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zbwLeibniz-InformationszentrumWirtschaft

LeibnizInformationCentreforEconomics

Güth,Werner;Kocher,Martin;Sutter,Matthias

WorkingPaper

Experimental'beautycontests'withhomogeneous

andheterogeneousplayersandwithinteriorand

boundaryequilibria

DiscussionPapers,InterdisciplinaryRearchProject373:QuantificationandSimulationof

EconomicProcess,No.2001,45

Providedincooperationwith:

Humboldt-UniversitätBerlin

Suggestedcitation:Güth,Werner;Kocher,Martin;Sutter,Matthias(2001):Experimental

'beautycontests'withhomogeneousandheterogeneousplayersandwithinteriorandboundary

equilibria,DiscussionPapers,InterdisciplinaryRearchProject373:Quantificationand

SimulationofEconomicProcess,No.2001,45,urn:nbn:de:kobv:11-10049913,

/10419/62714

Experimental’BeautyContests’withHomogeneous

andHeterogeneousPlayersandwithInteriorand

BoundaryEquilibria

WernerGüth∗,MartinKocher†,MatthiasSutter∗,†

June26,2001

Abstract

Westudybehaviorinexperimentalbeautycontestswith,first,boundaryand

interiorequilibria,and,cond,homogeneousandheterogenoustypesofplayers.

Wefindquickerandbetterconvergencetothegame-theoreticequilibriumwith

interiorequilibriaandhomogeneousplayers.

JEL-classificationcode:C72,C91

Keywords:beautycontestexperiments,individualbehavior]

∗Humboldt-UniversityofBerlin,DepartmentofEconomics,InstituteforEconomicTheoryIII,Span-

dauerStr.1,D-10178Berlin,Germany.e-mail:gueth@

†UniversityofInnsbruck,InstituteofPublicEconomics,Universitaetsstras15,A-6020Innsbruck,

Austria.e-mail:@;@isthecorresponding

authorathisInnsbruckaddress.

1Introduction

The’beauty-contest’game-whichwaslikenedbyKeynes(1936)toprofessionalinvest-

mentactivity-hasinspiredmanyexperimentalstudies(e,forinstance,Nagel,1995,

DuffyandNagel,1997,Hoetal.,1998).Allbeauty-contestgameshaveincommonthat

imple

formthewinneristhedecisionmakerwhonumberisclosttoq∅(s),where∅(s)is

theaverageofallguessandqisarealnumberannouncedatthebeginningofthegame.

Duetoitsfavorableenvironmentanditssimplicity,thebeauty-contestgameor’guessing

game’,asitissometimesalsocalled1,hasmainlybeenudtoanalyzereasoningprocess

rmore,ithasgainedmoreandmore

importanceintestingvariouslearningtheories(e,forinstance,CamererandHo,1999).

Onereasonforthepopularityofthebeauty-contestgame,apartfromitssimplicity,is

itsobviousremblancewithdecisionmakinginfirreasonmight

betheflexibilityinttinggameparametersandmodifyingequilibriumchoicestocreate

theobviousmodificationsof

thebasicgamehavealreadybeenexplored.2Thelatterstatementdoesnotonlyholdtrue

forparameterchanges,butalsofortheeffectsofdifferentsubjectpools(al

studentexperimentsorreadersofnewspapers,eNageletal.,2000)orofinstitutional

changeslikedistinguishingsingleandteamplayers(KocherandSutter,2000).Acommon

featureofallhithertoperformedbeauty-contestgameexperimentsisthatonlythewinner

(i.e.,theoneclosttothebestguess)isrewardedandsubjectgroupsarehomogeneous

(i.e.,thebestguessisthesameforallparticipants).Nearlyallpriorstudies,furthermore,

relyonboundaryequilibria.

,inour

versionofthebeauty-contestgamepayoffarticipantreceives

amonetaryendowmenttopayafinewhosizeisdeterminedbyhowfarthechon

,weexploreandcomparebehaviorbothwith

teraregeneratedbyaddingaconstanttothe

1Thethirdterminuis’averagegame’,introducedbyMoulin(1986).

2ForanovervieweNagel(1999).

1

,inadditiontothecommoncawithhomogeneous

groupswhereallplayershavetoguessthesametargetnumberinagroupofndecision

makersweintroduceheterogeneousgroups,wheren

2

decisionmakershavetoguessq

i∅(s)

andn

2

decisionmakershavetoguessq

j=i∅(s).

Thechont-upremblesfinancialdecisionmuchmorethanthebasicbeauty-contest

,e.g.,reasoningprocessassociatedwithpickingstocksinstockmarkets.

Returnsare,obviously,continuousandnotdichotomousasinallpriorbeauty-contest

games,boundaryequilibriararelyexistandheterogeneousgroupsaretheruleandnot

theexception(e.g.,shortversuslongpositions),whichhasalsobeenneglectedinbeauty-

contestgames,modificationtothebasicgameallowstoexploreveral

lly,firstroundguessinthestandarddesignarequitefar

awayfromtheboundaryequilibrium(eHoetal.,1998).Bycomparingguessin

boundaryandinteriorequilibirumenvironmentswewanttoanalyzewhetherdeviations

lanationforsuchadifferencein

theresultsofthetwotreamtentsmightlieintheexistenceofadesiretochoointerior

insteadofextreme,boundarystrategies(eRubinsteinetal.,1997).Theintroduction

ofheterogeneoustypesofplayersallowstoinvestigatewhetheramorecomplexsituation

inducesparticipantstothinkharderaboutotherplayers’behavior,and,thus,promotes

our

paymentschemetobemoreappropriateifonelikensbeauty-contestgamestofinancial

decisionmaking,butwedonotexpectittoalterresultsqualitatively,althoughdeviation

fromoptimalityshouldbeslightlysmaller.

Inction2weintroduceour2x2-design,theequilibriaofourbeautycontestsandour

resultsaredescribedinction4beforeconcludinginction5.

2Theexperimentalbeautycontests

Letn(>2)denotethenumberofplayersi=1,...,ninthegamewhoallchooareal

number

s

i∈Si

=[0,100]fori=1,...,n.

2

Foranystrategyvectors=(s

1

,...,s

n

)let

∅(s)=1

n

nXi=1

s

i

eralformofthepayofffunctionu

i

(s)isgivenby

u

i

(s

i

)=C−c|s

i−qi

[∅(s)+d]|

where|r|denotestheabsolutevalueoftherealnumberr;disaconstantaddedtothe

averagenumber∅(s)andq

i∈(0,1)isthequotaof[∅(s)+d]whichdeterminesplayer

i’eridoesnotguesscorrectly,hemustpayafineofc(>0)forevery

unitofdeviationwhichhecanpayoutofhispositive(monetary)

2×2-factorialdesignofourexperiment(Table1)distinguishesd=0(leftcolumn)and

d=50(rightcolumn)aswellasq

i

=1/2foralli=1,...,n=4(upperrow)andq

i

=1/3

foronehalfandq

i

=2/3fortheotherhalfofthe4players(lowerrow).

quotaq

i

d=0d=50

q

i

=1/2∀is∗

i

=0,∀is∗

i

=50,∀i

q

i

=1/3fori=1,2s∗

i

=0s∗

i

=100/3ifq

i

=1/3

q

i

=2/3fori=3,4fori=1,...,4s∗

i

=200/3ifq

j

=2/3

Table1:The2×2-factorialdesignandequilibria.

Theequilibrias∗

i

Appendix,weshowhowthesolutionscaneitherbejustifiedastheuniqueequilibriumof

eachgameorbyrepeatedeliminationofstrictlydominatedstrategies.

Ourmainhypothescanbesummarizedasfollows:3

riorequilibrium(d=50)issuppodtoyieldsmallerdeviationsofthe

guessfromthegame-theoreticequilibriumthanaboundaryequilibrium,since

participantsoftentrytoavoidextremechoices(e,forinstance,Rubinsteinetal.,

1997).

3Ourcontinuouspaymentschemeshouldnotcaudifferentbehaviorthanunderthestandardwinner-

takes-allrule(e,forinstance,BolleandOckenfels,1990,Cubittetal.,1998).

3

ucingheterogeneityofplayersshouldinducesubjectstothinkmorethor-

oughlyaboutthestrategiesoftheotherplayertypesinordertomakeareasonable

scomplexsituation,withhomogeneousplayers,itmightnotbethat

o-

fore,fromabehavioralpointofview,wewouldexpectparticipantsinheterogeneous

groupstobeclortoequilibriumthanparticipantsinhomogeneousgroups.

3ExperimentalProcedure

TheexperimentswereruninDecember2000andJanuary2001attheHumboldt-University

allparticipantswerestudentsattendinganundergraduatecourinmi-

croeconomics4,,for

mostparticipantsitshouldhavebeentheirfirstexperiencewithexperimentaleconomics.

Thesoftwareofthecomputerizedexperimenthasbeendevelopedwiththehelpofz-Tree

(Fischbacher,1998).

Inea

averagetimeneededtorunassionwas40minutes(about15forreadinginstructions

andaskingprivatelyforclarificationsand25minutesforplaying10rounds).Participants

weredividedintotwo(matching)retoldthattheyarematchedrandomly

tmentswithheterogeneous

groupssubjectsweretoldthatineachroundtherewouldbetwosubjectsofeachtypein

atinspiteofrepeatedinteraction,theaverageofassion’smatching

groupsqualifiesasanindependentobrvation.

Wetc=0.05DMandC=2DMinthepayofffunction.5Table2showsaverage

earningsparatelyforthefirstandthecondfiveroundsforplayerswithq

i

=1/2

(intreatmentswithhomogeneousgroups),orq

i

=1/3andq

i

=2/3,respectively(in

4Thetopicofthiscourisgeneralequilibriumtheoryanddidnotyetintroducegametheoretic

concepts.

5Ifasubjectwasmorethan40unitsawayfromhertargetvalue,ppenedto

5outof160subjectsinthefiubjectswereinformedthatlosscouldbebalancedand

gainsaccumulatedintheroundstofollow.

4

caofheterogeneousgroups).Ingeneral,weobrvethatsubjectarnedonaverage

alwaysmoreinthecondfitsinhomogeneousgroups

withtheinteriorequilibrium(d=50)earnedmost,namely18.41DMintotal(outof

amaximumof20DM).Subjectswithq

i

=2/3andtheboundaryequilibrium(d=0)

earnedthesmallestaverageamount.

d=0d=50

quotas1−56−101−56−10

q

i

=1/28.169.268.659.76

q

i

=1/3,7.559.108.109.39

q

i

=2/36.848.707.348.64

eearnings(inDM)pertypeofplayer

4Results

Figures1and2showtheaverageguessineachtreatmentinthecouroftheexperiment

aswellasthecorrespondingequilibria.6Intheheterogenoustreatmentswesplitthedata

forthetwotypesofplayerswithq

i

=1/3andq

i

=2/3,een,

guessconvergesteadilytowardstheequilibrium7,inparticularinourtreatmentswith

interiorequilibria(d=50).

Wecanconfirmourfirsthypothesis,statingthatinteriorequilibriatriggermoreequilibrium-

tingthishypothesis,werelyontheaverages

ofrounds1to5,rounds6to10,ingtohomogeneousgroups

wefindthatgroupswithaninteriorequilibrium(d=50)aresignificantlyclortothe

equilibriumthangroupswithaboundaryequilibrium(d=0).8Heterogeneousgroups

6Adatafilewithssion,respectivelytypeaveragesisincludedintheAppendix.

7Deviationsfromequilibriumgetsignificantlysmallerfromroundttoroundt+1inanyca(con-

sideringbothhomogeneousandheterogeneousgroups),withtheexceptionofround9to10withd=0,

andround3to4and9to10withd=50(Wilcoxonsignedrankstest,p<0.05).

8p<0.1forrounds1-5;p<0.05forrounds6-10,and1-10,respectively(U-test,two-sided).

5

TreatmentAverages(d=0)

0

10

20

30

40

round

q=1/2

q=1/3

q=2/3

s*=0

Figure1:

TreatmentAverages(d=50)

20

40

60

80

round

q=1/2

q=1/3

q=2/3

s*(1/2)

s*(1/3)

s*(2/3)

Figure2:

6

guesstheequilibriumsolutionmoreclolywhentheequilibriumisinterior,however,this

holdsonlyforthefirstfiverounds.9

Furthermore,exactequilibriumguessaremuchmorefrequentintheinteriorequilibrium

treatmentswithhomogeneousgroups.49.25%ofallguessinthetreatmentwithhomoge-

neousgroupsandaninteriorequilibriumareexactlyattheequlibrium(s∗

i

=50),whereas

thecorrespondingfigureincaoftheboundaryequilibrium(s∗

i

=0)is27.75%.10The

frequencyofequilibriumchoicesis,however,

thatinNagel(1995),whoalsohadhomogeneousgroupsandeitherq=1/2orq=2/3,

only3outof115subjectschoexactlyzero(theequilibriumchoice).Weethecontinu-

ouspaymentschemeasthedrivingforcebehindthehighfrequencyofequilibriumchoices.

Lookingatgroupswithheterogeneousplayerswefindequilibriumchoicestobemuchless

frequent(7.5%incaofaninteriorequilibriumand3.5%withboundaryequilibrium,

respectively)andnosignificantdifferenceinequilibriumchoicesbetweenbothtypesof

specttoprofits,subjectsfacinganinteriorequilibriumearn-ceteris

paribus-significantlymorethanthofacingaboundaryequilibrium.11

Astoourcondhypothesis,namelythatheterogeneityofplayersshouldtriggermore

thoroughdeliberationsand,thus,moreequilibriumlikedecisions,wetestwhetherdevi-

ationsfromequilibriumaresmallerinheterogeneousthaninhomogeneousgroups,given

thateitherd=0ord=rytoourexpectations,wefindthathomogeneous

=0,ssionaverages

withhomogeneousgroupsaresignificantlysmaller(and,thus,clortoequilibrium)than

averageguessinheterogeneousgroupsineachofthefirstvenrounds,intheaverages

ofthefirstfiverounds,andintheaverageoveralltenrounds.12Inourd=50treatments,

homogeneousgroupsareclortotheequilibriumthanheterogeneousgroupsinallrounds

butrounds2and4.13We,therefore,believethatthecomplexitygeneratedbytwotypes

9p<0.05(U-test,two-sided).

10Thedifferenceisstatisticallysignificantforthefirstroundandtheaverageofthefirstfiverounds,

takingssionsasindependentobrvations(p<0.05,U-test,two-sided).

11p<0.05(U-test,two-sided).

12p<0.05forrounds2,5,and6,andfortheaverageofrounds1-5.p<0.1inallothercas.(U-test,

two-sided).

13p<0.05forrounds5,7,8,9,and10,andfortheaverageofrounds6-10.p<0.1inallothercas.

(U-test,two-sided).

7

ofplayersmakesitmoredifficulttoapproachequilibriumbehaviorandthatsubjectsdo

notreasonmorethoroughlywhentherearedifferenttypesofgroupmembers.

Next,weexplorewhethertherearesystematicdifferencesbetweenthedifferenttypesof

playersinheterogeneousgroups.q

i

=1/3-playersmightconvergefastertothegame-

theoreticequilibriumbecauofthefastereliminationofweaklydominatedstrategies.14

InFigure1(d=0)wefindthat,onaverage,guessofq

i

=2/3-playersarehigher

thanthoofq

i

=1/r,takingssionaveragesasthe

onlyindependentobrvationswedonotfindasignificantdifferencebetweentheguess

ofbothtypesofplayers,neitherinanysingleroundnorconsideringtheaveragesof

ingtoFigure2(d=50)weethataverageguessof

q

i

=2/3-playersareslightlyfurtherawayfromequilibriumthanthoofq

i

=1/3-players.

Yet,thedifferenceisdrivenbyasinglesubjectwhohaddifficultiesinunderstandingthe

experimentandwhochonumbersintherangefrom0to10inallrounds.15Therefore,

deviationsfromequilibrium(33.33and66.67,respectively)doalsonotdifferbetween

playertypesincaofd=,therefore,concludethatheterogeneityofplayers

doesaffectgroupbehaviorbyincreasingdeviationsfromequilibrium,comparedwith

r,heterogeneityaffectsbothtypesofplayersand

bothtypesdonotsystematicallydifferintheirdeviationsfromoptimality.

5Conclusion

Wehaveexploredbehaviorinfourdiff

findthatdecisionsareclortothegame-theoreticequilibriumwhentheequilibriumis

itioningroupsofhomogeneousplayersalsopromotesconvergencetothe

mplexitythroughheterogeneousplayers,however,isdetrimentalfor

profitsaswellasforconvergencetotheequilibrium.

14SeetheAppendixfordetails.

15Thissubjectevenendedupwithaloss.

8

References

[1]Bolle,FriedelandOckenfels,Peter(1990):Prisoners’dilemmaasagamewithincom-

lofEconomicPsychology,March1990,11(1),pp.69-84.

[2]Camerer,ColinandHo,Teck-Hua(1999):Experience-weightedattractionlearning

etrica,July1999,67(4),pp.827-74.

[3]Cubitt,RobinP.;Starmer,ChrisandSugden,Robert(1998):Onthevalidityofthe

mentalEconomics,1(2),pp.115-132.

[4]Duffy,JohnandNagel,obustnessofbehaviourinexperimental

’beautycontest’icJournal,November1997,107(445),pp.1684-700.

[5]Fischbacher,U.(1999):Z-tree:Zurichtoolboxforreadymadeeconomicexperiments,

WorkingpaperNo.21,InstituteforEmpiricalRearchinEconomics,Universityof

Zurich.

[6]Ho,Teck-Hua;Camerer,ColinandWeigelt,eddominanceanditer-

atedbestresponinexperimental’p-beautycontests’.AmericanEconomicReview,

September1998,88(4),pp.947-69.

[7]Keynes,eraltheoryofinterest,:

Macmillan,1936.

[8]Kocher,MartinandSutter,Matthias(2000):Whenthe’decisionmaker’matters:

Individualversusteambehaviorinexperimental’beauty-contest’sion

paper2000/4,InstituteofPublicEconomics,UniversityofInnsbruck.

[9]Moulin,k,NewYorkPress,1986.

[10]Nagel,linginguessinggames:an

EconomicReview,December1995,85(5),pp.1313-26.

[11]Nagel,yonexperimentalbeautycontestgames:Boundedratio-

:Budescu,David;Erev,IdoandZwick,Rami(eds.).Games

andhumanbehavior:y,Lawrence

ErlbaumAssoc.,Inc.,1999,pp.105-42.

9

[12]Nagel,Romarie;Bosch-Domènech,Antoni;Satorra,AlbertandGarcía-Montalvo,

,two,(three),infinity:Newspaperandlabbeauty-contestexperiments.

Workingpaper,UniversitatPompeuFabra,Barcelona,2000.

[13]Rubinstein,Ariel;Tversky,AmosandHeller,Dana(1997):Naivestrategiesincom-

:Albers,.(eds).Understandingstrategicinteraction:Es-

berg,Springer,pp.394-402.

10

Appendix

briumstrategiesinthebeautycontestgames

Ford=0thebestreplytos∗

j

=0forj=iisobviouslys∗

i

=0duetoq

i

<

provesthatthebehaviorintheleftcolumnofTable1definesastrictequilibriuminboth

itsuniquenessassumeanequilibriums∗with∅(s∗)>ebest

reply-condition

s∗

i

=q

i∅(s∗)fori=1,...,n

itfollowsthat

nXi=1

s∗

i

=∅(s∗)

nXi=1

q

i

fori=1,...,n

or

n=

nXi=1

q

i

dueto

nXi=1

s∗

i

=n∅(s∗).

Thiscontradictsq

i

<1fori=1,...,,ford=0thebehaviorinTable1istheonly

equilibrium.

Solvingthecad=0byrepeatedeliminationofstrictlydominatedstrategiesproceeds

bythefollowingalgorithm:

•Setmj

=100forj=1,...,n.

•Foralli=1,...,nallstrategiessi

>q

i

n

nPj=1

m

j

areeliminated.

•Denotefori=1,...,nbym0

i

themaximals

i

remainingafterthepreviousstep.

Substitutefori=1,...,nthepreviousm

i

bym0

i

andrepeatthecondstep.

Inthefirsteliminationstepthixcludesallstrategiess

i

>q

i·100fori=1,...,

thequenceofthem

j

-valuesaremonotonicallydecreasinguntiltheyfinallyreachm

j

=0

forj=1,...,,however,thatinthelowerrowofTable1ford=0thisimplies

amuchfastereliminationforq

i

=1/3-playersthanforq

i

=2/3-players:Aq

i

=1/3-

playerwouldimmediatelyeliminatealls

i

>100/3whereasaq

i

=2/3-playerstartsby

eliminatingonlys

i

>200/tailscanbefoundinTableA1illustratingindetail

theimplicationsofthefirstfoureliminationsteps.

11

numberofeliminatedstrategies

eliminationq

i

=1/2,d=0q

i

=1/2,d=50d=0d=50

stepfori=1,...,4fori=1,...,4q

i

=1/3q

i

=2/3q

i

=1/3q

i

=2/3

1s

i

>50s

i

<25,s

i

>75s

i

>100

3

s

i

>200

3

s

i

<50

3

s

i

<100

3

s

i

>50

2s

i

>25s

i

<37.5,s

i

>62.5s

i

>50

3

s

i

>100

3

s

i

<25s

i

<50

s

i

>125

3

s

i

>125

3

3s

i

>12.5s

i

<43.75,s

i

<56.25s

i

>25

3

s

i

>50

3

s

i

<350

12

s

i

<350

6

s

i

>450

12

s

i

>450

6

4s

i

>6.25s

i

<46.875,s

i

>53.125s

i

>25

6

s

i

>25

3

s

i

<550

12

s

i

<550

6

s

i

>1700

48

s

i

>1700

24

.

.

.

∞si

>0s

i=50si

>0s

i

>0s

i=100

3

s

i=200

3

TableA1:Repeatedeliminationofstrictlydominatedstrategiesforour2×2-factorial

design

Ford=50oneprovesinthesamewaythatthebehaviorintherighthand-columnof

Table1istheunique(strict)ationofstrictlydominatedstrategies

i

=1/2fori=1,...,4,forinstance,notonly

thestrategiess

i

>75butalsothowiths

i

<25havetobeeliminatedinthefirststep.

Fortheasymmetriccaplayersiwithq

i

=1/3eliminatecorrespondinglys

i

>50and

s

i

<50/3inthefirsteliminationstepwhereasforq

i

=2/3onlythestrategiess

i

<100/3

canbeeliminatedinthefiainillustrateshowtheasymmetriccaforces

participantstoengageinquitedifferentconsiderationswhenanticipatingotherswitha

differentquotaandwhenanticipatinganotherwiththesamequotawhodeliberations

shouldbesimilartoone’sown.

ctions

Thewritteninstructionsdifferonlyinthedescriptionofthequotas(either1/2forall4

playersortwoplayerswith1/3andtwowith2/3)aswellasoftheconstant(whichwas

onlymentionedincaofd=50).Weprovidethetranslatedinstructionsford=0and

erinstructionsareavailableuponrequest.

12

Rulesoftheexperiment(originallyinGerman)

Welcometoourexperimentandthankyouforparticipating!

rsonofthisgrouphastochooanumber

x

i

betweenzero(0)andonehundred(100).

isnotnecessarytochooaninteger,however,numberswithmorethantwodecimalsare

excluded.

Yourpayoffintheexperimentdependsonhowcloyournumberistoatargetnumber.

Thecloryournumbertoatargetnumber,thehigheryourpayoff.

numbers

oftypeAandtypeBparticipantsarediffreatypeAyourtargetnumberis

one-third(factorf

A

)re

atypeByourtargetnumberistwo-thirds(factorf

B

)oftheaverageofallfournumbers

choninyourgroup.

Thepayoffofeachrounddependsonthedifferencebetweenyourchonnumberandthe

chonnumberandthetargetnumberareidentical,youreceive

(roundedup)unitdifferenceleadstoadeductionof0.05DM.

Formally,yourperroundpayoffiscalculatedasfollows:

payoffperround=2.00−0.05|x

i−fA,BPx

j

4|.

,though,

pyour

type(eitherAorB)inallrounds.

Aftereachroundyoureceiveinformationonthegroupaverage,yourtargetnumberand

yourpayoff.Payoffsareaccumulatedoverallroundsandpaidincashandprivatelyat

theendoftheexperiment.

13

file

Sessionaverages

Guess

SessionRounds

q1--56--101--10

d=010,518,568,154,313,131,980,660,360,240,206,447,221,584,40

20,520,576,982,320,540,060,010,006,251,025,366,092,534,31

30,523,4412,306,522,881,310,990,770,420,310,329,290,564,92

40,521,1912,847,764,602,341,041,050,4012,583,739,753,766,75

50,530,1819,7116,8012,529,456,064,252,461,460,5517,732,9610,34

d=50

10,535,6342,1946,0153,6550,0249,8349,7249,5849,7749,8445,5049,7547,62

20,535,9439,2542,8445,5048,1549,0849,9650,3450,2750,3442,3450,0046,17

30,547,3855,6948,1249,3848,6849,3649,3649,7249,9749,8849,8549,6649,75

40,542,6352,4447,7048,6348,9349,0349,5149,7449,7849,7748,0649,5648,81

50,549,0648,9449,4549,8749,8749,9050,0050,0050,0050,0049,4449,9849,71

d=01*29,9825,8624,7720,6112,226,383,401,860,910,4122,692,5912,64

2*26,5012,917,115,062,892,241,541,141,070,6710,891,336,11

3*22,1424,9816,7512,3910,207,616,073,072,4714,9717,296,8412,06

4*44,4931,9125,7719,7216,4217,8011,679,598,286,8527,6610,8419,25

5*41,1224,3911,527,315,192,541,430,700,320,0717,911,019,46

d=50

1*36,6339,9634,9038,1236,8437,0237,5938,3740,2541,1437,2938,8738,08

2*57,0054,3454,3652,7253,9653,3650,8151,2350,1651,2754,4851,3752,92

3*49,6350,2250,7251,6151,8251,4651,0650,4450,1450,0350,8050,6350,71

4*46,8152,7048,2450,3948,4450,4450,4146,6049,4450,1049,3249,4049,36

5*52,5656,5655,4962,3650,9453,3851,7252,3951,7852,8155,5852,4154,00

*Averagesofbothtypesinheterogeneousgroups

Typeaveragesinheterogeneousgroups

Session

q1--56--101--10

d=010,3323,4614,5818,4710,496,953,801,420,730,440,1214,791,308,04

10,6736,5037,1431,0830,7417,498,955,382,991,390,6930,593,8817,23

20,3331,2511,568,486,383,252,532,251,631,380,9112,181,746,96

20,6721,7514,255,753,752,531,950,830,650,760,449,610,925,26

30,3311,5211,7014,7511,289,146,234,231,601,560,7611,682,877,28

30,6732,7538,2518,7513,5011,259,007,904,543,3929,1822,9010,8016,85

40,3333,7523,3922,1616,0511,098,108,656,968,196,3921,297,6614,47

40,6755,2340,4329,3823,3921,7527,5014,6912,218,387,3034,0314,0224,02

50,3336,3824,907,654,503,381,880,990,510,220,0715,360,738,05

50,6745,8623,8915,3910,127,003,201,870,880,430,0820,451,2910,87

d=50

10,3345,2536,6623,8029,4328,3629,0329,6928,7229,2528,7232,7029,0830,89

10,6728,0043,2546,0046,8145,3145,0145,5048,0151,2553,5641,8748,6745,27

20,3349,5038,6937,4735,2736,1735,5634,0133,9133,9033,8339,4234,2436,83

20,6764,5070,0071,2570,1771,7571,1767,6268,5466,4268,7269,5368,4969,01

30,3345,0035,9635,7236,4736,2235,5434,6033,8533,3232,7237,8734,0035,94

30,6754,2564,4865,7266,7567,4267,3967,5267,0466,9667,3363,7267,2565,49

40,3340,1336,3333,7533,3332,3033,3133,3026,1032,9132,4635,1731,6133,39

40,6753,5069,0862,7367,4564,5967,5867,5167,1165,9767,7463,4767,1865,32

50,3340,5035,0035,8651,8434,5034,8833,9335,4034,9134,2539,5434,6737,10

50,6764,678,175,172,967,471,969,569,468,771,471,6370,1670,89

14