areinterestedinobservingtowhichextenttheabsenceofastrict preferenceover alternativesleadstoindifferenceorchoice-aversion.

Theneutralityattitudetowardchoiceshouldbeexpectedtobelargerforset

{

^h,b

}

^because^of^the^objectivesuperiorityof hoverb.However,weshouldalsoexpectsuchanattitudeinanyothertwofoldsetwhere,forwhateverreason,thesubject valuesoneofthealternativesmuchmorethantheother.

3. Results 3.1. Overview

Table1showstheaveragevaluethatsubjectsgavetosets

{

^b

}

^,

{

^h

}

^,

{

^k

}

^and

{

^l

}

ⁱⁿ^the^choice^set^valuation^stage.^This^can

benaturallyinterpretedastheaveragevaluethattheygavetothecorrespondingalternatives.

Result1. Theaveragevalueof

{

^b

}

^is^the^lowest^of^the^foursingletons.

The valuesgiven to the alternatives are ordinally consistent with what was expected butnot cardinally. That is,the average values received by the alternatives correspond to the relative diﬃculty of the tasks: Alternatives witha higher

3 LT20 call “choice aversion” to what we call here “betweenness”.

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Table 2

Difference between the set valuation and the valuation of the maximal alternative of the set.

Average V ({^{h, k, b}}) − max {^{h, k, b}} ^-1.07^∗∗∗

V ({ h, l}) − max { h, l} -0.72 ^∗∗∗

V ({ k, b}) − max { k, b} -0.89 ^∗∗∗

V ({ h, k }) − max { h, k } -0.54 ^∗∗∗

Table 3

Difference between the set value and the average value of the alternatives of the set.

Average V ({^{h, l}}) − a v g{^{h, l}} ^0.13 V ({ h, k }) − a v g{ h, k } -0.03 V ({ k, b}) − a v g{ k, b} 0.07 V ({ h, k, b}) − a v g{ h, k, b} 0.08

mean andstandard deviationreceived a lower value. In the caseof alternative l,although it could turn to have a huge standarddeviation(inanycaselowerthanthedeviationofb),itcouldalsoresulttobeaveryeasytask(0.7sd),sothatits valuationwasalsoexpectedtobehigherthanthevaluationofb.

However, the valuationsdo notcorrespond to theexpectedearnings associated toeach alternative. Inparticular, con- sidering that the prize was €15 minus the error andthat b hada so highstandard deviation(€237,664,721) we should not expectthat anysubjectwouldbe willingtopayforhavingtheoptionto makean estimationofit.Thissuggeststhat subjects,afterlookingattheinformationinthetables,wereabletoperceivetherelativediﬃcultyofthetasks,butnot to translate thenumerical informationintoexpectedearnings andintothecorresponding valuation giventhewaythe prize wascalculated.

Result2. Onaverage,thevalueofachoicesetislowerthanthatofitspreferredelement.

Table2showsthatthethevalueofthemaximalelementofasetishigherthanthevalueoftheset.⁴

Accordingto thedatainTable2,onaverage,addingsuboptimalalternativesreducesthevalue oftheset.Thisresultis consistent withthe ﬁndings inLT20.Givenour classiﬁcationof attitudes,thiscould be consistent withageneric choice- aversionattitudeorwithagenericbetweennessattitude.

As alreadyexplained,alternatives aredesignedinsuchawaythatsome attitudescould beassumeddependingonthe presumed attractivenessofthealternatives thatcompoundtheset.However, weobservethat,onaverage,thefourchoice sets are valued lower than their maximal alternative. Thishappens, even, in thecase ofset

{

^k,b

}

^, ^where^we ^could ^pre-

sumethat, b beingsoclearlyinferior,thedifference betweenV(

{

^k,b

}

) ^and^V(

{

^b

}

) ^should^be ^very^small^because^{b can}^be discarded withoutanykindofdeliberation.Something similarcanbesaidaboutset

{

^h,l

}

^.^When^valued^alone,^{l is}^a ^risky

alternative,however,whenbeingpartoftheopportunityset

{

^h,l

}

^,^due^to^its^special^contingentcharacteristics,l becomesa veryattractivealternativethatincorporatesﬂexibilitytotheuniqueavailabilityofh.Thatis,weshouldexpectthevalueof

{

^h,l

}

^to^be^greater^than^the^value^of

{

^h

}

ⁱⁿ^this^case.^However,^we^also^observe^that

{

^h,l

}

^hasônâverageâ^lower^valuation

than max

{

^h,l

}

^.Âll ^the^factsâbove ^reinforce ^LT20’s^view ^that ^the^subjects ^perform^some ^kindôfâverage ^valuationôf ^the

alternativesintheset,eveniftheyconsistof“opportunity” sets.

Table3givesmoreinformationaboutthisissuebyshowingthedifferencebetweenthesetvalueandtheaverage⁵value ofitsalternatives.

Thedatashowthatthereisnotstatisticalevidencetorejectthehypothesisthat thevaluationofasetistheaverageof itsitemsvaluations.

Asaconclusion,observingthetwotableswecandiscardthatallsubjectsareeitherchoice-neutral,choice-averseorthat they havepreferenceforchoice,i.ewe candiscardanyofthoseattitudesasgenericallyapplicabletoeveryset.Infactwe candiscardanyofthoseattitudesforeachset.AmongtheattitudesdescribedinSection2.1onlybetweennessremainsas anacceptablehypothesisforthegenericattitudetowardchoiceofthesubjects.

Allthedataaboveiscalculatedonaverage,sotheycouldbetheresultofoppositeattitudestowardchoiceacrossagents (e.g.choice-preferenceandchoice-aversion).Thenextsubsectionexaminesthispossibility.

4∗ represents the level of signiﬁcance for the Wilcoxon signed-ranked test being ^∗∗∗ 0.01 level, ^∗∗0.05 level and ^∗ 0.10 level. All the statistical tests related to the results are available in the Appendix.

5 For a generic set A , a v gA denotes the average value of its alternatives.

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Table 4

Attitudes distribution among choice sets.

Control Preference Aversion Between Neutrality Indif

{ h, l} 14% 14% 38% 17% 16%

{^{h, k}} ^22% ^33% ^13% ^5% ^27%

{ k, b} 16% 29% 25% 14% 16%

{ h, k, b} 19% 27% 27% 14% 13%

3.2. Attitudesdistribution

Intheprevioussectionwehavediscardedthatallsubjectshavethesamegenericattitudetowardchoice(exceptforthe betweennesscase)andthatanychoicesetgeneratesthesameattitudeinallthesubjects(again,exceptforthebetweenness case). Inthissubsectionwe studywhetherthe averageresultsabove are provokedby ageneralbetweennessattitude or, differently, they arethe resultofthecoexistence ofdiverseattitudes towardchoice.In thelattercasewe check whether certain attitudesaremorepronounced dependingofthesetsandifthedistributionofattitudesisinconsonancewithour conjectures.

Thenexttablerepresents,foreachofthesets,thepercentageofsubjectsthatshowedeachoftheattitudesdescribedin theprevioussection.Weusethesampleofthe63subjectswhoansweredcorrectlyallcontrolquestions.

Choice-preference

Wedonotobservedifferencesintheprevalenceofthechoice-preferenceattitudeamongchoicesets.Ourconjecturewas that suchanattitudewouldariseinthecaseofset

{

^h,l

}

^due^to^the^ﬂexibility^thatl introduceswithrespectto

{

^h

}

^.^When

readingtheinstructionsitwascarefullyexplainedthat theuncertaintyattachedtol wasgoingtobeclearedupbefore the final alternativehadtobechosenfromtheset.Thisinvolvedamorecomplexdescriptionofl:first,naturedeterminedthe outcome associatedto l, andthen the subjecthad to comparethe resulting outcome with hin orderto make the final choice.Eventhoughundertheappropriatereasoningalternativel shouldnotinducedecisioncoststothedecisionmaker,it seems thattherelativelycomplextreestructureofthedecisionprocess wasnegativelyperceived bythesubjects,ignoring theadvantagethatl involvedwhenpresentedwithh.

Choice-aversion

Result3. Theproportionofsubjectsthatshowchoice-aversionislowerin

{

^h,l

}

Weobservethattheonlysetinwhichtheproportionofsubjectsthatshowschoice-aversionissigniﬁcantlyloweris

{

^h,l

}

^,

whichispreciselythesetforwhichweexpectedahigherchoice-preferenceattitude.Giventhatchoice-aversionandchoice- preferenceare opposite attitudes,thisfactis, inprinciple,inlinewithourconjectures. However, thelower prevalenceof aversioninset

{

^h,l

}

^is^notsubstitutedbyahigherprevalenceofchoice-preference,asitcouldbeexpected,butbyahigher prevalenceofthebetweennessattitude.

Apossibleexplanationofthisisthatsubjectsperceivedset

{

^h,l

}

^not^as^atwo-optionsset,butasathree-possibleresults set (thevalue ofhandthetwo contingent valuesofl),making anaverage evaluationofthethree.Giventhat wedonot havetheinformationoftheevaluationofthetwo possiblevaluesofl by separate,we cannotcheck whetherthisiswhat actuallyhappened.

Betweenness

Result4. Theproportionofsubjectsthatshowbetweennessisthelowestin

{

^h,k

}

^.

Thisresultshowsthattheonlysetthatcontainstwoincomparablealternativesistheonewiththelowestproportionof subjectsthatshowbetweenness.

The lower prevalenceof betweennessinset

{

^h,k

}

^might^be ^explained^by ^the^smaller^difference ^between^the^maximal

andtheminimalvaluesoftheelementsintheset.Thisargumentwouldbe correctifthevaluationofthe(non-singleton) sets hadbeenrandom, sothattheprobability ofobservingvaluesintherangebetweenthemaximumandtheminimum would belower when thisrangewere smaller.However, the Friedmantest forrelatedsamples stateswitha signiﬁcance levelof0.019thatthedistributionsofthevaluesofthefourchoicesetsarenotequal.Thatis,asfarasthereisa positive differencebetweenthemaximumandtheminimumvaluesofthealternativesinthesetthereisroomfortheindividualto exhibitanyofthepossibleattitudestowardchoice,independentlyofthecardinalityofthosevalues.

Wegodeeperintotheinterpretationofthisresultinthelastdiscussionsection,wherewecanrelateittootherﬁndings.

Indirectutility

Result5. Thelowestproportionofsubjectsshowingneutralityisobservedin

{

^h,k

}

^(whereîtîs^not^supposed^to^beâ^clear

preferencerelation).Furthermore,onlyinthissettheproportionofsubjectsthatshowindifference-neutralityissigniﬁcantly higherthantheproportionofsubjectsthatshowneutrality.

“Neutrality” and“Indifference-neutrality” (thetwolastcolumnsinTable4)arebothconsistentwiththestandardindirect utility ruletovalue opportunitysets.Wedonotﬁnddifferencesinthesumofpercentageofthetwocolumnsforall sets,

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Table 5

Range of attitudes distribution among sets in LT20.

Preference Aversion Between Neutrality Indifference-neutrality

Highest % 23% 13% 48% 22% 42%

Lowest % 11% 2% 24% 7% 22%

which suggeststhat thereis arelatively constant percentageofpeople that valuatesets bymeans ofits best alternative.

Theyidentifythebestelement,andthentheyvaluethesetbymeansofthatelement.

Result5conﬁrmsthenthat thesimilaraveragevaluation ofthealternatives handk obeystothe factthatahighpro- portionofsubjectshaveactuallyperceived thesealternativesassimilar.Thissimilarityhasleadtoasmallpresenceofthe neutralityattitude.

3.2.1. Attitudesheterogeneitywithoutpresumedpreferences

As pointedout intheintroduction,an importantdifference betweenourdesignandLT20isthatinourcasethereisa presumed objectivepreferenceforsome alternatives oversome others.In LT20alternatives aredesignedasfinal consum- ablegoods overwhichthereisnot anypresumed preference.Inourexperimentalternatives aremeanstoobtainearnings because they provideusefulquantitative informationto make thenecessary calculations. Inourdesign some alternatives provideclearlymorechancestomakeamoreaccuratecalculationinthethirdstageandthereforeahigherchancetoobtain agreaterprize.Inaddition,alternativel isdifferenttotheothersregardingitsinformationalnature.Thisallowstopresume differentattitudes towardchoicedependingontheobjectiveattractiveness ofthealternativesthat compoundtheset.We havefoundthatthereiscertainconsistencybetweentheresultsandourconjecturesabouthowthepresumedpreferences affected thechoice attitudes,at leastfromtheperspective ofthe distributionofattitudes amongsets. Inthissection we try to contrastour results by comparisonwith the results in LT20. In particular, we take the raw data provided in the supplementarymaterialofLT20andapplytothemourfive-attitudesclassification.

The tablebelowcontainstheproportionofpeoplewho showedourﬁveattitudestowardchoice inLT20’sexperiment.

Foreach attitude,“Highest” (“Lowest”)isthemaximum(minimum)ofthepercentagesofsubjectsthatshowedthecorre- spondingattitudeacrossthedifferentsetsinLT20.Wetestiftheproportionofsubjectsthatshowsomeattitudeinagiven setofourexperimentissigniﬁcantlylower(higher)thanthelowest(highest)proportionobservedintheirexperiment.This willallowtoknowmoreabouttheeffectofthepresenceofobjectivepreferencesoverthealternativesinourexperiment.

According to the data in Table 5 we do not ﬁnd differences between the percentages of subjects showing choice- preferenceinourexperimentandinLT20.Inourexperimentthispercentageisintherange[14%,22%](seeTable4)while intheirexperimentthisrangeis[11%,23%].Thisreinforcestheideathat,eventhoughl wasdesignedtoinducepreference forﬂexibility,ithasbeeninfactperceivedasadominatedoption.

Result6. Thepercentageofsubjectsshowingchoice-aversionisabovetherangeofLT20inallsetsexcept

{

^h,l

}

^.

Inourresulttherangeofthepercentageofsubjectsshowingchoice-aversionis[14%,33%]([2%,13%]inLT20).Thissug- geststhatthetime pressureimposedintheexperimentcreatedan environmentthatfavoredthechoice aversionattitude.

Giventhispressure,subjectspreferredtoavoidtheactofchoiceandtofocusonthegiventask.

Result7. Onlyinset

{

^h,k

}

^,^the^percentage^of^people^that^showbetweennessisbelowtherangeofLT20.

Regarding the betweenness attitude,the percentage of people showingbetweenness in our experimentis within the rangeinLT20’sexperiment([24%,48%])exceptforset

{

^h,k

}

^(13%),^where^such^percentage^is^remarkably^lower.^This^reinforces

Result 4,inthesensethat thebetweennessattitudeisinverselyrelatedwiththediﬃculty tocomparethealternatives in theset.

Result8. Theproportionofsubjectswhoshowindifference-neutralityinourexperimentisbelowtherangeofLT20except inset

{

^h,k

}

^,^where^it^is^within^the^range.

In this result we have used only the doubletons in LT20 as a benchmark.⁶ The reason is that, in order to observe indifference-neutrality,allthealternativesmustbeequallyvalued, andthisismorelikelytohappen whentherearefewer alternatives.

Accordingtoourhypothesisthereisanobjectivesuperiorityofbothkandhoverb in

{

^k,b

}

^andⁱⁿ

{

^h,k,b

}

^,^while

{

^h,k

}

^is

theonlysetwherethealternativescanbeinterpretedasequallyattractivebyanyone.Incontrast,inLT20’sexperimentthere isnot anyaprioristic preferencefromsomealternatives overothers:some individualsmayprefer(anddoitinfact)some options andother individuals others.Forthisreasonwe could expect the proportionofsubjects whoshow indifference- neutrality inourexperimentfor

{

^k,b

}

^and

{

^h,k,b

}

^to^be^below^LT20’s^range,^but^at^the^same^time ^we^should^expect^that

proportiontobeabove(andnotwithin)therangeLT20’srangeforset

{

^h,k

}

^.

6 For the other attitudes we have used all the sets in LT20 as a benchmark. If we take only the doubletons the remaining results do not change.

Attitudes toward choice with incomplete preferences: an experimental study

Journal of Economic Behavior and Organization