Modeling Horizontal Shareholding with Ownership Dispersion

The dominant formulation for modeling the objective function of the rms manager in the presence of horizontal shareholding has been critiqued for producing the result that it may solely reect the interests of a small number of shareholders even if, collectively, those shareholders do not have full control of the rm. We show that this issue can be avoided with an alternative formulation. This formulation is derived from a probabilistic voting model in which shareholders may di¤er in their evaluation of the amount of resources the manager will divert from the rm for personal use, which yields the result that the manager maximizes a control-weighted sum of the shareholdersrelative (rather than absolute) expected returns. JEL Classication: L13, L41 Keywords: Horizontal Shareholding, Ownership Dispersion, Manager Objective Function, Proportional Control, Banzhaf Control We would like to thank R icardo Gonçalves, Torsten Persson , M ichele Polo , G iacomo Ponzetto , Vasco Rodrigues, M artin Schmalz, Guido Tab ellin i and Kyle W ilson , as well as partic ipants at the 2019 International Industria l O rganization Conference, the 13th Annual M eeting of the Portuguese Econom ic Journal and the 46th Annual Conference of the Europ ean Asso ciation for Research in Industria l Econom ics, and the sem inar audience at Universidade Católica Portuguesa, for help fu l comments and suggestions. Duarte Brito gratefu lly acknow ledges nancia l support from Fundação para a C iência e a Tecnologia (U ID/ECO/04007/2019) and FEDER/COMPETE (POCI-01-0145-FEDER-007659). R icardo R ib eiro gratefu lly acknow ledges nancia l support from Fundação para a C iência e a Tecnologia (U ID/GES/00731/2019). Helder Vasconcelos gratefu lly acknow ledges nancia l support from Fundação para a C iência e a Tecnologia (U ID/ECO/04105/2019). Duarte Brito is w ith Universidade Nova de L isb oa, Facu ldade de C iências e Tecnologia and Center for Advanced Studies in Managem ent and Econom ics (CEFAGE), 2829-516 Caparica , Portugal. E -mail address: dmb@ fct.un l.pt. E iner E lhauge is Petrie Professor of Law at Harvard Law School, 1563 Massachusetts Avenue, Cambridge, MA 02138, USA . E -mail address: e lhauge@ law .harvard .edu . R icardo R ib eiro (corresp onding author) is w ith Universidade Católica Portuguesa, Católica Porto Business School and Research Centre in Managem ent and Econom ics (CEGE), Rua de D iogo Botelho 1327, 4169-005 Porto, Portugal. E -mail address: rrib eiro@porto.ucp .pt. Helder Vasconcelos is w ith Universidade do Porto, Facu ldade de Econom ia and Center for Econom ics and F inance (CEF.UP), Rua Dr. Rob erto Frias, 4200-464 Porto, Portugal. E -mail address: hvasconcelos@ fep .up .pt. A ll remain ing errors are of course our own. Electronic copy available at: https://ssrn.com/abstract=3264113


Introduction
Horizontal shareholding exists when shareholders own partial …nancial rights in several horizontal competitors in an industry. Such horizontal shareholding can induce a con ‡ict in the …rm-speci…c interests of shareholders, wherein horizontal shareholders in any given …rm want that …rm to pursue a less competitive strategy than the strategy desired by non-horizontal shareholders. 1 Hence, …rm managers must weigh the eventual con ‡icting (…nancial) interests of di¤erent shareholders according to their relative (control) in ‡uence over the decisionmaking of their …rm.
We identify six desirable properties for the weighting scheme (objective function) used by managers. The …rst …ve properties follow (a selection of) the proposals in Schmalz (2018a): (i) absent horizontal shareholding, managers would not weigh the expected pro…t of rival …rms, which implies that managers would decide the strategy of their …rm to maximize own expected pro…t; (ii) with non-in…nitesimal horizontal shareholding, managers would weigh the interests of horizontal shareholders by assigning a positive weight to the expected pro…t of rival …rms when those shareholders have control rights in the …rm and …nancial rights in both …rms, which implies that managers would internalize the impact of their …rm's strategy on the expected pro…t of rival …rms; (iii) the weight that managers assign to the expected pro…t of rival …rms would be continuous on the …nancial and control rights of the shareholders that have …nancial rights in the …rm; (iv) the weight that managers assign to the expected pro…t of rival …rms would be one when all the shareholders that have …nancial rights in the …rm are fully diversi…ed across rivals, which implies that managers would maximize the expected pro…t of the industry; and (v) the elements of the weight that managers assign to the expected pro…t of rival …rms would have empirical counterparts, i.e., would be measurable. The …nal property is motivated by the discussions in Gramlich and Grundl (2017), O'Brien and Waehrer (2017), Crawford et al. (2018) and Schmalz (2018a). It addresses the in ‡uence of shareholders over managers in the presence of ownership dispersion: (vi) managers would not weigh -solely -the interests of horizontal (non-horizontal) shareholders when those shareholders do not have full control, even if the ownership of each non-horizontal (horizontal) shareholder is dispersed among a collection of in…nitesimal identical shareholders.
The dominant formulation of the objective function of managers is due to O'Brien and Salop (2000, henceforth O&S) who, incorporating features from both Rotemberg (1984) and Bresnahan and Salop (1986), assume that the interests of each shareholder can be captured 1 Although non-horizontal shareholders may favor a di¤erent …rm-speci…c strategy, that does not mean they are harmed by horizontal shareholding because horizontal shareholding also reduces the competitiveness of rival …rms, and non-horizontal shareholders bene…t from a reduction of competition between the …rm and its rivals (please see Schmalz, 2018b for a formal model).
by the expected return from her …nancial investments and, as such, the manager would decide the strategy of the …rm to maximize a control-weighted sum of the …rm's shareholders expected returns. Because those returns are a function of the pro…ts of the …rms in which shareholders hold …nancial rights, this implies that the manager would decide the strategy of the …rm to maximize a weighted sum of the expected pro…ts of all the …rms in the industry over which controlling shareholders have …nancial rights in. Azar (2016Azar ( , 2017 and Brito et al. (2018) show that this formulation can be microfounded through a voting model in which shareholders vote to elect the manager from two potential candidates, an incumbent and a challenger, with conceivably di¤ering strategy proposals to the …rm. 2 Shareholders are assumed to care about the expected returns that result from the di¤erent strategy proposals and to have a pro…t-irrelevant bias for (or against) the challenger -since the credibility (or lack of credibility) of the incumbent, being already in o¢ ce, is known to shareholders, while that of the challenger is not. Voting is probabilistic in the sense that this bias, while known to voters, is unobserved by candidates, who treat it as random.
However, this dominant formulation fails, as Gramlich and Grundl (2017), O'Brien and Waehrer (2017) and Crawford et al. (2018) discuss, property (vi). As the ownership of each non-horizontal (horizontal) shareholder becomes dispersed (among a collection of in…nitesimal identical shareholders), the objective function of managers would weigh solely the interests of horizontal (non-horizontal) shareholders, even if those horizontal (non-horizontal) shareholders do not have full control of the …rm. In other words, the dominant formulation yields that the objective function of managers may solely re ‡ect the interests of a small number of shareholders even if, collectively, those shareholders do not have full control of the …rm.
In this paper, we propose an alternative formulation. In the lines of Azar (2016Azar ( , 2017 and Brito et al. (2018), we also use a probabilistic voting model, in which shareholders are assumed to care about the expected returns that result from the di¤erent strategic proposals and to have a bias for (or against) the challenger. However, in contrast with that literature, we assume that the bias is the result of a di¤erence in shareholders expectations regarding their returns, which implies it is pro…t-relevant. We root this bias in the evaluation of shareholders regarding the amount of resources candidates will, once elected, divert from the …rm for personal use. 3 This key, distinctive assumption implies that the relevance of the bias -from the perspective of candidates, who treat it as random -will now be proportional to the …nancial stakes of each shareholder -since the diversion of resources from the …rm a¤ects each shareholder in proportion to her share of …nancial rights. As a consequence, all the determinants of shareholders voting behavior -the expected pro…ts of …rms (gross of diversion) and diversion by managers -will now be proportional to the …nancial stakes of each shareholder. We show that the equilibrium of this probabilistic voting model can be entirely equivalent to a setting in which the manager would decide the strategy of the …rm to maximize a control-weighted sum of the …rm's shareholders relative expected returns (with respect to their …nancial stakes in the …rm), which capture the shareholders ideal expected pro…t weights (regarding the …rm). 4;5 This proposed alternative formulation satis…es property (vi) in the sense that the objective function of managers would solely weigh the interests of horizontal (non-horizontal) shareholders as the ownership of each non-horizontal (horizontal) shareholder becomes dispersed (among a collection of in…nitesimal identical shareholders) if horizontal (non-horizontal) shareholders do have full control of the …rm. Furthermore, it is similar in nature to the formulation in Crawford et al. (2018) who, to address this property, consider normalizing the expected returns of shareholders by their overall …nancial investments, but we microfound our proposal through a probabilistic voting model.
The remainder of the paper is organized as follows. Section 2 presents the theoretical framework under which the proposed alternative formulation for the objective function of managers is derived, Section 3 discusses extensions to -and competition policy applications of -this theoretical framework and Section 4 concludes.

The Theoretical Framework
This section introduces the theoretical framework under which the proposed alternative formulation for the objective function of managers is derived. The general setting combines features from Brito et al. (2014), Azar (2016Azar ( , 2017 and Brito et al. (2018) as follows.
First, we consider that there are K shareholders, external to the industry, indexed by k, and N single-product …rms, indexed by j, whose total stock is composed of voting stock and non-voting (preferred) stock. Both give the holder the right to a share of the …rm's pro…ts, but only the former gives the holder the right to vote in the …rm's shareholder assembly. The holdings kj 2 [0; 1] of total stock of shareholder k in …rm j, regardless of whether it be voting or non-voting stock, capture her …nancial rights to the …rm's pro…ts. The holdings kj 2 [0; 1] of voting stock of shareholder k in …rm j, capture her voting rights in the …rm. These voting rights may not necessarily coincide with her control rights in the …rm, kj 2 [0; 1], which refer to her rights to in ‡uence the decisions of …rm j, to be discussed below. 6 Second, we consider that the ownership structure is such that a subset of shareholders can hold …nancial and voting rights in multiple …rms of the industry. This horizontal shareholding can induce a con ‡ict in the …rm-speci…c interests of shareholders, which managers must weigh according to the corresponding relative in ‡uence over the decision-making of their …rm.
Finally, we consider that the pro…ts of each of the di¤erent …rms in the industry is a function not only of the strategies of all the …rms but also of a state of nature. This implies that …rms'pro…ts and, consequently, shareholders'returns -since they are a function of the pro…ts of the …rms in which they hold …nancial rights -are random.

Dominant Formulation
As discussed above, the dominant formulation of the objective function of managers is due to O&S, who assume that the interests of each shareholder can be captured by the expected return from her …nancial investments and, as such, the manager would decide the strategy of the …rm to maximize a control-weighted sum of the expected returns of the …rm's shareholders, as follows: (1) where denotes the set of existing shareholders, j denotes the subset of shareholders that hold …nancial rights in …rm j, x j denotes the strategy of …rm j and, …nally, R k denotes the return of shareholder k, which is a function of the pro…ts of all the …rms in which shareholder k holds …nancial rights: R k = P g2= kg g , with = denoting the set of existing …rms and g denoting the pro…t of …rm g. This implies that the manager of any …rm j would maximize a weighted sum of the expected pro…ts of (potentially) all the …rms in the industry: which, normalizing the weight on own expected pro…t to one, is entirely equivalent to: As a consequence, the weight w jg that the manager of …rm j assigns to the expected pro…t of rival …rm g is, under this dominant formulation, thus, given by w jg = P k2 j kj kg P k2 j kj kj 0 for any j; g 2 = and j 6 = g. 7 Azar (2016Azar ( , 2017 and Brito et al. (2018) microfound this formulation through a voting model in which shareholders vote to elect the manager from two potential candidates with conceivably di¤ering strategy proposals to the …rm. They show that if candidates choose strategy proposals to maximize their expected utility from corporate o¢ ce, control rights would be endogenously measured by Banzhaf (1965)'s power index while if they choose strategy proposal to maximize their expected vote share, control rights would be endogenously measured by voting rights. Proposition 1 establishes the properties of the weighting scheme under this dominant formulation using the results in Azar (2016Azar ( , 2017 and Brito et al. (2018).
Proposition 1 Using the corporate control measures derived in Azar ( 2016Azar ( , 2017 and Brito et al. ( 2018), the objective function of managers under the dominant formulation satis…es properties (i) to (v). However, it fails property (vi).
Proof. See Appendix.
Proposition 1 makes clear, as discussed in Gramlich and Grundl (2017), O'Brien and Waehrer (2017) and Crawford et al. (2018), that the dominant formulation fails property (vi). The reason is rooted on the fact that, under this formulation, a dispersion of ownership from a large non-horizontal (horizontal) shareholder to a collection of small identical nonhorizontal (horizontal) shareholders that is equally large in aggregate decreases the relative in ‡uence over the manager of the …rm, independently of the impact of the dispersion on the distribution of the …rm's control rights. As a consequence, when the ownership of each non-horizontal shareholder becomes dispersed among a collection of in…nitesimal identical shareholders, P k2 j kj kj tends to re ‡ect solely the interests of the (the non-dispersed) horizontal shareholders as the summation referent to the non-horizontal shareholders approximates zero. As such, the manager weighs solely the interests of the horizontal shareholders, even when their voting rights do not induce full control of the …rm. Similarly, when the ownership of each horizontal shareholder becomes dispersed among a collection of in…nitesimal identical shareholders (a) P k2 j kj kj tends to re ‡ect solely the interests of the non-horizontal shareholders as the summation referent to the horizontal shareholders approximates zero; and (b) P k2 j kj kg approximates zero. As such, the manager weighs solely the interests of non-horizontal shareholders, even when their voting rights do not induce full control of the …rm. 7 In order to see why the weights w jg are non-negative, note that kj 0, kj > 0 and kg 0 for all k 2 j and all j; g 2 =.

Proposed Alternative Formulation
We follow Azar (2016Azar ( , 2017 and Brito et al. (2018) in microfounding the proposed alternative formulation through a voting model in which shareholders vote at the …rm's shareholder assembly to elect the manager from two potential candidates, an incumbent a j and a challenger b j , with the candidate receiving the majority of voting rights being elected manager of the …rm. Candidates are assumed to be opportunistic in the sense their only motivation is to hold o¢ ce. To do so, they compete for the voting rights of shareholders by -simultaneously and noncooperatively -proposing a strategy for the …rm, which is assumed binding in line with the literature on electoral competition (Down, 1957;Lindbeck and Weibull, 1987;Polo, 1998;and Persson and Tabellini, 2000). Shareholders are assumed to care about the utility derived from their expected returns and, as such, vote sincerely -simultaneously and noncooperatively -for the candidate whose strategy proposal maximizes their expected returns, randomizing between the two in case of indi¤erence. Shareholders expectation regarding their returns may have, however, a bias for (or against) the challenger. We assume the following regarding this bias.
Assumption 1 The bias of shareholders for (or against) the challenger is pro…t-relevant, rooted in the evaluation of shareholders regarding the amount of resources candidates will, once elected, divert from the …rm for personal use.
Assumption 1 constitutes implies that we root the pro…t-relevant bias in the disloyalty of candidates to shareholders (as in Bebchuk and Jolls, 1999;Pagano and Immordino, 2012;Amess et al., 2015;Noe et al., 2015;and Goshen and Levit, 2019). In particular, we assume that the two candidates may di¤er in their ability or willingness, once elected, to divert resources from the …rm for personal use. 8 This diversion is assumed a permanent trait of the candidates that cannot be credibly modi…ed or communicated to shareholders (as in Lindbeck and Weibull, 1987;and Persson and Tabellini, 2000) and over which shareholders may have di¤erent evaluations.
Although we root the bias in the evaluation of shareholders regarding the amount of resources candidates will, once elected, divert from the …rm for personal use, other microfoundations would be possible. Assuming that the two candidates may di¤er in the amount of resources diverted for personal use (disloyalty) is equivalent to assuming that the two candidates may di¤er in their incompetence or in their cost to exert e¤ort (see, for example, Gomes, 2000;and Goshen and Levit, 2019). Further, assuming that shareholders may di¤er in their evaluation of the amount of resources candidates will divert, once elected, for personal use is equivalent to assuming that …rms have a governance mechanism to detect managerial diversion (as in Desai et al., 2007;Pagano and Immordino, 2012;Amess et al., 2015;Noe et al., 2015;and Li and Li, 2018) and shareholders may di¤er in their evaluation of (and/or may not all be equally informed about) the e¤ectiveness of this mechanism in deterring illicit diversion (and enforce its reimbursement), modelled to depend on the tenure of the candidate within the …rm.

Figure 1
Timing In our framework, the time of events for each …rm j, depicted in Figure 1, is as follows. First, the two candidates propose a strategy for the …rm. Let x a j 2 j and x b j 2 j denote the strategy proposals of the incumbent and the challenger, respectively, and j denote the strategy space available to the candidates, which can refer to any decision variable(s) -e.g., quantity, price, R&D investment, etc. -of the …rm. Candidates know the distribution from which the evaluation of shareholders regarding managerial diversion is drawn, but not the realized values. Second, the actual evaluation of each shareholder regarding the diversion of resources by candidates is realized. Let kj = a kj b kj 7 0 denote the realized evaluation of shareholder k regarding the di¤erence in the amount of resources to be diverted by the incumbent and the challenger. 9 We follow the literature on electoral competition (Lindbeck and Weibull, 1987;Persson and Tabellini, 2000;Ponzetto, 2011;Matµ ejka and Tabellini, 2017) in allowing this realized evaluation to be disaggregated into two independently drawn components: kj =~ j +~ kj , where~ j denotes a component common to all shareholders of …rm j and~ kj denotes a component speci…c to shareholder k, which is independent and identically drawn across the shareholders of …rm j. Let H j ( ) and G j ( ) denote the cumulative distribution functions from which these common and speci…c components, respectively, are drawn. Third, the shareholder assembly is held and the candidate that receives the majority of the …rm's voting rights is elected manager. Let m j fa j ; b j g denote the identity of the 9 The assumption that the amount of resources diverted for personal use is …xed, in the sense it does not depend on the pro…t of the …rm (and consequently on the strategy proposal of the candidates), while contrasting with some literature that has assumed this amount to be a proportion of the pro…ts (or cash- ‡ows) of the …rm (see, for example, Gomes, 2000;Desai et al., 2007;Li and Li, 2018;Iacopetta et al., 2019), it is far from uncommon (see, for example, Bebchuk and Jolls, 1999;Pagano and Immordino, 2012;Amess et al., 2015;Noe et al., 2015;Goshen and Levit, 2019). elected manager. Finally, the elected manager implements the winning strategy proposal x j fx aj ; x bj g.

Shareholders Voting
We begin by addressing the equilibrium regarding the voting behavior of shareholders. As discussed above, shareholders are assumed to care about the utility derived from their expected returns. We follow O&S in assuming that the utility u k of each shareholder k is a linear function of the expected return from her …nancial rights, which -in the above framework -will be a function of -both -the winning strategy proposals in all the …rms of the industry and the identity of the corresponding elected managers: where m = (m 1 ; : : : ; m j ; : : : ; m N ) > denotes the N 1 vector of elected managers for all the …rms in the industry and x = (x 1 ; : : : ; x j ; : : : ; x N ) > denotes the corresponding N 1 vector of winning strategy proposals. Assumption 1 implies that the expected pro…t of each …rm g for shareholder k can be written as the sum of two components: (a) the …rm's expected gross pro…t, E[ g (x)], which captures the expected pro…t before manager diversion, which is a function of the winning strategies in all the …rms of the industry and assumed to be publicly generated -by, for example, the documentation distributed and discussed in the shareholder assemblies of the di¤erent …rms -so that every shareholder has, conditional on the candidates proposals, the same expectation; and (b) the realized evaluation of shareholder k regarding the amount of resources to be diverted from the …rm by the incumbent, a kg , and the challenger, b kg , which is a function of the elected manager. As a consequence, we can write the utility u k of each shareholder k as follows: ] denotes the expected gross return of shareholder k, and D k (m) = P g2= kg (1(m g = a g ) a kg + 1(m g = b g ) b kg ) denotes the evaluation of shareholder k regarding the amount of resources to be diverted. 10 As discussed above, we assume that shareholders vote sincerely, in each …rm's shareholder assembly, for the candidate whose strategy proposal, given their evaluation of managerial diversion, maximizes their utilities, randomizing between the two in case of indi¤erence. Following Alesina and Rosenthal (1995), Azar (2016) and Brito et al. (2018), we also assume the following regarding this voting behavior.
Assumption 2 Shareholders are conditionally sincere.
Assumption 2 implies that the vote of shareholders is, conditional on the equilibrium strategy proposals of the candidates to the remaining …rms, deterministic, as follows: shareholder k will vote for …rm j's incumbent with probability , and will randomize between the two candidates with equal probability if x a = (x 1 ; : : : ; x aj ; : : : ; x N ) > , x b = (x 1 ; : : : ; x bj ; : : : ; x N ) > , m a = (m 1 ; : : : ; a j ; : : : ; m N ) > and m b = (m 1 ; : : : ; b j ; : : : ; m N ) > condition on the equilibrium strategy proposals of the candidates to the remaining …rms.

Candidates Strategy Proposals
We now address the choice of strategy proposals by candidates. As discussed above, candidates are opportunistic in the sense their only motivation is to hold o¢ ce. Since at this stage, candidates know the distribution of the evaluation of shareholders regarding managerial diversion, but not the realized values, from their perspective, voting by shareholders is probabilistic. We follow Azar (2016Azar ( , 2017 and Brito et al. (2018) in considering candidates choose their strategy proposals under two alternative assumptions.
Assumption 3a Candidates choose strategy proposals to maximize their expected utility from corporate o¢ ce.
Assumption 3b Candidates choose strategy proposals to maximize their expected vote share.
We begin by addressing the choice of strategy proposals by candidates under Assumption 3a. In this setting, candidates choose strategy proposals to maximize the product of the probability that they are elected and the utility obtained from the rent associated with corporate o¢ ce they expect to accrue conditional upon being elected. Let Pr(m j = a j jx a ; x b ) and Pr(m j = b j jx a ; x b ) denote the probability that the incumbent and the challenger, respectively, are elected and let a j and b j denote the utility that the incumbent and the challenger, respectively, expect to accrue conditional upon being elected (and includes the resources each expects -in e¤ect -to divert for personal use). As such, the incumbent chooses x a j so to solve: while the challenger chooses x b j so to solve: Since , it is straightforward to see that the solution to the maximization problem of the two candidates to …rm j is symmetric. As such, we solve -for simplicity of exposition -solely the incumbent's problem. To do so, we must beforehand derive the probability that, in the candidates'perspective and given the common component~ j , each shareholder k votes for the incumbent, which is given by: This result makes use of the fact that P g2=;g6 =j kg (1(m g = a g ) a kj + 1(m g = b g ) b kj ) enters the utility obtained from both candidates and that, as discussed above, kj = a kj b kj and kj =~ j +~ kj . This implies that, given the common component~ j , the vote share of the incumbent is given by the sum, across all shareholders, of the product of the probability that each shareholder votes for them and the corresponding shareholder's voting rights: As this vote share depends on the realized value of~ j , it is, from the perspective of candidates, probabilistic. In turn, this implies that the probability Pr (m j = a j jx a ; x b ) with which the incumbent is elected manager of the …rm is given by: which captures the probability with which she receives the majority of voting rights. As a consequence, the incumbent chooses x a j so to solve: We now address the choice of strategy proposals by candidates under Assumption 3b. In this setting, candidates choose strategy proposals to maximize the expected sum (since it depends on the realized value of~ j ), across all shareholders, of the product of the probability that each shareholder votes for them and the corresponding shareholder's voting rights. As such, the incumbent chooses x a j so to solve: while the challenger chooses x b j so to solve: where the …rst expected value denotes the expected value with respect to the common com-ponent~ j . Since Pr kb j (x a ; m a ; x b ; m b ;~ j ) = 1 Pr ka j (x a ; m a ; x b ; m b ;~ j ) for each shareholder k, it is straightforward to see that the solution to the maximization problem of the two candidates to …rm j is symmetric. As such, we solve -for simplicity of exposition -solely the incumbent's problem, who chooses x a j so to solve: The objective functions (10) and (13), constructed under Assumptions 3a and 3b, respectively, make clear that candidates, when designing strategy proposals, pay attention not to the absolute expected (gross) returns of each shareholder k, E[R k ( )], but to her relative expected (gross) returns, , which establish the weights shareholder k would desire the manager to associate to the expected (gross) pro…t of each …rm in the industry -relative to the expected (gross) pro…t of the …rm.

Nash-Equilibrium
Having described the maximization problem of the candidates, we now address the purestrategy Nash equilibrium for the candidates strategy proposals'game. To do so, we follow Azar (2016Azar ( , 2017 and Brito et al. (2018) in making the following technical assumptions regarding the strategy space j available to the candidates of each …rm j, the expected gross return E[R k (x)] of each shareholder k and the cumulative distribution functions H j ( ) and G j ( ).
Assumption 4 The strategy space j available to the candidates of each …rm j is a nonempty compact subset of <.
Assumption 5 The expected gross return E hR k (x) i of shareholder k is (a) continuous and twice di¤erentiable in x, with continuous second derivatives; and (b) strictly concave in …rm j's strategy x j 2 x a j ; x b j , conditional on the strategies of the remaining …rms.
Assumption 6 H j ( ) and G j ( ) are the cumulative distribution functions of uniform distributions over the range 1 2 ' j ; 1 2 ' j and 1 2 j ; 1 2 j , respectively, with ' j ; j su¢ ciently large for each …rm j. 11;12 Assumptions 1 to 6 ensure the existence of a pure-strategy Nash equilibrium for the candidates strategy proposals'game (x a 1 ; x b 1 ; : : : x a j ; x b j ; : : : ; x a N ; x b N ), characterized as follows.
Proposition 2 Under Assumptions 1, 2, 3a or 3b, 4, 5 and 6, there exists a pure-strategy Nash equilibrium for the candidates strategy proposals'game that is entirely equivalent to the pure-strategy Nash-equilibrium from the case in which each candidate maximizes the following 11 Assumption 6 requires the support of H j ( ) and G j ( ) to be su¢ ciently large (relative to their argument -not to gross pro…ts) so to rule out corner solutions for probabilities (8) and (9). This technical assumption is standard and explicit in the electoral competition literature (Persson and Tabellini, 2000;Ponzetto, 2011;Matµ ejka and Tabellini, 2017) so that the behavior underlined by probabilities (8) and (9) is not perfectly predictable on the basis of strategy proposals. It is also standard -although implicitly -in the literature that microfounds the O&S formulation (Azar, 2016(Azar, , 2017Brito et al., 2018). 12 The requirements regarding the cumulative distribution function H j ( ) can be relaxed under Assumption 3b. In this scenario the only requirement regarding the cumulative distribution function H j ( ) is that E(~ j ) = 0 for each …rm j. objective function: where E[R k ] kj denotes the relative expected return of shareholder k and kj is measured by the voting rights of shareholder k in …rm j: kj = kj .
Proof. See Appendix.
Proposition 2 establishes that the two candidates would choose the same strategy proposal for each …rm j, conditional on the strategies of the candidates to the remaining …rms. Further, it establishes also, in contrast with O&S, that candidates (and thus the elected manager) would decide the strategy of the …rm to maximize a weighted sum of the …rm's shareholders relative expected returns, which establish their ideal expected pro…t weights. Furthermore, it establishes that the weights kj , which capture the control of shareholders over the decision-making of the …rm, would be endogenously measured by their voting rights. Finally, it implies that the manager of each …rm j would still maximize a weighted sum of the expected total pro…ts of (potentially) all the …rms in the industry: where the weight that the manager of …rm j assigns to the expected pro…t of rival …rm g is, under this proposed new, alternative formulation of the objective function of managers, thus, given by w jg = P k2 j kj kg kj 0 for any j; g 2 = and j 6 = g. 13 The key, distinctive assumption driving this alternative formulation is Assumption 1. 14 A pro…t-relevant bias implies that all the determinants of shareholders voting behavior -the expected pro…ts of …rms (gross of diversion) and diversion by managers -will be proportional to the …nancial stakes of each shareholder and, in turn, that candidates (and thus the elected manager) would decide the strategy of the …rm to maximize a weighted sum of the shareholders relative expected returns.
Proposition 3 establishes the properties of the weighting scheme under this proposed alternative formulation.
Proposition 3 Using the corporate control measures derived in Proposition 2, the objective function of managers under the proposed alternative formulation satis…es properties (i) to 13 The weights w jg are non-negative by the same arguments as were used in footnote 7. 14 Assumptions 2 to 5 are similar to the literature that microfounds the O&S formulation (Azar, 2016(Azar, , 2017Brito et al.,2018). Although Assumption 6, in contrast to that literature, allows for a correlation in the bias of shareholders, we show below that it is not this correlation that is driving the proposed alternative formulation. This makes clear that the key, distinctive assumption is Assumption 1.
Proposition 3 makes clear that the proposed alternative formulation satis…es property (vi), addressing the critique in Gramlich and Grundl (2017), O'Brien and Waehrer (2017) and Crawford et al. (2018) regarding the dominant formulation. The reason is rooted on the fact that, under this proposed alternative formulation, a dispersion of ownership from a large non-horizontal (horizontal) shareholder to a collection of small identical non-horizontal (horizontal) shareholders that is equally large in aggregate does not impact the relative in ‡uence over the manager of the …rm, 15 unless such dispersion impacts the distribution of the …rm's control rights. 16 As a consequence, the interests of horizontal (non-horizontal) shareholders will only disappear from the objective function of the manager when the voting rights of the non-horizontal (horizontal) shareholders do induce full control of the …rm.

Shareholders Independent Evaluation
We have assumed a framework in which the bias of shareholders for or against the challenger -rooted in their evaluation regarding managerial diversion -may be correlated across shareholders. This contrasts with the literature that microfounds the dominant formulation of the objective function of managers (Azar, 2016(Azar, , 2017Brito et al., 2018), which assumes this bias to be independent across shareholders. In this section, we derive our proposed alternative formulation for the setting in which~ j = 0 for each …rm j. This implies that the evaluation of shareholders regarding managerial diversion in each …rm coincides with the idiosyncratic component, which is independently drawn across shareholders. In this setting, shareholder k votes for the incumbent with probability Pr ka j (x a ; m a ; x b ; m b ), which is given by: We can then use this result to derive the probability Pr(m j = a j jx a ; x b ) with which the incumbent is elected manager of the …rm. To do so, let`j denote the number of shareholders 15 Since the dispersion is among identical shareholders, it does not impact relative returns. 16 The weight w jg is -for a given distribution of control rights -constant with respect to the dispersion of the ownership of each shareholder (among a collection of in…nitesimal identical shareholders). However, that does not imply that such dispersion does not have an impact on w jg . It does, but solely via the control rights which are induced by the distribution of the …rm's voting rights. with voting rights in …rm j, } j denote all the 2`j 1 possible subsets of those shareholders that can award the majority of votes to a candidate and { j 2 } j denote a particular subset of those shareholders. Given that the election of the incumbent is ensured with the votes of the shareholders in each subset in } j , we have that Pr(m j = a j jx a ; x b ) just sums the probabilities with which she is elected in each subset { j , Pr(m j = a j jx a ; x b ; { j ), as follows: The independence assumption implies that Pr m j = a j jx a ; x b ; { j is just the product of the voting probabilities of the corresponding shareholders: (18) As such, the probability with which the incumbent is elected manager of the …rm is given by: where the last equality makes use of probability (16), with d k taking the value one if shareholder k 2 { j and takes the value zero otherwise. This implies that the incumbent's problem can be rewritten as follows. Under Assumption 3a, we have that: (20) while, under Assumption 3b, we have that: The pure-strategy Nash equilibrium (x a 1 ; x b 1 ; : : : x a j ; x b j ; : : : ; x a N ; x b N ) for the candidates strategy proposals'game is, then, characterized as follows.
Proposition 4 If~ j = 0 for each …rm j, under Assumptions 1, 2, 3a or 3b, 4, 5 and 6, there exists a pure-strategy Nash equilibrium for the candidates strategy proposals' game that is entirely equivalent to the pure-strategy Nash-equilibrium from the case in which each candidate maximizes the following objective: denotes the relative expected return of shareholder k. Under Assumption 3a, kj is measured by the normalized Banzhaf ( 1965) power index of shareholder k in …rm j: , where p kj denotes the number of subsets of j that can award victory to a candidate in which shareholder k is pivotal. Under assumption 3b, kj is measured by the voting rights of shareholder k in …rm j: kj = kj .
Proof. See Appendix.
Proposition 4 establishes that it is not the correlation in the bias of shareholders that is driving the result that candidates (and thus the elected manager) would decide the strategy of the …rm to maximize a weighted sum of the …rm's shareholders relative expected returns. The exact same formulation is obtained under the independence assumption. Moreover, Proposition 4 also establishes that the independence assumption is, as expected, instrumental in the derivation (by the literature that microfounds the dominant formulation of the objective function of managers) of the Banzhaf (1965)'s power index as an endogenous measure of corporate control.

Extensions and Applications
In this section, we introduce and discuss extensions to the framework discussed above as well as competition policy applications of the proposed alternative formulation for the objective function of managers.

Shareholders Inattention
We have assumed a framework in which shareholders are fully attentive to the strategy proposals of candidates. In this section, we examine the robustness of the proposed alternative formulation by discussing an extension framework in which shareholders can either be attentive or inattentive to those proposals. In particular, we follow Gilje et al. (2019) in considering that each shareholder k is attentive to the strategy proposals of …rm j's candidates with probability kj and inattentive with probability 1 kj . If attentive, as discussed above, shareholder k will vote for the incumbent with probability 1 if u k (x a ; m a ) > u k (x b ; m b ), will vote for the challenger with probability 1 if u k (x a ; m a ) < u k (x b ; m b ), and will randomize between the two candidates with equal probability if u k (x a ; m a ) = u k (x b ; m b ). If inattentive, shareholder k will, irrespective of the strategy proposals of the candidates, vote for the incumbent with probability k and will vote for the challenger with probability 1 k . In this setting, it is relatively straightforward to show that the weights of the proposed alternative formulation would be given by w jg = P k2 j kj kj kg kj 0 for any j; g 2 = and j 6 = g, 17 a weight that is qualitatively similar to the measure proposed by Gilje et al. (2019) to capture the impact of common ownership on managerial incentives. The attention probabilities kj can be modelled to be a function of a multitude of observed …rm and shareholder factors (for example, the importance of …rm j in shareholder k's investment portfolio) and estimated using voting data (see Gilje et al., 2019 for an illustrative example and the references therein). The extension of this framework to endogeneize the determinants of the attention probabilities seems to be a very interesting avenue for future research.

Cross-Ownership Structures
We have assumed a framework in which the ownership structure is such that shareholders are external to the industry. In this section, we examine the robustness of the proposed alternative formulation by discussing an extension framework in which shareholders can include rival …rms of the industry. In particular, consider that there are K shareholders, indexed by k 2 f1; : : : ; N; : : : Kg, who may include not just shareholders that are external to the industry (and can engage in common-ownership), but also shareholders from the subset of …rms that are internal to the industry (and can engage in cross-ownership), both of which can hold …nancial and voting rights in multiple …rms of the industry.
In this setting, it is relatively straightforward to show that the weights of the proposed alternative formulation would be given by w jg = P k2 j u kj u kg u kj 0 for any j; g 2 = and j 6 = g, where u kj and u kj denote the ultimate …nancial and control rights, respectively, of external shareholder k in …rm j, which can be computed following the algorithm in Brito et al. (2018).

Measuring Anti-Competitive E¤ects
We now discuss competition policy applications of the proposed alternative formulation. We highlight two, related to the quanti…cation of the unilateral anti-competitive e¤ects associated to partial horizontal acquisitions. This alternative formulation can be straightforwardly incorporated into (a) the generalized HHI and GUPPI indicators proposed in Brito et al. (2018); and (b) the structural empirical methodology proposed in Brito et al. (2014).

Conclusions
We propose an alternative formulation to model the objective function of managers in the presence of horizontal shareholding. In this alternative formulation, managers would decide the strategy of the …rm by maximizing a weighted sum of the …rm's shareholders relative (rather than absolute) expected returns. We do not claim it to be preferred to O&S's formulation. We solely propose it as a microfounded alternative which avoids an allegedly unattractive feature of the O&S's formulation: that it may solely re ‡ect the interests of a small number of shareholders even if, collectively, those shareholders do not have full control of the …rm. Future empirical testing might help establish which formulation more accurately predicts …rm behavior.

Mathematical Appendix
In this mathematical appendix, we present the proofs of Propositions 1 to 4.

Proof of Proposition 1
First, absent horizontal shareholding, the manager of each …rm j would maximize the expected own-pro…t E [ j ], since kg = 0 for the subset of shareholders k who hold …nancial rights in …rm j and all j; g 6 = j. This implies w jg = 0 for all j,g 6 = j and, thus, that property (i ) holds.
Second, with non-in…nitesimal horizontal shareholding, the manager of each …rm j would internalize the impact of her …rm's strategy on the expected pro…t of rival …rm g when the shareholders that have …nancial rights in both …rms have also control rights in the …rm, since if kj 6 = 0 and kg >> 0 for at least one shareholder k, we have that w jg 6 = 0 for all j,g 6 = j. This implies that the manager of each …rm j would maximize E [ j ] + w jg E [ g ] and, thus, that property (ii ) holds.
Third, the weight w jg that the manager of each …rm j assigns to the expected pro…t of rival …rm g is continuous in kj , kj and kg for the subset of shareholders k with …nancial rights in …rm j, since the product, sum and quotient, respectively, of continuous functions is continuous. This implies that property (iii ) holds.
Fourth, the manager of each …rm j would maximize the expected pro…t of the industry when all shareholders that have …nancial rights in the …rm are fully diversi…ed across rivals, since if those shareholders are fully diversi…ed, for the subset of shareholders k who hold …nancial rights in …rm j, we have kj = kg = k and kj = kg = k for all j,g 6 = j. This implies w jg = P k2 j k k P k2 j k k = 1 for all j,g 6 = j and that the manager of each …rm j would maximize E [ j ] + P g2=;g6 =j E [ g ]. As such, property (iv ) holds.
Fifth, the weight w jg that the manager of each …rm j assigns to the expected pro…t of rival …rm g can be computed with the control rights of the shareholders in the …rm and their …nancial rights in the two …rms. The …nancial rights have clear empirical counterparts and can thus be measurable. The control rights have less clear empirical counterparts, but Azar (2016Azar ( , 2017 and Brito et al. (2018) show that they can be endogenously measured (depending on the particular assumptions) by the shareholders Banzhaf (1965)'s power index or by their voting rights, both of which have clear empirical counterparts and can thus be measurable.
Finally, the objective function of the manager of …rm j will approximate a weighted sum of (solely) the interests of the …rm's horizontal (non-horizontal) shareholders when non-horizontal (horizontal) shareholders are highly dispersed, for any given value of the control rights of the horizontal (non-horizontal) shareholders. In order to see why, let the subset of shareholders that hold …nancial rights in …rm j, j , be divided in two smaller subsets: the subset of horizontal shareholders, h j , and the subset of non-horizontal shareholders, nh j . This implies that the objective function of the manager of …rm j can be written as follows: As the ownership of each non-horizontal shareholder becomes dispersed among a collection of in…nitesimal identical shareholders, we have that P k2 nh j kj kj ! 0. As such, the objective function of the manager will weigh solely the interests of horizontal shareholders, even when the voting rights of those horizontal shareholders do not induce full control of the …rm: Similarly, as the ownership of each horizontal shareholder becomes dispersed among a collection of in…nitesimal identical shareholders, we have that P k2 h j kj kj ! 0 and P k2 h j kj kg ! 0. As such, the objective function of the manager will weigh solely the interests of non-horizontal shareholders (yielding an objective function proportional to the expected own-pro…t), even when the voting rights of those non-horizontal shareholders do not induce full control of the …rm: This implies a failure of property (vi ).

Proof of Proposition 2
The structure of this proof follows three steps.
First, we show that the objective function of the incumbent is strictly concave conditional on the strategy proposal of the challenger to the …rm and on the strategy proposals of the candidates to the rival …rms. Given that strategy proposals are, under Assumption 4, de…ned in a convex set, this implies that the incumbent's maximization problem has a unique maximum conditional on the strategy proposal of the challenger to the …rm and on the strategy proposals of the candidates to the rival …rms. Given the symmetry of the solution to the maximization problem of the two candidates to the …rm, we have that they will choose best-response functions that are, conditional on the strategy proposals of the candidates to the remaining …rms, symmetric with respect to the strategy proposal of the opponent candidate. This implies that the two candidates will choose the same strategy proposal for the …rm, conditional on the strategies proposals of the candidates to the rival …rms, i.e., they will choose the same best-response function to the strategy proposals of the candidates to the rival …rms. Since this common best-response function achieves, conditional on the strategies proposals of the candidates to the rival …rms, the unique maximum of the objective functions of the two candidates to the …rm, there are no unilateral incentives to deviation.
Second, we show that this common best-response function is the same as the best-response function that would arise from maximizing a weighted average of the relative expected returns of the …rm's shareholders conditional on the strategy proposals of the candidates to the rival …rms.
Finally, given that the strategy proposal of each candidate to the di¤erent …rms is, under Assumption 4, de…ned in a convex set and the expected gross returns (and since managerial diversion is …xed, also the expected returns) of the …rm's shareholders is, under Assumption 5, continuous, the best-response functions of the candidates to the di¤erent …rms are guaranteed to be upper-hemicontinuous, which implies that we can apply Kakutani's …xed point theorem to ensure that the Nash equilibrium exists.
We now address the sub-proof of the remaining points: (a) that the objective function of the incumbent is strictly concave conditional on the strategy proposal of the challenger to the …rm and on the strategy proposals of the candidates to the rival …rms; and (b) that the common best-response function is the same as the best-response function that would arise from maximizing a weighted average of the relative expected returns of the …rm's shareholders conditional on the strategy proposals of the candidates to the rival …rms. We …rst present these sub-proofs under Assumption 3a and then under Assumption 3b.

Under Assumption 3a
We begin by showing that the objective function of the incumbent is strictly concave conditional on the strategy proposal of the challenger to the …rm and on the strategy proposals of the candidates to the rival …rms.
a j denote the objective function of the incumbent under Assumption 3a. Under Assumption 2, shareholders are conditionally sincere, which implies that the incumbent of …rm j can choose her strategy proposal taking the strategies of the candidates to the remaining …rms as given.
The …rst order condition of this problem is, thus, given by: which makes use of the fact that, under Assumption 6: In turn, the second order condition is given by: which implies, given Assumption 5, that the objective function of the manager is strictly concave in xa j , conditional on the strategy proposal of the challenger to the …rm and on the strategy proposals of the candidates to the rival …rms.
We now show that the common best-response function is the same as the best-response function that would arise from maximizing a weighted average of the relative expected returns of the …rm's shareholders conditional on the strategy proposals of the candidates to the rival …rms, with voting rights as weights. This is established by the …rst order condition above: where kj = kj denotes the weight assigned by …rm j's manager to the relative expected return of shareholder k, measured by the voting rights of shareholder k in …rm j.

Under Assumption 3b
We again begin by showing that the objective function of the incumbent is strictly concave conditional on the strategy proposal of the challenger to the …rm and on the strategy proposals of the candidates to the rival …rms.
kj ~ j kj denote the objective function of the incumbent under Assumption 3b. Under Assumption 2, shareholders are conditionally sincere, which implies that the incumbent of …rm j can choose her strategy proposal taking the strategies of the candidates to the remaining …rms as given.
The …rst order condition of this problem is, thus, given by: which makes use of the fact that, under Assumption 6, E G j = 0 (note that this makes clear that, under Assumption 3b, we can relax Assumption 6 regarding the cumulative distribution function H j ( ): the only requirement is that E ~ j = 0 for each …rm j). In turn, the second order condition is given by: which implies, given Assumption 5, that the objective function of the manager is strictly concave in xa j , conditional on the strategy proposal of the challenger to the …rm and on the strategy proposals of the candidates to the rival …rms.
We now show that the common best-response function is the same as the best-response function that would arise from maximizing a weighted average of the relative expected returns of the …rm's shareholders conditional on the strategy proposals of the candidates to the rival …rms, with voting rights as weights. This is established by the …rst order condition above: where kj = kj denotes the weight assigned by …rm j's manager to the relative expected return of shareholder k, measured by the voting rights of shareholder k in …rm j.

Proof of Proposition 3
First, properties (i ) to (iii ) hold by the same arguments as were used in the proof presented for Proposition 1.
Second, the manager of each …rm j would maximize the expected pro…t of the industry when all shareholders that have …nancial rights in the …rm are fully diversi…ed across rivals, since if those shareholders are fully diversi…ed, for the subset of shareholders k who hold …nancial rights in …rm j, we have kj = kg = k and kj = kg = k for all j,g 6 = j.
This implies w jg = P k2 j k k k = P k2 j k = 1 for all j,g 6 = j and that the manager of each …rm j would maximize . As such, property (iv ) holds.
Third, the weight w jg that the manager of each …rm j assigns to the expected pro…t of rival …rm g can be computed with the control rights of the shareholders in the …rm and their …nancial rights in the two …rms. The …nancial rights have clear empirical counterparts and can thus be measurable. Proposition 2 shows that the control rights can be endogenously measured by the shareholders voting rights, which have clear empirical counterparts and can thus be measurable.
Finally, the objective function of the manager of …rm j will weigh solely the interests of the …rm's horizontal (non-horizontal) shareholders as the ownership of each non-horizontal (horizontal) shareholder becomes dispersed (among a collection of in…nitesimal identical shareholders) when the voting rights of the horizontal (non-horizontal) shareholders do induce full control.
In order to see why, let the subset of shareholders that hold …nancial rights in …rm j, j , be divided in two smaller subsets: the subset of horizontal shareholders, h j , and the subset of non-horizontal shareholders, nh j . This implies that the objective function of the manager of …rm j can be written as follows: As the ownership of each non-horizontal shareholder becomes dispersed among a collection of in…nitesimal identical shareholders, the objective function of the manager will weigh solely the interests of horizontal shareholders when the voting rights of those horizontal shareholders do induce full control of the …rm, i.e., when P k2 nh j kj = 0. Similarly, as the ownership of each horizontal shareholder becomes dispersed among a collection of in…nitesimal identical shareholders, the number of shareholders in h j increases, but the ratio kg kj of each new in…nitesimal identical shareholder will be identical to the corresponding ratio of the previous (non-dispersed) shareholder. As a consequence, the objective function of the manager will weigh solely the interests of non-horizontal shareholders when the voting rights of those non-horizontal shareholders do induce full control of the …rm, i.e., when P k2 h j kj = 0. As such, property (vi ) is satis…ed.

Proposition 4
The structure of this proof follows the same three steps as the proof presented for Proposition 2. As such, we just have to address the sub-proof of the remaining points: (a) that the objective function of the incumbent is strictly concave conditional on the strategy proposal of the challenger candidate to the …rm and on the strategy proposals of the candidates to the rival …rms; and (b) that the common best-response function is the same as the best-response function that would arise from maximizing a weighted average of the relative expected returns of the …rm's shareholders conditional on the strategy proposals of the candidates to the rival …rms. We …rst present these sub-proofs under Assumption 3a and then under Assumption 3b.

Under Assumption 3a
We begin by showing that the objective function of the incumbent is strictly concave conditional on the strategy proposal of the challenger to the …rm and on the strategy proposals of the candidates to the rival …rms.
1 2 when xa = x b , both for all k. This …rst-order condition can, in turn, be rewritten as: where jk denotes the number of subsets in } j in which shareholder k enters and 2`j 1 jk denotes the number of subsets in } j in which shareholder k does not enter. Finally, consider that jk can be divided in two terms: the number of subsets in } j in which shareholder k enters and is pivotal, p jk , and the number of subsets in } j in which shareholder k enters and is not pivotal, hj > 0:5. The number of subsets in } j in which shareholder k enters and is not pivotal is, by construction, equal to the number of subsets in } j in which shareholder k does not enter. This implies that p jk = 2`j 1 jk and that the …rst-order condition can be rewritten as: where p jk 2`j 1 denotes the Banzhaf power index associated to shareholder k in …rm j. This establishes that the common bestresponse function is the same as the best-response function that would arise from maximizing a weighted average of the relative expected returns of the …rm's shareholders conditional on the strategy proposals of the candidates to the rival …rms: where kj =

Under Assumption 3b
Proposition 2 holds in this case. As discussed in the proof above, under Assumption 3b, the only requirement regarding the cumulative distribution function H j ( ) is that E ~ j = 0 for each …rm j, which is satis…ed if~ j = 0 for each …rm j, establishing the result.