Blockholder voting

Blockholder Voting

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Blockholder Voting Heski Bar-Isaac, Joel Shapiro PII: DOI: Reference:

S0304-405X(19)30284-3 https://doi.org/10.1016/j.jfineco.2019.11.005 FINEC 3140

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Journal of Financial Economics

Received date: Revised date: Accepted date:

15 December 2017 17 May 2019 11 June 2019

Please cite this article as: Heski Bar-Isaac, Joel Shapiro, Blockholder Voting, Journal of Financial Economics (2019), doi: https://doi.org/10.1016/j.jfineco.2019.11.005

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Blockholder Voting✩ Heski Bar-Isaaca,1 , Joel Shapirob a

University of Toronto, CEPR, and CRESSE b University of Oxford

Abstract Blockholders play a large role in corporate governance. We examine their voting behavior by adding a voter with many votes, i.e., a blockholder, to a standard voting model. A blockholder may not vote with all of her shares. This is efficient because it prevents her from drowning out the information in others’ votes. This effect holds even when shares may be traded. Consequently, regulations prohibiting abstention will harm information aggregation, though such regulations may promote information acquisition by blockholders. We extend the model by allowing: the blockholder to announce her information upfront, the possibility of blockholder bias, and asymmetric priors. Keywords: Blockholder, Shareholder voting, Corporate governance JEL: G34, D72

✩ We thank the Editor (Ron Kaniel) and an anonymous referee for extensive and helpful comments. We also thank Patrick Bolton, Philip Bond, Susan Christoffersen, Vicente Cunat, Steven Davidoff Solomon, Ignacio Esponda, Stephen Hansen, Peter Iliev, Alexander Ljungqvist, Frederic Malherbe, Daniel Metzger, Thomas Noe, Frank Partnoy, Miriam Schwartz-Ziv, Scott Winter, Moqi Xu, Yishay Yafeh and participants of seminars at UC Dublin, Lancaster, London FIT, London FTG, Goethe University, U. Utrecht, U. Nottingham, Tel Aviv U., Universidad Carlos III Madrid, EFA and the third IO Theory and Empirics Research Conference for helpful comments. We thank Zheng Gong for outstanding research assistance. Bar-Isaac thanks SSHRC (435-2014-0004) for financial support and the London Business School for its hospitality in hosting him as a visitor while some of this paper was written. 1 corresponding author; email: [email protected]; Tel: ++1 416 978 3626; Fax: ++1 416 978 5433

Preprint submitted to Journal of Financial Economics

November 22, 2019

1. Introduction Shareholder voting has a stark difference from voting in the political context: a shareholder may have many shares and, thus, many votes. Blockholders, that is, shareholders with a large fraction (often defined as 5%) of the shares of a firm, are ubiquitous. In a sample of representative U.S. public firms, Holderness (2009) finds that 96% have at least one blockholder, and the average stake of the largest blockholder of a firm is 26%.2 The identity of blockholders may vary substantially, from activist investors, to passive index funds, to even the managers or directors of the firm. Voting by blockholders has been the subject of increased regulatory scrutiny. The U.S. Securities and Exchange Commission (SEC) mandated that investment advisers (including mutual funds) ensure that their votes are in the best interests of their clients, and they must publicly report their votes (SEC, 2003). Large investors must report their stakes and intentions when they reach 5% ownership of a firm. We analyze a strategic voting model [as in Feddersen and Pesendorfer (1996) or Maug (1999) in the corporate setting] with a blockholder. The blockholder adds asymmetry to the model; she differs from other shareholders in both the number of shares she holds and the amount of information she has. Otherwise, the voting model in this paper is standard. All shareholders (including the blockholder) want to improve the value of the firm and receive a signal about which of two decisions will increase value. A vote wins by a majority, and voters may vote strategically; voting strategically means that the voter may not vote in line with her signal (either abstaining or voting for a different option) and she does this in order to improve the outcome of the vote. Our first result is that blockholders may prefer not to vote with all of their shares, and this is efficient.3 As the blockholder wants to maximize the value of the firm, if she does not have precise information, she will prefer that other shareholders’ information not be drowned out by her votes. The blockholder is not wasting her unvoted shares; she is acting optimally to improve the efficiency of the vote. 2

He also finds that this distribution is not markedly different from that in the rest of the world. 3 We define efficiency in the text as maximizing the probability that the vote matches the true state of the world.

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We then allow for information acquisition by the blockholder and sharetrading for all shareholders. This provides a robustness check for the result above as it is natural to ask why the blockholder would not trade shares that she does not vote. We show that the blockholder might face a “lemons” discount when she sells shares, as investors may infer that the blockholder did not acquire information. The blockholder will then prefer to retain the shares rather than sell them. The result that the blockholder may prefer not to vote all of her shares implies that a regulation that prohibits abstention might be inefficient. The 2003 SEC rule for investment advisers does not explicitly prohibit abstention, but it may be interpreted as doing so.4 Nevertheless, we demonstrate that there might be a rationale for prohibiting abstention, as it might provide a blockholder with incentives to acquire information. Therefore, a regulation that prohibits abstention will harm information aggregation but may promote information acquisition. However, we show that this argument relies on impediments to trade; when the blockholder can trade shares, no such new incentives arise with this type of regulation. Shareholder voting is an important source of corporate governance (see, e.g., McCahery, Sautner, and Starks, 2016). Maug and Rydqvist (2009) provide evidence from aggregate voting results that shareholders vote strategically in corporate elections. Our results above have empirical implications for when blockholders will abstain, when minority shareholders will abstain, how share trading before votes will work, and the impact of the 2003 SEC rule. Nevertheless, to examine whether blockholders and shareholders behave as predicted in our model, we would need detailed shareholder voting data. The existing data on shareholder voting are far from satisfactory for our purposes; we describe the implications and challenges in the text. We consider several extensions of the model: First, we allow the blockholder to publicly announce information before other shareholders vote. This could represent an activist investor or a large pension fund using public statements to communicate its position. In this case, if this blockholder is well informed, some of the shareholders may ignore their information and vote with the blockholder. It is perfectly rational to 4

As we discuss in the text, the rule at one point suggests that abstention may not be in the best interest of clients. The requirement that votes by investment advisers be disclosed also creates a reputational cost for abstaining (which we do not model).

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ignore their information; if the blockholder has considerable information but insufficient shares to express that information, the shareholders essentially supplement the blockholder’s shares with their own. It can be efficiency enhancing to coordinate shareholders to support the blockholder. Our result, then, suggests that some regulatory concerns about coordination by groups of investors such as “wolf packs” (see, e.g., Coffee and Palia, 2016) may be overstated. Second, we consider the situation in which the blockholder may be biased. We define bias here as the blockholder supporting one side of the proposal regardless of what information she receives. For example, the blockholder could support management’s position because she is a manager or has business dealings with the firm that depend on good relationships with management.5 In this case, some shareholders may ignore their information to counter the blockholder’s bias. Lastly, we allow the prior to be asymmetric, i.e., more tilted towards one of the two states. In this case, the shareholders will tilt their votes in the direction of the prior, to incorporate its information. Our analysis is kept simple to present the effects clearly and maintain tractability. We recognize a few directions in which we have kept the model uncomplicated. First, the precision of all shareholders’ and the blockholder’s information is common knowledge. Second, we select two types of equilibria to examine: the most informative equilibrium and symmetric equilibria. We discuss these selection criteria in further detail in the text, but as in all strategic voting models, there are many other equilibria. In the following subsection, we review the related theoretical literature. In Section 2, we set up the model. In Section 3, we analyze the model. In Section 4, we allow the blockholder to acquire information and trade shares. In Section 5, we examine the effect of regulations that prohibit abstention and look at the empirical implications of the model. In Section 6, we extend the model in three directions: first to allow the blockholder to communicate before the vote, second to allow for the possibility that the blockholder may 5

For example, a mutual fund may support management to preserve business ties (see Davis and Kim, 2007, Ashraf, Jayaraman, and Ryan, 2012, and Cvijanovic, Dasgupta, and Zachariadis, 2016). Another example of blockholder bias is documented in Agrawal (2012), where labor union pension funds may vote for labor-friendly directors. Such bias may also arise as a result of a blockholder’s holdings in other firms, as in Azar, Schmalz, and Tecu (2018).

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be biased, and third to study the effect of an asymmetric prior. In Section 7, we conclude. The proofs for Sections 3-5 are in the Appendix. The proof for the Extensions Section (Section 6) are in an Internet Appendix. 1.1. Theoretical literature We are only aware of one other paper on blockholder voting. The paper is Dhillon and Rossetto (2015), which provides a rationale for the presence of multiple mid-sized blockholders in a setting where shareholders are riskaverse and have private values.6 Following Austen-Smith and Banks (1996), who highlighted its importance, the paper in the political economy literature that began the analysis of strategic voting was Feddersen and Pesendorfer (1996). They analyzed the swing voter’s curse, where uninformed voters abstain to allow more informed voters to sway the vote. We demonstrate that in a model where an agent has many votes (i.e., the blockholder), this logic can lead to abstention on a fraction of the agent’s votes. Furthermore, we also study share/vote trading, information acquisition, communication, and bias. A few other papers have strategic voting in contexts related to ours. Eso, Hansen, and White (2014) study empty voting7 and find that uninformed shareholders and even biased shareholders may sell their votes (at a zero price) to informed shareholders to improve the outcome. Note that our analysis demonstrates that activists do not necessarily have to resort to empty voting strategies to gain votes if they have credibility, as other shareholders will support them. Maug (1999), Maug and Rydqvist (2009), and Persico (2004) find that informed voters/shareholders may ignore their information in response to different voting rules.8 Furthermore, Maug (1999) and Maug and Yilmaz (2002) find that informed voters/shareholders may ignore their information to counter the bias of other shareholders. These effects are analogous to our results when we study a biased blockholder. Beyond differences in 6

Edmans (2014) and Edmans and Holderness (2017) provide surveys on blockholders. Dhillon and Rosetto (2015) is the only theoretical paper they cite about voting—in the rest of the cited literature, blockholders make costly interventions in a firm and/or trade the firm’s shares. Yermack (2010) surveys the literature on shareholder voting and also does not cite any such research. 7 Brav and Mathews (2011) also examine empty voting in a model of a hedge fund who may both trade shares and buy votes, while all other voting is random. 8 Bhattacharya, Duffy, and Kim (2014) observe a similar effect when symmetric voters receive signals of different precision regarding one state of the world versus another.

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the questions we address and the asymmetries (the presence of a blockholder with multiple shares and more precise information) present in our model, our model is different from these papers in that we allow for abstention. The literature has focused primarily on simultaneous voting. One might expect that communication prior to a vote might affect voting outcomes; certainly, this will be the case with non-strategic voting, and as Coughlan (2000) illustrated, voters might (efficiently) choose to vote non-strategically with even fairly minimal communication under the unanimity rule. Dekel and Piccione (2000), like Coughlan (2000), assume symmetry among voters (in contrast to our paper) and focus on strategic voting under general majoritarian rules; they suggest that sequential voting does no better than simultaneous voting. In contrast, in our model, if the blockholder makes announcements before a vote and has better information than other shareholders, the voting outcome can more accurately aggregate information; if the blockholder has few votes, more information is incorporated into the vote when some shareholders ignore their information to vote with the blockholder. Malenko and Malenko (2019) consider a common value model where shareholders can acquire information and/or purchase it from a proxy advisory service. The authors demonstrate that although this proxy advisor may provide useful information, the outcome may be less efficient than in the absence of the advisor9 since the advisor’s presence may crowd out independent investment in information acquisition. They focus on an environment where shareholders and their strategies are symmetric, which is critically different from our asymmetric environment (with a large blockholder). However, our result that regulations that force a blockholder to vote all of her shares in line with her information may be inefficient is related; it is an inefficiency in information aggregation due to over-reliance on one signal (akin to the over-reliance on the signal of the proxy adviser). While we study the role of information aggregation in corporate elections, there is a literature that focuses on preference aggregation. Perhaps most relevant to our study, Harris and Raviv (1988) and Gromb (1993) examine whether it is optimal to allocate uniform voting rights for all shares 9

We note that there are two spellings: adviser and advisor. Both spellings are used for investment adviser, but adviser is the more commonly used one (see www.sec.gov). Similarly, both spellings are used for proxy advisors, but advisor is the more commonly used one. We adhere to common use spellings in our paper.

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(one shareholder, one vote). Cvijanovi´c, Groen-Xu, and Zachariadis (2017) examine the participation decision in a corporate setting where voters are partisan.10 2. Model The model setup is similar to that in the strategic voting literature (e.g., Feddersen and Pesendorfer, 1996), with the novel departure that one voter (shareholder) has more votes (shares) than the others—the blockholder. Most of the literature relies on symmetry to pin down equilibria. However, in the corporate environment, it is natural to consider asymmetry in the number of votes participants have. We will also allow for asymmetry in strategies (for shareholders who have an equal number of shares). There are two types of agents who own shares in the firm. There is a blockholder (B) who has 2b shares and 2n + 1 other shareholders (each denoted by S), each of whom owns one share. Henceforth, we use the term shareholder only to refer to one of the 2n + 1 who each hold only a single share. The assumption that the blockholder has an even and shareholders an odd number of shares allows us to disregard ties for many of the cases we study. For simplicity, we assume that there is no trading of shares, although we relax this assumption in Section 4. We also assume that the blockholder does not own a majority of the shares: Assumption A1: n ≥ b. A proposal at the shareholder meeting will be implemented if there are more votes in favor than against. If there is a tie, we will assume that each of the two possible decisions will be implemented with probability 0.5, as is standard in the literature. There are two states of the world θ, management is correct (θ = M ), or management is incorrect (θ = A), which are both ex ante equally likely to occur. Let decision d denote whether management wins (d = M ) or loses (d = A). Shareholders and the blockholder have common values, i.e., they both prefer the choice that maximizes the value of the firm. We will relax this assumption in Section 6.2, where we allow the blockholder to possibly be biased and strictly favor one decision. The payoff per share 10

Bolton, Li, Ravina, and Rosenthal (2019) and Bubb and Catan (2019) examine preference aggregation in an empirical context.

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for both the blockholder and shareholders u(d, θ) from a vote depends on the decision and the state of the world. If the decision matched the state of the world, the payoff is one for each share: u(M, M ) = u(A, A) = 1. If the decision did not match the state of the world, the payoff is zero: u(M, A) = u(A, M ) = 0. The blockholder and shareholders receive imprecise individual signals s ∈ {m, a} about what the correct state of the world is. The signals are independent conditional on the state. This precision is given by π i (θ | s), where i ∈ {B, S}. π B (M | m) = π B (A | a) = q π S (M | m) = π S (A | a) = p The probability that the blockholder infers the correct state from the signal is q, and the corresponding probability for the shareholder is p. We assume that q ≥ p > 0.5. This indicates that signals are informative, as their precision is above 0.5, and that the blockholder receives a more precise signal than the shareholder. The blockholder presumably receives more precise information because she is a larger investor and possibly has (i) more contact with the firm, (ii) more infrastructure in place to gather information, and (iii) more incentives to gather information. In the analysis, we discuss what happens when the blockholder’s precision varies, and we endogenize information acquisition by the blockholder in Section 4. We allow the blockholder and shareholders to vote strategically. Given a particular signal, they may vote for M or A or abstain. We say that a shareholder or blockholder votes sincerely if they vote with their signal. To gain some tractability and focus on the key trade-offs of the model, we restrict the voting behavior of the blockholder as follows: Assumption A2: The blockholder can vote only an even number of shares and cannot simultaneously vote for both sides of the proposal. Assuming that the blockholder votes with an even number of shares (like the assumption that the blockholder holds an even number of shares and other shareholders an odd number) allows us to disregard ties for many of the cases we study. This does not affect the results but simplifies the analysis and presentation. Preventing the blockholder from voting for both sides of the proposal is more a question of presentation, as what will matter in that case would be the net votes the blockholder produces for one side, which 8

could be replicated by voting only a fraction of her votes and abstaining with the rest (which we allow).11 The precision of the signals and the structure of the game is common knowledge. Finally, the voting setting leads naturally to multiple equilibria. We use two equilibrium selection criteria. First, it is natural to focus on the equilibria that lead to the best outcome. Note that since all shareholders and the blockholder have identical preferences and differ only in their information, they all agree on what “best” means here—the equilibria that lead to the highest probability of selecting the decision that matches the state. We will refer to this as the most informative equilibrium. We will focus on the most informative equilibrium in pure strategies,12 but there will be cases in which the equilibrium is the most informative for any type of strategy, which we will highlight in the text. Second, the literature, which has considered symmetric voters, has focused on symmetric strategies. In our environment, a natural counterpart would be to restrict the (symmetric) shareholders to symmetric strategies. For our main result, the equilibrium that is most informative involves symmetric pure strategies for shareholders, and so these two equilibrium refinements are not in conflict. In other cases, we present equilibria that satisfy both criteria. In the extension with a blockholder who may be biased (in Section 6.2), the first criterion is no longer the best for all agents, and we use the second.

3. Analysis In this section, we analyze the model outlined above. Examining how much information the blockholder holds relative to shareholders is critical to understanding what the most informative equilibrium is in this context. 11

This choice of presentation might raise the question of whether quorum rules are satisfied. Often, these require a majority of shares to be present at a vote. Abstention by those present may be permitted. Alternatively, by voting on both sides of an issue, and thus replicating abstention, shareholders in our model can ensure compliance with a quorum requirement. 12 Persico (2004) restricts attention to pure strategies. Esponda and Pouzo (2012) argue that in the voting environment we consider, pure strategy equilibria are stable in a particular way, whereas mixed strategy equilibria are not.

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Consider two examples. In the first example, the blockholder has imprecise information, say, equal to the precision of an individual shareholder’s information. Nevertheless, in this example, the blockholder owns 40% of the shares, while each individual shareholder holds less than 1%. If the blockholder voted with all of her shares in this scenario, the final vote would mostly reflect the blockholder’s information, and the blockholder’s vote would determine the election even in many cases where a large majority of shareholders voted in opposition to the blockholder. If instead the blockholder did not vote all of her shares, the final vote would reflect the information of all the shareholders. Because the blockholder cares about maximizing the informativeness of the vote, as that will raise the likelihood of increasing firm value, it is natural to posit that the blockholder prefers not to vote all of her shares. In the second example, the blockholder is perfectly informed, i.e., q = 1 , but only owns 5% of the shares of a firm. Individual shareholders have imprecise information. In this case, if all individual shareholders vote, the blockholder’s vote will have little impact. However, if the individual shareholders could delegate the vote to the blockholder, they would prefer to do so, as it would maximize the informativeness of the vote. They can accomplish this by abstaining. These two examples lead us to two equilibrium outcomes that might maximize informativeness. We prove in Proposition 1 that the most informative equilibrium outcome in pure strategies is unique and generalizes the intuition of the examples.13 Proposition 1. Define 2b∗ as defining the optimal vote threshold for the blockholder, where b∗ is defined as: (  2b+1 ) p q ≤ . (1) b∗ := min b ∈ N| 1−q 1−p 13

While these involve symmetric shareholder behavior, we make no such restriction. As the proof of the proposition suggests, mixed strategy equilibria cannot be more informative for b ≥ b∗ , since, in this case, the voting rule corresponds to aggregating information efficiently—that is, to updating according to Bayes’ rule when using all of the agents’ signals. Moreover, we conjecture that mixed strategy equilibria cannot be more informative for b < b∗ since they lead to costly miscoordination that asymmetric pure strategy equilibria can avoid, but we do not prove this.

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Furthermore, define Condition C1 as:
p i P2n+1 2n+1 ( 1−p ) q i=n+b+1 i . Pn−b 2n+1
p i > 1 − q ( ) i=0 i 1−p

(C1)

An equilibrium of the following form always exists: (i) When b ≥ b∗ , all shareholders vote sincerely, and the blockholder votes sincerely with 2b∗ shares. (ii) When b < b∗ and Condition C1 holds, all shareholders vote sincerely and the blockholder votes sincerely with 2b shares. (iii) When b < b∗ and Condition C1 does not hold, only the blockholder votes, and all shareholders abstain. There are many equilibria that are outcome-equivalent to this. Such an equilibrium is the most informative equilibrium within the class where all shareholders employ a pure strategy. We now provide intuition for the results in Proposition 1. If we were to assume that voting were sincere rather than strategic, Nitzan and Paroush (1982) show that, to maximize the informativeness of a vote, a planner should weight the votes of voters with heterogeneous information according to how precise their signals are (that is, in relation to the likeliq p and 1−q ); essentially, these reflect the weights a Bayesian hood ratios 1−p decision maker would assign to the signals. Our definition of the optimal vote threshold 2b∗ reflects these weights, where the blockholder’s weight is scaled to reflect that a shareholder has a single vote, and accounts for the blockholder voting an even integer number of shares. In our model, in contrast to Nitzan and Paroush (1982), shareholders need not vote sincerely. Strategic voting implies they may vote against their signals and/or abstain. We therefore prove that the strategies in Proposition 1, part (i) are indeed equilibrium strategies (rather than the planner’s choice). Further, we must also consider the case in which the blockholder has fewer than 2b∗ shares. We first demonstrate that the equilibria we propose in this case are, in fact, equilibria. We then show that they are the most informative equilibria among those that involve voters choosing pure strategies, where Condition C1 determines which of the two is most informative. Hence, when the blockholder has too many shares relative to her information, she will not vote them all. If she were to vote them all, she would drown out useful information from the other shareholders, impairing the effectiveness of the overall vote. Because the blockholder and the shareholders have 11

a common interest in maximizing the value of the firm, the blockholder internalizes this effect and only votes a fraction of her shares. As mentioned in the introduction, for investment advisers (such as mutual funds), the requirement to act in the best interest of investors may be interpreted as prohibiting the blockholder from abstaining. This result demonstrates that there is an efficiency loss from such a policy. We discuss this policy further in Section 5.1. As is well understood following Condorcet (1785), having more shareholders leads to a more accurate vote. This reasoning suggests that having all shareholders vote sincerely dominates having an intermediate number vote sincerely. This is consistent with the behavior characterized in Proposition 1 (i) and (ii). However, when the blockholder has high quality information, few votes, and there are relatively few shareholders, then rather than drown out the blockholder’s signal with their relatively noisy votes, it may be optimal for shareholders to simply “get out of the way” and abstain. This is clearly the case for q = 1, for example. In this way, when the blockholder has too few shares relative to her information—that is, b < b∗ — there are two equilibria that may be the most informative. If there are sufficiently many shareholders, the standard Condorcet reasoning applies and makes their contribution sufficiently valuable: We obtain an equilibrium in which the blockholder votes all of her shares sincerely, and the other shareholders also vote sincerely. Instead, if there are few shareholders with relatively low quality information, then their votes act primarily to drown out the more informative signal of the blockholder. Here, the most informative equilibrium involves all shareholders abstaining and allowing the blockholder’s superior information to determine the vote. Given these two equilibria, Condition C1 determines which is more informative. It is clear that the number of shareholders, 2n + 1, the precision of the signals (q and p), and the number of shares the blockholder has (2b) affect whether Condition C1 holds. Consider an extreme case in which there are many shareholders (n is high). In this case, Condition C1 holds, and in the extreme case in which n → ∞, the shareholders’ collective vote will be accurate with probability approaching one, as is well understood given Condorcet (1785). Therefore, having the shareholders vote is helpful. Instead, when n is small, there is scope for noise and mistakes. In this case, Condition C1 fails, and the most informative equilibrium involves only the blockholder voting since the collective vote of all shareholders is relatively inaccurate. More broadly, we can consider whether the parameters make Condition C1 12

more or less likely to hold. Lemma 1. If Condition C1 is satisfied for some (p, n, b, q) then it is also satisfied when p, n, or b are higher or q is lower. Consequently, when b < b∗ , the most informative equilibrium involves shareholders voting sincerely the higher p, n, or b, or the lower q is. These results and the Condorcet reasoning above also provide some intuition on how the ownership structure affects the accuracy of the vote (and therefore shareholder value). Specifically, Condorcet reasoning suggests that increasing the number of small shareholders n should always improve the accuracy of the vote. However, for Proposition 1 (iii) in the range where Condition C1 fails, there is no effect. Interestingly, when b < b∗ and C1 holds, increasing the blockholder’s number of shares 2b leads to a more accurate vote (as follows from the proof of Proposition 1(ii), which shows that in this range a higher value of b leads to a more accurate vote), but for b < b∗ where C1 doesn’t hold and for b ≥ b∗ , there is no effect. Fixing the overall number of shares and transferring shares from the blockholder to shareholders is strictly efficiency enhancing when b > b∗ , strictly harmful when b < b∗ and C1 holds, and has no effect when b < b∗ and C1 doesn’t hold. 4. Information acquisition and trading We extend the model to allow the blockholder an opportunity to acquire information and highlight that the blockholder will underinvest in information acquisition. As we discuss in Section 5.1, this can provide a simple rationale for regulations that prohibit abstention. We then extend the model to allow the blockholder to freely trade shares before voting in a simple model of trade with endogenous share prices. The opportunity to trade can lead to an outcome that is more efficient since trading allows votes to move to where they have the largest informational value; in particular, the equilibrium will involve no abstention. Finally, we allow for both information acquisition and trading. The interaction can reverse the result on trading, and, thereby, provides an explanation for why the blockholder holds shares that she does not vote. Since potential buyers cannot be certain whether the blockholder has invested in information acquisition, there is a possibility of a lemons discount if the blockholder trades her shares, i.e., shareholders do not believe the blockholder has acquired information and therefore discount the shares being sold. This discount may 13

incentivize the blockholder to keep her shares even if she would abstain on them. 4.1. Information acquisition We suppose that the blockholder initially has equally precise information as a shareholder, i.e., precision p. The blockholder may then pay a fixed cost c to acquire information, which boosts the precision of her information to q > p. If the blockholder invests, then her higher quality information, which will be incorporated at least to some extent into a vote, will lead to a better outcome and thus a higher payoff. Indeed, this higher payoff would be enjoyed by all shareholders, who all benefit from a vote that more accurately captures 2b of the shares, the underlying state. The blockholder owns a fraction 2n+2b+1 and thus, enjoys only this fraction of any gain from a more accurate vote. Consequently, her private incentives to acquire information will be below the socially efficient level. 4.2. Trading shares We examine a simplified trading game where we assume that the blockholder trades publicly. The price at which a share trades is determined by the blockholder making simultaneous, observable, take-it-or-leave-it offers. Consequently, the price will reflect the expected value of the share. This expectation, in turn, reflects shareholders’ beliefs about the quality of the blockholder’s information and the blockholder’s post-trade holdings, which determine the extent to which information is efficiently incorporated into the vote. We assume the blockholder can sell to new shareholders who buy one share each and have precision of information p. The observability of the blockholder’s offers simplifies the inference problem of the shareholders and might reflect the fact that the blockholder is likely to face limits to her ability to disguise large trades. After trade occurs the vote proceeds as in Section 3. To focus on the most relevant case, we will restrict our analysis to the situation in which the blockholder initially has more shares than the optimal vote threshold, i.e., 2b > 2b∗ . We assume that this is the case throughout our analysis. Based on our previous analysis, it is in this case that the blockholder is most likely to have shares with which she does not vote. We assume the following timing: 14

1. The blockholder begins with a number of shares 2b. There are 2n + 1 existing shareholders. The blockholder and each of the shareholders have information of precision p. 2. The blockholder can pay c to improve the precision of her information to q. This investment decision is not observed. 3. The blockholder can simultaneously and publicly make take-it-or-leaveit offers to sell to potential new shareholders or to buy from existing shareholders. After trading, the number of shares that the blockholder holds is given by 2bb and the number of shareholders is 2ˆ n + 1 = 2n + b 1 + 2b − 2b. 4. The blockholder and shareholders observe the realization of their private signals and votes are cast. We suppose that voting in stage 4 corresponds to the most informative equilibrium and so the voting in that stage will be as characterized in Proposition 1, given the shareholdings that result from stage 3.14 Note that the price at which shares trade will depend on all agents’ expectations about the blockholder’s quantity of shares held after trading, since this will affect the likelihood of voting correctly. The price will also, of course, reflect shareholders’ beliefs about the blockholder’s information precision. We begin by considering the case of trade when the blockholder’s information precision is exogenous (i.e., the cost of information acquisition c = 0, so that the blockholder always acquires information) as in the baseline model, before allowing for information acquisition in Section 4.2.2. 4.2.1. Trading with exogenous information The blockholder’s trading decision will affect the quality of the voting decision (since it will affect the quantity of information that feeds into the decision and how it does so). Consequently, this affects the price at which shares trade; the price will simply reflect the likelihood that the vote correctly identifies the state of the world. Note that the blockholder will also value the shares that she does not trade in the same way: every share’s value corresponds to the likelihood that the vote correctly identifies the state of the world. Thus, she trades so as to maximize the overall expected value 14 A slight subtlety here is to consider voting off-equilibrium. We suppose that shareholders, given their beliefs about whether or not the blockholder has invested in information, vote as outlined in Proposition 1, and that the blockholder best responds to this behavior.

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of the firm (which she will capture in part through her shareholdings, and in part through the price of shares that she sells). In the model, a share has the highest value in the hands of the agent who can best use it to vote informatively, and so the blockholder wants to trade so as to maximize the vote’s accuracy. Following the proof of Proposition 1, it is immediate that this is the case when the blockholder trades so as to hold 2b∗ , yielding the following result. Proposition 2. In the case of exogenous information and trade where b > b∗ , the blockholder will trade so as to hold exactly 2b∗ shares. In particular, the blockholder will not hold shares with which she will abstain. 4.2.2. Trading and information acquisition In this subsection, we incorporate both trading and information acquisition. Prior to trade and prior to voting, we allow the blockholder to acquire information at a cost (stage 2 of the timing listed above). We assume that the information acquisition decision is unobservable. Given this, we are able to explain why, if a blockholder does not vote all of her shares, she does not sell the shares with which she does not vote, i.e., effectively, we reverse the result of Proposition 2. The key intuition is that the blockholder may not be able to trade shares easily, as new potential shareholders may be dubious about whether the blockholder acquired information if she is attempting to dump shares on the market and, hence, may have a low willingness to pay for them. Thus, the equilibrium price offered for a share may be very low, and the blockholder might therefore prefer not to sell shares even if doing so would lead to a more accurate voting outcome (conditional on the information acquisition decision). The sequential investment and trading game that we consider admits numerous subgames, and the equilibria will depend on the off-equilibrium restrictions made, as we discuss further below. We proceed by backward induction. We assume that given the shareholdings resulting from stage 3, in stage 4 (the voting game), the shareholders hold degenerate beliefs (that is, the blockholder has acquired information with probability one or zero) and play the most informative equilibrium (whether on or off the equilibrium path) as given by Proposition 1.15 To further limit the number of cases we consider, 15

Note that Proposition 1 holds for q ≥ p, so the most informative equilibrium will be

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we impose a parametric restriction: that Condition C1 holds regardless of how much trading occurs in stage 3. Next, we must consider beliefs at stage 3 when the blockholder proposes trades. Here, beliefs must be defined both off and on the equilibrium path. We will make the following restriction on beliefs: shareholders consider the blockholder and shareholder holdings in the case in which trade goes through and believe that the blockholder has acquired information if and only if the blockholder earns more with these shareholdings when she acquires information than when she does not acquire information. In particular, this restriction implies that beliefs (off- or on-equilibrium) are degenerate—shareholders either believe with probability one or zero that the blockholder has acquired information. The refinement is based on forward induction logic where the beliefs considered “reasonable” depend on the “type” that is likely to benefit from subsequent induced play. This equilibrium refinement is formalized in the Appendix. With this belief restriction, we prove in Proposition 3 that the blockholder will find it worthwhile to invest in information and will hold shares on which she abstains if (i) the minimal holding that induces investment is greater than 2b∗ (formalized as Condition (A.6) in the Appendix), and (ii) the blockholder would earn more by making the investment than she would by selling all her shares at a price that assumes no investment (formalized as Condition (A.7) in the Appendix). The proof of Proposition 3, which appears in the Appendix, formally characterizes the required conditions and belief restriction, and shows that there are parameters which satisfy them. Proposition 3. There exist equilibria where: i) Beliefs off the equilibrium path satisfy the restriction characterized in Definition 1 (stated in the Appendix), ii) Conditions (A.6)and (A.7), as characterized in the appendix are satisfied, and iii) Condition C1 holds for all bb ≤ b and n b such that 2b n + 1 = 2n + 1 + summarized by that proposition when the blockholder has acquired (not acquired) information and shareholders believe the blockholder has acquired (not acquired) information. Given our refinement (defined below), non-degenerate beliefs are not possible. However, the blockholder and shareholders may hold different degenerate beliefs when there are deviations. We show in the proof of the proposition below that the most informative equilibrium still takes the form described in Proposition 1.

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2b − 2bb. In such equilibria the blockholder invests in improving her information, and, after trading, holds strictly more than 2b∗ shares so that she abstains with some of her shares in the vote. This proposition (and the required conditions) essentially rely on the observation that large trades by the blockholder will suffer a lemons discount by shareholders. The discount comes from the belief that the blockholder would not have invested in acquiring information if she were planning to trade so many shares. 5. Policy and empirical implications In this section, we first examine regulations that prohibit abstention and then turn to consider empirical implications of the analysis above. 5.1. Regulations that prohibit abstention The 2003 SEC rule was promulgated in the wake of a major breach of fiduciary duty by an investment adviser and an environment of corporate scandals (Taub, 2009). The rule itself emphasized that (i) investment advisers must vote in the best interests of clients and (ii) provide data on voting.16 What effect do these two stipulations have on voting behavior? We argue that they make abstention very costly. An investment adviser must be able to justify its abstention to the regulator and, perhaps more importantly, to its clients (who can now observe how the adviser votes). Explanations for abstention such as the investment adviser has poor information and/or the investment adviser is allowing other more informed shareholders to decide the vote may be costly in terms of reputation. The SEC rule recognizes this possible effect, explicitly stating “We do not suggest that an adviser that fails to vote every proxy would necessarily violate its fiduciary obligations.”17 Nevertheless, the SEC rule later states that “We believe an adviser that has assumed the responsibility of voting client proxies cannot fulfill its fiduciary responsibilities to its clients by merely refraining from voting the proxies. Such proxies would not be voted in the best interest 16

We discuss these data in detail in Section 5.3. The rule also made investment advisers provide access to their voting policies and procedures. 17 SEC (2003) Section II.A.2.a.

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of the clients.”18,19 Iliev et al. (2015) note that there appears to be a small amount of abstention by mutual funds, “in 6.6% of the director election votes, the institutions did not vote.”20 In order to analyze the effect of regulations such as the SEC rule on voting, we take a simplified approach by assuming abstention is prohibited, i.e., the regulation requires the blockholder to vote with all her shares in one direction. Requiring the blockholder to vote all of her shares in one direction can reduce the efficiency of information aggregation in voting by prohibiting strategic behavior. However, if such a regulation led to more information acquisition, the cumulative effect could actually enhance efficiency. We now formalize this intuition. For ease of exposition, we also assume that shareholders vote sincerely, as would be the case when (C1) is satisfied.21 We begin by supposing that shares cannot be traded, but we then extend the analysis. Incorporating a regulation prohibiting abstention into the model implies that the value per share when the blockholder invests in information acquisition may be lower since information is less efficiently aggregated into the vote. However, this regulation also implies that the value per share when the blockholder does not invest in information acquisition is also lower. This second effect may be stronger, as the blockholder might be incorporating very poor information into the vote. This can skew the blockholder’s decision towards acquiring information. This will be socially beneficial when the blockholder would not have acquired information without the regulation, even though that would have maximized social welfare. Of course, if the requirement were insufficient to induce the blockholder to acquire information or the blockholder would have acquired information even without the rule, then its sole effect would be to lead to an inefficient 18

SEC (2003), Footnote 23. Maug and Rydqvist (2009) interpret the rule along these lines. They state, “some investors are required to vote. Under the Employee Retirement Income Security Act (ERISA) pension funds must vote according to two letters issued in 1988 (Avon Letter) and 1990 (ISSI letter), and an amendment of the Investment Advisers Act (1940) as of 2003 requires that investment advisers vote.” 20 Iliev et al. (2015) in Footnote 7. 21 Note that this condition guarantees that shareholders vote sincerely (rather than abstain) in the absence of the SEC rule. 19

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aggregation of the available information and, in this way, would be detrimental. Note that given the result in Proposition 1, in the case in which the blockholder acquires information and b ≤ b∗ , the requirement that the blockholder vote with all her shares does not reduce the efficiency of the vote: she would in any case choose to vote with all her shares. Thus, the regulation would reduce the value of not investing in information, with no impact on her value if she does invest. Consequently, this increases the blockholder’s incentives to invest and welfare will be (weakly) higher if she does so.22 The case in which b > b∗ is more involved since, conditional on the blockholder acquiring information, a requirement to vote with all her shares would be detrimental for the decision and, thus, for the blockholder. Proposition 4. When b ≤ b∗ , the regulation weakly enhances efficiency when condition (C1) is satisfied and the blockholder acquires information if there is regulation. When b > b∗ , the regulation enhances efficiency when it leads to information acquisition, whereas no such acquisition would arise in the absence of regulation and where information acquisition is socially desirable. These conditions appear as (A.12),(A.13), and (A.14) in the Appendix. There is a non-empty set of parameters for which they are satisfied simultaneously. The proof defines the incentive constraints and parameters for which the regulation enhances efficiency when b > b∗ . In Fig. 1, we illustrate where the regulation improves welfare for some given parameters (n = 30, p = 0.52, and q = 0.64) and allow the cost of information c and the number of shares the blockholder owns 2b to vary. The dotted line defines b∗ , and the graph examines b > b∗ . To the left of the dot-and-dashed line (that is, with sufficiently low cost), a blockholder would invest in information acquisition even in the absence of the regulation. To the right of the solid line (with high enough cost), even with the regulation, the blockholder would not invest. Thus it is only between these two lines that the regulation would change the blockholder’s investment behavior. The final dashed line reflects whether it is socially efficient to induce the blockholder to invest (to the left of the line) or not (to the right) reflecting that the regulation 22 Welfare is weakly higher because the regulation may not change the blockholder’s investment decision, in which case welfare is the same.

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itself is welfare enhancing when the cost is lower but that its dependence on the blockholder’s shareholdings is nonlinear, as evidenced by Eq. (A.14) in the Appendix. Overall, therefore, the shaded region corresponds to the region where the regulation induces the blockholder to invest in information acquisition when she would not do so otherwise, and such a change raises social welfare.

Fig. 1. Region where regulation is beneficial, when n = 30, p = 0.52, q = 0.64. Allowing for trading: With trading, the beneficial role of the regulation described above is eliminated. Suppose that the blockholder did not acquire information in the absence of the regulation. Then, she would gain by trading all of her shares (since she has the same information precision p as other shareholders, and so, there are shareholders who would improve the value of the vote by buying the shares and voting informatively). The introduction of the regulation does not affect how much the blockholder receives from trading all her shares and therefore does not affect her value when she does not invest in information acquisition. Therefore, the regulation no longer worsens the blockholder’s payoff when she does not invest in information acquisition. The rule then does not make information acquisition more likely and may make information aggregation less efficient (by forcing the blockholder to vote her 21

2b > 2b∗ shares if they are not traded away, as in Section 4.2.2). Of course, there are several reasons, not modeled here, why a blockholder might not want to engage in trading and, so, might suggest that the rule has bite. One is that new potential shareholders might not have useful information to incorporate into the vote, reducing their willingness to pay (e.g., liquidity traders). A second is that the blockholder is assumed to obtain the full surplus from any trade, which is unlikely to be true. Finally, there may be practical impediments to trading. This could arise if the blockholder were an index fund that needed to track the index23 or if the blockholder had to worry about having too much of a price impact from selling shares. Regulation and proxy advisors: Our analysis suggests that the regulation can improve incentives for information acquisition by blockholders. One method of information acquisition is paying information intermediaries to do the work. Indeed, in the wake of the SEC rule, the proxy advisory industry gained prominence and expanded substantially. As a source of additional information, proxy advisors can improve efficiency. However, Malenko and Malenko (2019) show that information provided by one source (the proxy advisor) can crowd out information gathered by shareholders. In our model, where the shareholders may be small and the blockholder large, it is natural to consider the case in which only the blockholder purchases information from the proxy advisor. In this case, there would be no crowding-out. Our analysis above suggests an additional concern with the SEC rule that is related to Malenko and Malenko’s (2019) analysis but is not about the proxy advisory industry per se: the implicit requirement that the blockholder vote with all her shares can worsen information aggregation through over-reliance on one (the blockholder’s) signal. 5.2. Empirical implications The model makes several predictions that can be tested empirically: 1. Minority shareholders are more likely to abstain when their information is less precise, when there are fewer of them (and hence their aggregate information is noisier), and when the blockholder has fewer shares combined with more precise information. [Proposition 1 and Lemma 1] 23 Given the results above, this offers one explanation for why there is a rule for investment advisers rather than blockholders in general.

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2. The blockholder is more likely to partially or fully abstain when she holds many shares and her information is less precise. [Proposition 1] 3. Trading may occur before a vote with shares being traded to well informed shareholders from blockholders with many shares but imprecise information. [Propositions 2 and 3] 4. If the 2003 SEC rule was interpreted by market participants as a prohibition of abstention, we would expect information acquisition to increase right after its adoption. [Proposition 4] 5. There is an interdependence in shareholders’ voting behavior. That is, shareholders will condition their votes on the precision of other voters’ information and the ownership structure (the number of minority shareholders and the number of shares that minority shareholders and blockholders own). [Proposition 1 and Lemma 1] In Section 5.3, we discuss the limitations of current data sources. Even given great data sources, observation 5 is difficult to test given the endogeneity problem and the fact that these variables may be highly correlated. It would be of interest to see how activist campaigns affect nonactivists. We discuss this in further detail in Section 6.1, where we extend the model to allow the blockholder to communicate information. A possible source of exogeneity for ownership structure changes could be to look at deaths in firms where families have large shares. A key variable of interest in the predictions above is the precision of shareholders’ information. There have been several related measures in the empirical literature. Iliev and Lowry (2015) posit that informed funds have a tendency to place less weight on Institutional Shareholder Services (ISS) recommendations and are less likely to vote in a “one-size-fits-all” manner. They find that these funds are part of larger fund families, are larger, are located near other funds, have low turnover, and have higher alpha. Iliev, Kalodimos, and Lowry (2019) demonstrate that investor views of SEC Electronic Data Gathering, Analysis, and Retrieval (EDGAR) filings are a good proxy for research effort. Kacperczyk and Seru (2007) show that more skilled/privately informed fund managers are less responsive to changes in public information, and that this responsiveness can be used as a measure of skill/private information. Bushee and Goodman (2007) show that institutions that have larger investments and for which an investment represents a larger part of their portfolio trade with more private information. Lastly, a key result of our model is that a blockholder and shareholders 23

will attempt to align the weight that the blockholder has on the vote with how precise her information is. In the model this occurs through abstention and trading, but in reality it could occur by other means. First, giving shares to those with information to express may motivate share grants to executives. Second, there are other markets for shares (and votes) that we do not formally analyze. One is the stock loan market (see Christoffersen et al., 2007; and Aggarwal et al., 2015). Another is Initial Public Offerings (IPOs), which create a market for shares. In addition, poorly informed investors may prefer to own cash flow rights in firms but delegate voting rights to more informed investors. Delegation can occur explicitly with brokers and implicitly through investing in mutual funds or Exchange Traded Funds (ETFs). However, agency issues may obstruct such efforts—brokers vote more often for management, and funds may or may not acquire useful information. 5.3. Empirical challenges Assessing the theoretical results empirically is challenging as it is difficult for researchers to observe voting behavior at the voter level. The only available shareholder-level voting data in the U.S. were made possible by the 2003 SEC rule discussed above. The rule makes votes by investment advisers public information; investment advisers must submit Form N-PX annually.24 These data, while very useful in several applications, are not particularly useful in our context for several reasons. First, the form does not require the fund to report the number of shares voted, and it says nothing about what to report if a fraction of shares were not voted in the same direction as others.25,26 Second, Form N-PX is only for investment advisers. Even 24

The data from this form can be downloaded from the SEC; ISS has compiled all of these data for sale. 25 An SEC rule proposed in 2010 requires that these data be collected but has not been implemented. A comment on the rule by the fund industry body, the Investment Company Institute, reads as follows: “Specifically, we oppose expanding the types of disclosure that funds are required to provide on Form N-PX to include the precise number of shares they were entitled to vote and the precise number of shares that were actually voted. This data is not required by Section 951, would be difficult to compile (and in fact, may be impossible to produce with the degree of precision required), is of no value to most fund investors, and is generally unnecessary to achieve the purposes of proxy vote disclosure.” (https://www.ici.org/pdf/24721.pdf) 26 Mutual fund families sometimes delegate a vote to the funds themselves, which may result in a “split vote” for the family, where funds vote in different directions. Technically,

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if these data were helpful, any results obtained from these data may not generalize to other blockholders; Holderness (2009) states that only 29% of blockholders in the U.S. are financial, which includes mutual funds, banks, and pension funds. Finally, the 2003 SEC rule specifically requires investment advisers to vote in the best interest of their clients and makes the votes public. As we discussed earlier, this may lead investment advisers feeling pressured not to abstain. The empirical literature has also documented a reliance of institutional investors on proxy advisors for information and voting recommendations, which indicates how liability has influenced incentives.27 6. Extensions In this section, we consider several extensions to the baseline model. In part this is to show robustness of the central intuition that strategic blockholders and shareholders might not vote with all their shares or in line with their signals. Instead, they vote in a way that appropriately weights the quality of the underlying information to the extent possible. In addition, these extensions provide further applied insight as we discuss below. 6.1. Active blockholder In this section, we suppose that the blockholder can also observably announce information (cheap talk) before other shareholders vote but otherwise maintain the framework outlined in Section 2. In particular, all shareholders, including the blockholder, still vote simultaneously. We call this the case of the active blockholder, in contrast to the analysis in previous sections where a “passive” blockholder makes no such announcement. In this common interest game, communication before voting can lead to a better vote outcome. Communication among many dispersed shareholders is generally infeasible, and voting is used to aggregate information. Nevertheless, blockholders may easily be more visible and scrutinized. Any shareholder with over 5% of the shares of a publicly traded firm must publicly disclose this to the SEC. Some blockholders are very public about their activities, such as activist investors. Activist investors make public statements the result would be the same as a partial abstention. However, the underlying reasons for this behavior may be quite different. 27 McCahery, Sautner, and Starks (2016) and Iliev and Lowry (2015) demonstrate the use of proxy advisors for this purpose.

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about their intentions and often discuss their intentions with other investors (see Coffee and Palia, 2016). The analysis of this sequential game proceeds by backward induction, and the equilibrium concept we apply is Perfect Bayesian Equilibrium: the shareholders observe the blockholder’s statement, drawing inferences on the blockholder’s signal, and then choose their votes; the blockholder, anticipating the equilibrium behavior of the shareholders in response to their signals and to her own statement, will choose her voting strategy. Consider a blockholder with few votes but very precise information. This blockholder cannot represent all of her information in her vote. A shareholder with an imperfect signal who has observed the blockholder’s voting intention may, then, prefer to ignore its information and vote in line with the blockholder (to support the blockholder’s information). Indeed, some voters should mimic the blockholder’s vote to make the weight of the blockholder’s information correspond to the optimal threshold 2b∗ , with the remainder voting sincerely.28 Given this, the blockholder prefers to communicate truthfully.29 We show that this behavior is an equilibrium and is, in fact, the most informative equilibrium in pure strategies. Proposition 5. Given an active blockholder who makes an announcement before the vote, there is an equilibrium in which the blockholder communicates truthfully, votes sincerely with min[2b, 2b∗ ] shares, and (i) if b ≥ b∗ , all shareholders vote sincerely, or (ii) if b < b∗ , 2b∗ − 2b shareholders vote in an identical fashion to the blockholder, and the rest of the shareholders vote sincerely. Such an equilibrium is the most informative equilibrium within the class where all shareholders employ a pure strategy. The proof is in the Internet Appendix. It is natural to compare outcomes and the informational efficiency of the equilibria in the non-active (passive) blockholder case to that of the active blockholder case. When b ≥ b∗ , the outcome is the same. When b < b∗ , 28

This is similar to the way that some voters in Persico (2004), Maug (1999), and Maug and Rydqvist (2009) ignore their information to bring a general (non-majoritarian) voting rule in line with an optimal one or to counter conflicts of interest in Maug (1999) and Maug and Yilmaz (2002). 29 There are, of course, other equilibria involving, for example, babbling. Sincere communication will be a feature of the most informative equilibrium.

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in the case of the non-active (passive) blockholder, there is no scope for shareholders to allow the blockholder’s signal to carry more weight because the blockholder’s information is unobservable. This leads to an outcome where either all shareholders vote with their signal or all abstain. Instead, when the active blockholder’s intention is observed, shareholders can condition their vote on the blockholder’s communication, with some ignoring their information to vote with the blockholder.30 As each shareholder seeks to maximize the value of the firm, the fact that some shareholders decide to condition their vote implies that welfare can be strictly higher in the active blockholder case.31 It is noteworthy that when b < b∗ , for the non-active (passive) blockholder, a bangbang solution is most informative, with either all shareholders voting sincerely or abstaining. This is in contrast to the case of the active blockholder, which has only a fraction of shareholders voting sincerely. This contrast can be understood by realizing that in Proposition 5, the shareholders who do not vote sincerely vote along with the blockholder, whose announcement they observe. Thus as increasing numbers of them vote with the blockholder, they give the blockholder’s signal increasing weight in the decision relative to any individual shareholder who votes sincerely. Instead, in Proposition 1, as shareholders do not observe the blockholder’s information, they may abstain—as more shareholders abstain, the blockholder’s signal’s weight remains constant relative to any individual shareholder who votes sincerely. 31 This contrasts with Dekel and Piccione (2000), who assume symmetry among voters and suggest that sequential voting does no better than simultaneous voting. Note that in our model, the blockholder is different in two ways: she has a different signal precision and a different number of votes. It is the first difference that is key to the contrast with Dekel and Piccione (2000): if it were the case that p = q, then b∗ = 0 and so, necessarily b > b∗ , and Proposition 5(i) would lead to an identical outcome to that of Proposition 1. Instead, when q > p and b∗ > b, the blockholder does not have enough votes to ensure that her relatively precise information is incorporated into the vote; consequently, a sequential vote that allows some shareholders to ignore their own (relatively imprecise) information to bolster the impact of the blockholder’s information in the vote leads to a more efficient outcome. Further, note that in contrast to herding models (e.g., Bikhchandani et al., 1992; Banerjee, 1992) shareholders get no utility from their own actions directly. Instead, they get utility from the outcome of the vote, which depends on the behavior of all the voters. Consequently, if they anticipate that the information from the blockholder’s signal is already appropriately incorporated into the vote, they optimally vote to incorporate their own additional information rather than to reflect their updated prior (which relies on the information that is in any case incorporated into the final decision). If a shareholder’s payoffs depended on his own action directly then herding effects could arise; one example is if they preferred to vote with the majority as in Callander (2007). 30

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Note that another contrast with Proposition 1 is that the most informative equilibrium can involve asymmetric shareholder behavior, as some fraction of shareholders ignore their information to support the blockholder’s information and maximize the probability that the vote matches the state. Specifically, the most informative equilibrium involves (2b∗ − 2b) shareholders ignoring their information; this clearly leads to a coordination problem. Moreover, the shareholders only want to coordinate because the blockholder has moved first. This suggests that leadership and coordination among shareholders may enhance efficiency.32 Activist investors are often very public about their positions. For example, Bill Ackman, of the Pershing Square Hedge Fund, appeared “almost daily on CNBC to take his case directly to investors” (George and Lorsch, 2014). Direct communication with other shareholders was permitted by the SEC’s Rule 14a-12 in 1999. Communication enters a gray area when investors have sizable stakes in the firm—an individual or group stake of 5% or more must be declared publicly to the SEC under Section 13(d). The gray area is what constitutes a group, i.e., do communication and parallel actions constitute coordination or not? The purpose of this public declaration to the SEC is to make other shareholders aware of changes in control of the firm (Lu, 2016). The downside of such unobservable coordination is that trading profits are made by the insiders in a “wolf pack,” while other shareholders are unaware. However, the trading profits may be needed to incentivize such behavior. Our model demonstrates that there can be benefits from having such a “wolf pack,” i.e., having (i) a lead informed activist and (ii) subsequent coordination among some other investors to support the activist’s position. Because we do not incorporate coordinated trading into the model, we cannot address the costs of such behavior.33 A different type of coordination might be made feasible by having shareholders invest in a fund, with the fund taking on the coordination role. Appel, Gormley, and Keim (2019) find that passive mu32 Allowing for heterogeneity among the shareholders in the precision of their signals creates a natural coordination device: the lower-precision shareholders would be those who ignore their information and follow the blockholder. 33 Brav, Dasgupta, and Mathews (2016) provide a very different motivation for coordination by a wolf pack in a model that does not include voting. A sufficiently large bloc of shareholders can make a value-enhancing change in a firm. These activists have complementarities in their costly decision to take a stake because they receive reputation benefits from a successful activism campaign. The complementarities lead to a coordination game.

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tual funds have facilitated activism by supporting activists with large blocks of shares and that this has led to changes in activist presence, tactics, and overall outcomes. In this way, our result speaks to the observation in Edmans (2009) that blockholders in the U.S. tend to be small and that this leads to a puzzle regarding how they can exert governance. He suggests this can be resolved by the possibility of exit. Our model provides an alternative answer to this puzzle—communication allows a small blockholder to exert outsized influence and potentially justify information acquisition costs. This illustrates that the free-rider problem for information acquisition can be partially overcome by allowing for communication. Brav et al. (2008) note that activist hedge funds are small (the median maximum ownership stake in their sample is approximately 9.1%) and take many forms of public action.34 Of course, there are settings in which the coordination implicit in Proposition 5(ii) might appear unrealistic; for this reason, much of the literature has focused on symmetric strategies.35 Indeed, Proposition 5(i) involves symmetric strategies, with all shareholders voting sincerely for the case in which b ≥ b∗ . The case in which b < b∗ is where coordination is relevant; nevertheless, it is intuitive that even considering symmetric shareholder strategies, the blockholder should vote with all her shares even though she does not have “enough” influence. We now analyze a mixed strategy equilibrium in which the shareholders play symmetric strategies and mix to provide the blockholder with more influence. In this way, even without direct coordination between shareholders a small blockholder can still exert influence. Other mixed strategy equilibria may also exist. This particular equilibrium has similar properties to that described in Proposition 5(ii) in that shareholders vote in a way that provides the blockholder with more say.36 34

This includes seeking board representation without a proxy contest or confrontation with management (11.6%), making formal shareholder proposals or publicly criticizing the company (32.0%), threatening to wage a proxy fight to gain board representation or sue (7.6%), launching a proxy contest to replace the board (13.2%), and suing the company (5.4%). 35 Maug and Rydqvist (2009), who consider both, argue that pure strategy equilibria (which involve similar shareholders behaving asymmetrically) and symmetric mixed equilibria deliver qualitatively similar results. We demonstrate the same. 36 It is intuitive that the equilibrium characterized in Proposition 5 is more informative than that described in Proposition 6 and indeed any equilibrium that involves mixed strategies. As mixed strategies can lead to coordination failures, equilibria that involve

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Proposition 6. When b < b∗ , there is a symmetric equilibrium in mixed strategies for large enough n (n > 2b∗ − b) such that (i) the blockholder votes all of her shares with her signal and communicates her signal truthfully, and (ii) shareholders vote with the blockholder with probability 1 when they receive the same signal as the blockholder and vote with a mixed strategy between the other alternative and voting with the blockholder when they receive a different signal. The proof is in the Internet Appendix. 6.2. Biased blockholder Thus far, we have considered votes where the blockholder and shareholders have had identical objectives. In this section, we suppose that the blockholder may be biased; bias here means that she prefers that the action M be chosen regardless of the underlying state. For example, this bias may arise because the blockholder is part of management or directly tied to it through business dealings. To study this, we assume that there is no communication as in our baseline model.37 In this situation, the agents do not have common values and a focus on the most informative outcome is less clear as a selection criterion. We therefore study a specific symmetric strategy equilibrium. We model bias as follows. Suppose that the blockholder may be completely biased, in which case she strictly prefers M , or completely unbiased (like the shareholders) and prefers to maximize the value of the firm. There is uncertainty over the preferences of the blockholder. Specifically, let β denote such strategies will be less informative than similar purified strategies that involve no coordination failure, and since Proposition 5 is the most informative among those equilibria that involve pure strategies, the conclusion is immediate. 37 Note that with communication, there is an equilibrium outcome identical to the equilibrium outcome in the case of no communication. Indeed, this may be the most natural equilibrium. Since preferences are not fully aligned here, a biased blockholder would attempt to make statements that influence shareholders, and shareholders may be skeptical of the blockholder’s statements. Consequently, there will, at best, be coarse information transmission, and as in Crawford and Sobel (1983), there always exist “babbling” equilibria. In such an equilibrium, the blockholder’s ability to make statements before the vote has no effect at all, and the outcome would be identical to that in the case with no communication that we consider in Proposition 7.

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the probability that the blockholder is biased towards M . With the complementary probability, the blockholder has the same preferences as shareholders. It is immediate that in any equilibrium, a biased blockholder prefers to vote all of her shares for her preferred position: her choice is not observed and can have no effect on the choices made by shareholders. Since more votes in favor of her preferred position increase the likelihood that this position is adopted, taking as given the behavior of shareholders, she will vote all of her 2b shares for M . Intuitively, the possibility of such bias leads shareholders to employ strategies that offset it; that is, shareholders vote A more often than their signals suggest to offset the expected votes for M that do not reflect information and reflect only bias. This is similar to the way in which shareholders support an unbiased blockholder who does not have sufficient votes, as we describe in Propositions 5 and 6, and similar to the analysis of Maug (1999) and Maug and Yilmaz (2002), who consider shareholders with conflicts of interest. Perhaps somewhat less intuitively, this causes an unbiased blockholder to tilt in the direction of M to offset the shareholders’ behavior. We characterize such behavior in a specific equilibrium in which shareholders do not coordinate and employ symmetric strategies. Proposition 7. Consider the case of b > b∗ and values µ, bm , and ba that satisfy (IA1), (IA2), and (IA3), respectively, which appear in the Internet Appendix; then, the following strategies constitute an equilibrium of the simultaneous voting game in which there is a probability β that the blockholder is biased and strictly prefers the M outcome: (i) A biased blockholder votes for M with all 2b shares regardless of signal received. (ii) An unbiased blockholder with signal a votes for A with 2ba shares, where ba < b∗ , (and if the characterization in (IA3) gives a negative value of ba , we interpret this as voting for M ). (iii) An unbiased blockholder with signal m votes for M with 2bm shares, where bm ≥ b∗ . (iv) Shareholders vote with symmetric mixed strategies: each votes for M with probability µ and votes for A with probability 1 − µ when receiving the m signal, and votes for A with probability one when receiving the a signal. Moreover, there are parameters for which such an equilibrium exists. The proof is in the Internet Appendix. 31

In Proposition 7, shareholders partially ignore their signals. They do so to cancel out the bias of the blockholder and allow the sincere voting that otherwise arises to provide an informative vote. This is similar to the case with the unbiased active blockholder, but here, it is done to oppose the blockholder rather than support her. In Proposition 7, parts (ii) and (iii), the unbiased blockholder acts differently than the unbiased blockholder in Proposition 1, where the blockholder is known to be unbiased. In that case, the blockholder votes a fixed number of shares in the direction of the signal, for both signals. Here, the possibility that the blockholder might be biased leads her to behave differently. When she has an m signal, she votes more for M than the optimal vote threshold, and when she has an a signal, she votes less for A than the optimal vote threshold. This asymmetric behavior is a type of feedback effect; because the shareholders will react to the possibility of bias by weighting their votes towards option A, the blockholder acts to counter this overweighting of option A. This result is closely related to the results of Maug and Rydqvist (2009), who study a common value model with strategic voting where shareholders each have one vote and there may be different majority voting rules. In their equilibrium with asymmetric pure strategies (i.e., where identical shareholders may have different strategies), they find that a voting rule that is not optimal will induce a subset of shareholders to ignore their information and vote to restore the optimality of the voting rule, allowing all other shareholders to vote informatively. Here, we find a similar result: the biased blockholder, by voting for M without using information, imposes a higher threshold for A to win. Shareholders ignore their information to some extent to counteract this effect. 6.3. Asymmetric prior A central result that emerges in various forms throughout this paper is that the number of votes that the blockholder or shareholders control may differ from the optimal number of votes given their signals. Indeed, as we have argued before, if 2b > 2b∗ in our baseline case, there is an equilibrium where the vote aggregates information fully efficiently; that is, in the same way that a Bayesian with access to all the signals directly would do. An asymmetric prior (e.g., one that assigns probability r > 12 to state M ) can be viewed as an additional signal with respect to the baseline case where the

32

prior assigns equal likelihood to either state.38 Thus, if there were a way to put sufficient weight on this prior without detracting at all from the other signals, this would clearly be beneficial.39 Our analysis identifies two such cases. The first is the case where the blockholder has substantially more shares than the optimal threshold (b >> b∗ ). Given our result in Proposition 1(i), with symmetric priors the blockholder abstains on votes beyond the optimal threshold. With an asymmetric prior, the blockholder can therefore vote M with an appropriate number of shares on which it would have abstained and allocate 2b∗ in line with her signal.40 The second is the case characterized in Proposition 1(iii) where, with a symmetric prior, all shareholders abstain. In an equilibrium where shareholders coordinate, an appropriate number would vote for M with the rest abstaining.41 More generally, there is an informational cost of allocating votes to the prior rather than to the signals of the agents. Trivially, if the prior is sufficiently close to the symmetric prior, a shareholder should allocate a vote in line with her signal rather than with the prior. However, if there are no “free” votes available and the prior is sufficiently informative then shareholders should clearly put weight on the prior. We now analyze a case with an asymmetric prior. The prior must be sufficiently asymmetric that it becomes relevant for voting: with probability r > p the state is M . Also, in order that only shareholders (and not the blockholder) want to divert votes towards the prior, we assume q > r. Additionally, we look at parameters where Condition C1 holds and b < b∗ , as in Proposition 1(ii). We prove that the following strategies constitute an equilibrium: the blockholder votes sincerely with all its shares; shareholders act symmetrically 38

Maug (1999) considers asymmetric priors, but in a symmetric environment where there is no blockholder. 39 Analogous considerations arise in the case of non-majoritarian (non-unanimous) voting rules and so we omit further discussion of them. 40 Specifically, following the logic of Nitzan and Paroush (1982), the asymmetric prior could be treated as an additional signal, and the appropriate way to use the signal would ln r−ln(1−r) ∗ be to allocate a number of votes to M corresponding to ln p−ln(1−p) ; call this integer r . ∗ ∗ Clearly, this is feasible if 2(b − b ) ≥ r as the blockholder will have extra shares that it will not use to express its vote. In this case, again following the logic of the proof of Proposition 1, this would aggregate information fully efficiently. 41 Again and trivially, the appropriate number of votes would be r∗ .

33

and vote for M with probability µ for some interior µ ∈ (0, 1), and otherwise vote sincerely. Hence, this demonstrates that shareholders incorporate the prior into the vote by voting for M with positive probability when they receive an a-signal. In this way, we show that our central intuitions extend qualitatively to this environment.  2b+1 1−p q Proposition 8. When Condition C1 holds, b < b∗ , q > r > p, and 1−q > p  n+b+1 p 1−q r > 1−p , there is an equilibrium such that (i) the blockholder 1−r q votes all of her shares with her signal, and (ii) shareholders play symmetric mixed strategies; voting for M with probability µ ∈ (0, 1) and otherwise voting sincerely. The proof is in the Internet Appendix. 7. Conclusion Our paper extends a standard voting environment by introducing a voter who has multiple votes: the blockholder. This is natural in a corporate setting and leads to new results. Contrary to the common wisdom promoted by regulators, we demonstrate that allowing for abstention can increase the informativeness of a vote. A blockholder who wants to maximize the value of the firm and has imprecise information but many shares would prefer not to vote all of her shares to allow other shareholders’ information to be used in the vote. We also demonstrate that allowing shareholders to act strategically can also increase the informativeness of a vote to support an unbiased blockholder’s information or to counter a biased blockholder’s vote. In our analysis, we have made many simplifications to demonstrate the basic driving forces when a blockholder is voting. It would be of interest to further extend the model to allow for a more realistic environment. Additional avenues that could be pursued are allowing for a richer model of share-trading and considering multiple blockholders.

34

Appendix A. Proofs Appendix A.1. Proof of Proposition 1 (Blockholder voting) We prove our result through a series of lemmas. We begin by examining equilibrium behavior, demonstrating that the equilibria we describe are indeed equilibria. Lemma 2. There is always an equilibrium in which the blockholder votes sincerely with min{2b∗ , 2b} votes and all shareholders vote sincerely. Proof. We begin by proving that a shareholder’s best response when all other shareholders are voting sincerely and the blockholder is voting sincerely is to vote sincerely. We then prove that the blockholder’s best response to all shareholders voting sincerely is to vote min{2b∗ , 2b} shares sincerely. A shareholder votes sincerely Describe the probability that the blockholder votes 2b shares for xB ∈ {M, A} given state θ ∈ {M, A} as PB (xB | θ) ≥ 0. We examine the choice of one shareholder. Fix the other 2n shareholders as having a symmetric strategy, and describe the probability that they vote for xS ∈ {M, A} given state θ as PS (xS | θ) ≥ 0. We will impose that they vote sincerely below. Define the signal that a shareholder i receives as si ∈ {m, a}. The probability of the shareholder being pivotal π S depends on both alternatives having the same number of votes. π S (piv | 2b, θ) = PB (A | θ) +PB (M | θ)

2n! PS (A | θ)n−b (PS (M | θ))n+b (n − b)!(n + b)!

2n! PS (A | θ)n+b (PS (M | θ))n−b (n + b)!(n − b)!

We first will prove that π S (piv | 2b, A) = π S (piv | 2b, M ) given (i) sincere voting by the blockholder and 2n shareholders and (ii) the probability that the state of the world is M and equal to the probability that the state of the world is equal to A (and therefore 21 ). Note that: PB (A | θ) = Pr(a | θ) Pr(A | a) + Pr(m | θ) Pr(A | m).

35

(A.1)

We write: Pr(a

| =

Pr(a) Pr(θ | a) Pr(θ) 1 Pr(θ | a) 2

θ) =

1 2

= Pr(θ | a) where in the second line, Pr(θ) = 12 by the assumption that both states have probability 21 , and Pr(a) = Pr(a | A) Pr(A)+Pr(a | M ) Pr(M ) = p 21 +(1−p) 21 . This technique also implies that Pr(m | θ) = Pr(θ | m). We can now rewrite Eq. (A.1) as: PB (A | θ) = Pr(θ | a) Pr(A | a) + Pr(θ | m) Pr(A | m). Using our definitions in Section 2 of the paper and the assumption that the blockholder votes sincerely implies the following: PB (A | A) = q(1) + (1 − q)(0) = q. Similarly, PB (A | M ) = 1 − q, PB (M | M ) = q, and PB (M | A) = 1 − q. We can perform similar calculations for a shareholder who votes sincerely, which gives us PS (A | A) = p, PS (A | M ) = 1 − p, PS (M | M ) = p, and PS (M | A) = 1 − p. Taking this and rewriting the pivotal probability π S (piv | 2b, A): π S (piv | 2b, A) = q

2n! 2n! pn−b (1−p)n+b +(1−q) pn+b (1−p)n−b . (n − b)!(n + b)! (n + b)!(n − b)!

Similarly, we can write the pivotal probability π S (piv | 2b, M ) as: π S (piv | 2b, M ) = (1−q)

2n! 2n! (1−p)n−b pn+b +q (1−p)n+b pn−b . (n − b)!(n + b)! (n + b)!(n − b)!

Therefore, π S (piv | 2b, A) = π S (piv | 2b, M ). We denote these as π S (piv | 2b). We can now define the expected utility of a shareholder from voting his share for choice M over abstaining given signal m: EUS (M, ∅

1 1 si = m) = pπ S (piv | 2b, M ) − (1 − p)π S (piv | 2b, A) 2 2 1 = (2p − 1)π S (piv | 2b) > 0. 2 |

36

Therefore, if the shareholder has signal m, he prefers to vote for M rather than abstain when all other shareholders and the blockholder are voting sincerely. We can now define the expected utility of a shareholder of voting his share for choice A over abstaining, given a signal a: EUS (A, ∅

1 1 si = a) = pπ S (piv | 2b, A) − (1 − p)π S (piv | 2b, M ) 2 2 1 = (2p − 1)π S (piv | 2b) > 0. 2 |

Therefore, the shareholder prefers to vote for A rather than abstain when he receives signal a. Now we analyze the choice of a shareholder between voting for A over M given a signal si = a. ˜ = pπ S (piv | 2b, a) − (1 − p)π S (piv | 2b, m). EUS (A, M | σ S = A) ˜ ) is positive. This is clearly positive. Similarly, EUS (M, A | σ S = M This implies that the shareholder will vote sincerely. Note that this proof works given any number of shares 2b. In addition, it allows for a mixed strategy on the part of the marginal shareholder but finds that this shareholder has a pure strategy to vote sincerely. It also does not depend on the parameters q and p, aside from the fact that both are larger than 21 . A blockholder votes sincerely with all of her shares We begin by assuming that the blockholder votes sincerely and examine how many shares she will vote with. Assume without loss of generality that the blockholder received an m signal. Suppose that the blockholder votes 2˜b shares; then, her payoff per share owned is: V (n, ˜b, p, q) = q

  n+˜b  n−˜b  X X 2n + 1 2n + 1 2n+1−i i p (1−p) +(1−q) p2n+1−i (1−p)i . i i i=0

i=0

The blockholder prefers to vote two more shares for M if and only if: V (n, ˜b, p, q) < V (n, ˜b + 1, p, q) 37

     Pn+˜b+1 2n + 1 2n + 1 2n+1−i i p2n+1−i (1 − p)i + p (1 − p) +   q i=0  q i=0 i i <     ⇐⇒     Pn−˜b−1 2n + 1 Pn−˜b 2n + 1 2n+1−i i (1 − q) i=0 (1 − q) i=0 p2n+1−i (1 − p)i p (1 − p) i i     2n + 1 2n + 1 ˜ ˜ n+˜b+1 n−˜b ⇐⇒ (1 − q) p (1 − p)
Pn+˜b



This leads to the definition of b∗ in (1): the blockholder prefers to vote more shares sincerely if b < b∗ and vote fewer shares sincerely if b > b∗ . The result is the same if the blockholder receives an a signal due to symmetry. Finally, we demonstrate that the blockholder prefers to vote all of her shares (when b ≤ b∗ ) sincerely rather than vote against her signal. That is, for all ˜b:        Pn+˜b 2n + 1 Pn−˜b 2n + 1 2n+1−i i p (1 − p) +   q i=0 p2n+1−i (1 − p)i +  q i=0 i i  >        Pn−˜b 2n + 1 Pn+˜b 2n + 1 2n+1−i i (1 − q) i=0 p (1 − p) (1 − q) i=0 p2n+1−i (1 − p)i i i     n+˜b n+˜b X X 2n + 1 2n + 1 2n+1−i i ⇐⇒ q p (1−p) > (1−q) p2n+1−i (1−p)i i i i=n−˜b+1

which is true given q > 12 .

i=n−˜b+1

⇐⇒ q > 1 − q

We must also consider the case in which shareholders abstain. Lemma 3. There is always an equilibrium in which the blockholder votes sincerely with all of her votes (or two or more votes) and all shareholders abstain. Proof. It is immediately clear that if shareholders do not vote, then the blockholder votes sincerely (and is indifferent regarding the number of votes with which she does so). If the blockholder votes sincerely with two or more votes and all shareholders abstain, then a single shareholder can never be pivotal, so abstaining is a best response.

38

   



  

Taken together, Lemmas 2 and 3 establish that there is always a multiplicity of equilibria. Given the observation above, it is clear that they cannot be ranked unambiguously when b < b∗ . Instead, their relative efficiency depends on parameter values. It is immediate that if b > b∗ , the first type of equilibrium is most informative given that the definition of b∗ in (1) corresponds to the finding in Nitzan and Paroush (1982), Theorem 1, that if all votes are sincere, it is optiq and each shareholder a weight mal to give the blockholder a weight of ln 1−q p of ln 1−p . These weights in turn simply reflect the weights that a Bayesian decision maker would assign to each signal. Inspection of (1) shows that 2b∗ corresponds to the weights of Nitzan and Paroush (1982) incorporating that each shareholder has a single vote and the vote share of the blockholder ln q−ln(1−q) ) must be modified to reflect that b∗ is (which would otherwise be ln p−ln(1−p) an integer. Note that an equilibrium in which two shareholders abstain is outcomeequivalent to an equilibrium in which one shareholder disregards its information and votes for M while another shareholder disregards its information and votes for A. Therefore, any equilibrium with shareholders abstaining implies that there are other equilibria with an identical outcome where shareholders cancel out each others’ votes. Therefore, the candidate equilibrium in which only the blockholder votes is outcome-equivalent to many other equilibria. When b < b∗ , we have found two classes of equilibria that are candidates for being the most informative equilibrium: one that is outcome-equivalent to only the blockholder voting and another in which the blockholder votes all 2b shares and all shareholders vote sincerely. The first question we must answer is whether there are any more candidate equilibria to consider. Other potential equilibria: Consider a possible equilibrium in which the blockholder votes zB shares sincerely, zS shareholders vote sincerely, and the remaining shareholders abstain, where zB < zS < 2n + 1. From our result in Lemma 1 (which we prove in the next subsection and implies that if shareholders can affect the outcome of the vote then having more of them voting can only help the accuracy of the vote), if this equilibrium is more informative than that in which only the blockholder votes, it will be less informative than one in which the blockholder votes zB shares and all shareholders vote sincerely. Again invoking a finding of Lemma 1, this in turn is dominated by having the blockholder vote with more shares: up to 2b∗ . This implies that it is less informative than one

39

of the two candidates we have found. If any of the shareholders who abstain in this possible equilibrium were to instead vote against their signal or vote for one alternative irrespective of their signal, it would be less informative. The only possible candidates left are an equilibrium taking the form of (i) the blockholder sincerely voting zB shares and fewer than zB shareholders voting (sincerely or otherwise) or (ii) shareholders vote regardless of their signal z > 2b net42 shares for proposal J = {M, A}, and fewer than z − 2b other shares from shareholders are cast sincerely. Equilibria that take the form of (i) are informationally equivalent to the equilibrium in which only the blockholder votes, so we will select that equilibrium. Equilibria that take the form of (ii) are informationally inferior to the equilibrium in which the blockholder votes 2b shares sincerely and all shareholders vote their shares sincerely. Therefore, the two candidate equilibria we summarized initially are the only two candidates for the most informative equilibrium. Comparing the two equilibria: An equilibrium in which only the blockholder votes will lead to an outcome that matches the state with a probability (and delivers the expected utility) of q. An equilibrium where the blockholder votes all of her shares and all shareholders vote sincerely matches the state with probability:   2n+1 X 2n + 1 2n + 1 i 2n+1−i q p (1 − p) + (1 − q) pi (1 − p)2n+1−i . i i i=n−b+1 i=n+b+1 (A.2) This expression corresponds to n + b + 1 or more votes being cast in favor of the true underlying state. The first expression corresponds to the blockholder’s signal matching the state (with probability q), with the summation indicating that n − b + 1 or more of the 2n + 1 shareholders have signals that match the state. The second expression instead represents the likelihood that the blockholder’s signal does not match the state but that n + b + 1 or more of the shareholders have signals that do. We now compare the probability with which each equilibrium matches the state. All shareholders voting is a more informationally efficient equilibrium 2n+1 X

42

Net means that if x shareholders are voting for M regardless of their signal and y shareholders are voting for A regardless of their signal, z = x − y > 0 are voting for M regardless of their signal.

40

when:   2n+1 X 2n + 1 2n + 1 i 2n+1−i q p (1−p) +(1−q) pi (1−p)2n+1−i > q i i i=n−b+1 i=n+b+1 2n+1 X

   n−b  X 2n + 1 i 2n + 1 i 2n+1−i ⇐⇒ (1 − q) p (1 − p) >q p (1 − p)2n+1−i i i i=0 i=n+b+1
P2n+1 2n+1 p i ) ( 1−p q i=n+b+1 i . ⇐⇒ Pn−b 2n+1
p > 1−q ( 1−p )i i=0 i 2n+1 X

This is Condition C1 in the text.

Appendix A.2. Proof of Lemma 1 (How parameters affect Condition C1) The result with respect to q is trivial, we consider the other parameters in turn. First, it is convenient to simplify expressions slightly by introducing p the notation r := 1−p . Thus, Condition C1 can be written as:


P2n+1

2n+1 i r i=n+b+1 Pn−b 2n+1i
i r i=0 i

(i) Consider
P2n+1 2n+1 d i=n+b+1 i ri Pn−b 2n+1
i dr r i=0 i Pn−b 2n+1
i P2n+1 r i=0 i=n+b+1 i =

>

q . 1−q


i−1 P2n+1 ir − i=n+b+1 Pn−b 2n+1
i 2 ( i=0 r). i

2n+1 i


i Pn−b r i=0

2n+1 i




2n+1 i

The denominator is positive, but the numerator is positive if and only if:
i P2n+1 Pn−b 2n+1
i 2n+1 ir ir i=n+b+1 i i
> Pi=0 . P2n+1 n−b 2n+1
i 2n+1 i r r i=n+b+1 i=0 i i Note that P (n − b) n−b i=0

P2n+1 i=n+b+1
2n+1 i r. i




2n+1 i

iri > (n+b+1)

41

P2n+1


i P r and n−b i=0

2n+1 i=n+b+1 i

iri−1




2n+1 i

iri <

Given that

P 2n+1 i (n+b+1) 2n+1 i=n+b+1 ( i )r P2n+1 2n+1 i i=n+b+1 ( i )r

>

P 2n+1 i (n−b) n−b i=0 ( i )r Pn−b 2n+1 i reduces i=0 ( i )r

to n+b+1 >

n − b, which is certainly true, the result follows. (ii) Define the probability of reaching the correct decision when all of the shareholders vote sincerely as: 2n+1 2n+1 X 2n + 1 X 2n + 1 i 2n+1−i F (n) = q p (1−p) +(1−q) pi (1−p)2n+1−i . i i i=n−b+1 i=n+b+1

Given that q is the probability of reaching the correct decision when only the blockholder votes, we want to show that when F (n) ≥ q, F (n + 1) > q to prove the result. We begin by writing out F (n + 1). 2n+3 X 2n + 3 F (n + 1) = q pi (1 − p)2n+3−i i i=n−b+2 2n+3 X 2n + 3 +(1 − q) pi (1 − p)2n+3−i . i i=n+b+2

We can write the first term of F (n + 1) as: 2n+3 X 2n + 3 pi (1 − p)2n+3−i i i=n−b+2

P P2n+1 2n+1 i 2n+1 i p2 2n+1 p (1 − p)2n+1−i + 2p(1 − p) p (1 − p)2n+1−i i=n−b i=n−b+1 i i
P = 2n+1 i +(1 − p)2 2n+1 p (1 − p)2n+1−i i=n−b+2 i

P2n+1 2n+1 i p (1 − p)2n+1−i + p2 2n+1 pn−b (1 − p)n+1+b i=n−b+1 i n−b
n−b+1 = 2 2n+1 −(1 − p) n−b+1 p (1 − p)n+b
P2n+1 2n+1 i p (1 − p)2n+1−i i=n−b+1 i  2n+1

= 2n+1 + n−b p − n−b+1 (1 − p) pn−b+1 (1 − p)n+b+1
P2n+1 2n+1 i p (1 − p)2n+1−i i=n−b+1 i  p  = (2n+1)! 1−p − n−b+1 + (n−b)!(n+b)! pn−b+1 (1 − p)n+b+1 n+b+1
P2n+1 2n+1 i p (1 − p)2n+1−i i=n−b+1 i = . (2p−1)(n+1)−b (2n+1)! + (n+1−b)(b+n+1) (n−b)!(n+b)! pn−b+1 (1 − p)n+b+1

This implies that the second term of F (n+1) can be simplified as follows:
P2n+1 2n+3 2n+1 i X 2n + 3 p (1 − p)2n+1−i + i=n+b+1 i 2n+3−i i p (1−p) = (2p−1)(n+1)+b (2n+1)! n+b+1 . p (1 − p)n−b+1 i (n+1−b)(b+n+1) (n−b)!(n+b)!

i=n+b+2

42

Therefore: q F (n + 1) =

=

P2n+1

i=n−b+1


P2n+1 2n+1 i p (1 − p)2n+1−i pi (1 − p)2n+1−i + (1 − q) i=n+b+1 i (2p−1)(n+1)−b (2n+1)! +q (n+1−b)(b+n+1) pn−b+1 (1 − p)n+b+1 (n−b)!(n+b)! (2p−1)(n+1)+b (2n+1)! +(1 − q) (n+1−b)(b+n+1) pn+b+1 (1 − p)n−b+1 (n−b)!(n+b)!


2n+1 i

(2p−1)(n+1)−b (2n+1)! F (n) + q (n+1−b)(b+n+1) pn−b+1 (1 − p)n+b+1 (n−b)!(n+b)! . (2p−1)(n+1)+b (2n+1)! +(1 − q) (n+1−b)(b+n+1) pn+b+1 (1 − p)n−b+1 (n−b)!(n+b)!

Note that this is greater than F (n) as long as the following holds. (2n+1)! (2p−1)(n+1)−b q (n+1−b)(b+n+1) pn−b+1 (1 − p)n+b+1 (n−b)!(n+b)! > 0. (2p−1)(n+1)+b (2n+1)! +(1 − q) (n+1−b)(b+n+1) pn+b+1 (1 − p)n−b+1 (n−b)!(n+b)!

This can be simplified to: (2p − 1)(n + 1) − b p 2b 1 − q +( ) > 0. (2p − 1)(n + 1) + b 1−p q Note that this expression is increasing in n. Therefore, if there exists an n for which F (n − 1) < q and F (n) > q, this will imply that F (m) > q for all m > n, which is what we need for our result. We now demonstrate that there must exist such an n. The function F (n) is well defined given our assumptionP that 2b < 2n + 1, 2b+1 2b+1
i and the minimum value n can take is b. At n = b, F (n) = q i=1 p (1− i p)2b+1−i , which is smaller than q. Thus, it is not possible that F (n) > q for all n. Furthermore, it is not possible that F (n) < q for all n, since as n approaches infinity, F (n) > q. This is true, since Condorcet’s Jury Theorem states that in our model, if q = p, F (n) → 1 as n → ∞, it cannot be smaller for q > p.43 We have now demonstrated that there must exist an n for which F (n − 1) < q and P F (n) > q. 2n+1 i n+2b i=n−2b+1 ( i )r is increasing in b iff: (iii) The term P n−2b 2n+1 i i=0 ( i )r
Pn+2b+2 2n+1
i Pn+2b 2n+1 i r r i=n−2b−1 i=n−2b+1 i i Pn−2b−2 2n+1
i > Pn−2b 2n+1
i r r i=0 i=0 i i 43

One proof of Condorcet’s Jury Theorem is in Ladha (1992).

43

This holds iff: n+2b+2 n+2b X n−2b X 2n + 12n + 1 X n−2b−2 X 2n + 12n + 1 i+j r > ri+j i j i j i=n−2b−1 j=0 i=n−2b+1 j=0 This is immediate—the lower bound of the summation on the left-hand side is lower than that on the right-hand side, while the upper bounds of the summation on the left-hand side are greater than that on the right-hand side. Otherwise, all of the terms are identical and positive. Appendix A.3. Proof of Proposition 2 (Trading with exogenous information) Proceeding by backward induction: The voting behavior given that the blockholder holds 2bb (and shareholders consequently hold 2ˆ n + 1 = 2n + 1 + b 2b − 2b) shares after trade is determined according to Proposition 1. In particular, this implies that at the voting stage, the expected value per share can be written as a function of the blockholder’s holdings v(2bb), noting that the blockholder’s holdings post-trade (2bb) also determine the shareholders’ post-trade holdings (2n + 1 + 2b − 2bb). Working backwards, the highest price that a new shareholder would pay for the blockholder’s share (correctly anticipating that in equilibrium all the 2(b − bb) offered shares are purchased by new shareholders) is v(2bb). Consequently, this is the price that the blockholder would offer to sell at when selling 2(b − bb). In turn, this implies that the blockholder would offer to sell 2(b − bb) in order to maximize 2(b−bb)v(2bb)+2bbv(2bb) where the first quantity corresponds to the value from the sale of shares offered, and the second the expected value of post-trading shareholdings. This is simply equal to 2bv(2bb) and the maximization problem is equivalent to maximizing v(2bb). As in the proof of Proposition 1 and following Nitzan and Paroush (1982) it is immediate that this is maximized when bb = b∗ and, again, following Proposition 1, in this case the blockholder votes with all her shares. Appendix A.4. Proof of Proposition 3 (Trading and information acquisition) First, we must formalize the characterization of beliefs and the restriction on shares described in the text. To do so we introduce several definitions. First, we can define the expected value per share when the blockholder is uninformed VU (b n) and when 44

the blockholder is informed VI (b n) given that the blockholder holds 2bb shares and the shareholders hold 2b n +1 = 2n+1+2b−2bb shares (these will naturally be interpreted as shareholdings after any trades take place given the initial allocations 2n + 1 and 2b). These values can be written as: VU (b n) =

2b n+1 X

i=b n+1

  2b n+1 i p (1 − p)2bn+1−i , i

where this value reflects the fact that if the blockholder does not invest, then her optimal choice is to simply abstain with all her shares,44 and VI (b n) = q

2b n+1 X

i=b n−b∗ +1

  2b n+1 i p (1−p)2bn+1−i +(1−q) i

2b n+1 X

i=b n+b∗ +1



 2b n+1 i p (1−p)2bn+1−i , i

(A.3) where these values reflect optimal voting as characterized in Proposition 1 (under the assumption that Condition C1 holds). With these definitions, if the investment decision was observed, the blockholder would prefer to invest when the value of post-trade shareholdings net of investment costs, that is, 2bbVI (n + b − bb) − c is greater than the value without investment, 2bbVU (n + b − bb); equivalently, c VI (n + b − bb) − VU (n + b − bb) ≥ . 2bb

(A.4)

We can now define the set of shareholdings under which the blockholder would find it worthwhile to invest in information acquisition as follows:   c b b b IN V := b|VI (n + b − b) − VU (n + b − b) ≥ . (A.5) 2bb

Because trading offers are observable, potential buyers can anticipate what the overall holdings of shareholders and of the blockholder will be after trade and whether these holdings would justify the blockholder’s investment in information acquisition, that is, whether the proposed post-trade bb is an element of IN V . Underlying our belief restriction is the idea that since 44

This would be the case whether or not shareholders anticipated investment since in either case they vote sincerely.

45

investment is unobserved it would not affect the price at which shares trade and so these gains are “sunk.” Consequently, the investment decision should reflect the blockholder’s incentive for information acquisition based only on the anticipated post-trade holdings.45 Thus, we now have notation to formally state our restriction on offequilibrium beliefs. Definition 1. Off-the-equilibrium-path beliefs satisfy the “belief restriction on investment”: shareholders assign probability zero to the blockholder having invested if, after the proposed trade, the resulting holdings, eb, are not an element of IN V and assign probability one to the blockholder having invested if, after the proposed trade, the resulting holdings are in IN V . The rationale for these beliefs is that shareholders posit that the blockholder would acquire information only if it was rational to do so, contingent on her trade offer going through (which is indeed what occurs on the equilibrium path).46 The off-equilibrium belief defined in Definition 1 ensures that there are only two possible (degenerate) beliefs that can arise on- or off-equilibrium, namely, that the blockholder has acquired information or has not done so. Moreover, the belief restriction ensures that the blockholder can anticipate how her proposed trading behavior will determine the shareholder beliefs. 45

To see this directly, consider the ex ante investment decision:

pIb−bb (2b

Value from investing ≥ Value from not investing b − 2b) + 2bbVI (n + b − bb) − c ≥ pU (2b − 2bb) + 2bbVU (n + b − bb), b−b b

where pIb−bb is the price at which shares trade when selling 2(b − bb) shares and investing, and pU is the price at which shares trade when selling 2(b − bb) and not investing. Since b−b b

the investment decision is unobservable pIb−bb = pU , and this equation reduces to Eq. b−b b (A.4). 46 If the timing of the game were different so that information acquisition occurred in between stage 3 and stage 4, such a belief would be sequentially rational. There are no refinements in the literature that pin down beliefs in our environment where types are an endogenous choice and senders may take an action after choosing their type, but our focus on such beliefs can be seen as related to the reordering invariance introduced by In and Wright (2018).

46

We must now consider the blockholder’s optimal trading (and hence shareholding) strategy given (i) her decision to acquire information and (ii) what shareholders believe about her decision given the trading offer. This simplifies into four cases, which we illustrate in Fig. 2. Within each case there is a range of possible trading offers (choices of b − ˆb); we find the optimal one for the blockholder. Subsequently, we look at deviations from our proposed equilibrium to trade offers in the other cases.

Fig. 2. Blockholder information acquisition choice and trade offer. Case I : The blockholder acquires information and makes an offer to trade b − bb that induces shareholders to believe that she acquired information (i.e., bb ∈ IN V ). In this case, the precision of the blockholder’s information is q. In the continuation game, Proposition 1 holds, and we have assumed that Condition C1 holds for all shareholdings. Consider the optimal trading strategy among the set of trade offers that induce the belief that the blockholder has acquired information. First, we begin by introducing notation for the minimal blockholder holding bmin in the set IN V as follows: n o bmin = min bb ∈ IN V . Our first condition that is needed to prove our result is: bmin > b∗ .

(A.6)

Condition (A.6) is key to our result that there exist parameters for which the blockholder will not sell some shares on which it will abstain. The condition states that the minimum number of post-trade shares for which the blockholder would have credibly acquired information is larger than the optimal vote threshold b∗ . Note that implicit in (A.6) is that bmin is well defined, 47

which requires that IN V is non-empty (this is trivially the case for low enough c). For any accepted trade offer b − bb, where bb ∈ IN V , the ex ante value of the blockholder’s shareholding is given by 2bVI (n + b − bb). This is because (i) both the price at which shares trade and the value of owning a share reflect the expected value per share when the blockholder is informed at the resulting shareholding and (ii) the blockholder will vote with b∗ shares given that bb ≥ bmin > b∗ . Consequently, the optimal trading strategy among the set of trade offers that induce the belief that the blockholder has acquired information is the choice of bb ∈ IN V that maximizes 2bVI (n + b − bb). Standard Condorcet reasoning (see, e.g., Boland, 1989) implies that vote accuracy (and hence the value of the firm) increases when the blockholder’s unvoted shares are traded to new informed shareholders. Thus, the optimal trading strategy among those that induce the belief that the blockholder has acquired information is to choose bmin . Case II: The blockholder acquires information and makes an offer to trade b−bb that induces shareholders to believe that she has not acquired information (i.e., bb ∈ / IN V ). In this case, the precision of the blockholder’s information is q but the belief of the shareholders is that the blockholder has precision p. This difference in beliefs cannot constitute part of an equilibrium, but must be analyzed as a possible deviation for the blockholder. In the continuation game, the shareholders behave as though their beliefs are correct; the blockholder knows their beliefs and best responds. Therefore there is an equilibrium where the shareholders all vote sincerely and the blockholder votes sincerely with min[bb, b∗ ] shares. This equilibrium is the most informative equilibrium given Proposition 1 and the assumption that C1 holds. We denote the optimal choice of shares to retain by the blockholder as bII . We do not solve explicitly for this shareholding. Nevertheless, we will show below that this shareholding is dominated and can’t be a profitable deviation. Case III: The blockholder does not acquire information and makes an offer to trade b − bb that induces shareholders to believe that she acquired information (i.e., bb ∈ IN V ). In this case, the precision of the blockholder’s information is p but the belief of the shareholders is that the blockholder has precision q. Once again, a difference in beliefs cannot constitute part of an equilibrium, but must be an48

alyzed as a possible deviation for the blockholder. In the continuation game, the shareholders behave as though their beliefs are correct; the blockholder knows their beliefs and best responds. Therefore there is an equilibrium where the shareholders all vote sincerely and the blockholder abstains with all shares that it holds (since its information precision is p). This equilibrium is the most informative equilibrium given Proposition 1 and the assumption that C1 holds.47 The optimal amount of shares for the blockholder to retain are bmin as that allows the blockholder to sell as many shares as possible and still be perceived as having acquired information precision q. Selling shares is beneficial because (i) their price is inflated due to the perception that the blockholder is informed, and (ii) when the blockholder’s information precision is p, the blockholder will abstain with all of her shares. Case IV: The blockholder does not acquire information and makes an offer to trade b − bb that induces shareholders to believe that she has not acquired information (i.e., bb ∈ / IN V ). In this case, the precision of the blockholder’s information is p. In the continuation game, Proposition 1 holds setting the blockholder’s precision to p and Condition C1 holds. Given Proposition 1, it follows that regardless of how many shares she holds after trading, the blockholder will abstain with them all, and all shareholders vote sincerely. In this case, as each share traded goes to a new shareholder with independent information and increases the aggregate value of the vote, the new shareholders are willing to pay more than the value of the share if the blockholder were to hold on to it. Clearly, the blockholder would sell all her shares.48 We now examine possible deviations from the hypothesized equilibrium (Case I: the blockholder chooses bmin ). Possible deviation to Case IV: 47

We note that Proposition 1 holds setting the blockholder’s precision to p, and given that Condition C1 holds for all shareholdings when the blockholder’s precision is q, it also holds for all shareholdings when the blockholder’s precision is p (as q ≥ p implies q p 1−q ≥ 1−p ). 48 Further note that trivially, such a trading strategy according to Definition 1, is indeed consistent with the belief that the blockholder does not invest in information acquisition.

49

The second condition needed for our result is: 2bVI (n + b − bmin ) − c > 2bVU (n + b).

(A.7)

Condition (A.7) ensures that the blockholder prefers to acquire information and choose bmin rather than not invest in information acquisition and sell all of her shares (and be perceived as not having acquired information), i.e., the best deviation for Case IV. Note that (A.7) is not immediate from bmin ∈ IN V . Formally, one can observe that this is a different condition by noting that the argument of VI (.) on the left-hand side of Condition (A.7) incorporates post-trading shareholdings (n + b − bmin ), while the argument of VU (.) on the right-hand side is based on pre-trade shareholdings (n + b). Possible deviation to Case II: We denoted the optimal choice of the blockholder in Case II as bII . We proceed with four steps to prove that the blockholder will not find it profitable to deviate to bII . Step 1: Being informed and choosing shareholdings bII where the blockholder is perceived as uninformed is dominated by being informed, choosing bII , and being perceived as informed (this is hypothetical, as bII ∈ / IN V ). Step 2: However, being informed, choosing bII , and being perceived as informed is dominated by being uninformed, choosing bII , and being perceived as uninformed (this is the definition of IN V ). Step 3: Being uninformed, choosing bII , and being perceived as uninformed is dominated by being uninformed, selling all of your shares, and being perceived as uninformed (this is discussed in Case IV above). Step 4: Being uninformed, selling all of your shares, and being perceived as uninformed is dominated by being informed, choosing bmin , and being perceived as informed (by Condition (A.7)). This then proves that a deviation from bmin to bII is not profitable. Possible deviation to Case III: In order for a deviation to Case III to be unprofitable, the following condition must hold: 2bVI (n+b−bmin )−c > (2b−2bmin )VI (n+b−bmin )+2bmin VU (n+b−bmin ) (A.8) In Case III, the payoff of the uninformed blockholder who chooses bmin and is perceived to be informed consists of two parts. First, the blockholder sells 2b − 2bmin shares at an elevated price; the shareholders believe that the blockholder is informed and pay VI (n + b − bmin ) per share. Second, 50

the blockholder retains 2bmin shares and knows that they are only worth the uninformed value VU (n + b − bmin ). Rewriting Eq. (A.8) yields: 2bmin VI (n + b − bmin ) − c > 2bmin VU (n + b − bmin ).

(A.9)

This condition must be true by the definitions of bmin and IN V . Summary: Above we demonstrate that there are no profitable deviations as long as Condition (A.7) holds. Finally, since bmin > b∗ , the blockholder must abstain in the voting stage. Existence: Finally, it can be verified that the conditions characterize a non-empty set of parameters. In particular, p = 0.51 and q = 0.53 imply that b∗ = 1. Using these values of p and q , taking initial shareholdings (b, n) = (2, 2) and c = 0.05 yields bmin = 2 so that (A.6) holds. In addition, the blockholder prefers to invest in information and votes with fewer shares than she holds rather than selling all her shares at a price that reflects no information acquisition— that is, (A.7). In addition, it can be verified that Condition (C1) holds throughout. Appendix A.5. Proof of Proposition 4 (Regulations prohibiting abstention can enhance efficiency) The value per share when the blockholder does not invest in information acquisition corresponds to the probability that the outcome of the vote matches the state. This can be written as: 2n+1 2n+1 X 2n + 1 X 2n + 1 i 2n+1−i p p (1 − p) + (1 − p) pi (1 − p)2n+1−i . i i i=n−b+1 i=n+b+1 (A.10) Note that in writing down this expression, we interpret the SEC regulation as implying that the blockholder must vote all of her 2b shares. The blockholder’s signal matches the state with probability p, in which case she needs the votes of n − b + 1 or more shareholders (whose signals match the state) to have a majority. The blockholder’s signal does not match the state with the complementary probability, in which case n + b + 1 or more shareholders must vote for the correct state for the majority to be in favor of the true state. 51

Similarly, the value per share when the investment c is made is:   2n+1 X 2n + 1 2n + 1 i 2n+1−i q p (1 − p) + (1 − q) pi (1 − p)2n+1−i . i i i=n−b+1 i=n+b+1 (A.11) The blockholder invests in information acquisition under the regulation when the value of her 2b shares with investment is higher than their value without investment by at least as much as the cost c of investment, or equivalently, " 2n+1  #  2n+1 X X 2n + 1 2n + 1 i c (q−p) p (1 − p)2n+1−i − pi (1 − p)2n+1−i > . i i 2b i=n−b+1 i=n+b+1 (A.12) In the absence of the regulation, the blockholder would not vote with all b shares but instead with b∗ shares in the case of investment and none (since the blockholder is required to vote an even number) without it. It follows that the blockholder would not invest in information acquisition when: 2n+1 X


 2n+1 P 2n+1 i X 2n + 1 c q 2n+1 p (1
− p)2n+1−i ∗ +1 i=n−b i P2n+1 pi (1−p)2n+1−i < . 2n+1 i 2n+1−i − +(1 − q) i=n+b∗ +1 i p (1 − p) 2b i i=n+1 (A.13) The regulation can lead to a socially desirable outcome (by spurring investment) relative to the scenario in which there is no regulation (and no investment) if:




 P 2n+1 2n+1 i X 2n + 1 c q 2n+1 p (1 − p)2n+1−i i=n−b+1 P2n+1 i 2n+1
i > pi (1−p)2n+1−i . − +(1 − q) i=n+b+1 i p (1 − p)2n+1−i i 2n + 2b + 1 i=n+1 (A.14) Note that the regulator takes into account the impact on all 2n + 2b + 1 shares. Thus, the regulation improves efficiency when (i) there would be no investment in the absence of the SEC rule (Eq. A.13), (ii) the SEC rule leads to investment (Eq. A.12), and (iii) this is socially efficient (Eq. A.14). This is interesting as long as the set of parameters for which this is true is non-empty. This is the case, for example, when b = 15, c = 2, n = 20, p = 0.51, and q = 0.6. Note that b∗ = 5 in this case. 

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