Modelling the Objective Function of Managers in the Presence of Overlapping Shareholding

The objective function of managers in the presence of overlapping shareholding may differ from the traditional own-firm profit maximization, as they may internalize the externalities their strategies impose on other firms. The dominant formulation of the objective function in such cases has, however, been critiqued for yielding counter-intuitive profit weights when the ownership of non-overlapping shareholders is highly dispersed. In this paper, we examine this issue. First, we make use of a probabilistic voting model (in which shareholders vote to elect the manager) to microfound an alternative formulation of the objective function of managers, which solves the above-mentioned criticism. Second, we apply the two formulations to the set of S&P500 firms. We show that ownership dispersion of non-overlapping shareholders is, in fact, a relevant empirical issue, which may induce an over-quantification of the profit weights computed from the dominant formulation, particularly under a proportional control assumption.


Introduction
The assumption of own-firm profit maximization is key, (at least) since Fisher (1930)'s separation theorem, to most literature in corporate finance and industrial organization. However, the validity of this assumption has been recently questioned due to the increase, documented for a multitude of industries and economies, particularly since 2000, of overlapping shareholding (Azar, Schmalz and Tecu, 2018; Newham, Seldeslachts and Banal-Estañol, 2019; Azar, Raina and Schmalz, 2021; Backus, Conlon and Sinkinson, 2021a). 1 The reason being that if firms impose externalities on one another, overlapping shareholding may imply a failure of the competitiveness condition, established by Hart (1979) to be essential for shareholders, regardless of their preferences, to unanimously agree on own-firm profit maximization. 2 In order to see why, note, for example, that if firm A imposes a negative externality on firm B, a shareholder of firm A who also holds shares in firm B typically wants the manager of firm A to pursue a less aggressive strategy than the strategy desired by a shareholder with no holdings in firm B.
The managers of firms with overlapping shareholders, rather than maximizing own profit, may therefore weigh the eventual conflicting objectives of their shareholders. This implies that they may internalize (to some degree) the externalities their strategies impose on other firms (Rotemberg, 1984;Hansen and Lott, 1996). This internalization can decrease the incentives to compete and can naturally lessen market competition. 3 In order to empirically examine the impact of overlapping ownership on market outcomes, we must quantify the above-mentioned induced internalization. To do so, the literature proposes three different approaches (see Backus, Conlon and Sinkinson, 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 (see Appendix B in Gilje, Gormley and Levit, 2020 for a review). 4 The second approach places additional structure and 1 An increase very much driven by the growth in transient and quasi-indexer institutions, which responded to the demand of the general public for a low-cost and convenient way to hold the market portfolio, as suggested by the equilibrium results from the modern portfolio theories (Azar, 2020). 2 Although 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, 2018b for a formal model). 3 For example, Bresnahan and Salop (1986), Dietzenbacher, Smid and Volkerink (2000), Shelegia and Spiegel (2012), and Brito, Ribeiro and Vasconcelos (2019), among others, show that the internalization induced by overlapping ownership among firms with horizontal relationships (and thereby are likely to impose a negative externality on each other) can directly lead to higher product prices and lower output levels. The structure placed by the second and third approaches (above) is instrumental in deriving economically meaningful claims. And under both those approaches, the formulation of the objective function of managers is key. This formulation is, however, non-trivial. In order to see why, consider, for example, that firm A has four shareholders, each holding 25% of the firm, and that one of those shareholders also holds 20% of firm B. If firm A imposes an externality on firm B, what would the mathematical formulation of the objective function of the manager of firm A be? What weight would the manager of firm A assign to the profit of firm B?
The dominant formulation of the objective function of managers in the presence of overlapping shareholders is due to O'Brien and Salop (2000). Incorporating features from both Rotemberg (1984) and Bresnahan and Salop (1986), they assume that the manager of a firm with overlapping shareholders would decide the strategy of the firm to maximize a controlweighted sum of the expected returns of the firm's shareholders. Azar (2016Azar ( , 2017,  and Moskalev (2019) 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 differing strategy proposals to the firm.
Candidates are assumed to care about holding office. 5 In turn, shareholders are assumed to care about the returns that result from the different strategy proposals and to have an additive profit-irrelevant bias for (or against) the challenger. 6 Voting is probabilistic in the sense 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. 5 Azar (2017) considers the case in which candidates choose strategy proposals to maximize their vote share while Azar (2016),  and Moskalev (2019) consider the case in which candidates choose strategy proposals to maximize their expected utility from corporate office. 6 Azar (2016Azar ( , 2017 and  consider the case in which this bias is independent (and identically) distributed across shareholders while Moskalev (2019) considers the case in which the bias can be correlated across shareholders.
that the bias, while known to voters, is unobserved by candidates, who treat it as random.
This microfoundation (a) is consistent with empirical evidence establishing that shareholders voting impacts the objective function of managers (Aggarwal, Dahiya and Prabhala, 2019) and that managerial wealth is more sensitive to own performance in the absence of overlapping ownership (Antón et al., 2021); and (b) provides an endogenous measure of shareholders corporate control within the firm. 7 The dominant formulation, although heavily used in the literature, has also been critiqued for yielding counter-intuitive profit weights when the ownership of non-overlapping shareholders is highly dispersed (see, for example, Gramlich  ownership when the ownership of non-overlapping shareholders is highly-dispersed, then empirical studies that use that dominant formulation will suffer from attenuation bias that will tend to create empirical results that understate the magnitude and statistical significance of the marginal effect of overlapping ownership on market competition. This could cause analysts to incorrectly reject or underestimate an empirical connection between overlapping shareholding and anticompetitive effects (Elhauge, 2020). Further, even if one conclude that empirical studies that use current formulations validly find an effect despite having to overcome such attenuation bias, using a more accurate formulation should enable studies to predict effects better than other formulations and enable antitrust enforcers to better distinguish markets where overlapping shareholding is anticompetitive from those markets where it is not (Elhauge, 2020).
In this paper, we examine this issue, from both a theoretical and empirical perspective.
From a theoretical perspective, we examine the role of the profit-irrelevance assumption of the bias of shareholders on the derivation of the objective function of managers in the presence 7 Azar (2017)  while maintaining the remaining assumptions of the literature. 8 We show that the profitirrelevance assumption plays a key role on how ownership dispersion (of both overlapping and non-overlapping shareholders) is mapped into profit weights. 9 In particular, we show that a profit-relevant bias microfounds an alternative formulation of the objective function of managers in the presence of overlapping shareholders in which the manager of a firm with overlapping shareholders would decide the strategy of the firm to maximize a controlweighted sum of the relative expected returns of the firm's shareholders. Under this proposed alternative formulation, the weight assigned by the manager to the profit of other firms (in which overlapping shareholders hold shares in) will never reflect solely the interests of the (the non-dispersed) overlapping shareholders, unless the dispersion yields overlapping shareholders the full control of the firm. As such, it solves the above-mentioned criticism regarding the dominant formulation.
From an empirical perspective, we apply the two formulations to the set of S&P500 firms (following Backus, Conlon and Sinkinson, 2021a) from 1999 to 2017. We show that the dispersion of shareholders' ownership, particularly of non-overlapping shareholders, is a relevant empirical issue. We show also that, in those cases, the dominant formulation may, in fact, over-quantify profit weights, particularly under a proportional control assumption.
This empirical evidence shows that the theoretical issue noted above is indeed of theoretical relevance. The dispersion of non-overlapping shareholders is sufficiently low that empirical studies that rely on the dominant formulation could well result in attenuation bias that causes 8 We 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. 9 The 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 control rights).
them to underestimate the magnitude and statistical significance of the marginal effect of overlapping shareholding on market outcomes. If so, using the alternative formulation proposed above could be a preferable option that could result in more accurate estimates of the effects of overlapping shareholding. This alternative formulation would also be particularly relevant for competition agencies because it suggests that using the alternative formulation could lead to more accurate conclusions in specific markets about whether overlapping shareholding is likely to produce anticompetitive effects. If so, the alternative formulation will accurately conclude in more markets that the effects are not likely to be anticompetitive, thus more accurately distinguishing those markets from other markets where the effects are more likely to be anticompetitive. This issue is also particularly relevant for antitrust policy because overlapping ownership "has stimulated a major rethinking of antitrust enforcement" (Elhauge, 2016; Scott Morton and Hovenkamp, 2018; Hemphill and Kahan, 2020), which must naturally be based on empirical evidence as accurate as possible.
The remainder of the paper is organized as follows. Section 2 introduces the generalized probabilistic voting model used to derive the objective function of managers and the implied profit weights. Section 3 applies the profit weights established in Section 2 to the S&P 500 index constituents and discusses policy implications of the results. Section 4 presents extensions to the generalized probabilistic voting model to account for shareholders inattention and cross-ownership structures. Section 5 concludes.

Theoretical Framework
This section introduces the generalized probabilistic voting model used to derive the object-

Setup
There are K shareholders, indexed by k ∈ Θ ≡ {1, . . . , K}, and N firms, which impose externalities on one another, indexed by j ∈ ≡ {1, . . . , N }, whose total stock is composed of voting stock and non-voting (preferred) stock. Both stock give the holder the right to a share of the firm's profits, but only the former gives the holder the right to vote in the firm's shareholder assembly. The holdings φ kj ∈ [0, 1] of total stock of shareholder k in firm j, regardless of whether it be voting or non-voting stock, capture her financial rights to the firm's profits. The holdings υ kj ∈ [0, 1] of voting stock of shareholder k in firm j, capture her voting rights in the firm. These voting rights may not necessarily coincide with her control rights in the firm, γ kj ∈ [0, 1], which refer to her rights to influence the decisions of firm j, to be discussed below. 10 The ownership structure of the different firms is such that a subset of shareholders can hold general financial and voting rights in multiple firms. 11 This overlapping shareholding can induce a conflict in the firm-specific interests of shareholders, which managers must weigh.
Finally, the profit of each of the different firms is assumed to be a function not only of the strategies of all the firms but also of a state of nature. This implies that firms' profits and, consequently, shareholders' returns -because they are a function of the profits of the firms in which they hold financial rights -are random.

Voting Model
We Shareholders and candidates are assumed to play the following two-stage game. In the first stage, candidates to all firms, who are assumed to be opportunistic in the sense their only motivation is to hold office, compete for the voting rights of shareholders by -simultaneously and noncooperatively -proposing a strategy for their firm, which is assumed binding in line with the literature on electoral competition (Downs, 1957;Lindbeck and Weibull, 1987;Polo, 1998;and Persson and Tabellini, 2000). Let x a j ∈ Ω j and x b j ∈ Ω j denote the strategy proposals of the incumbent and the challenger to firm j, respectively, where Ω j denotes the strategy space available to the candidates, which can refer to any decision variable(s) -e.g., quantity, price, R&D investment, etc. -of firm j. In the second stage of the game, the shareholder assemblies of all firms are simultaneously held and shareholders vote to elect the manager of each firm. Let m j ≡ {a j , b j } denote the identity of the manager elected to firm j.
Naturally, this voting model constitutes a reduced form model of the decision making process and the knowledge structure within the firm. The manager may not be elected directly by shareholders and operational decision variable(s) may be often decided, not by top managers, but by middle managers, who may not know the extent of the holdings of the firm's shareholders in other firms. Antón et al. (2021) show that, even in those cases, managerial incentives can serve as a mechanism (which requires no communication or 10 Short-sales are not allowed and so financial, voting and control rights are non-negative. 11 We 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 4. coordination between the different players) that links overlapping ownership with operational decision variable(s).

Shareholders Voting
We begin by addressing the equilibrium regarding the voting behavior of shareholders. Shareholders are assumed to care about the utility derived from their expected returns and, as such, vote -simultaneously and noncooperatively -for the candidate whose strategy proposal maximizes their expected returns, randomizing between the two in case of indifference.
We consider that the utility u k (x, m) of each shareholder k to be a function of her expectation regarding the return from her financial rights, which will be a function of the winning strategy proposals in all the firms x = (x 1 , . . . , x j , . . . , x N ) and the identity of the corresponding elected managers m = (m 1 , . . . , m j , . . . , m N ) : where E k (R k (x, m)) denotes the expectation of shareholder k regarding her return R k (x, m).
We model this expectation to be the sum of two components: a common component and (b) an additive shareholder-specific expectation bias, as follows: where E (R k (x)) = j∈ (φ kj E (Π j (x))) denotes the common component, rooted on a common expectation E (Π j (x)), across shareholders, regarding the profits Π j (x) of each firm j, assumed to be publicly generated by, for example, the documentation distributed and discussed in the shareholder assemblies of the different firms, and B k (m) denotes the expectation bias of shareholder k for (or against) the challengers of (potentially) each firm. . However, while in a political electoral setting it is reasonable to assume that voters have an idealogical bias towards a candidate, in a corporate finance setting, it may be less so.
Although one can argue that the assumption can be rooted on the fact that the credibility (or lack of credibility) of the incumbent, being already in office, is known to shareholders, while that of the challenger is not, one can also argue that if shareholders indeed care about the returns that result from the different strategy proposals (and, thereby, ultimately care about profits), there is not an obvious reason why the bias should not be profit-relevant.
We contribute to the literature by considering a more general formulation than the one the challengers of (potentially) each firm is given by: where ξ kj ≶ 0 denotes the bias of shareholder k for (or against) the challenger of firm j and 1(m j = b j ) denotes a dummy variable that takes the value 1 if the challenger is elected manager of firm j. λ ∈ {0, 1} controls the profit relevance of the bias. When λ = 0, we have that B k (m) = j∈ 1 (m j = b j ) ξ kj , which implies that: and, as a consequence, that the bias is profit-irrelevant, as in Azar (2016, 2017), Brito et al.
(2018) and Moskalev (2019). When λ = 1, we have that B k (m) = j∈ φ kj 1 (m j = b j ) ξ kj , which implies that: and, as a consequence, that the bias is profit-relevant (as it impacts the shareholder's specific expectation regarding the firm's profit). 12 We assume that shareholders vote, in each firm's shareholder assembly, for the candidate whose strategy proposal, given their bias, maximizes their utilities, randomizing between the two in case of indifference. Following Alesina and Rosenthal (1995), Azar (2016) and , we also assume the following regarding this voting behavior.
Assumption 1 implies that the vote of shareholders is, conditional on the equilibrium strategy proposals of the candidates to the remaining firms, deterministic, as follows: shareholder k will vote for firm j's incumbent with probability , and will randomize between the two candidates with equal probability if

Candidates Strategy Proposals
Having described the second stage of the game, we now address the first stage, in which candidates simultaneously choose strategy proposals. To do so, we follow Lindbeck and whereξ j denotes a component common to all shareholders of firm j, which induces a correlation among the biases of all shareholders of the firm, andξ kj denotes a component specific to shareholder k and firm j.ξ j is independently drawn across firms from a distribution with density function h j (·), cumulative distribution function H j (·) and support in the interval , whileξ kj is independently drawn across shareholders and firms from a distribution with density function g j (·), cumulative distribution function G j (·) and support in the We begin by addressing the choice of strategy proposals by candidates under Assumption 2. In this setting, the incumbent chooses x a j to solve: while the challenger chooses x b j so to solve: 13 This 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 j to be a shareholder-specific weighted sum of M j common biases. 14 We do not allow τ j = 0, i.e, that ξ kj =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. 15 We 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 P r ka j (x a , m a , x b , m b ) and P r kb j (x a , m a , x b , m b ) denote the probability that, in the candidates perspective, shareholder k votes for the incumbent and the challenger of firm j, to see that the solution to the maximization problem of the two candidates to firm j is symmetric. As such, we characterize -for simplicity of exposition -solely the incumbent's problem. To do so, we must derive P r ka j (x a , m a , x b , m b ). Using the law of total probability, we can write P r ka j (x a , m a , x b , m b ) as follows: where P r ka j x a , m a , x b , m b ,ξ j denotes the probability that, in the candidates perspective, shareholder k votes for the incumbent of firm j conditional on the common component of the biasξ j , which, in turn, is given by: 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 (m g = b g ) ξ kg ), enter the utility obtained from both candidates.
We now address the choice of strategy proposals by candidates under Assumption 3. In this setting, the incumbent chooses x a j so to solve: while the challenger chooses x b j so to solve: where P r(m j = a j |x a , x b ) and P r(m j = b j |x a , x b ) denote the probability that the incumbent and the challenger, respectively, are elected while Ξ a j and Ξ b j denote the utility that the incumbent and the challenger, respectively, expect to accrue conditional upon being elected.
, it is straightforward to see that the solution to the maximization problem of the two candidates to firm j is symmetric.
As such, we characterize -for simplicity of exposition -solely the incumbent's problem. To do so, we must derive P r(m j = a j |x a , x b ). Let j denote the number of shareholders with voting rights in firm j, ℘ j denote all the 2 j −1 possible subsets of those shareholders that can award the majority of votes to a candidate and Θ ı j ∈ ℘ 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 P r(m j = a j |x a , x b ) just sums the probabilities with which she is elected by each subset Θ ı j , P r(m j = a j |x a , x b , Θ ı j ), as follows: where the last equality uses the law of total probability to write P r m j = a j |x a , x b , Θ ı j in terms of P r m j = a j |x a , x b , Θ ı j ,ξ j , which conditions on the common component of the bias ξ j . Further, given that conditional onξ j , the shareholders-specific biasesξ kj for k ∈ Θ j are independently distributed, we can write P r m j = a j |x a , x b , Θ ı j ,ξ j as the product of the voting probabilities of the corresponding shareholders, as follows:

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  strictly concave in firm j's strategy x j ∈ x a j , x b j , conditional on the strategies of the remaining firms.
is the cumulative distribution function of an uniform distribution over the range [− ψ j /2, ψ j /2] for each firm j.
is the cumulative distribution function of an uniform distribution over the range [− τ j /2, τ j /2], with τ j sufficiently large for each firm j.
Assumptions 5, 6 and 7 can ensure that the objective function of candidates is strictly concave conditional on the strategy proposals of other candidates (to the firm and to other firms). 16 As strategy proposals are, under Assumption 4, defined in a convex set, this implies that the maximization problem of candidates has, conditional on the strategy proposals of other candidates, a unique maximum. Given the symmetry of the solution to the maximization problem of the two candidates to the firm, we have that they will choose best-response functions that are, conditional on the strategy proposals of the candidates to the remaining firms, 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 firm, conditional on the strategies proposals of the candidates to the other firms, i.e., they will choose the same best-response function to the strategy proposals of the candidates to the other firms.
Proposition 1 below characterizes the pure-strategy Nash equilibrium for the candidates Proposition 1. There exists a pure-strategy Nash equilibrium for the candidates strategy proposals' game that is entirely equivalent to the pure-strategy Nash-equilibrium in the case in which each candidate maximizes the following objective function: where Θ j denotes the subset of shareholders that hold financial rights in firm j and w jg denotes the (normalized) weight that candidates assign to the expected profit of firm g for 16 Assumption 7 requires the support of G j (·) to be sufficiently large (relative to their argument -not to profits) so to rule out corner solutions for probabilities any j, g ∈ : Under Assumptions 1, 2, 4, 5 and 7, γ kj is measured by the voting rights of shareholder k in firm j: γ kj = υ kj . Under Assumptions 1, 3, 4, 5, 6 and 7, γ kj is measured by the following power index of shareholder k in firm j: γ kj = β kj / h∈Θ j β hj , with β kj given by: where ℘ p kj denotes the subsets of Θ j that can award victory to a candidate of firm j in which shareholder k enters and is pivotal, #Θ ı j denotes the number of shareholders in Θ ı j , C #Θ ı j −1 i denotes the number of combinations of i elements taken from a set with #Θ ı j − 1 elements, and m j −#Θ ı j +i denotes the j − #Θ ı j + i th raw moment of an uniformly distributed variable over the range Proof. See Appendix.
Proposition 1 establishes that the two candidates would choose the same strategy proposal for each firm j, conditional on the strategies of the candidates to the remaining firms. In particular, they would choose the strategy proposal of each firm j to maximize a weighted sum of the expected profits of (potentially) all the firms. We now address the mathematical properties and the empirical applicability of this (unique) objective function of managers.
Corollary 1 establishes the mathematical properties. (iii ) The weight that managers assign to the expected profit of other firms is continuous on the financial and control rights of the shareholders that have financial rights in the firm.
(iv ) The weight that managers assign to the expected profit of other firms is one when all the shareholders that have financial rights in the firm are fully diversified across firms. Properties (i ) to (iv ) are satisfied independently of the assumptions regarding the objective function of candidates and the bias of shareholders for (or against) the challenger.
In contrast, property (v ) is satisfied solely if the bias of shareholders for (or against) the challenger is profit-relevant (λ = 1). 17,18,19 In order to illustrate why, we now describe the objective function of the manager (and the implied profit weights) under the two assumptions. When the bias is profit-irrelevant (λ = 0), as in Azar (2016Azar ( , 2017,  and Moskalev (2019), Proposition 1 establishes that managers would decide the strategy of the firm to maximize a control-weighted sum of the expected returns of the firm's shareholders, as follows: where the (normalized) weight that they assign to the expected profit of firm g for any j, g ∈ is given by: which then replicates the dominant formulation. Dividing the subset of shareholders Θ j that hold financial rights in firm j in two smaller subsets: the subset of overlapping shareholders Θ o j and the subset of non-overlapping shareholders Θ no j , we can rewrite this profit weight as: dispersed among a collection of infinitesimal identical shareholders that is equally large in aggregate, w jg tends to reflect solely the interests of the non-overlapping shareholders as 17 This implies that the assumption regarding the profit-irrelevance or profit-relevance of the bias of shareholders for (or against) the challenger is the only key assumption with a direct impact on how ownership dispersion (of both overlapping and non-overlapping shareholders) is mapped into profit weights. It does not imply, however, that the assumptions regarding the non-correlation or correlation of the bias of shareholders for (or against) the challenger does not have such an impact. They do, but it is indirect via the impact of ownership dispersion on the (endogenous) measure of control rights. 18 If we allow λ ∈ [0, 1] so that the bias is a weighted average of the two elements, this result remains valid, with property (v ) failing for λ < 1. 19 A profit-relevant bias is also instrumental in obtaining a formulation that is invariant to the distribution of ownership among non-overlapping shareholders, which may constitute an important empirical advantage. In turn, when the bias is profit-relevant (λ = 1), Proposition 1 establishes that managers would decide the strategy of the firm to maximize a control-weighted sum of the relative expected returns of the firm's shareholders, as follows: 20 where E(R k (x)) /φ kj denotes the relative expected return of shareholder k and the (normalized) weight that they assign to the expected profit of firm g for any j, g ∈ is given by: When candidates choose strategy proposals to maximize their expected utility from corporate office, the endogenous measure of the control rights of shareholders depends on the assumption regarding the non-correlation or correlation of the bias of shareholders for (or 21 Further, no assumption regarding the distribution and support of the common component of the bias is required to derive this result. 22 As we may expect a shareholder who holds 10% of the voting rights in a firm to have effective control if each of the remaining shareholders hold a tiny amount of the firm's voting rights.
where the first equality makes use of the fact that m j −#Θ ı j +i = 1 2 j −#Θ ı j +i under ψ j = 0, the second equality makes use of the binomial theorem, which establishes that

Data Description
We combine data from multiple sources. First, we use historical quarterly ownership data, Backus Conlon and Sinkinson (2021a), we consolidate the holdings of all BlackRock entities in each firm and quarter. 26 Further, we drop, in each quarter, all the firms in which institutional shareholders, in aggregate, report holding more than 100% of shares outstanding.
Finally, we use historical constituent information of the S&P 500 from CRSP, which we 23 Backus, Conlon and Sinkinson (2021a) lightly cleaned the data by dropping, in each quarter, firms in which (a) a single institutional shareholder reports holding more than 50% of shares outstanding; and (b) institutional shareholders, in aggregate, report holding more than 120% of shares outstanding. 24 We use this data instead of the competing Thomson Reuters S34 database since the latter under-reports institutional ownership in the 13F filings (as described by Ben-David et al., 2018), even when using the (July 2018) update provided by Wharton Research Data Services (as described by Backus, Conlon and Sinkinson, 2021a). 25 In this database, each firm in each quarter can be identified by both nCUSIP and PERMNO. 26 The CIK identifiers of the consolidated BlackRock entities is: 913414, 1003283, 1006249, 1060021, 1085635, 1086364, 1305227 and 1364742.
merge, for each firm (identified by PERMNO) and quarter, with the resulting data from above. This constitutes our S&P 500 final data.

Economy-Wide Analysis
We use the data above to compute the profit weights that the managers of S&P 500 firms assign to the expected profit of each of the remaining S&P 500 firms, according to the objective function of managers established in Proposition 1 for both λ = 0 and λ = 1. As the S&P 500 is designed to reflect the U.S. economy, this mirrors an economy-wide analysis.  In order to examine whether shareholder ownership dispersion is an issue for S&P 500 firms, we apply a (common) measure of concentration, the Herfindahl-Hirschman index, to shareholder's financial rights, which following Backus, Conlon and Sinkinson (2021a) we label IHHI. We do so for each S&P 500 firm j and quarter, decomposing the IHHI on the components associated to the ownership of non-overlapping and overlapping shareholders, as follows:   ownership of non-overlapping shareholders is below the 5th and 25th percentiles, the overall average profit weight reflects 98.7% and 95.6%, respectively, of the interests of overlapping shareholders, which seems counter-intuitive as, on aggregate, those overlapping shareholders only hold, on average, 51.4% and 54.1%, respectively, of the control rights. 28 If we con-28 Note that Figure 3, Panel A1 and B1 consider the case in which the control rights of shareholders are measured by their voting rights, a measure of control rights that does not converge to 100% as the voting rights of a shareholder approach 50%. As such, when on aggregate, overlapping shareholders hold, on average, 51.4% or 54.1% (as described above) of the control rights, they do not have full control of the firm, even if pooled into a single shareholder.    29 The plots also show that the fraction of the interests of overlapping shareholders reflected in the dominant formulation has increased slightly over time, as both the dispersion of the ownership of non-overlapping shareholders and the aggregate corporate control of overlapping shareholders have also increased slightly over time.

Within-Industry Analysis
The above economy-wide analysis can, naturally, be critiqued because it encompasses all cross-pairs of S&P 500 firms and not solely of those firms that impose externalities on one another. In order to address this issue, we now perform a within-industry analysis, considering solely firms with horizontal relationships and, thus, are likely to impose a negative externality on one another.
To do so, we use historical annual horizontal relatedness scores from Hoberg and Phillips

Policy Implications
The results above suggest that (a) shareholder ownership dispersion, particularly of non-  2018; Hemphill and Kahan, 2020), which must naturally be based on empirical evidence as accurate as possible.

Extensions
In this section, we introduce and discuss extensions to the theoretical framework used to derive the objective function of managers established in Proposition 1.

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 objective function of managers established in Proposition 1 by discussing an extension framework in which shareholders can either be attentive or inattentive to those proposals. In particular, we follow Gilje, Gormley and Levit (2020) in considering that each shareholder k is attentive to the strategy proposals of firm j's candidates with probability δ kj and inattentive with probability 1−δ 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 , 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 ρ k and will vote for the challenger with probability 1 − ρ k .
In this setting, it is relatively straightforward to show that the (normalized) weight that the manager of firm j would assign to the expected profit of firm g for any j, g ∈ would be given by: where γ a kj denotes the control rights of shareholder k in firm j, which now incorporate (additionally) the attention probabilities δ kj . Under Assumption 2 and λ = 0, we obtain γ a kj = υ kj δ kj , yielding a weight that is qualitatively similar to the measure proposed by Gilje, Gormley and Levit (2020) to capture the impact of overlapping ownership on managerial incentives. Although the attention probabilities δ kj do not have a direct empirical counterpart, they can be modeled to be a function of a multitude of observed firm and shareholder factors (for example, the importance of firm j in shareholder k's investment portfolio) and estimated using observed voting behavior (see Gilje, Gormley and Levit, 2020 for an illustrative example and the references therein).

Cross-Ownership Structures
We have assumed a framework in which shareholders are external, in the sense that we In this setting, it is relatively straightforward to show that the (normalized) weight that the manager of firm j would assign to the expected profit of firm g for any j, g ∈ would be given by: where φ u kj and γ u kj denote the ultimate financial and control rights, respectively, of external shareholder k in firm j, which can be computed following the algorithm in .

Conclusions
We examine the objective function of managers in the presence of overlapping shareholding.
We do so, from both a theoretical and empirical perspective. From a theoretical perspective, we make use of a probabilistic voting model in which shareholders vote to elect the manager from two potential candidates (the incumbent and a challenger) with conceivably different strategy proposals to microfound a proposed alternative formulation of the objective function of managers in which (in contrast to the dominant formulation) the manager of a firm with overlapping shareholders would decide the strategy of the firm to maximize a controlweighted sum of the relative expected returns of the firm's shareholders. In particular, to do so, we generalize the probabilistic voting model typically used in the literature to allow the bias of shareholders for (or against) the challenger to be both profit-irrelevant or profitrelevant. We show that a profit-relevant bias microfounds an alternative formulation which can cope better with ownership dispersion in the sense that (in contrast to the dominant formulation) it will never reflect solely the interests of a set of (non-dispersed) shareholders, unless the dispersion yields those shareholders the full control of the firm.
From an empirical perspective, we apply the two formulations to the set of S&P500 firms.
We show that the dispersion of shareholders' ownership, particularly of non-overlapping shareholders, is a relevant empirical issue. We show also that the dominant formulation, in such cases, may, in fact, over-quantify profit weights, particularly under a proportional control assumption. This, in turn, suggests that, when shareholders' ownership is highly dis-

Mathematical Appendix
In this mathematical appendix, we present the proofs of Proposition 1 and Corollary 1.

Proof of Proposition 1
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 firm and on the strategy proposals of the candidates to the other firms. Given that strategy proposals are, under Assumption 4, defined 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 firm and on the strategy proposals of the candidates to the other firms. Given the symmetry of the solution to the maximization problem of the two candidates to the firm, we have that they will choose best-response functions that are, conditional on the strategy proposals of the candidates to the remaining firms, 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 firm, conditional on the strategies proposals of the candidates to the other firms, i.e., they will choose the same best-response function to the strategy proposals of the candidates to the other firms. Because this common best-response function achieves, conditional on the strategies proposals of the candidates to the other firms, the unique maximum of the objective functions of the two candidates to the firm, 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, conditional on the strategy proposals of the candidates to the other firms, the objective function established in the proposition.
Finally, given that the strategy proposal of each candidate to the different firms is, under Assumption 4, defined in a convex set and the common expectation of each shareholder is, under Assumption 5, continuous, the best-response functions of the candidates to the different firms are guaranteed to be upperhemicontinuous, which implies that we can apply Kakutani's fixed 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 firm and on the strategy proposals of the candidates to the other firms; and (b) that the common best-response function is the same as the best-response function that would arise from maximizing, conditional on the strategy proposals of the candidates to the other firms, the objective function established in the proposition. We do so considering, in turn, Assumptions 2 and 3.
Consider, first, Assumption 2. We begin by addressing (a) and show that the objective function of the incumbent is strictly concave conditional on the strategy proposal of the challenger to the firm and on the strategy proposals of the candidates to the other firms. Let aj denote the objective function of the incumbent, as follows: which makes use of the fact that, under Assumption 7, G j (·) is the cumulative distribution function of an uniform distribution over the range [− τj /2, τj /2] with τ j sufficiently large such that: for allξ j ∈ [− ψj /2, ψj /2], x am ∈ Ω m and x bm ∈ Ω m ∀j, m ∈ .
Under Assumption 1, shareholders are conditionally sincere, which implies that the incumbent of firm j can choose her strategy proposal taking the strategies of the candidates to the remaining firms as given. The first order condition of this problem is, thus, given by: while the second order condition is given by: which implies that, because under Assumption 5 we have that ∂ 2 E(R k (xa)) /∂x 2 a j < 0, the objective function of the manager is strictly concave in x aj , conditional on the strategy proposal of the challenger to the firm and on the strategy proposals of the candidates to the other firms.
We now address (b) and show that the common best-response function is the same as the best-response function that would arise from maximizing, conditional on the strategy proposals of the candidates to the other firms, the objective function established in the proposition. To do so, note that because the two candidates will choose the same best-response function, in equilibrium, we have E (R k (x a )) = E (R k (x b )) = E (R k (x)) for all k ∈ Θ j . As a consequence, we have from the first order condition above that: where γ kj = υ kj is measured by the voting rights of shareholder k in firm j.
Consider, now, instead, Assumption 3. We begin by readdressing (a) and show that the objective function of the incumbent is strictly concave conditional on the strategy proposal of the challenger to the firm and on the strategy proposals of the candidates to the other firms. Let aj denote the objective function of the incumbent, as follows: Under Assumption 1, shareholders are conditionally sincere, which implies that the incumbent of firm j can choose her strategy proposal taking the strategies of the candidates to the remaining firms as given. In order to compute the first order condition of this problem, it will help decompose, with reference to any given shareholder k, the set ℘ j of all the 2 j −1 possible subsets of those shareholders that can award the majority of votes to a candidate into three subsets: (a) the subsets where shareholder k enters and is pivotal: ℘ p kj , (b) the subsets where shareholder k enters and is not pivotal: ℘ np kj , and (c) the subsets where shareholder k does not enter: ℘ n kj . As such, we have, for any given shareholder k, that ℘ j = ℘ p kj ∪ ℘ np kj ∪ ℘ n kj . This implies that we can write Θ ı j ∈℘j P r m j = a j |x a , x b , Θ ı j ,ξ j as follows: Because all the subsets in ℘ n kj can award the majority of votes to a candidate, if we add shareholder k to ℘ n kj , the corresponding new subsets will be able to award as well the majority of votes to a candidate, with shareholder k not being pivotal. As such, we have that, for all Θ ı j ∈ ℘ n kj , Θ ı j ∪ {k} ∈ ℘ np kj , with the number of subsets in ℘ n kj being equal to the number of subsets in ℘ np kj . This implies -letting, for notation compactness, for the purposes of this proof, P r c raj denote P r raj x a , m a , x b , m b ;ξ j -that: where the second equality factors out the (conditional) probability associated to shareholder k. As a con-sequence, we can write Θ ı j ∈℘j P r m j = a j |x a , x b , Θ ı j ,ξ j as follows: Having this result in mind, we can now address the first order condition of the problem. We can write this condition as follows: Because there is always at least one shareholder that is pivotal, we can rewrite this condition, using the result above for Θ ı j ∈℘j P r m j = a j |x a , x b , Θ ı j ,ξ j , as follows: which makes use of the fact that the term Θ ı j ∈℘ n kj r∈Θ ı j P r c raj r / ∈Θ ı j ,r =k 1 − P r c raj does not include the (conditional) probability associated to shareholder k. As a consequence, the second order condition is given by: In order to examine the sign of ∂ 2 aj /∂x 2 a j , we begin by addressing ∂ 2 P r c ka j /∂x 2 a j . Under Assumption 7, G j (·) is the cumulative distribution function of an uniform distribution over the range [− τj /2, τj /2] with τ j sufficiently large such that: for allξ j ∈ [− ψj /2, ψj /2], x am ∈ Ω m and x bm ∈ Ω m ∀j, m ∈ . This implies that: ∂x aj , which in turn implies, because under Assumption 5 we have ∂ 2 E(R k (xa)) /∂x 2 a j < 0, that: We now address ∂P r(mj =aj |xa,x b ,Θ ı j ,ξj ) /∂P r c ka j . P r m j = a j |x a , x b , Θ ı j ,ξ j = r∈Θ ı j P r c raj r / ∈Θ ı j 1 − P r c raj for Θ ı j ∈ ℘ p kj . As such, because under Assumption 7, 0 < P r c raj < 1, for allξ j ∈ [− ψj /2, ψj /2], x am ∈ Ω m and x bm ∈ Ω m ∀j, m ∈ , we have that: ∂P r m j = a j |x a , x b , Θ ı j ,ξ j ∂P r c kaj = r∈Θ ı j ,r =k P r c raj r / ∈Θ ı j 1 − P r c raj > 0.
Combining the three results above, we have that the objective function of the manager is strictly concave in x aj , conditional on the strategy proposal of the challenger to the firm and on the strategy proposals of the candidates to the other firms.
We now readdress (b) and show that the common best-response function is the same as the best-response function that would arise from maximizing, conditional on the strategy proposals of the candidates to the other firms, the objective function established in the proposition. To do so, note that because the two candidates will choose the same best-response function, in equilibrium, we have E (R k (x a )) = E (R k (x b )) = E (R k (x)) for all k ∈ Θ j . This implies that P r c raj = 1 /2 −ξ j /τj and: ∂P r m j = a j |x a , x b , Θ ı j ,ξ j ∂P r c kaj = r∈Θ ı j ,r =k As a consequence, the first order condition of this problem can be written as follows: where β kj = Θ ı j ∈℘ p kj ( 1 /ψj)´1 2 ϕj − 1 2 ϕj 1 /2 −ξ j /τj #Θ ı j −1 1 /2 +ξ j /τj j −#Θ ı j dξ j . This makes use of the fact that, distribution over the range [τ j − ψ j , τ j + ψ j ] and density 1 /2ψj. As such, we can rewrite β kj as follows: maximize E (Π j (x)) + w jg E (Π g (x)) and, thus, that property (ii ) holds.
Third, the weight w jg that the manager of each firm j assigns to the expected profit of firm g is continuous in φ kj , γ kj and φ kg for the subset of shareholders k with financial rights in firm j, because the product, sum and quotient, respectively, of continuous functions is continuous. This implies that property (iii ) holds.
Fourth, the manager of each firm j would maximize the sum of the expected profits when all shareholders that have financial rights in the firm are fully diversified across firms, because if those shareholders are fully diversified, for the subset of shareholders k who hold financial rights in firm j, we have φ kj = φ kg = φ k and γ kj = γ kg = γ k for all j,g = j. This implies that: for all j,g = j and that the manager of each firm j would maximize E (Π j (x)) + g∈ ,g =j E (Π g (x)). As such, property (iv ) holds.
Finally, if λ = 1, the objective function of the manager of firm j will weigh solely the interests of the firm's overlapping (non-overlapping) shareholders as the ownership of each non-overlapping (overlapping) shareholder becomes dispersed (among a collection of infinitesimal identical shareholders) when the voting rights of the overlapping (non-overlapping) shareholders do induce full control. In order to see why, note that the objective function of the manager of firm j can be written as follows: ) .
As the ownership of each non-overlapping shareholder becomes dispersed among a collection of infinitesimal identical shareholders, the objective function of the manager will weigh solely the interests of overlapping shareholders when the voting rights of those horizontal shareholders do induce full control of the firm, i.e., when γ kj = 0 for each k ∈ Θ no j . Similarly, as the ownership of each overlapping shareholder becomes dispersed among a collection of infinitesimal identical shareholders, the number of shareholders in Θ o j increases, but the ratio φ kg/φ kj of each new infinitesimal identical shareholder will be identical to the corresponding ratio of each of the previous (non-dispersed) shareholders. As a consequence, the objective function of the manager will weigh solely the interests of non-overlapping shareholders when the voting rights of those non-overlapping shareholders do induce full control of the firm, i.e., when γ kj = 0 for each k ∈ Θ o j . As such, property (vi ) is satisfied.
Note that the same would not be true if λ = 0. In this case, the objective function of the manager of firm j will approximate a weighted sum of (solely) the interests of the firm's overlapping (non-overlapping) shareholders when non-overlapping (overlapping) shareholders are highly dispersed, for any given value of the control rights of the overlapping (non-overlapping) shareholders. In order to see why, let the subset of shareholders that hold financial rights in firm j, Θ j , be divided in two smaller subsets: the subset of overlapping shareholders, Θ o j , and the subset of non-overlapping shareholders, Θ no j . This implies that the objective function of the manager of firm j can be written as follows: max xj k∈Θ o j γ kj φ kj E (Π j (x)) + k∈Θ no j γ kj φ kj E (Π j (x)) + g∈ ,g =j k∈Θ o j γ kj φ kg E (Π g (x)) .
As the ownership of each non-overlapping shareholder becomes dispersed among a collection of infinitesimal identical shareholders, we have that k∈Θ no j γ kj φ kj → 0. As such, the objective function of the manager will weigh solely the interests of overlapping shareholders, even when the voting rights of those overlapping shareholders do not induce full control of the firm: Similarly, as the ownership of each overlapping shareholder becomes dispersed among a collection of infinitesimal identical shareholders, we have that k∈Θ o j γ kj φ kj → 0 and k∈Θ o j γ kj φ kg → 0. As such, the objective function of the manager will weigh solely the interests of non-overlapping shareholders (yielding an objective function proportional to the expected own-profit), even when the voting rights of those non-overlapping shareholders do not induce full control of the firm: max xj k∈Θ no j γ kj φ kj E (Π j (x)) .