Signaling with costly acquisition of signals

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ContentslistsavailableatScienceDirect

Journal

of

Economic

Behavior

&

Organization

jo u r n al ho me p ag e :ww w . e l s e v i e r . c o m / l o c a t e / j e b o

Signaling

with

costly

acquisition

of

signals

Ennio

Bilancini

a,∗

_,

_Leonardo

_Boncinelli

b

a_Dipartimento_di_Economia_“Marco_Biagi”,_Università_degli_Studi_di_Modena_e_Reggio_Emilia,_Viale_Berengario_51,₄₃_ovest,₄₁₁₂₁

Modena,Italy

b_Dipartimento_di_Scienze_per_l’Economia_e_l’Impresa,_Università_degli_Studi_di_Firenze,_Via_delle_Pandette_9,₅₀₁₂₇_Firenze,_Italy

a

r

t

i

c

l

e

i

n

f

o

Articlehistory: Received5July2016 Receivedinrevisedform 10September2017 Accepted30October2017 Availableonline11November2017 JELclassiﬁcation:

D82 D83 Keywords: Costlycognition Costlysignalacquisition Pooling

Equilibriumreﬁnements Forwardinduction Trembles

a

b

s

t

r

a

c

t

Inthispaperweinvestigatetheconsequencesofintroducingacosttoobservethesignal inanotherwisestandardsignalinggame.Beyondidentifyingequilibria,whichwecontrast withthoseofastandardsignalinggame,westudytheirrobustnesstotwoimportantclasses ofreﬁnements:actingthroughrestrictionsonout-of-equilibriumbeliefsandthrough trem-bles.Ourresultssuggestthatmoreprominenceshouldbegiventothepoolingoutcomeon theminimumsignal.

1. Introduction

Signalingisapervasivephenomenonineconomicinteractions,emerginginmanysituationswherethereare

informa-tionasymmetries.Manysignalingmodelshavebeendevelopedandstudied,makingtheclassofsignalinggamesaquite

prominentoneineconomics(seeRiley,2001,foracomprehensivesurvey).Animportantcharacteristicofsignalinggames

isthattheytypicallyshowmanyequilibriawithratherdistinctfeatures:equilibriainwhichsender’stypespooltogetherby

sendingthesamesignalandequilibriawheresender’stypesseparatefromeachotherbysendingdifferentsignals.

Inappliedresearch,signalingmodelsareoftenusedwiththefocusonthebestseparatingequilibrium,alsocalledthe

Rileyequilibrium,i.e.,theequilibriumwhereallsender’stypesseparatebuttheyincurtheminimumnecessarysignaling

costtodoso.ThisisingoodpartduetoanimportantstreamofliteraturethathasshowntheprominenceoftheRiley

equilibriumwhenplayersaresupposedtopossessasufﬁcientdegreeofforwardinduction:typically,plausiblerestrictions

onout-of-equilibriumbeliefsareappliedtoreﬁneawaymanyequilibria(seeSobel,2009,foraninstructivesurvey).

Despitetherelevanceofsignalingmodels,theliteraturehassofargivenrelativelyscarceattentiontothepossibility

thattheacquisitionofthesignalbythereceivermightbeadeliberateandcostlyactivity.Istheassumptionofautomatic

acquisitionofsignalsinnocuous?Inthispaperweshowthatitisdeﬁnitelynotso.Indeed,evenaverysmallcostofsignal

∗ Correspondingauthor.

E-mailaddresses:ennio.bilancini@unimore.it(E.Bilancini),leonardo.boncinelli@uniﬁ.it(L.Boncinelli). https://doi.org/10.1016/j.jebo.2017.10.022

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acquisitioncanmakeagreatdifferenceintermsoftherobustness(andplausibility)ofequilibria.Inparticular,weshowthat

inthepresenceofcoststoacquirethesignalthepoolingofsender’stypescanonlyhappenontheminimum(leastcostly)

signal,whileallotherpoolingoutcomescannotbesustainedasequilibria.Atthesametime,thepoolingequilibriumonthe

minimumsignalisnotreﬁnedawaybyanyrestrictiononbeliefsatout-of-equilibriumsignals,sothatitbecomesatleastas

prominentastheRileyequilibriumasoutcomeofasignalinggame.Theseresultsaredrivenbytheemergenceofastrategic

complementarity:ifthereceiverchoosesnottoincurtheacquisitioncost,thenallsender’stypesﬁnditoptimaltopoolon

theminimumsignaland,atthesametime,ifthedifferenttypesofthesenderpoolonthesamesignal,thenthereceiverﬁnds

itoptimalnottoincurtheacquisitioncost.So,acomplementaritynaturallyarisesbetweenthereceiver’sincentivetocostly

acquirethesignalandthesender’sincentivetoengageinthecostlysignalingactivity.Thisalsoimpliesthattwodeviations

arerequired,startingfromapoolingequilibriumontheminimumsignal,foranon-equilibriumsignaltobeobservedbythe

receiver:adeviationbythesenderonlywouldnotevenbeennoticedbythereceiver.Thisintuitivelyexplainstherobustness

torestrictionsonout-of-equilibriumbeliefs.

Sinceourmodelisnotapuresignalinggame,asitalsodealswithinformationacquisitionbythereceiver,westudythe

robustnessofequilibriawhenout-of-equilibriumplayisnotduetointentionaldeviations,buttomistakesthatplayersmake

whenchoosingequilibriumstrategies.Ourmainﬁndingisthatthepoolingoutcomeontheminimumsignalisrobustto

tremblesonplayers’strategies,sincethebeneﬁtsofchangingstrategyarenegligiblewhentremblesareverysmall,while

bothsignalingandsignalacquisitionimplynon-negligiblecostsforplayers.

Ourresultssuggestthatnewattentionshouldbegiventopoolingoutcomes,inparticulartopoolingontheminimum

signal.Thiscouldhavefar-reachingimplications,especiallyinthelightofthewidespreadrelianceonseparatingequilibria

inappliedmodels.

Inthispaperwealsoconsiderseparationoutcomes.Inparticular,weshowthatthesetofseparatingequilibriaisthe

samewhensignalacquisitionisautomaticandwhen,instead,isdeliberateandcostly,providedthattheacquisitioncost

issufﬁcientlysmallrelativelytothevalueofinformation.Thesameresultholdswhenwerestrictattentiontoseparating

equilibriareﬁnedthroughrestrictionsonbeliefsatout-of-equilibriumsignalsandtremblesonplayers’strategies.

Thepaperisorganizedasfollows.InSection2wereviewtheliteratureoncostlyacquisitionofinformation.InSection

3weintroducesignalinggameswithcostlyacquisitionofsignalsbymeansofanexamplethatisavariantoftheclassical

Spence’ssignalingmodel.InSection4wedeﬁneageneralclassofsignalinggameswithcostlyacquisitionofsignals.In

Section5wegiveourresultsontheequilibriaofthesegames,andhowtheyarereﬁnedbymeansofrestrictionsonbeliefsat

out-of-equilibriumsignalsandbymeansoftrembles.InSection6weexploretherobustnessofourresultsalonganumber

ofdimensions.Section7summarizesourcontributionandprovidesﬁnalremarksonwelfare.AformaldeﬁnitionofPerfect

Bayes-NashequilibriumcanbefoundinAppendixA.ProofsarecollectedinAppendixB.

2. Relatedliterature

Theideathattheacquisitionofinformationisastrategicchoicewhichcomesatacostisreceivingincreasingattention

ineconomics.SincetheseminalcontributionbyGrossmanandStiglitz(1980),whereitisshownhowcostlyinformation

canimpairtheefﬁcientfunctioningofmarkets,severalmodelswiththisfeaturehavebeeninvestigated,butjustafewof

themarecloselyrelatedtoourmodel.Infact,mostofthesemodelsdonotconsiderasender-receiversetup,andthosewho

considerasignalingframeworkdonotexplorethepossibilitythatinformationacquisitioninvolvesthesignalitself.1,2

Tothebestofourknowledge,theonlypapersconsideringcostlyacquisitionofinformationinsignalinggamesareBilancini

andBoncinelli(2016b),whostudythecasewherethereceiverisananalogicalreasoner,andBilanciniandBoncinelli(2016a),

whoconsiderthecaseofareceiverthathastoincuracognitivecosttofullyandpreciselyelaborateinformationonsender’s

type.AnimportantdifferencebetweenthemodelsinBilanciniandBoncinelli(2016a,b)andtheonedevelopedinthepresent

paperisthatintheformerstheacquisitioncostispaidtoacquirehardinformationonthestateoftheworld,whileinthe

presentpaperthecostispaidtoacquirethesoftinformationembodiedbythesignal.

ApapermorecloselyrelatedtooursisDewatripontandTirole(2005)whichdevelopsatheoryofcostlycommunication

whereboththesenderandthereceiverhavetoincuracostinordertocommunicate.3_The_model_can_be_seen_as_a_standard

cheaptalkmodel,withtheadditionalfeaturethatinorderforcommunicationtooccurboththesenderandthereceiver

havetoincuracost.4_Due_to_this,_a_form_of_strategic_{complementarity}_arises_–_that_is_similar_to_the_one_emerging_in_our

model–whichgivesrisetoarobustbabblingequilibriumwherethemessagesentbythesendercontainsnoinformationand

1 _See_for_instance_the_literature_on_strategic_pricing_and_information_acquisition_of_product_quality_from_a_third_party_started_with_Bester_and_Ritzberger

(2001)(seealsoGertz,2014andMartin,2015).

2 _Costly_acquisition_of_information_has_been_recently_studied_in_a_number_of_different_settings:_costly_cognition_(Gabaix_et_al.,_2006;_Caplin_and_Dean,

2014);strategicinteractiontoinﬂuencethirdparties(Liu,2011;Brocasetal.,2012)orthesubsequentoutcome(MorathandMünster,2013;Colomboetal., 2014);auctions(Shi,2012);voting(Oliveros,2013);expertsandauditstructure(Argenzianoetal.,2014;MenichiniandSimmons,2014).

3 _We_observe_that,_while_costs_are_paid_by_both_the_sender_and_the_receiver,_{communication}_remains_one-sided._For_a_recent_contribution_where

communicationistwo-sidedandcostless,seeEs ˝oandWallace(2014).

4 _The_{informativeness}_of_{communication}_in_a_cheap_talk_setting_has_been_recently_studied_by_Chen_and_Gordon_(2015),_who_generalize_the_comparative

staticsanalysisintheseminalpaperofCrawfordandSobel(1982),andbyShimizuandIshida(2015),whoshowthattheextentofcommunicationis severelylimitedasthereceiverbecomesmoreinformed.

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thereceiverdoesnotacquireit.ThemaindifferenceconcerningthesettingisthatDewatripontandTirole(2005)consider

signalsthathaveallthesamecost,whileinourmodelhighersignalsaremorecostly.Besidesconsideringadifferentsetting,

wealsocarryoutadifferentanalysisthatfocusesontherobustnessofequilibriatoreﬁnementsactingthroughrestrictions

onbeliefsatout-of-equilibriumsignalsandthroughreﬁnementsthatarebasedonplayers’trembles.Intuitively,ourresults

abouttherobustnessofpoolingequilibriaontheminimumsignalextendinthesettingofDewatripontandTirole(2005)to

babblingequilibria.However,thefactthatdifferentsignalsentaildifferentcostsforthesenderiscrucialfortheelimination

ofpoolingequilibriathatarenotontheminimumsignal.

Inanapparentlyverydifferentsetting,SolanandYariv(2004)obtainresultsthatpartlyoverlapwithours.Theyconsider

gameswithespionage:startingfromasettingwhereeachoftwoplayershastomakeadecisionwithoutknowingthechoice

madebytheopponent,theyendowoneoftheplayerswiththepossibilitytoobservetheopponent’sactionatacostpriorto

takinghisowndecision.Amongotherthings,theyshowthatanequilibriumoftheoriginalgamewheretheopponentplaysa

purestrategywillbemaintainedincasethepossibilityofespionageisgiventotheotherplayer,whohoweverpreferstosave

espionagecostssinceinequilibriumthereisnoinformationtobeacquired.Suchresultisbasicallythesameasobservingin

oursettingthatthepoolingproﬁleontheminimumsignalisequilibriumforanyleveloftheacquisitioncost.5_Differently

fromSolanandYariv(2004),wefocusonaspeciﬁcclassofgames,i.e.,signalinggames.Thisleadsustocomparethecase

ofpositiveacquisitioncostswiththecasewithzeroacquisitioncosts,whichcanbeseenasthestandardsignalinggame;

onthecontrary,SolanandYariv(2004)’sbasisforcomparisonisthecaseofinﬁnitecosts,whennoespionageoccurs.This

allowsustoshowthatpoolingproﬁleswhicharenotontheminimumsignalceasetobeequilibriawhenweintroducecostly

acquisitionofsignals.Furthermore,ouranalysisexplorestherobustnessofthepoolingproﬁleontheminimumsignalto

reﬁnementsactingthroughrestrictionsonbeliefsatout-of-equilibriumsignals,andtoreﬁnementsactingthroughplayers’

tremblesaswell,whereasSolanandYariv(2004)onlyconsiderespionageitselfasapotentialreﬁnementtool(forgames

whereoriginallythereisnoespionage).

3. Amotivatingexample

ConsiderthefollowingsimplevariantoftheclassicalmodelbySpence(1973).ThereisoneemployerEthatwantsto

hireaworkerW.Therearetwotypesofworkers,distinguishedbytheirproductivity∈{1,2},whichisaworker’sprivate

information;Ehasaprior0<p<1thatWishighlyproductive,i.e.,that=2.Technologyandmarketconditionsaresuch

thatE’snetproﬁtsaregivenby−wifaworkerishired,withwthewagepaidtothehiredworkerandhisproductivity,

whileotherwiseproﬁtsare0.

Moreover,Wcanacquireeducationbyincurringacostthatistype-dependent.Inparticular,supposethatWcomesfroma

foreigncountryandthathehastomovetoE’scountryinordertobehired.SupposealsothatWcanonlyacquireeducationin

theforeigncountry,andthattheonlyavailablealternativesareagoodschoolGandabadschoolB,whicharenotpreviously

knowntoE.Fortheprospectiveworkeroftype,thecostofattendingGis2/andthecostofattendingschoolBis1/.

So,attendingschoolGismorecostlythanattendingschoolB,anditisrelativelymoresoforthelowtype(i.e.,=1).This

providesWwithacostlysignalx∈{G,B}thatpotentiallyallowsW’stypestoseparate.

Sofar,thereisnosubstantialdifferencefromSpence’smodel.However,whatifE,inordertoassessthequalityofthe

schoolingsignalxsentbyW,hastoactivelyacquiretheinformationonwhatschoolWhasattendedinthecountryhecomes

from,andwhatattendancecostshavebeenpaid?Thisinformationcanwellnotcomeforfreeandourpointisthatthiscan

makethedifference.Weobservethatthecostsofacquiringsuchinformationcanbeinterpretedasduetothematerialand

thecognitiveeffortwhichisnecessarytoretrieveandelaboratetherelevantdataonx.Onthematerialside,Emighthaveto

searchandcollectinformationonGandB,andmaybealsopaytotranslatedocumentsthatwouldbeotherwiseinaccessible.

Onthecognitiveside,Emighthavetoexertefforttoelaboratethecollectedinformationinordertoestablishthatoneschool

isGwithcosts2andtheotherisBwithcostsandtoassignthesignalxtoeitherGorB.Ifno(materialorcognitive)effort

isexerted,thenschoolsarefundamentallyindistinguishabletoE;inthiscase,itisreasonablethatEdoesnotconditionher

decisiononx.Tomodelthis,supposethatEhastopayacostc>0toacquirethesignalxsentbyW.Inparticular,ifEdoes

notincurthecostc,thensignalxremainsunknowntoE.

Considernowthefollowingsituation:WchoosesBindependentlyofhistype,i.e.,x(1)=x(2)=B,andEdecidesnotto

acquirethesignalx.Itiseasytocheckthatthisisanequilibriuminthepresentexample(whateveristhevalueforEof

knowingW’stypes):bothtypesofWstrictlylosebyswitchingtothemorecostlyGasthisdoesnotgrantadifferentwage,

andEstrictlylosesbyacquiringthesignalbecauseitcostscandprovidesnonewinformation.Weobservethatsuchan

equilibriumisverysimilartothepoolingequilibriumwithlowestsignalthatemergesinSpence’smodel.However,thereis

animportantdifference:becauseofacquisitioncostssuchpoolingequilibriumismuchmorerobusttoreﬁnementsthanthe

oneinSpence’smodel.Firstly,wenotethatrestrictionsonbeliefsatout-of-equilibriumsignalsareineffectivetoreﬁneaway

thisequilibrium,sinceEdoesnotacquireanysignal,andhenceWcannotuseout-of-equilibriumsignalstocommunicate

withE;indeed,a deviationbyWonlywouldnotevenbeennoticedbyE.Inparticular,evenifW’shightypedeviates

fromx(2)=Btox(2)=G,whenEdecidesnottoacquirethesignalthereisnowaytoletEknow–orevenimagine–about

suchadeviation.So,argumentsbasedonthereasonablenessofout-of-equilibriumbeliefscannotreﬁneawaythispooling

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equilibrium.Secondly,thispoolingequilibriumisrobustevenwhendeviationsfromequilibriumbehaviorisnotintentional

buttheresultofmistakes.Indeed,whentheprobabilityofmistakesbecomessufﬁcientlylow,thepossiblebeneﬁtofchanging

strategyisnotworthitscostforbothplayers,sincechoosingGismorecostlythanchoosingBforW,andacquiringthesignal

iscostlyforEaswell.

Thereisanotherimportantdifference.Inthisexamplethepoolingoutcomeassociatedwiththelowestsignalpooling

equilibriumistheonlypoolingoutcomethatcanbesustainedinequilibrium,whichisincontrastwithSpence’smodel

wheremultiplepoolingoutcomesarepossibleinequilibrium.Thishappenswithoutanyadditionalreﬁnementofequilibria.

Toseewhyitisso,considerthecasewherebothtypesofWpoolonG,i.e.,x(1)=x(2)=G.GiventhisbehaviorbyW,Eﬁnds

itstrictlyproﬁtablenottoincurtheacquisitioncost,asacquiringthesignalprovidesnonewinformation.ButifEdoesnot

acquirethesignalx,thenthechoiceofx(1)=x(2)=GcannotbesustainedinequilibriumsinceeachofW’stypewouldstrictly

gainbyswitchingfromGtoB,asthisallowstosaveonthecostofsignalingwithoutadverselyaffectingE’sbeliefs.

4. Themodel

Wenowintroducethemoregeneralgameofsignalingwithcostlyacquisitionofsignals(SCAS).ThereisonesenderSand

onereceiverR(sometimesreferredtoas“he”and“she”,respectively).ThesenderSobserveshisowntypet∈T,withTa

ﬁnitesetofcardinalityn,andthenchoosesasignalx ∈X=R+.6ThereceiverRcanexertcostlyeffortandacquirethesignal

x,orsaveoneffortandobservenothing.Wedenotewiths_∈_{s1,s2_}suchachoice,wheres1meansthatxisnotacquired

ands2meansthatxisacquiredandeffortisexerted.7_In_any_case,_then_R_has_to_choose_an_action_y_∈_Y₌_R._The_prior_beliefs

heldbyRonTaregivenbyp=(p1,...,pn)∈T,whereptdenotestheprobabilitythatSisoftypet∈T,andTdenotesthe

unit(n−1)-simplex.TheposteriorbeliefsheldbyR,onceasignalxhasbeensentandaninformationacquisitionchoices

hasbeenmade,aredenotedbyˇ(x,s)=(ˇ1(x,s),...,ˇn(x,s))∈T,whereeachˇt(x,s)istheprobabilitythatSisoftypet

conditionalon(x,s).Consistentlywiththeideathatnosignalisobservedincases=s1,weassumethatˇ(x,s1)=ˇ(x_,_s1)_for

allx,x∈X.

UtilityforSisU:T×X×Y→R,andutilityforRisV:T×X×Y×{s1,s2}→R.WeassumethatUisstrictlyincreasingin

y,andstrictlydecreasinginx.WealsoassumethatVexhibitsaﬁxedpositivecostofacquiringthesignal,sothatthenotation

canbesimpliﬁedasfollows:V(t,x,y,s1)=

v

(t,x,y)andV(t,x,y,s2)=

v

(t,x,y)−c,forallt∈T,x∈X,y∈Y.Moreover,we

assumethatargmax

y∈Y

t∈Tˇt

v

(t,x,y)isasingleton,forallˇ∈T,andallx∈X.Finally,thesingle-crossingpropertyis

assumedtohold:U(t,x,y)≤U(t,x,y),withx>x,impliesthatU(t,x,y)<U(t,x,y)forallt>tandy,y∈Y.

Welimitouranalysistopurestrategies.AstrategyforSisafunction:T→X;wedenotewithMthesetofallpossible

.AstrategyforRisapair(s,˛)wheres∈{s1,s2}and˛:X×{s1,s2}→Yisafunctionsuchthat˛(x,s1)=˛(x_,_s1)_for_all_x,

x∈X,i.e.,R’sactionisunconditionalonxwhenevers=s1ischosen;wedenotewithAthesetofallsuchfunctions.8

Lastly,inordertobettercontrastourresultswiththeresultsonstandardsignalinggamesweﬁnditusefultodeﬁnea

classofgamesthatareequivalenttostandardsignalinggamesandthatcanbeobtainedfromaSCASgame.Thiscanbedone,

startingfromaSCASgame,byforcingRtoplays=s2.Werefertosuchgameassignalingwithforcedacquisitionofsignals

(SFAS).NotethataSFASgamewithutilitiesUand

v

isactuallythestandardsignalinggame–i.e.,withnocoststoacquire

thesignal–thatcanbeobtainedfromaSCASgamewithutilitiesUand

v

andanyacquisitioncostc>0.

5. Results

5.1. Equilibria

AsasolutionconceptwefocusonPerfectBayes-Nashequilibria(seeAppendixAforaformaldeﬁnitionappliedtoSCAS

gamesandSFASgames).Apoolingequilibriumisaproﬁle(,(s,˛))suchthat(t)=(t₎_for_all_t,_t_∈_T._A_pooling_equilibrium

ontheminimumsignalisaproﬁle(,(s,˛))suchthat(t)=0forallt∈T.A(fullyorpartially)separatingequilibriumisa

proﬁle(,(s,˛))suchthat(t) /=(t₎_for_some_t,_t_∈_T.

InaSFASgame,givenasender’sstrategy,wedeﬁnethevalueofacquiringinformationonthesignalas:

()=max ˛∈A

t∈T pt

v

(t,(t),˛((t),s2))−max ˛∈A

t∈T pt

v

(t,(t),˛((t),s1)). (1)

6 _We_have_chosen_to_use_continuous_action_spaces_because_this_is_the_most_common_setting_for_signaling_models._We_stress_that_our_results_might_be

stated,withfewstraightforwardadjustments,usingdiscreteactionspacestoo.

7 _This_labeling_owes_to_the_{classiﬁcation}_of_elaboration_processes_as_“System_1”,_or_S1,_which_is_fast,_cheap_and_intuitive,_and_“System_2”,_or_S2,_which

isslow,costlyandanalytical(see,e.g.,Kahneman,2003).Westressthisinterpretationbasedoncognitiveeffortbecausewethinkthatitappliestomany relevantcasesofsignalacquisition.Ofcourse,otherinterpretationsarepossiblewherethecostofacquiringthesignalisduetonon-cognitivefactors.

8 _Here_we_adopt_a_{speciﬁcation}_of_strategies_for_R_that_is_more_compact_but_perhaps_less_{straightforward}_than,_e.g.,_˛(s,_˛s1_,˛s2₎_where˛s1∈_Y_is_the

actionchosenifs=s1and˛s2_:_X→_Y_is_the_action_chosen_if_s₌_s₂_._We_stress_that_the_two_formulations_are_equivalent_in_the_sense_that_we_can_go_from_one

totheotherbycoalescing(Thompson,1952)theinformationsetthatRreacheswhens=s1withtheonewhereRchoosess,althoughinthisgamethereis

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ThesetofPerfectBayes-NashequilibriaofaSCASgameisingeneraldifferentfromthesetofPerfectBayes-Nashequilibria

ofatypicalsignalinggame(aSFASgameinoursetting).Proposition1belowstatesthatpoolingequilibriaontheminimum

signalhaveaparticularroleinSCASgamesbecause,whiletheyalsoexistinordinarySFASgames,theyaretheonlypooling

equilibriainSCASgames.Also,Proposition1statesthatthesetofseparatingequilibriaoftheSCASgameisincludedinthe

setofseparatingequilibriaoftheSFASgame:indeed,aseparatingequilibriumoftheSCASgameisaseparatingequilibrium

oftheSFASgamewiththeadditionalrequirementthatthecosttoacquirethesignalisnotlargerthanthevalueofacquiring

informationonthesignal.

Proposition1. ConsideraSCASgameanditsassociatedSFASgame.

(a)ApoolingequilibriumontheminimumsignalexistsbothintheSCASgameandintheSFASgame;moreover,intheSCASgame

nopoolingequilibriumexistswhichisnotontheminimumsignal.

(b) If(,(s,˛))isaseparatingequilibriumintheSCASgame,thenitisalsoaseparatingequilibriumintheSFASgame;if(,(s2,

˛))isaseparatingequilibriumintheSFASgame,thenitisaseparatingequilibriumintheSCASgameifandonlyifc≤().

Weobservethatitfollowsfromstandardresultsonsignalinggameswithoutacquisitioncoststhatpoolingequilibriacan

existintheSFASgamewherethesignalchosenbyallS’stypesisnottheminimum.Therefore,inasensethesetofseparating

equilibriaofaSCASgameconvergestothatoftheassociatedSFASgameasctendsto0,whereasthesameisnottrueof

thesetofpoolingequilibria.Wealsoobservethatascgetslargerandlarger,thesetofequilibriaofaSCASgameshrinks.

Indeed,aseparatingproﬁle(,(s2,˛))whichisequilibriumofaSCASgameceasestosuchwhencexceedsitsvalue()

ofacquiringinformationonthesignal.Further,sincethevalueofacquiringinformationonthesignalonlydependsonits

informativenessaboutsender’stypes,allseparatingequilibriawiththesamevalueceasetoexistatthesametime.Toﬁx

ideas,considerasignalingmodelwithonlytwotypesofsenders:theentiresetofseparatingequilibriaoftheSFASgameis

maintainedintheSCASgamewhencdoesnotexceedthevalueforthereceivertoacquiretheinformationonthesender’s

currenttype,whichisthesameforallseparatingstrategiesofthesender;however,whencexceedssuchvalue,allseparating

equilibriaceasetoexistinthisSCASgame.Finally,fromProposition1wecanalsoconcludethatseparatingequilibriawhere

aremoreinformativewilldisappearathighercostlevelsthanthosewhicharelessinformative(think,forinstance,ofafully

separatingequilibriumcomparedtoapartiallyseparatingequilibrium).Moreprecisely,asignalingstrategyfortheSender

canbeseenasaninformation-revealingexperiment,andaccordingtotheBlackwell’stheorem(Blackwelletal.,1951,1953),

themoreinformativeanexperimentthehigheritsvalueforanyexpected-utilitymaximizingdecision-maker.

5.2. Reﬁnements

Giventhemultiplicityofequilibriain asignalinggames(aSCASgame isnotanexceptiontothis),somenotionof

equilibriumreﬁnementisoftenappliedtogetridofimplausibleequilibria.So,evenifthesetofequilibriaofaSCASgame

differsfromthatofaSFASgame,anequivalencemightberestoredthroughreﬁnements.Inthefollowingwearguethatthis

isnotthecase.

Animportantclassofreﬁnementsappliedtosignalinggamesfollowtheideathatout-of-equilibriumbeliefsshouldnot

betotallyfree,butneedtosatisfysomecriterionofreasonableness.9_With_the_aim_of_extending_such_class_of_reﬁnements

toSCASgames,wedeﬁnethenotionofrobustnesstorestrictionsonbeliefsatout-of-equilibriumsignals,andinpoint

(a)ofProposition2weshowthatinaSCASgametheequilibriumwhereallsender’stypespoolontheminimumsignal

satisﬁessuchrobustnesscriterion.Indeed,reﬁnementsactingthroughrestrictionsonout-of-equilibriumbeliefsrelyonthe

possibilitythatadeviationbythesendertriggersapathofplayalongwhichthereceivergetssomepieceofinformationthatis

unexpectedalongtheequilibriumpath;butinapoolingequilibriumofaSCASgamethereceiverdoesnotacquirethesignal,

sothatthispossibilitydoesnotexist.Thisiswhyreﬁnementsactingthroughrestrictionsonbeliefsatout-of-equilibrium

signalsdonothaveabiteinsuchcase.Wenowformalize,forSCASandSFASgames,theconceptofanequilibriumproﬁle

thatisrobusttorestrictionsonbeliefsatout-of-equilibriumsignals.Givenasender’sstrategy,wedenotewithXe₍₎_the

setofsignalsthatarechosenbyatleastonesender’stype,i.e.,Xe₍₎₌_{x_∈_X_:

_∃

_t_∈_T,_(t)₌_x_}._Similarly,_we_denote_with

Xo₍₎_the_set_of_signals_that_are_not_chosen_by_any_sender’s_type,_i.e.,_Xo₍₎₌_X_\_Xe_._Moreover,_given_a_sender’s_strategy_,

arestrictiononbeliefsatout-of-equilibriumsignalsisacollectionofsets{B(x,s2)}x∈Xo₍₎,withB(x,s)⊆T,B(x,s)/=∅.10_An

equilibriumissaidtoberobusttoarestrictiononbeliefsatout-of-equilibriumsignalsifitiscompatiblewiththerestricted

setofadmissiblebeliefs.11

Wearereadyforthefollowingpropositionontherobustnessofequilibriatorestrictionsonbeliefsatout-of-equilibrium

signals.

9_A_{non-exhaustive}_list_of_reﬁnements_acting_through_restrictions_on_{out-of-equilibrium}_beliefs_include:_the_Intuitive_Criterion_(Cho_and_Kreps,_1987),

Divinity(BanksandSobel,1987),D1(ChoandKreps,1987),D2(ChoandKreps,1987),theUndefeatedEquilibrium(Mailathetal.,1993).

10_We_observe_that_even_a_signal_x_∈_Xe₍₎_triggers_an_{out-of-equilibrium}_belief_if_the_receiver_deviates_from_s1_to_s2._However,_such_beliefs_are_already

restrictedbythedeﬁnitionofPerfectBayesNashequilibriumgiveninAppendixA(seetheE3.2requirement).

11_More_formally,_it_must_satisfy_the_deﬁnition_of_a_perfect_Bayes-Nash_equilibrium_with_a_{strengthening}_of_the_E3.3_requirement_(see_Appendix_A),_that

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Proposition2. ConsideraSCASgameanditsassociatedSFASgame.

(a)IntheSCASgame,givensuchthat(t)=0forallt∈T,foranyrestrictiononbeliefsatout-of-equilibriumsignalsthereexists

(s,˛)suchthat(,(s,˛))isapoolingequilibriumrobusttotherestriction.

(b)If(,(s,˛))isaseparatingequilibriumthatisrobusttoagivenrestrictiononbeliefsatout-of-equilibriumsignalsinthe

SCASgame,thenitisalsoaseparatingequilibriumthatisrobusttothesamerestrictionintheSFASgame;if(,(s2,˛))isa

separatingequilibriumoftheSFASgamethatisrobusttoagivenrestrictiononbeliefsatout-of-equilibriumsignals,thenitis

aseparatingequilibriumoftheSCASgamethatisrobusttothesamerestrictionifandonlyifc≤().

Weremarkthat,inpoint(b)ofProposition2,reﬁnementsactingthroughrestrictionsonbeliefsatout-of-equilibrium

signals,evenifineffectiveagainstpoolingequilibriaontheminimumsignal,allowtogetridofmanyseparatingequilibriaof

theSCASgame,inamannersimilartowhattheydoinasignalinggamewithoutcostlyacquisitionofsignals.Indeed,inany

separatingequilibriumRmustbeplayings2,sothatadeviationbySallowstoreachanout-of-equilibriuminformationset

ofR,wherethefreedominthechoiceofbeliefsisnowrestrictedbythereﬁnementunderconsideration,possiblyinducing

RtoplayinawaythatisfavorabletoS.

Thereisanotherclassofreﬁnementswhichconsidersout-of-equilibriumplaynotasintentionaldeviations,butasthe

resultofmistakesthatplayersmakewhenchoosingequilibriumstrategies.12_These_reﬁnements_require_players’_strategies

tobeoptimalagainstcompletelymixedstrategyproﬁlesthatapproachtheequilibriumproﬁle.Proposition3establishes,

inpoint(a),therobustnessofthepoolingequilibriumontheminimumsignalwithrespecttoanykindoftrembles.The

intuitionisasfollows.Startingfromthepoolingequilibriumontheminimumsignal,ifRisforcedtochooses2withsome

smallprobabilityduetotrembles,thenSmayhavesomeincentivetooptforasignalthatisnottheminimum;however,

sincesignalingisacostlyactivity,ifthetrembleissufﬁcientlysmallthenthebeneﬁtfromsignalingfallsshortofthecost.

AnanalogousreasoningappliesifweconsiderR’soptimalbehaviorwhenS’schoiceistrembling,duetothefactthats2is

morecostlythans1.

Toformalizetheideaofrobustnesstoanykindoftrembles,wemakeuseofthenotionoftrulyperfectequilibrium

(Kohlberg,1981).13_To_present_it_formally,_we_have_ﬁrst_to_modify_our_setting_by_considering_a_ﬁnite_set_of_{signals ˆ}_X,_with

0∈ ˆX,andaﬁnitesetofactions ˆY,sothatthesetofstrategiesisalsoﬁniteforbothSandR,whichwedenotewith ˆM and

{s1,s2}× ˆA.14_Moreover,_we_have_to_introduce_mixed_strategies_to_model_players’_trembles_in_the_choice_of_strategies._We

denotebySamixedstrategyforS,withS()indicatingtheprobabilityattachedtopurestrategy.Similarly,wedenote

byRamixedstrategyforR,withR(s,˛)indicatingtheprobabilityattachedtopurestrategy(s,˛).WenotethatinaSFAS

gameR(s1,˛)=0forall˛∈ ˆA,sinces1cannotbechosenbyReitherbymistake.Givenaproﬁle(S,R),werefertoa

trembleasasequenceofcompletelymixedstrategyproﬁles(_Sk,k

R)_k=1,...,∞suchthat(Sk,Rk)→(S,R)ask→∞.Given

adegeneratemixedstrategyproﬁle(S,R)whereS()=1andR(s,˛)=1,andatremble(Sk,Rk)_k=1,...,∞,wesaythat

(,(s,˛))isrobusttothetrembleifthereexists ˆk suchthat,forallk≥ ˆk,isbestreplyagainstk

R and(s,˛)isbestreply

againstk

S;furthermore,wesaythat(,(s,˛))isatrulyperfectequilibriumif(,(s,˛))isrobusttoeverytremble.Finally,to

simplifythestatementofProposition3,weintroducethefollowingdeﬁnition:givenadegeneratemixedstrategyproﬁle,a

trembleintheSCASgame(_Sk,k

R)_k=1,...,∞,andatrembleintheSFASgame(ˆSk,ˆRk)_k=1,...,∞,wesaythatthesetwotrembles

areequivalentif(i)forevery∈ ˆM,theexpectedutilityforSofplayingisthesameagainstk

RandˆRk,forallk,and(ii)

forevery(s,˛)∈{s1,s2}× ˆA,theexpectedutilityforRofplaying(s,˛)isthesameagainstk

SandˆSk,forallk.

Wearereadyforthefollowingpropositionontherobustnessofequilibriatoplayers’trembles.

Proposition3. ConsideraSCASgamewithﬁnitesetsofsignalsandactions,anditsassociatedSFASgame.

(a)IntheSCASgame,anypoolingequilibriumontheminimumsignalisatrulyperfectequilibrium.

(b)If(,(s,˛))isaseparatingequilibriumthatisrobusttoagiventrembleintheSCASgame,thenitisalsoaseparatingequilibrium

thatisrobusttoanequivalenttrembleintheSFASgame;if(,(s2,˛))isaseparatingequilibriumthatisrobusttoagiven

trembleintheSFASgame,thenitisaseparatingequilibriumthatisrobusttoanequivalenttrembleintheSCASgameifc<().

Weremarkthat,inpoint(b)ofProposition3,similarlytowhathappensforrestrictionsonbeliefsatout-of-equilibrium

signals,thesetofseparatingequilibriathatarerobusttoagiventrembleisthesameinaSCASgameandintheassociated

SFASgame,providedthatacquisitioncostsaresmallenough.15_On_the_whole,_we_can_reinforce_an_observation_already_made

afterProposition1:itisnotonlythesetofseparatingequilibriaofaSCASgame,butthesetoftheseparatingequilibria

12 _Another_possibility_is_to_consider_{perturbations}_in_payoffs,_rather_than_in_the_choice_of_strategies._A_reﬁnement_based_on_this_idea_is_the_essential

equilibrium(Wen-TsunandJia-He,1962).

13 _Indeed,_a_truly_perfect_equilibrium_is_perfect_(Selten,_1975),_proper_(Myerson,_1978),_strictly_perfect_(Okada,_1981),_strictly_proper_(Van_Damme,_1991).

Moreover,atrulyperfectequilibriumcanbeseenasaone-pointsetthatisKohlberg-Mertensstable(KohlbergandMertens,1986).

14 _The_results_in_Proposition₁_hold_in_this_setting_with_ﬁnite_strategy_sets_as_well.

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thatarerobusttoreﬁnements(belief-basedortremble-based),whichconvergestotheanalogoussetoftheassociatedSFAS

gameasctendsto0.

Onefurtherremarkthatmaybeworthdoingconcernstheinabilityoftremble-basedreﬁnementstorestrictR’sbehavior

atinformationsetswhereanon-minimumsignalhasbeenchosen,andRhasalreadychosens2.Weobservethatsuch

informationsetscannotbereachedbymeansofS’smistakesonly;indeed,amistakebyRwhenchoosingbetweens1and

s2isalsorequired.Inordertorecoversomereﬁningpowerfortremblesattheabove-mentionedinformationsets,wemay

resorttotheagent-normalformgameandapplytremblesattheagentlevel,withtheresultthatoptimalbehaviorwould

berequiredatsuchinformationsetsaswell.InthiscaseastatementsimilartotheoneinProposition2applies:forany

sequenceoftremblesthereexistsapoolingequilibriumontheminimumsignalthatiscomprisedofstrategiesthatarebest

repliesagainsteachsufﬁciently-large-indexedelementofthesequence.

Onelastremarkisabouttheroleoftheassumptionthatsignalacquisitionisadeliberateandcostlychoicefortheresults

inPropositions2and3.Havingachoiceforthereceiverbetweens1ands2iswhatreallymatterstoobtainthatthepooling

outcomeontheminimumsignalisrobusttorestrictionsonbeliefsatout-of-equilibriumsignals;indeed,ifRchoosess1

thenheswitchesoffherlisteningability,andhenceanysender’sdeviationisnotobservable.Therobustnesstoreﬁnements

basedonplayers’trembles,instead,dependsnotonlyonRchoosingtolisten(s2)ornottolisten(s1),butalsoonthefact

thats2ismorecostlythans1,whichmakess1strictlybetterthans2whenSischoosingapoolingstrategyandtremblesare

smallenough.

6. Discussion

Inthissectionwediscusstherobustnessofourresultsifweassumeasmoothprocessofinformationacquisitionbythe

receiver(Section6.1),ifthesenderhasthepossibilitytoinvitethereceivertoacquirethesignal(Section6.2),ifthesignalis

notpurelycostlytothesender(Section6.3),andifthereceivercancommittoacquirethesignal(Section6.4).Thediscussion

iskeptataninformallevel.

6.1. Smoothacquisitioncosts

InthispaperwehaveconsideredamodelthatisreminiscentofReis(2006):acquisitionofinformationisalways

all-or-nothing,andsignalacquisitionisbinary,i.e.,Reitherpaystheacquisitioncostandacquiresxwithcertainty,orpaysnothing

andacquiresnothing.Onecaninsteadthinkoftheprocessofsignalacquisitionasasmoothoruncertainone:thegreater

thecostincurredtoacquirethesignal,thegreatertheacquisitionofthesignalcontentorthelikelihoodthatacquisitionis

successful.Inthisrespect,anaturalquestiontoaskiswhetherseparationofsender’stypesbecomesmorelikelyundersuch

asmoothoruncertainprocess.Ingeneral,theliteraturehasshownthattheinformationacquisitiontechnologydoesmatter

fortheselectionofequilibria(seeYang,2015andDenti,2016forrecentcontributionsontheissue).Whatwearguehereis

thatthepoolingoutcomeretainsitsprominencealsowhensignalacquisitionisnotall-or-nothingorisnotbinary.

Asimplewaytomodelasmoothprocessofsignalacquisitionthatisnotbinaryistoconsiderastochasticacquisition

wheretheprobabilityofacquiringxisanincreasingfunctionofthecostpaid.SupposeRhasthepossibilitytochoosealevel

ofacquisitionefforte∈[0,1],whichreplacesthechoiceofs∈{s1,s2};also,withprobability1−enosignalisacquired,

whilewithprobabilityethesignalisacquired.

Asimplewaytomodelasmoothprocessofsignalacquisitionthatisnotall-or-nothingistohavethesignalxalways

acquiredbutwithsomeblurringnoisewhoseimpactnegativelydependsontheacquisitioneffort,sothatthesignalobserved

isequaltothesignalsentplusanoisyterm.Wedenotewith∈[0,+∞)theprecision(i.e.,theinverseofthevariance)of

thenoisyterm,whichischosenbythereceiver.

Weobservethatthepoolingequilibriumontheminimumsignalisstillsupportedbyaformofstrategiccomplementarity

inbothmodelssketchedabove:ifSchoosesx=0forallt∈T,thenRhasnoreasontospendinsignalacquisition,therefore

R’soptimalchoiceise=0(thesignalisneveracquired)or=0(thesignaliscompletelyuninformative);ifRchoosese=0,

or=0,thenShasnoreasontospendincostlysignaling,thereforeS’soptimalchoiceisx=0forallt∈T.Moreover,note

thatwhene=0noout-of-equilibriuminformationsetofRcanbereachedbymeansofasingledeviationbyS,andwhen

=0noout-of-equilibriuminformationsetexists,inthatzeroprecisionofthesignaldoesnotconstrainthesetofpossible

signals.InbothcasesintentionaldeviationsbySareineffectiveattriggeringadifferentpathofplaybyR;thisintuitively

leadstotherobustnessofthepoolingoutcomeontheminimumsignaltoreﬁnementsactingthroughrestrictionson

out-of-equilibriumbeliefs.Finally,notethatnobeneﬁtcanbeobtainedbyunilateraldeviationstoe>0,or_>_0,_and_to_x_>_0,_while

suchdeviationswouldentailastrictlylargercost;thisallowsustoconcludethatthepoolingoutcomeontheminimum

signalisalsorobusttoreﬁnementsbasedonplayers’trembles,inaversionofthesegameswithﬁnitestrategysets.

6.2. Invitingtoacquirethesignalthroughfurthersignaling

ItseemsnaturaltoaskwhethertheprominenceofseparationisrestoredifShasthepossibilitytocommunicatetoRthat

heisactuallysendinganinformativesignal–i.e.,asignalthatseparates(atleastpartly)types–andthatthereforethesignal

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Onecanthinkofmanysituationswhereindeedthesendercansend,togetherwiththemainsignalx,anaccompanying

costlysignal,sayz,thatactsasaninvitationforthereceivertoengageinthecostlyacquisitionofx.Weargueherethat,in

fact,notmuchcanberestoredbytheuseofz.

Wenotethatinthissettinginformationontypescanbetransmitted,i.e.,separationcanoccur,eitheronxoronz,but

notonboth.Thisissobecause,ifseparationisattainedonx(oronz)andthereceiveracquiresx(orz),thenalltypeswould

strictlyprefertosaveoncostsandpoolonanullz(orx),andsimilarlythereceiverwouldstrictlyprefernottoincurthecost

ofacquiringz(orx),sinceitsacquisitionwouldaddnousefulinformation.So,supposethatseparationiseffectivelyattained

onzonly.Inorderforthiskindofseparationtobemorerobustthanapoolingequilibrium,itisnecessarythatthesender’s

utilityfunctionsatisﬁesanequivalentofthesingle-crossingpropertyontypesandz,whichisnotguaranteedingeneral.

Evenifsuchanecessaryconditionholds,torestoretheprominenceofseparationthereceivermustacquirezautomatically.

Infact,iftheacquisitionofzrequiresadeliberatechoiceandiscostlytoR,thenthekindofstrategiccomplementarity

betweensignalingandacquiringthesignalthatwehaveillustratedfortheSCASgameisatworkhereaswell;moreover,

out-of-equilibriuminformationsetsarenotreachablethroughdeviationsbySonlywhenstartingfromthepoolingoutcome

ontheminimumsignal,andalsounilateraldeviationsfromitwouldbestrictlycostlyforbothSandR.Therefore,results

similartothosefortheSCASgamecanbeintuitivelyobtainedinthissetupaswell.

6.3. Signalnotpurelycostlytothesender

TheSCASgamestudiedinthispaperaccommodatescaseswherethesignalxispurelydissipative–itisalwaysanetcost

forSandofnointrinsicutility(orsomedisutility)forR–aswellascaseswherethesignalxisofsomeintrinsicvaluetothe

receiver.However,themodeldoesnotaccommodatethecasewherexisnotapurenetcostforthesender.

Inourmodel,allsender’stypesstrictlyprefer,otherthingsbeingequal,tosetx=0:

x∗_(t,_y)_≡_arg_max

x∈X

U(t,x,y)=0, forall t ∈T,y∈Y. (2)

ItturnsoutthatthekindofstrategiccomplementaritythatsupportsthepoolingoutcomeinaSCASgamemaynotexistif(2)

doesnothold.However,westressthatwhatiscrucialtoourresultsisnotthat0isthecommonbestsignalforallsender’s

typesintheabsenceofasignalingvalue–anassumptionwhich,infact,caneasilybesubstitutedwithacommonoptimal

x*_>₀_for_all_t_∈_T;_what_really_matters_for_the_existence_of_the_needed_strategic_{complementarity}_is_that_a_common_best_signal

existsforalltypes.Toseewhy,considertheextremecasewherex*_(t,_y)_is_one-to-one_in_t_for_any_given_y._This_implies_that,_in

thecasethatRchoosess=s1,eachsender’stypeﬁndsitoptimaltochooseadistinctx.Iftheinformationaboutthesender’s

typeissufﬁcientlyvaluabletoR,itbecomesimpossibleforaproﬁlewithnosignalacquisitiontobeanequilibriumbecause

typesseparateindependentlyofR’sbehavior,andthereforeRalwaysﬁnditoptimaltoacquirethesignal.Therefore,inthe

absenceofacommonsetsignalforalltypes,thepoolingoutcomeontheminimumsignalwouldevenfailtobeaPerfect

Bayes-Nashequilibrium.

Letusconcludewithafewremarksthat,inouropinion,indicatethatacquisitioncosts–andingeneraltheanalysis

conductedinthispaper–mightberelevantevenwhen(2)doesnothold.

Oneremarkregardsthereﬁnementpotentialofarbitrarilysmallacquisitioncostsinastandardsignalinggame.Note

thatifx*_(t,_y)_is_one-to-one_in_t_for_any_given_y,_then_the_incentive_for_S’s_types_to_separate_does_not_come_from_the_fact

thatRacquiresthesignal,butfromthefactthateachtypehasitsownpreferredx.Thisrulesoutallpoolingequilibriain

aSCASgame,butitdoesnotsointheassociatedSFASgame(i.e.,inastandardsignalinggame).Infact,inaSFASgame

Ralwayschoosess=s2,andinparticularitdoessoalsowhenallS’stypespoolonthesame ¯x;thisallowsforbeliefsat

out-of-equilibriumsignalsonthepartofRthatharshlypunishtypeswhodeviatefrom ¯x,sustainingthepoolingequilibrium.

InaSCASgame,instead,Rwouldswitchfroms=s2tos=s1,leavingeachtypet∈Tfreetoswitchtohispreferredx*_(t,

y).Perhapsinterestingly,thisargumentshowsthatanarbitrarilysmallacquisitioncostrulesoutallpoolingequilibriain

signalinggameswheretypesstrictlypreferdifferentsignallevels.

Anotherremarkregardsthepotentialbackﬁringofmandatorydisclosurepolicies.ConsideraSCASgamewherex

repre-sentscostlydisclosureofsomecharacteristiconthepartofthesender,andsupposethatapublicauthoritywantstokeepx

aboveacertainthreshold.Ifx*_(t,_y)_is_one-to-one_in_t_for_the_relevant_range_of_y,_then_some_disclosure_will_certainly_happen

asnopoolingcanbesustainedinequilibrium.However,ifthepublicauthorityimposesaminimum ¯x,thenitcanhappen

thatseparationcollapsesandapoolingequilibriumon ¯xwithnosignalacquisitionemerges.Inparticular,thiscanhappen

whenever ¯x≥maxt∈Tx∗(t, ¯y),where ¯y isthebestactionforRunders1whenalltypespoolon ¯x (forsmallervaluesof ¯x a

partialpoolingcanemerge,instead).Thismayleadtoalossintermsofinformationtransmissionthatmorethanoffsetsthe

targetedbeneﬁtsofahighx.

6.4. Publiclycommittingtoacquirethesignal

IntheSCASgamestudiedinSections4and5,aswellasinthevariantsdiscussedinSections6.1,6.2,and6.3,there

isnopossibilityforthesender,priortochoosingthesignal,toobservethechoiceofthereceiverbetweenacquiringand

notacquiringthesignal.ThisneglectedcaseisstrategicallyequivalenttoasituationwherethereceiverRmustpublicly

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complicatedtoseethatifRcanpubliclycommittoagivens∈{s1,s2},andinformationissufﬁcientlyvaluabletoher,then

theprominenceofseparationisrestored.ConsideravariantoftheSCASgamewhereRmust16_initially_commit_herself_to

choose ¯s∈{s1,s2}andsupposealsothatsuchcommitmentisobservedbySbeforehechooseswhatsignaltosend.This

conﬁguresanadditionalinitialstageofthegamewhereRannounces ¯s,followedbyasecondstageofthegamewhereS

choosesthesignalx,andthenathirdstagewhereRplays ¯s andchoosesanactiony.Inthissetup,Scanconditionthechoice

ofthesignalon ¯s,sothathisstrategyisnowrepresentedbyfunction¯ :T×{s1,s2}→X.Notethatthissetupconﬁgures

twodistinctsubgames:onesignaling(sub)gameinwhichRhascommittedtos1andSknowsthathissignalwillneverbe

acquiredbyR(basically,signallingisimpossible),andanothersignaling(sub)gamewhereRhascommittedtos2andSknows

thathissignalwillalwaysbeacquired(basically,theassociatedSFASgame).Suchdynamicstructureofthegamenaturally

callsforanequilibriumconceptthatentailsbackwardreasoning.Asimplewaytodosoistolookforsubgameperfection

afterhavingreﬁnedtheequilibriaofeachsignalingsubgameusingstandardreﬁnementsactingthroughout-of-equilibrium

beliefs.

InthesignalingsubgamewhereRhascommittedtos1,thereisjustonePerfectBayes-Nashequilibrium:alltypesofS

poolonx=0,sincex>0iscostlyandSiscertainthatRwillneverobservex.DenoteS’sstrategyinthissubgamewithP_,

withP_(t)₌₀_for_all_t._In_the_signaling_subgame_where_R_has_committed_to_s2,_there_are_many_Perfect_Bayes-Nash_equilibria,

bothpoolingandseparating.Sincethecostcofchoosings2issunk,inthissubgamethereareexactlythesameequilibriaof

theassociatedSFASgame,i.e.,theyarethesameofastandardsignalinggame.

Weconsiderherethetypicalcasewherethebestfullyseparatingequilibriumstandsupasmostprominent,i.e.,wefocus

ontheequilibriumwhereallsender’stypesseparateandeachtypespendsonsignalxtheminimumrequiredforseparation

(Riley,2001).DenoteS’sstrategyinthisequilibriumwithS_._So,_S’s_strategy_in_the_full_game_can_be_written_as_¯ ₌₍P_,S_).

Usingbackwardreasoning,Ranticipatesthatbycommittingtos1shewillendupinapoolingequilibriumwhereshe

playstheoptimalyagainstthebeliefsassociatedwithnosignalacquisition,whilebycommittingtos2shewillendupin

afullyseparatingequilibriumwheresheplays,foreacht,y(t)∈argmaxy∈Yu(t,S(t),y).Iftheinformationconveyedby

S_is_valuable_to_R_in_the_SFAS_game,_then_it_is_{straightforward}_to_see_that_for_a_positive_acquisition_cost_c_which_is_small

enough,Rstrictlypreferstoendupinthefullyseparatingequilibrium,andthereforeshewillcommittos2.Insuchacase,

theprominenceoftheoutcomeoffullseparationisrestored.

Letusendthisdiscussionwithamoregeneralpointregardingtheactionsthatthereceivercanmaketofacilitate

commu-nication.Totheextentthatinformationtransmissionisvaluabletothereceiver,itisreasonabletoexpectthatthereceiver

acts(andevenincurscosts)inordertofacilitatetheemergenceofafullyseparatingequilibrium.Ifpubliccommitmenttothe

acquisitionofthesignalispossibleoriftheacquisitioncanbemadebeforethesignalissent,thenseparationisactuallythe

mostprominentoutcome.Buthowlikelyitisthatthisisthecase?Forthesignaltobeeffectivelyacquiredbeforethesender

sendsit,wemustbeinasituationwherethesenderandthereceivercommunicatethroughachannelthatthereceivercan

“switchon”atherwillandwhoseon/offstatusiseasilyobservable(i.e.,atnocost)bythesender.Inaddition,asthecostof

keepingthechannelswitchedonreasonablydependsonhowlongitisleftinsuchastate,anothercrucialrequirementisthat

thereceivermustknowapproximatelywhenthesenderisgoingtosendthesignal,otherwiseanothercoordinationproblem

arises.Whilethissetupcertainlyﬁtssomerealcasesofsignaling,ithardlyﬁtsmostofthem.Whensignalacquisitiontakes

theformofbuyingthenecessary“hearing”tools,orwhenittakestheformofmovingtothecorrect“listening”location,

thenpubliccommitmentsoundsreasonable.However,ifsignalacquisitionisamatterofcognitiveeffortorattention,then

thepossibilityofpubliccommitmentseemsfarlesslikely,especiallyforwhatconcernslettingthesenderknowaboutthe

commitment.

7. Conclusions

InProposition1wehaveshownthatinaSCASgamethepresenceofcoststoacquirethesignalrestrictsthefeasibilityof

poolingequilibriatothosewhereallsender’stypespoolontheminimumsignal,whileintheassociatedSFASgamethere

canbepoolingequilibriaonnon-minimumsignals;furthermore,anyseparatingequilibriumoftheSFASgameexistsinthe

SCASgameaswellifandonlyiftheacquisitioncostisnotlargerthanthevalueofacquiringinformationonthesignalforthe

receiver.Furthermore,thereisnoreﬁnementactingthroughrestrictionsonbeliefsatout-of-equilibriumsignals,orrelying

onplayers’trembleswhenchoosingstrategies,thatcangetridofthepoolingequilibriumontheminimumsignal,asitis

showninPropositions2and3,respectively.TheseﬁndingssuggestthatinaSCASgameapoolingoutcomeontheminimum

signalismuchmoreprominentthanitisintheassociatedSFASgame,andsuchprominencegrowsintheacquisitioncosts

(attheextreme,noseparatingequilibriumexistsforsufﬁcientlyhighacquisitioncosts).

Isthisgoodorbadnews?Fromawelfareperspective,apoolingequilibriumontheminimumsignalcanleadtoeithera

betterorworseoutcomewithrespecttoanalternativeseparatingequilibrium.Ofcourse,thisistrueforstandardsignaling

gamesaswell,sincethetransmissionofinformationwhichoccursinaseparatingequilibrium–whilesurelybeneﬁcialfor

thereceiver–maywellbedetrimentalforsometypesofthesender.Ontopofthis,westressthatinthepresenceofcosts

toacquirethesignal,apoolingequilibriumwherethereceiverexertsnoeffortinsignalacquisitionallowstosavethecosts

16_We_stress_that_if_R_is_not_forced_to_commit,_but_has_just_the_option_to_do_so,_the_main_thrust_of_the_argument_still_applies._The_reason_is_that,_as_long_as

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ofinformationacquisition.Withoutfurtherassumptionsontherelationshipbetweensender’stypesandreceiver’sactions,

itremainsundecidedwhethersuppressinginformationbutsavingininformationacquisitionisdesirableornot.

Acknowledgements

Thecurrentversionofthepapergreatlyimprovesontheoriginalworkingpaperversionthankstothevaluableindications

oftheCo-Editor(FriederikeMengel),aswellasthepreciouscommentsofanAssociateEditorandthreereferees,whohave

urgedustoincludetherobustnessanalysisagainstplayers’trembles,andSection6.1.WealsowanttothankGeorgeMailath

forinsightfulcommentswhichhavemotivatedtheadditionofSection6.3,andanadvisoryeditorforhavinginspiredSection

6.4.Also,wethankHeskiBar-Isaac,StefanoComino,AndreaGallice,andAntonioNicolò,aswellasallthepeoplewho

haveprovidedusefulcommentsduringthe2014G.R.A.S.S.workshophostedbyCollegioCarloAlbertoandthe2014EARIE

conferencehostedbyBocconiUniversity.WedeclarethatwehavereceivedsupportfromtheItalianMinistryofEducation,

UniversitiesandResearchunderPRINproject2012Z53REX“TheEconomicsofIntuitionandReasoning:a StudyOnthe

ChangeofRationalAttitudesunderTwoElaborationSystems(SOCRATES)”.

AppendixA. Supplementarydata

Supplementary data associated with this article can be found, in the online version, at

https://doi.org/10.1016/j.jebo.2017.10.022.

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