Modelling the objective function of managers in the presence of overlapping shareholding

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International Journal of Industrial Organization

journalhomepage:www.elsevier.com/locate/ijio

Modelling the objective function of managers in the presence of overlapping shareholding ^R

Duarte Brito

^a

, Einer Elhauge

^b

, Ricardo Ribeiro

^c,∗

, Helder Vasconcelos

^d

aUniversidade Nova de Lisboa, Faculdade de Ciências e Tecnologia and NOVASBE, Portugal

bHarvard Law School, United States

cUniversidade Católica Portuguesa, Católica Porto Business School and CEGE, Portugal

dUniversidade do Porto, Faculdade de Economia and CEF.UP, Portugal; Compass Lexecon, Spain

a rt i c l e i nf o

Article history:

Received 3 November 2021 Revised 4 December 2022 Accepted 10 December 2022 Available online 18 December 2022 JEL classification:

L13 L41 Keywords:

Manager objective function Overlapping shareholding Ownership dispersion Proportional control Banzhaf control

a b s t ra c t

Theobjectivefunctionofmanagersinthepresenceofoverlappingshareholdingmaydiffer fromthetraditionalown-firmprofitmaximization,astheymayinternalizetheexternal- ities theirstrategies imposeon otherfirms. The dominantformulation of theobjective function insuchcases has,however, beencriticised for yieldingcounter-intuitive profit weightswhentheownershipofnon-overlappingshareholdersishighlydispersed.Inthis paper,weexaminethisissue.First,wemakeuseofaprobabilisticvotingmodel(inwhich shareholdersvotetoelectthemanager)tomicrofoundanalternativeformulationofthe objectivefunctionofmanagers,whichsolvestheabove-mentionedcriticism.Second,we apply the two formulationstothe set ofS&P 500firms. We show thatownershipdis- persionofnon-overlappingshareholdersis,infact,arelevantempiricalissue,whichmay induceanover-quantificationoftheprofitweightscomputedfromthedominantformula- tion,particularlyunderaproportionalcontrolassumption.

© 2022TheAuthors.PublishedbyElsevierB.V.

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

1. Introduction

Theassumptionofown-firmprofitmaximizationiskey,(atleast)sinceFisher(1930)’sseparationtheorem,tomostliter- atureincorporatefinanceandindustrialorganization.However,thevalidityofthisassumptionhasbeenrecentlyquestioned duetotheincrease,documentedforamultitudeofindustriesandeconomies,particularlysince2000,ofoverlappingshare- holding(Azaretal.,2018;Newhametal.,2019;Backusetal.,2021b;Azaretal.,2022).Thereasonbeingthatiffirmsimpose externalitiesononeanother,overlappingshareholdingmayimplyafailureofthecompetitivenesscondition,establishedby Hart(1979)tobeessentialforshareholders,regardlessoftheirpreferences,tounanimouslyagreeonown-firmprofitmax- imization. Inordertoseewhy,note,forexample,thatiffirmAimposesanegativeexternalityonfirmB,ashareholderof

R We would like to thank the Editor, José L. Moraga, and three anonymous referees for their thorough and thoughtful reports. We would also like to thank Ricardo Gonçalves, Fiona Kasperk, Torsten Persson, Michele Polo, Giacomo Ponzetto, Vasco Rodrigues, Martin Schmalz, Guido Tabellini, Kyle Wilson and Alminas Žaldokas, as well as seminar and conference participants at EARIE, IIOC, PEJ and Universidade Católica Portuguesa for helpful comments and suggestions. We want to especially thank Amir Amel-Zadeh, Fiona Kasperk, and Martin Schmalz for sharing the data with us. Ricardo Ribeiro and Helder Vasconcelos gratefully acknowledge financial support from Fundação para a Ciência e a Tecnologia (UID/GES/0 0731/2019, UIDB/0 0731/2020 and UID/ECO/04105/2019). Einer Elhauge gratefully acknowledges financial support from Harvard Law School. All remaining errors are of course our own.

∗Corresponding author.

E-mail addresses: dmb@fct.unl.pt (D. Brito), elhauge@law.harvard.edu (E. Elhauge), rribeiro@ucp.pt (R. Ribeiro), hvasconcelos@fep.up.pt (H. Vasconcelos) .

https://doi.org/10.1016/j.ijindorg.2022.102905

0167-7187/© 2022 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

firm AwhoalsoholdssharesinfirmBtypicallywantsthemanageroffirmAtopursuealessaggressivestrategythanthe strategydesiredbyashareholderwithnoholdingsinfirmB.

Themanagersoffirmswithoverlappingshareholders,ratherthanmaximizingownprofit,maythereforeweightheeven- tual conflicting objectives oftheir shareholders.This impliesthat theymay internalize(to some degree)the externalities their strategiesimpose onotherfirms(Rotemberg,1984;Hansen andLott, 1996).Thisinternalization canalterthe incen- tivestocompeteandcannaturallyimpactmarketcompetition.^1,2

Inordertoempiricallyexaminetheimpactofoverlappingownershiponmarketoutcomes,wemustquantifytheabove- mentionedinducedinternalization.Todoso,theformulationoftheobjectivefunctionofmanagersiskey.³Thisformulation is,however,non-trivial.Inordertoseewhy,consider, forexample,thatfirmAhasfourshareholders,each holding25%of thefirm,andthatoneofthoseshareholdersalsoholds20%offirmB.IffirmAimposesanexternalityonfirmB,whatwould themathematical formulationoftheobjectivefunction ofthemanager offirmAbe?Whatweight wouldthemanager of firmAassigntotheprofitoffirmB?

The dominantformulationofthe objectivefunction ofmanagersinthepresenceofoverlappingshareholders isdueto O’BrienandSalop(2000).IncorporatingfeaturesfrombothRotemberg(1984)andBresnahanandSalop(1986),theyassume that themanagerofa firmwithoverlappingshareholderswoulddecidethestrategyofthefirmtomaximizeacontrol-weighted sumoftheexpectedreturnsofthefirm’sshareholders.Azar(2012,2016,2017),Britoetal.(2018a)andMoskalev(2019)show that thisformulationcan bemicrofoundedthroughavotingmodelinwhichshareholdersvotetoelectthe managerfrom two potential candidates, an incumbent and a challenger, with conceivably differing strategy proposals to the firm. This formulation, although heavily used in the literature, has alsobeen criticised foryielding counter-intuitive profit weights when the ownershipof non-overlapping shareholders is highlydispersed (see, forexample, Gramlich andGrundl, 2017, pages 9–13;O’BrienandWaehrer,2017,pages 760–761;Crawfordetal.,2018,supplementary material,pages12–13).⁴ The argumentcanbegeneralizedasfollows.Nomatterhowsmalltheownershipofoverlappingshareholders(andthecorporate control inducedfromtheir votingrights)isinafirm, asthedispersionoftheownershipofnon-overlappingshareholders increases,theweightassignedbythemanagertotheprofitofother firms(in whichoverlapping shareholdersholdshares in) tends to reflect solelythe interests of the (non-dispersed) overlapping shareholders. In other words, no matter how smalltheownershipofoverlappingshareholdersis,thedominantformulationyieldsthatthemanagerwouldweighsolely theinterestsoftheoverlappingshareholderswhenevertheremainingownershipofthefirmbecomesdiffuse,evenifsuch dispersiondoesnotyieldoverlappingshareholdersthefullcontrolofthefirm.

In thispaper,wetake the firststepto examinethisissue,fromboth atheoretical andanempirical perspective.From a theoreticalperspective, wemake useofaprobabilistic votingmodel(in thelinesofAzar, 2012;Azar,2016;Azar,2017;

Britoetal.,2018a;andMoskalev,2019)to microfoundanalternativeformulationoftheobjectivefunctionofmanagersin thepresenceofoverlappingshareholders.Accordingtothisalternativeformulation,themanagerofafirmwithoverlapping shareholderswoulddecidethestrategyofthefirmtomaximize acontrol-weightedsumoftherelativeexpectedreturnsof thefirm’sshareholders.Andtheweightassignedbythemanagertotheprofitofotherfirms(inwhichoverlappingshare- holders hold sharesin) will never reflectsolely theinterests of the(the non-dispersed)overlapping shareholders,unless thedispersionyieldsoverlappingshareholdersthefullcontrolofthefirm.Assuch,itsolvestheabove-mentionedcriticism regardingthedominantformulation.Froman empiricalperspective,we applythetwo formulationstothesetofS&P500

1Bresnahan and Salop (1986) , Dietzenbacher et al. (20 0 0) , Shelegia and Spiegel (2012) , and Brito et al. (2019) , among others, show that the internalization induced by intra-industry overlapping ownership (i.e., among firms with horizontal relationships which, in partial equilibrium, are likely to impose a negative externality on each other) can directly lead to higher product prices and lower output levels. Azar and Vives (2021a) show that the internalization induced by inter-industry overlapping ownership (i.e., among firms in different industries which, in general equilibrium, are likely to impose a positive externality on each other) increases the incentive for firms to expand production, reducing relative prices in their industry relative to other industries.

2Although non-overlapping shareholders may favor a different firm-specific strategy, that does not mean they are harmed by overlapping shareholding because overlapping shareholding may, for example, reduce the competitiveness of rival firms, and non-overlapping shareholders benefit from a reduction of competition between the firm and its rivals (please see Schmalz, 2018 for a formal model).

3The literature proposes three different approaches for this quantification (see Backus et al. 2020 for a review). The first approach measures (from different perspectives, but atheoretically) the extent to which shareholders hold shares in more than one firm. Consider two firms A and B. Examples of the atheoretical measures used in the literature to examine the impact of overlapping ownership in firms A and B on market outcomes include (a) the number of overlapping shareholders in firms A and B; (b) the sum (across overlapping shareholders) of the minimum holdings (in firms A and B) of each overlapping shareholder; (c) the sum (across overlapping shareholders) of the holdings of each overlapping shareholder in firms A and B, weighted by the market capitalization of each firm; and (d) the sum (across overlapping shareholders) of the product of the holdings of each overlapping shareholder in firms A and B. These measures are not, however, microfounded from any theoretical model. In that sense, they are atheoretical measures (see Appendix B in Gilje et al. 2020 for a review). The second approach places additional structure and maps shareholders overlapping ownership into the weight that managers would assign to the profit of other firms in which their firm’ s shareholders also hold shares. In other words, it maps shareholders overlapping ownership into the objective function of managers (see, for example, O’Brien and Salop, 20 0 0; Azar, 2012; Azar, 2016; Azar, 2017; Brito et al., 2018a;

Crawford et al., 2018; Moskalev, 2019 ; and Gilje et al., 2020 ). Finally, the third approach maps the objective function of managers into equilibrium market outcomes, yielding generalizations of the two most traditional indicators used to screen unilateral anti-competitive effects: the Herfindahl-Hirschman Index and the Gross Upward Price Pressure Index (see, for example, O’Brien and Salop, 20 0 0; Azar et al., 2018; Brito et al., 2018a ; and Azar et al., 2022 ). The structure placed by the second and third approaches is instrumental in deriving economically meaningful claims. And under both those approaches, the formulation of the objective function of managers is key.

4These authors illustrate the counter-intuitive profit weights with several examples, labelling them as “pathological scenarios”, “questionable implica- tions” or “counter-intuitive implications”.

firms(in thelinesofBackus etal., 2021b;andAmel-Zadehetal.,2022)from2003to2019.We showthatthe theoretical issuenotedaboveisindeedofempiricalrelevanceforthesetofS&P500firms.

The remainder ofthepaperisorganizedasfollows.Section2 reviewsthe relatedliteratureandclarifiesthe contribu- tion ofthepaper.Section 3introducesthe generalizedprobabilistic votingmodelusedto derivetheobjectivefunction of managersandtheimpliedprofitweights.Section 4appliesthe profitweights establishedinSection 3totheS&P 500in- dex constituents.Section 5 presentsextensions tothe generalizedprobabilistic votingmodelto account forshareholders inattentionandcross-ownershipstructures.Section6concludesanddiscussespolicyimplicationsoftheresults.

2. Relatedliteratureandcontribution

Azar(2012,2016,2017),Britoetal.(2018a)andMoskalev(2019)microfoundthedominantformulationoftheobjective function ofmanagers inthepresence ofoverlapping shareholders through avotingmodel inwhich shareholdersvoteto elect themanager fromtwo potential candidates,an incumbentanda challenger,withconceivably differingstrategy pro- posals tothefirm. Candidatesare assumedtocareaboutholding office.⁵ Inturn,shareholders areassumedtocareabout the returns that resultfromthe differentstrategy proposals andto havean additive profit-irrelevant bias for(or against) thechallenger.⁶Votingisprobabilisticinthesensethatthebias,whileknowntovoters,isunobserved bycandidates,who treatitasrandom.Thismicrofoundation(a)isconsistentwithempiricalevidenceestablishingthatshareholdersvotingim- pactsthe objectivefunction ofmanagers(Aggarwaletal., 2019) and(b)providesan endogenousmeasure ofshareholders corporatecontrolwithinthefirm.⁷

In thispaper, we examine the role ofthe profit-irrelevanceassumption of the biasof shareholders on thederivation ofthe objectivefunctionofmanagersinthepresence ofoverlappingshareholders.Thisassumption isborrowedfromthe electoral competitionliterature(LindbeckandWeibull,1987;PerssonandTabellini,2000)andwhileinapoliticalelectoral settingit isreasonableto assumethat votershaveanideologicalbias towardsacandidate,ina corporatefinancesetting, however,itmaybelessso.Althoughonecanarguethat theassumptioncanberootedonthe factthatthecredibility (or lackofcredibility)oftheincumbent,beingalreadyinoffice,isknowntoshareholders,whilethat ofthechallenger isnot, onecanalsoarguethatifshareholdersindeedcareaboutthereturnsthatresultfromthedifferentstrategyproposals(and, thereby,ultimatelycareaboutprofits),thereisnotanobviousreasonwhythebiasshouldnotbeprofit-relevant.Toexamine theroleofthisassumption,wegeneralizetheprobabilisticvotingmodelintheliteraturetoallowthebiasofshareholders for(oragainst)thechallengertobeprofit-irrelevant orprofit-relevant,whilemaintainingtheremainingassumptions ofthe literature.⁸Weshowthattheprofit-irrelevanceassumptionplaysakeyroleonhowownershipdispersion(ofbothoverlap- pingandnon-overlappingshareholders)ismappedintoprofitweights.⁹

Backusetal.(2021b)applythedominantformulationtothesetofS&P500firmsfrom1980to2017.Todoso,theyparse theSEC13-Fformfillingsbyinstitutionalshareholders(withover$100millioninassetsundermanagement)andshowthe average weight assigned by the managersof S&P 500 firms to the profitof the remaining S&P 500 firmshas increased from 0.2in1980to almost 0.7in2017. Amel-Zadeh etal.(2022)parsenot only theSEC 13-F formfillings,butalso five additionalSECformfilings,accountingforholdingsnotonlyofinstitutionalshareholders,butalsoofcorporateinsidersand blockholders.TheyapplythedominantformulationtothesetofS&P500firmsfrom2003to2019andshowthat oncewe account fortheholdings ofcorporateinsidersandblockholders,theaverageweightassignedby themanagersofS&P500 firmstotheprofitoftheremainingS&P500firmsis,infact,lower.

Inthispaper,weexaminetherole ofownershipdispersion(ofboth overlappingandnon-overlappingshareholders)on profitweights.Todoso,wemakeuseofAmel-Zadehetal.(2022)’sownershipdatatoapplythetwoformulationstotheset ofS&P500firms.Weshowthatthedispersionofshareholders’ownership,particularlyofnon-overlappingshareholders,isa relevantempiricalissueintheS&P500andthat,inthosecases,thedominantformulationmay,infact,over-quantifyprofit weights, particularly undera proportional control assumption.This issueisparticularly relevantbecause ifthe dominant formulation is over-quantifyingthe magnitude ofoverlapping ownershipwhen the ownershipof non-overlapping share- holders is highly-dispersed, then empirical studies that usethat dominantformulation will suffer fromattenuation bias thatwilltendtocreateempiricalresultsthatunderstatethemagnitudeandstatisticalsignificanceofthemarginaleffectof

5Azar (2012, 2017) considers the case in which candidates choose strategy proposals to maximize their vote share while Azar (2016) , Brito et al. (2018a) and Moskalev (2019) consider the case in which candidates choose strategy proposals to maximize their expected utility from corporate office .

6Azar (2012, 2016, 2017) and Brito et al. (2018a) consider the case in which this bias isindependent (and identically) distributed across shareholders while Moskalev (2019) considers the case in which the bias can be correlated across shareholders.

7Azar (2012, 2017) shows that the corporate control of shareholders can be microfounded to be endogenously measured by their voting rights (propor- tional control) while Azar (2016) , Brito et al. (2018a) and Moskalev (2019) show it can be microfounded to be endogenously measured by the normalized Banzhaf power indices that result from their voting rights.

8We allow candidates to choose strategy proposals to maximize their vote share or their expected utility from corporate office and the bias of shareholders for (or against) the challenger to be correlated or non-correlated across shareholders.

9The assumptions regarding the objective function of candidates and the non-correlation or correlation of the bias of shareholders for (or against) the challenger impact solely the (endogenous) measure of the control rights of shareholders (computed from their voting rights). As such, their influence on how ownership dispersion (of both overlapping and non-overlapping shareholders) is mapped into profit weights, is indirect (via the influence of ownership dispersion on the (endogenous) measure of control rights).

overlappingownershiponmarketcompetition.Thiscouldcauseanalyststoincorrectlyrejectorunderestimateanempirical connectionbetweenoverlappingshareholdingandanticompetitiveeffects(Elhauge,2020).

3. Theoreticalframework

Thissectionintroducesthegeneralizedprobabilisticvotingmodelusedtoderivetheobjectivefunctionofmanagers.The generalsettingcombinesfeaturesfromAzar(2012,2016,2017),Britoetal.(2014),Britoetal.(2018a)andMoskalev(2019).

3.1. Setup

There are K shareholders, indexedby k∈≡

{

¹,...,K

}

^{, andN firms,which impose} externalities on one another, in- dexedby j∈≡

{

¹,...,N

}

^{,whosetotalstockiscomposedofvotingstockandnon-voting}(preferred)stock.Bothstockgive the holder theright toa share ofthe firm’sprofits,but onlythe formergives the holderthe rightto voteinthe firm’s shareholderassembly.The holdings

φ

k j∈[0,1]oftotalstockofshareholderkinfirm j,regardlessofwhetheritbevoting ornon-votingstock,captureherfinancial rightstothefirm’sprofits.Theholdings

υ

k j∈[0,1]ofvotingstockofshareholder kin firm j,capturehervotingrightsinthefirm.Thesevotingrightsmaynotnecessarilycoincidewithhercontrolrightsin thefirm,

γ

k j∈[0,1],whichrefertoherrightstoinfluencethedecisionsoffirm j,tobediscussedbelow.¹⁰

Theownershipstructureofthedifferentfirmsissuchthatasubsetofshareholderscanholdgeneralfinancialandvoting rightsinmultiplefirms.¹¹ Thisoverlappingshareholdingcaninduceaconflictinthefirm-specificinterestsofshareholders, whichmanagersmustweigh.

Finally, theprofitofeach ofthedifferent firmsisassumedto bea function notonly ofthestrategies ofall thefirms butalsoofa stateofnature.Thisimpliesthat firms’profits and,consequently, shareholders’returns-becausethey area functionoftheprofitsofthefirmsinwhichtheyholdfinancialrights-arerandom.

3.2. Votingmodel

We followAzar(2012,2016, 2017),Britoetal.(2018a) andMoskalev (2019)inmicrofounding theobjectivefunction of managersthroughavotingmodelinwhichshareholdersofeachfirm j voteatthefirm’sshareholderassemblytoelectthe managerfromtwopotentialcandidates,anincumbenta_jandachallengerb_j,with the candidatereceivingthe majorityof votingrightsbeingelectedmanagerofthefirm.

Shareholdersandcandidatesareassumedtoplaythefollowingtwo-stagegame.Inthefirststage,candidatestoallfirms, who are assumedto be opportunistic in the sense their only motivation is tohold office, compete forthe voting rights ofshareholdersby -simultaneouslyandnoncooperatively-proposing astrategy fortheir firm,whichisassumedbinding in line withthe literature on electoral competition (Downs,1957; Lindbeck and Weibull, 1987; Polo,1998; Persson and Tabellini, 2000). Letx_a

j∈j andx_b

j∈j denotethe strategy proposals of the incumbent andthe challenger to firm j, respectively, wherej denotes the strategy space available tothe candidates, which can referto anydecisionvariable(s) - e.g.,quantity,price,R&D investment,etc.-offirm j.In thesecondstage ofthegame,theshareholderassembliesofall firmsaresimultaneouslyheldandshareholdersvotetoelectthemanagerofeachfirm.Letm_j≡

a_j,b_j

denotetheidentity ofthemanagerelectedtofirm j.

Naturally,thisvotingmodelconstitutesareducedformmodelofthedecisionmakingprocessandtheknowledgestruc- turewithinthefirm.Themanagermaynotbeelecteddirectlybyshareholdersandoperationaldecisionvariable(s)maybe oftendecided,not bytopmanagers, butby middlemanagers, whomaynotknowtheextentoftheholdingsofthefirm’s shareholdersinother firms.Antónetal.(2022)showthat,eveninthosecases,managerialincentivescanserveasamech- anism (whichrequiresnocommunicationorcoordinationbetweenthedifferentplayers)that linksoverlapping ownership withoperationaldecisionvariable(s).

3.2.1. Shareholdersvoting

Webeginbyaddressingtheequilibriumregardingthevotingbehaviorofshareholders.Shareholdersareassumedtocare about theutility derived fromtheir expected returnsand, as such, vote-simultaneously and noncooperatively -forthe candidatewhosestrategyproposalmaximizestheirexpectedreturns,randomizingbetweenthetwoincaseofindifference.

Weconsiderthattheutilityu_k(^x,m)^ofeachshareholderktobeafunctionofherexpectationregardingthereturnfrom herfinancialrights,whichwillbeafunctionofthewinningstrategyproposalsinallthefirmsx=

x₁,...,x_j,...,x_N

and theidentityofthecorrespondingelectedmanagersm=

m₁,...,m_j,...,m_N

: u_k

(

^x,m

)

=Ek

(

^Rk

(

^x,m

) )

,

10Short-sales are not allowed and so financial, voting and control rights are non-negative.

11We assume that shareholders are external in the sense that firms do not hold financial and voting rights in other firms. We extend the framework to internal shareholders in Section 5 .

whereEk(^Rk(^x,m))^denotestheexpectationofshareholderkregardingherreturnR_k(^x,m)^.Wemodelthisexpectationto bethesumoftwocomponents:acommoncomponentand(b)anadditiveshareholder-specificexpectationbias,asfollows:

Ek

(

^Rk

(

^x,m

) )

^=E

(

^Rk

(

^x

) )

^+Bk

(

^m

)

^,

whereE(^Rk(^x))=

j∈

φ

k jE

j(^x)

denotesthecommoncomponent,rootedonacommonexpectationE

j(^x)

,across shareholders,regardingtheprofitsj(^x)^{ofeachfirm j,assumedtobepubliclygeneratedby,forexample,thedocumenta-} tiondistributedanddiscussedintheshareholderassembliesofthedifferentfirms,andB_k(^m)^denotestheexpectationbias ofshareholderkfor(oragainst)thechallengersof(potentially)allfirms.

Azar (2012, 2016, 2017), Brito et al. (2018a) and Moskalev (2019) associate the expectation bias B_k(^m) ^{to a profit-} irrelevantdifferentiationofcandidates.Thisassumptionisborrowedfromtheelectoralcompetitionliterature(Lindbeckand Weibull,1987; Persson andTabellini,2000).However, whileina politicalelectoralsettingitis reasonableto assumethat votershaveanidealogicalbiastowardsacandidate,inacorporatefinancesetting,itmaybelessso.Althoughonecanargue that theassumption canbe rootedonthe factthat thecredibility (orlack ofcredibility)oftheincumbent,beingalready inoffice,isknowntoshareholders,whilethatofthechallengerisnot,onecanalsoarguethatifshareholdersindeedcare aboutthereturnsthatresultfromthedifferentstrategyproposals(and,thereby,ultimatelycareaboutprofits),thereisnot anobviousreasonwhythebiasshouldnotbeprofit-relevant.

We contributeto theliterature byconsidering amoregeneralformulationthan theoneinthe extantliterature which allows theexpectation bias tobe profit-irrelevant orprofit-relevant. A profit-relevant bias can, forexample,be rooted(a) on adifferenceofexpectation ofshareholders regardingthecompetenceorthecostto exerteffortofthetwo candidates, withanimpactonthefirm’sprofit(asinGomes,2000;andGoshenandLevit, 2020),or(b)onadifferenceofexpectation of shareholdersregardingthe (dis)loyaltyofthe two candidatesto shareholders,who maydivert, onceelected,resources fromthefirmforpersonaluse,withanimpactonthefirm’sprofits(asinBebchukandJolls,1999;PaganoandImmordino, 2012;Amessetal.,2015;Noeetal.,2015;andGoshenandLevit,2020),or(c)onadifferenceofexpectationofshareholders regarding the effectiveness of the firm’s governance mechanism in deterring illicit managerial diversion and enforce its reimbursement, withan impacton thefirm’sprofit(as inDesaietal., 2007;Pagano andImmordino, 2012;Amessetal., 2015;Noeetal.,2015;andLiandLi,2018).Assuch,theexpectationbiasofshareholderkfor(oragainst)thechallengersof (potentially)eachfirmisgivenby:

B_k

(

^m

)

⁼

j∈I

λφ

kj1

m_j=b_j

ξ

kg+

(

¹⁻

λ )

¹

m_j=b_j

ξ

kj

=

j∈I

1−

λ

⁺

λφ

kj

1

m_j=b_j

ξ

kj,

where

ξ

k j≶0denotestheexpectationbiasofshareholderkfor(oragainst)thechallengeroffirm jand1(^mj=b_j)^denotes adummyvariablethattakesthevalue1ifthechallengeriselectedmanageroffirm j.

λ

∈

{

^0,1

}

^{controlstheprofitrelevance}

ofthebias.When

λ

=0,wehavethatB_k(^m)=

j∈1

m_j=b_j

ξ

k j,whichimpliesthat:

Ek

(

^Rk

(

^x,m

) )

⁼_j∈

φ

k jE

j

(

^x

)

+1

m_j=b_j

ξ

k j

,

and, as a consequence, that the bias is profit-irrelevant, as in Azar (2012, 2016, 2017), Brito et al. (2018a) and Moskalev(2019).When

λ

=1,wehavethatB_k(^m)=

j∈

φ

k j1

m_j=b_j

ξ

k j,whichimpliesthat:

Ek

(

^Rk

(

^x,m

) )

⁼_j∈

φ

k j

E

j

(

^x

)

+1

m_j=bg

ξ

k j

,

and, asa consequence,that the bias isprofit-relevant (asit impacts the shareholder’s specific expectation regardingthe firm’sprofit).¹²

Weassumethatshareholdersvote,ineachfirm’sshareholderassembly,forthecandidatewhosestrategyproposal,given their bias, maximizestheir utilities,randomizing between the two incase of indifference.Following Alesina and Rosen- thal(1995),Azar(2012,2016)andBritoetal.(2018a),wealsoassumethefollowingregardingthisvotingbehavior.

Assumption1. Shareholdersareconditionallysincere.

Assumption 1impliesthat thevoteofshareholdersis, conditionalon theequilibriumstrategy proposalsofthe candi- dates to theremaining firms, deterministic,asfollows: shareholderk will voteforfirm j’s incumbentwith probability 1 if u_k(^xa,ma)>u_k(^xb,m_b)^{, willvoteforfirm j’s challengerwith}probability 1 ifinstead u_k(^xa,ma)<u_k(^xb,m_b)^{, andwill} randomizebetweenthetwocandidateswithequalprobability ifu_k(^xa,ma)=u_k(^xb,m_b)^,wherexa=

x₁,...,x_{a j},...,x_N

, x_b=

x₁,...,x_{b j},...,x_N

,ma=

m₁,...,a_j,...,m_N

andm_b=

m₁,...,b_j,...,m_N

conditionontheequilibriumstrategy proposalsofthecandidatestotheremainingfirms.Assumption1ispresentedforsimplicity.Itcanberelaxedinlinewith Moskalev(2019)byconsideringanequilibriumrefinementthatexcludesweaklydominatedstrategiesofshareholders.

12The discreteness of λdoes not impact the results. They remain virtually unchanged even if we allow λ∈ [ 0 , 1 ] so that the bias is a weighted average of the two elements.

3.2.2. Candidatesstrategyproposals

Having describedthe second stage of the game, we now address the first stage, in which candidates simultaneously choosestrategyproposals.Todoso,wefollowLindbeckandWeibull(1987),Azar(2012,2016,2017),Britoetal.(2018a)and Moskalev (2019)inassumingthat thebias ofshareholdersfor(oragainst)thechallenger,whileknowntoshareholders,is unobserved by candidates, whotreat it asarandom utility shock.As aconsequence, votingby shareholdersis,from the perspectiveofcandidates,probabilistic.

Azar(2012,2016,2017)andBritoetal.(2018a) assumethat candidatesconsiderthebiasesofshareholderstobeinde- pendent(andidentically) distributedacross firmsandshareholdersaccordingtoasymmetricprobability distributionwith meanzeroandcumulative distribution.However,thereisnotanobviousreasonwhythey shouldbedrawnseparately for each shareholderofthesame firm.Ifshareholders indeedcareabouttheutility derived fromtheirreturns (and,thereby, ultimately aboutprofits), andinformationabout theprofits ofa firm is, tosome extent, commonacross shareholders of the firm, then biasesmaybe correlated across shareholders(Moskalev,2019). To doso, we considerthe bias

ξ

k j ofeach shareholderkfor(oragainst)thechallengerofeachfirm j canbedisaggregated,followingtheliteratureonelectoralcom- petition, into two independentlydrawn components (Lindbeck and Weibull,1987; Persson andTabellini, 2000;Ponzetto, 2011;Mat˘ejkaandTabellini,2021):¹³

ξ

k j=

ξ

˜j+

ξ

˜k j,

where

ξ

˜jdenotesacomponentcommontoallshareholdersoffirm j,whichinduces acorrelationamongthebiasesofall shareholdersofthefirm,and

ξ

˜k jdenotesacomponentspecifictoshareholderkandfirm j.

ξ

˜jisindependentlydrawnacross firms from a distribution with density function h_j(·)^{, cumulative} distribution function H_j(·) ^{and support in the interval} [−^ψj/²,^ψj/²],while

ξ

˜k jisindependentlydrawnacrossshareholdersandfirmsfromadistributionwithdensityfunctiong_j(·)^, cumulative distribution functionG_j(·)^{andsupport inthe interval[}−^τj/²,^τj/²],where

ψ

j≥0and

τ

j>0. Weallow

ψ

j≥0 to control thecorrelation ofthebiasesofall shareholdersin afirm. When

ψ

j=0,wehave that

ξ

k j=

ξ

˜k j,whichimplies thatthereisnocommoncomponentandthebiasesareindependentlydistributedacrossfirmsandshareholders,asinAzar (2012, 2016, 2017)andBrito etal.(2018a).In turn,when

ψ

j>0,the biases combinethe twocomponents and, thereby, exhibitcorrelationamongtheshareholdersofthefirm.¹⁴

Weconsiderthatcandidateschoosetheirstrategyproposalsundertwoalternativeassumptions(Azar,2012;Azar,2016;

Azar,2017;Britoetal.,2018a;Moskalev,2019):¹⁵

Assumption2. Candidateschoosestrategyproposalstomaximizetheirvoteshare.

Assumption3. Candidateschoosestrategyproposalstomaximizetheirexpectedutilityfromcorporateoffice.

WebeginbyaddressingthechoiceofstrategyproposalsbycandidatesunderAssumption2.Inthissetting,theincumbent choosesx_a

j tosolve:

maxxaj

k∈j

Pr_kaj

(

^xa,ma,xb,m_b

) υ

k j,

whilethechallengerchoosesx_b

j sotosolve:

maxx_b j

k∈j

Pr_kbj

(

^xa,ma,xb,m_b

) υ

k j,

wherePr_ka

j(^xa,ma,x_b,m_b)^andPrkb_j(^xa,ma,x_b,m_b)^denotetheprobabilitythat,inthecandidatesperspective,shareholderk votesfortheincumbentandthechallengeroffirm j,respectively. BecausePr_kb

j(^xa,ma,x_b,m_b)=1−Pr_ka

j(^xa,ma,x_b,m_b)^, it is straightforward to see that the solution to the maximization problem ofthe two candidates to firm j is symmet- ric. As such, we characterize - for simplicityof exposition - solely the incumbent’s problem. To do so, we must derive Pr_ka

j(^xa,m_a,x_b,m_b)^{.Usingthelawoftotal}probability,wecanwritePr_ka

j(^xa,m_a,x_b,m_b)^asfollows:

Pr_kaj

(

^xa,ma,xb,m_b

)

⁼

1 2ψj

−¹2ψj

Pr_kaj

xa,ma,x_b,m_b,

ξ

˜^j

h

_˜

ξ

^j

d

ξ

˜^j,

13This two-component structure is presented for simplicity. It can be relaxed in line with Moskalev (2019) by considering the bias of each shareholder of firm jto be a shareholder-specific weighted sum of M jcommon biases.

14We do not allow τj= 0 , i.e, that ξk j= ξ˜ j, which would imply that the biases are independently distributed across firms, but perfectly correlated across all shareholders of each firm, so to rule out, as it will become apparent below, corner solutions for the voting (and election) probabilities.

15We are implicitly assuming that candidates do not derive any direct utility from the strategy proposal because, as established by Azar (2020) , doing so breaks down the equivalence, when shareholders are fully diversified across firms, between the equilibrium in monopoly and oligopoly settings.

where Pr_ka

j

x_a,m_a,x_b,m_b,

ξ

˜j

denotes the probabilitythat,in thecandidates perspective, shareholder k votesforthe in- cumbentoffirm jconditionalonthecommoncomponentofthebias

ξ

˜j,which,inturn,isgivenby:

Pr_kaj

xa,ma,x_b,m_b,

ξ

˜j

=Pr

(

^uk

(

^xa,ma

)

^>uk

(

^xb,m_b

) )

=Pr

E

(

^Rk

(

^xa

) )

^>E

(

^Rk

(

^xb

) )

⁺

1−

λ

⁺

λφ

^kj

ξ

^kj

=Pr

E

(

^Rk

(

^xa

) )

^>E

(

^Rk

(

^xb

) )

⁺

1−

λ

⁺

λφ

^kj

_˜

ξ

^j+

ξ

^˜kj

=Pr

ξ

˜^kj<E(Rk

₍

⁽₁^x₋^a)_λ^)−E_+λφ^(R_kj^k

₎

^(xb))−

ξ

^˜j

=G_j ^{E(Rk(xa))−E(Rk(xb))}

(

^{1−λ+λφkj}

)

⁻

ξ

^˜j

,

where the second equality makes use of the fact that the biases for (or against) the challenger of other firms, g∈,g=j

1−

λ

+

λφ

kg

1

mg=bg

ξ

kg

,entertheutilityobtainedfrombothcandidates.

We now address the choice of strategy proposals by candidates under Assumption 3. In this setting,the incumbent choosesx_aj sotosolve:

maxxaj

Pr

m_j=a_j

|

^xa,xb

^aj,

whilethechallengerchoosesx_b

j sotosolve:

maxxbj

Pr

m_j=b_j

|

^xa,xb

bj,

wherePr(^mj=a_j

|

^xa,x_b)^andPr(^mj=b_j

|

^xa,x_b)^denotetheprobabilitythattheincumbentandthechallenger,respectively, are elected while a_j and b_j denote the utility that the incumbent andthe challenger, respectively, expect to accrue conditional upon beingelected.Again, becausePr(^mj=b_j

|

^xa,x_b)=1−Pr(^mj=a_j

|

^xa,x_b)^{,it is}straightforwardto seethat the solution to the maximization problemof the two candidates to firm j is symmetric.As such, we characterize - for simplicityof exposition-solely theincumbent’s problem. Todoso, wemust derive Pr(^mj=a_j

|

^xa,x_b)^.Let j denotethe number of shareholders withvoting rights infirm j,℘j denote all the 2^j−1 possible subsets ofthose shareholders that canawardthemajorityofvotestoacandidateand^ıj∈℘jdenoteaparticularsubsetofthoseshareholders.Giventhatthe electionoftheincumbentisensuredwiththevotesoftheshareholdersineachsubsetin℘j,wehavethatPr(^mj=a_j

|

^xa,x_b) justsumstheprobabilitieswithwhichsheiselectedbyeachsubset^ı_j^,Pr(^mj=a_j

|

^xa,x_b,^ı_j)^,asfollows:

Pr

m_j=a_j

|

^xa,xb

= ^ıj∈℘j

Pr

m_j=a_j

xa,x_b,

^ıj

= ^ıj∈℘j

¹_2ψj

−¹2ψj

Pr

m_j=a_j

xa,x_b,

^ıj,

ξ

˜j

h

_˜

ξ

j

d

ξ

˜j,

where the last equality uses the law of total probability to write Pr m_j=a_j

|

^xa,x_b,^ı_j

in terms of Pr m_j=a_j

|

^xa,x_b,^ıj,

ξ

˜j

, which conditions on the common component of the bias

ξ

˜j. Further, given that conditional on

ξ

˜j,theshareholders-specificbiases

ξ

˜k jfork∈jareindependentlydistributed,wecanwritePr m_j=a_j

|

^xa,x_b,^ı_j,

ξ

˜j

astheproductofthevotingprobabilitiesofthecorrespondingshareholders,asfollows:

Pr

mj=aj

xa,x_b,

^ıj,

ξ

˜^j

=

k∈^ıj

Pr_kaj

xa,ma,x_b,m_b,

ξ

˜^j

k∈/^ıj

1−Pr_kaj

xa,ma,x_b,m_b,

ξ

˜^j

.

3.2.3. Nash-equilibrium

Havingdescribedthemaximizationproblemofthecandidates,we nowaddressthepure-strategyNashequilibriumfor thecandidatesstrategyproposals’game.Todoso,wefollowAzar(2012,2016,2017)andBrito et al. (2018a)in making the following technical assumptions regardingthe strategy spacej available to the candidates ofeach firm j,the common expectationE(^Rk(^x))^{ofthereturnofeach}shareholderk,andthecumulativedistributionfunctionsH_j(·)^andGj(·)^. Assumption4. Thestrategyspacejavailabletothecandidatesofeachfirm jisanonemptycompactsubsetof .

Assumption 5. Thecommonexpectation E(^Rk(^x))^{ofthereturnofeach} shareholderk is(a)continuousandtwicediffer- entiableinx,withcontinuoussecondderivatives;and(b)strictlyconcaveinfirm j’sstrategyx_j∈

x_aj,x_b

j

,conditionalon thestrategiesoftheremainingfirms.

Assumption6. H_j(·)^{isthecumulative}distributionfunctionofanuniformdistributionovertherange[−^ψj/²,^ψj/²]foreach firm j.