COMMUNICATION AMONG SHAREHOLDERS · 2019-01-17 · COMMUNICATION AMONG SHAREHOLDERS 5 terminology, authority) to an informed agent, in order reduce the informational loss caused by

COMMUNICATION AMONG SHAREHOLDERS

NEMANJA ANTIC∗ AND NICOLA PERSICO†

Abstract. This paper provides a model of information transmission among shareholders in

an investment venture. An expert shareholder chooses how much information to communi-

cate about the return on investment to a controlling shareholder who controls the investment

strategy. Relative to the cheap-talk literature there are two key innovations in our model:

first, that the shareholdings drive the incentive to communicate; second, that these share-

holdings are determined endogenously. We embed this new model into two settings. In the

first setting, equity ownership is determined by investors buying shares on a competitive

equity market. We provide conditions under which share-trading delivers perfect commu-

nication and full risk-sharing. The second setting is a principal-agent relationship where

the equity is granted as compensation by a principal (board) to an agent (CEO). Within

this setting, the model highlights a novel trade-off between incentivizing effort provision and

promoting information transmission. Also, the principal-agent setting delivers insights into

the effects of prudential regulation.

1. Introduction

There are many instances in which information needs to be communicated among share-

holders. Two examples follow.

Example 1: Activist Investing. Recently, so-called “activist investors”have been

eager to provide advice to corporate boards regarding strategy. The challenge for

activist investors has been to persuade the shareholders who control the board, which

Date: December 16, 2016.∗ Nemanja Antic: Managerial Economics and Decision Sciences Department, Kellogg School of Manage-

ment, Northwestern University, [email protected];

† Nicola Persico: Managerial Economics and Decision Sciences Department, Kellogg School of Manage-

ment, Northwestern University, and NBER, [email protected]

We would like to thank Olivier Darmouni, Michael Fishman, Henry Hansmann and Bilge Yilmaz and sem-

inar audiences at Calgary, Michigan, Northwestern and Simon Fraser for helpful comments.1

2 N. ANTIC AND N. PERSICO

are often large institutional investors such as pension funds, to follow their advice.

To be persuaded, the controlling shareholders claim that they need to see “skin in the

game,”i.e., the payoff-alignment (or lack thereof) that equity ownership is thought

to provide. Similarly, activist investors say that they want the board to have “skin

in the game.”1 Consistent with the importance of “skin in the game,”we provide a

model where the activist investor’s ability to persuade the controlling shareholder

depends partly on the alignment provided by their equity holdings. Furthermore, the

model allows the activist, and all other investors, to purchase shares in anticipation

of the communication game.

Example 2: Principal-Agent Model. We consider an agent (e.g., a CEO) who

provides advice to a principal (e.g., the board) regarding corporate strategy. In

addition, the agent also exerts costly non-contractible effort. Thus the setting fea-

tures information transmission and moral hazard. Equity can be used as part of the

agent’s compensation scheme to incentivize two different activities: effort provision

and information transmission. Thus the optimal compensation scheme will trade off

two traditional principal-agent considerations: risk sharing and moral hazard; and

a novel one: information transmission.

The common thread in these two examples is that the incentives to communicate infor-

mation among shareholders depend in part on the amount of equity they own. This paper

offers a model that features these incentives. In the model, after shares are acquired one of

the shareholders learns inside information about the return of a risky investment project.

This shareholder could be an activist investor or, in the principal-agent setting, the CEO;

we refer to him as the expert investor. Once the expert investor becomes private informed he

can no longer trade on the inside information, but he can communicate with the controlling

investor who controls the corporation’s investment strategy. The quality of communication

will depend in part on the shareholders’equity holdings. After the communication takes

place, the controlling investor chooses an investment level and payoffs are realized.

1For example, in a letter to auction house Sotheby’s investors, famed activist investor D. Loeb disagreedwith the board’s long-term strategy saying “Their lack of ‘skin in the game’ has led to a dysfunctionalcorporate culture overly focused on short-term metrics such as auction volumes at the expense of long-term investment in key areas”. Quoted from http://www.law360.com/articles/525502/loeb-s-third-point-asks-investors-to-ditch-sotheby-s-board .

COMMUNICATION AMONG SHAREHOLDERS 3

We embed this communication model into two settings. In the first setting, the share-

holders are investors who buy shares on a competitive equity market. The second setting

is a principal-agent relationship where the shares are granted as compensation by the con-

trolling shareholder (board, principal) to the expert shareholder (CEO, agent). In both

settings equity is acquired/allocated anticipating the communication stage that follows. We

ask whether, in either setting, the equilibrium share allocation will feature full risk-sharing

and perfect communication.

When investors acquire shares in a competitive stock market, we find that the incentives

to trade shares for risk-sharing can be remarkably aligned with those that govern infor-

mation transmission. We provide conditions such that the share allocation that optimizes

risk-sharing among investors also promotes maximal information transmission (Section 5.1).

However, an ineffi ciency lurks in the background: a competitive market for shares fails to

reward the positive externality that the expert investor provides to other investors by pur-

chasing shares. In some scenarios, this externality leads the expert investor to acquire too

few shares and to transmit too little information to the controlling investor (Section 5.2).

In the principal-agent setting (Section 5.3) we find that, if the space of contracts includes

a fixed salary plus shares in the enterprise, the optimal contract allocates too much equity to

the agent for full communication to occur. At the optimal contract the agent will feel that

the principal’s investment strategy is too aggressive and this will lead the agent to misreport

his information. In addition, as usual in a principal-agent problem, too much risk (equity)

is provided to the agent relative to the first-best risk sharing allocation.

The principal—agent setting described above lends itself to studying the effects of pruden-

tial regulation (Section 5.4). Prudential regulation has received increasing attention since

the global financial crisis. The premise of prudential regulation is that risk-taking by a single

firm has externalities on other stakeholders. We find that direct regulation of risk-taking

behavior (e.g., for financial institutions: capital requirements, stress tests, “too big to fail”

regulations, etc.) is beneficial for two reasons: first, it has a direct effect of making the con-

trolling investor behave in a more risk-averse way, which is the goal of prudential regulation;

second, by making the controlling investor behave in a more risk-averse way, direct pruden-

tial regulation effectively more-closely aligns the controlling investor’s preferences with those


of the executive, resulting in better information transmission by the executive and therefore

a more accurate investment strategy. (*****)We also discuss a new form of prudential reg-

ulation: indirect regulation through checks on executive compensation, specifically, reducing

its dependence on market valuation.2

We briefly discuss the scenario in which the expert investor’s information is correlated

with the size of his equity holdings (Section 5.5). We conclude by micro-founding the role

of the controlling investor in a collective decition-making model (Section 5.6). In most of

the analysis we assume that one of the investors is given the power to choose the investment

strategy. But if there is disagreement among investors, who is to be chosen as the controlling

investor? We show that, if the investment strategy is chosen collectively among shareholders

through “one share — one vote,” then the share-weighted median voter functions as the

controlling investor.

Relative to the cheap-talk literature there are two key innovations in our model: first,

that the shareholdings drive the incentive to communicate; second, that these shareholdings

are determined endogenously. Technically, our theoretical model builds on Crawford and

Sobel (1982). However, because our payoff functions are micro-founded in an investment

model, the functional forms that arise are different from the example originally analyzed by

Crawford and Sobel (1982) and still prevalent today in analyses of information transmission.

The analysis of our model is innovative from a technical perspective and, though it is mostly

relegated to the appendix, should be considered a significant technical contribution of this

paper. In particular, the results on Chebyshev polynomials in the appendix may be relevant

for other cheap talk models with similar microfoundations.3

Two seminal cheap-talk papers are conceptually related to our work. Gilligan and Krehbiel

(1987) demonstrate that a controlling agent may want to partially delegate/reduce their

ability to exercise control, in order to motivate the sender to acquire information. Dessein

(2002) shows that it can be in the interest of a controlling agent to delegate control (in his

2The second form of regulation, while not traditional, is now part of the regulatory arsenal; for example,the U.K.’s Financial Conduct Authority has set forth Remuneration Codes designed to curb risk-taking byfinancial institutions.3While economic models may result in general second-order difference equations, analytic solutions do notgenerally exist and thus obtaining comparative statics is diffi cult. The general linear equation, which may bestudied using the techniques presented here, seems like an important generalization of the literature basedon the example in Crawford and Sobel (1982) and a relevant direction for further research.


terminology, authority) to an informed agent, in order reduce the informational loss caused

by cheap talk. In both papers, the focus is on the allocation of control between sender

and receiver. In our paper, in contrast, the focus is on allocation of ownership. Thus our

paper explores a novel tool to achieve transparent information transmission. Whereas the

previous literature may be more relevant in environments where it is diffi cult to manipulate

ownership allocation (political settings, for example), our analysis may be more relevant in

environments where ownership can be flexibly adjusted (as in publicly traded corporations,

for example), but delegation may not be possible (boards have fiduciary responsibilities).

The corporate boards literature counts several papers based on cheap talk. Adams and

Ferreira (2007) analyze the consequences of the board’s dual role as advisor as well as monitor

of management. Harris and Raviv (2008) analyze the dynamics between inside and outside

directors. Adams et al. (2010) provides a review of this literature. To our knowledge, no

model in this literature allows shareholdings to drive the incentive to communicate; nor are

the shareholdings determined endogenously.

A somewhat related literature is the one on monitoring by large shareholders, whose

seminal papers include Shleifer and Vishny (1986) and Admati et al. (1994); see Gillan et al.

(1998) for a review. The monitoring activity in this literature shares some similarity with

the communication activity in our paper in that both are value-enhancing activities which

produce positive externalities for all shareholders.

2. Model

A firm is contemplating how much capital a to invest in a new venture. Investing a yields

profits aX. The rate of return X is a random variable with mean µX and variance σ2X . The

parameter σ2X is common knowledge to all investors (though µX is not).

The firm also has a mature business whose profits are given by Y, a random variable

with commonly know mean and variance µY and σ2Y . Note that the firm makes no decision

concerning Y. For analytical convenience we assume that Y is uncorrelated with X.


Aside from owning a fraction of the firm, every investor j may also own a non-tradable,

exogenously given background portfolio Zj. For analytical convenience we assume that Zj is

uncorrelated with X.

2.1. Players, Information Asymmetry, Ownership, and Control

A controlling investor C owns a fraction θC > 0 of the firm and has the right to choose a.

She does not know µX and has a prior belief that µX is uniformly distributed on [0, σ2X ].4 We

will think of the controlling investor as the (median voter in the) board of directors because

the board controls the firm’s major strategic decisions.

An expert investor E owns a fraction θE of the firm. He knows µX . In one setting the

expert investor may be an activist investor; in another, a CEO. In both cases, the expert

investor has private information and can advocate with the board, but cannot directly choose

a. Note that we allow θE to be negative, in which case the expert investor is a short-seller.

There are a number of noncontrolling investors i, who own fractions θi of the firm. They

too do not know µX and their prior on µX is uniformly distributed on [0, σ2X ].

2.2. Preferences and Payoffs

All investors have mean-variance preferences with risk-aversion parameter r. Thus, if an

agent with background portfolio Z owns a fraction θ of the firm, and the firm invests a, the

investor’s expected utility equals:

E (Z + θ (Y + aX))− r

2V ar (Z + θ (Y + aX)) .

After some algebra, and denoting ω ≡ µX/σ2X , this expression can be rewritten as:

−σ2X

2r(rθa− ω)2︸︷︷︸

new-venture payoff

+ θ (µY − Cov (Y, rZ))− r

2θ2σ2Y︸︷︷︸

mature-business payoff

, (2.1)

up to an additive constant which does not depend on θ or a.5 The investor’s payoff can

be decomposed into the portion originating from the new venture aX, and the portion

4The support’s lower bound being zero implies that every project returns on average more than the invest-ment; the upper bound being equal to σ2X is just a normalization.5Note that this step makes use of the assumption that X is uncorrelated with both Y and Z. The algebra isdemonstrated in Lemma A.1.


originating from the mature business Y. The first portion depends on the controlling investor’s

strategy. The second portion is independent of the other investors’behavior.

Henceforth, we normalize the controlling investor’s risk parameter rC = 1 and denote the

expert’s and the noncontrolling investors’risk parameters by rE and ri, respectively.

2.3. Information Transmission

Before the controlling investor chooses the investment level a, the expert investor may

transmit information to the controlling investor through a single round of cheap talk.6 This

cheap talk stage takes place under common knowledge of all the investors’share holdings.

2.4. Acquistion of Shares

Before the cheap talk stage, shares are acquired by all investors (for example, by trading

in a competitive market) and the investors’holdings become known to all.

2.5. Timing of Events

(1) All investors acquire shares in the firm.

(2) The expert investor learns µX and engages in cheap talk.

(3) The controlling investor directs the venture to invest a.

(4) X, Y, and the Zj’s are realized and payoffs accrue.

2.6. Discussion of Modeling Assumptions

2.6.1. Roles of Ownership In the Model. In our model, equity ownership serves three

roles. First, it is a source of wealth and risk for all investors. Second, equity represents “skin

in the game”that the expert investor needs to have in order to credibly transmit informa-

tion to the controlling investor; how much “skin” is needed will depend on the controlling

investor’s risk aversion and equity holdings. Third, equity ownership affects the controlling

investor’s choice of a.

6If the noncontrolling investors can also listen in to the cheap talk, nothing changes in the equilibrium. Seethe discussion in Section 2.6.3. We do not consider multiple rounds of cheap talk.


2.6.2. Heterogeneity Among Investors. Investors are potentially heterogeneous in two

dimensions. First, in their risk-aversion parameter; for given equity holdings, the risk aver-

sion of the expert and controlling investors determine their ability to communicate. Second,

investors may differ in their background portfolios. Their background portfolios do not affect

communication, but the portfolios’correlations with the firm’s profits create heterogeneous

diversification motives for owning equity in the company. For example, an activist investors’

background portfolio may include just a handful of companies, leading to a different ap-

petite for ownership (“skin in the game”) compared to a large institutional investor whose

background portfolio may be very diversified.

2.6.3. Modeling of Information and Timing of Trade. The information we have in

mind is venture-specific intelligence. The intelligence could pertain to a business reorganiza-

tion or to a strategic opportunity. This information can only be generated by a sophisticated

investor (in our model, the “expert investor”) after the investor is able to examine inside

information. After the inside information has been examined and the venture-specific intel-

ligence generated, the active investor can no longer trade on this inside information. This is

why, in our model, trading takes place before information about µX is available.

The assumption that trade takes place before the expert investor learns µX has several

consequences for modeling. First, it ensures that, when purchasing shares, the expert investor

cannot engage in insider trading. If shares were traded after the expert investor learned

µX , but before the cheap-talk stage, then the other investors would worry about adverse

selection. If shares were traded after the cheap-talk stage, then the cheap talk could be

used to manipulate the share market. In our framework, instead, investors do not have to

worry about trading under adverse selection and cheap talk is used solely to influence the

controlling investor’s investment choice.

The second consequence of trade taking place before cheap talk is that it does not matter

whether the noncontrolling investors become privy to cheap-talk communication. This is

because the noncontrolling investors play no further role in the model after they trade.

2.6.4. Source of Friction In Communication. Friction in the communication game arises

solely due to a “misallocation”of ownership. To see this, consider two investors with risk


parameters r, r′. If these investors hold shares θ, θ′, respectively, in the exact ratio θ′/θ = r/r′,

then their payoffs (2.1) are identical functions of a up to a linear affi ne transformation.

Therefore, at this holdings ratio there is no conflict of interest regarding the optimal choice

of a and hence we expect perfect communication.

3. Information Transmission with Fixed Equity Holdings

Fix all equity holdings θC , θE, θi . We analyze the game where the controlling investorchooses a after a single round of cheap-talk communication from the expert investor.

Definition 1 (Risk-adjusted holdings ratio). The ratio ρ = θC/rEθE will be called “risk-

adjusted holdings ratio.”

The risk-adjusted holdings ratio parameterizes the conflict of interest in the communication

game. If ρ = 1 then there is no conflict of interest, as anticipated in Section 2.6.4. If ρ < 1,

the conflict of interest arises from the controlling investor holding “too little”equity relative

to the expert investor; if ρ > 1, the conflict of interest arises from the controlling investor

holding “too much”equity. If ρ is negative the expert investor is a short-seller.

Proposition 1 (Characterization of cheap-talk equilibrium strategies). Fix a risk-adjusted

holdings ratio ρ. Take any equilibrium with partition cutoffs ω0, ω1, ..., ωN , where ω0 = 0

and ωN = 1.

(1) Upon learning that ω falls into partition (ωn, ωn+1), the controlling investor’s equilib-

rium action equals (ωn+1 + ωn) /2θC .

(2) Equilibrium cutoffs ωn solve the following difference equation:

ωn+2 − 2kρωn+1 + ωn = 0, (3.1)

where kρ = 2ρ− 1, ω0 = 0, and ωN = 1.

(3) Fix an equilibrium with N partitions. Equilibrium partitions have the form ωn =

αUn−1 (kρ), where Un is the order-n Chebyshev polynomial of the second kind, and

α = 1/UN−1 (kρ) .

(4) If ρ = 1 perfect communication is an equilibrium.


(5) Suppose ρ > 1, that is, the expert investor (sender) has less than the optimal risk-

sharing holding relative to the controlling investor (receiver).Then there are equilibria

with an arbitrarily large number of partitions, but none of these equilibria approaches

full information transmission.

(6) Suppose 0 ≤ ρ < 1, that is, the expert investor (sender) has more than the optimal

risk-sharing holding relative to the controlling investor (receiver). Then the maximal

number of partitions consistent with an equilibrium is an increasing step function

of ρ converging to infinity as ρ ↑ 1. No information can be communicated when

0 ≤ ρ < 3/4.

(7) If ρ < 0, that is, the expert investor (sender) is a short-seller, then no information

can be communicated in equilibrium.

Proof. See the Appendix.

Part 1 indicates that the controlling investor’s equity holdings scale her investment strategy

multiplicatively. Part 2 introduces the parameter kρ which encodes the conflict of interest

between the controlling and the expert investors. When kρ = 1, there is no conflict of

interest because ρ = 1. The parameter kρ enters equation (3.1) multiplicatively.7 Part 2

also provides the difference equation that generates equilibrium partitions. This difference

equation is famous because it generates the family of Chebyshev polynomials as its solution.

The order-n Chebyshev polynomial of the second kind, Un (k) , is a polynomial function of

k defined as the unique solution to the following functional difference equation:

U−1 (k) = 0,

U0 (k) = 1,

Un (k)− 2kUn−1 (k) + Un−2 (k) = 0. (3.2)

7Equation (3.1) differs subtly from difference equation (21) in Crawford and Sobel (1982). That equationreads

an+2 − 2an+1 + an − 4b = 0. (CS)In equation (CS), the conflict of interest is captured by the parameter b, which is additive.


The most common expression for Un (k), and the one that will best serve our purposes is

Un (k) =

sin((n+1) arccos k)

sin(arccos k)if |k| ≤ 1

sinh((n+1) arccosh k)sinh(arccosh k)

if k > 1,

where arccosh k is nonnegative (refer to expressions 1.4 and 1.33b in Mason and Handscomb,

2003). Because equation (3.2), which generates the family of Chebyshev polynomials, is the

same as equation (3.1), there must be a strong connection between the equilibrium of our

cheap-talk game and the family Un (k) .8 The differences are in the initial conditions and

the fact that cheap-talk partitions, ωn, are required to be monotonic, while Chebyshevpolynomials, Un, are not. Chebyshev polynomials are generated by specifying two initial

conditions, whereas the requirements for a cheap-talk equilibrium only specify one initial

condition, ω0 = 0; therefore, ω1 is a free parameter (subject to the entire sequence ωnsatisfying monotonicity). Varying this free parameter generates the plethora of cheap-talk

equilibria. Another way to index the family of cheap-talk equilibria is as follows. Equation

(3.1) is linear in ω, so if a sequence ωn solves (3.1) then, for any real number α, the sequenceαωn also solves (3.1). This means that the plethora of cheap-talk equilibria must, in thismodel, be indexed by a single scaling factor.

Part 3 confirms that, within a given cheap-talk equilibrium, partition cutoffs are indeed

proportional to Chebyshev-polynomial functions of kρ. The factor α denotes the scaling

factor. An intriguing possibility is that, by choosing a suffi ciently small scaling factor α,

one might be able to generate cheap-talk equilibria with an arbitrarily large number of

partitions. The following parts of Proposition 1 show that: (a) it is not always possible to

generate cheap talk equilibria with an arbitrary number of partitions; and (b) when it is

possible, these equilibria are not arbitrarily informative.

Part 5 of proposition 1 indicates that if the conflict of interest arises from the expert

investor owning too small a share relative to the controlling investor, there is no upper bound

for the number of equilibrium partitions. In this case there are cheap-talk equilibria with

any number of partitions, but none of these equilibria approaches full communication. In

these equilibria, low realizations of ω will be communicated very accurately to the controlling

8Pursuing this connection, one could view the cheap-talk equilibrium thresholds ωn (kρ) as polynomial func-tions of the conflict-of-interest parameter kρ, just as Un (k) is a polynomial function of k.


investor (receiver), but high realizations of ω will not. Part 6 indicates that if the conflict of

interest arises from the expert investor owning too large a share relative to the controlling

investor, then no cheap talk equilibrium can have more partitions than an upper bound.

This upper bound is a result of the periodic nature of the sine function as there is a maximal

number of terms in the sequence which satisfy the monotonicity restriction.9 This upper

bound grows without bound as the conflict of interest vanishes.

We now turn to computing expected payoffs. Any investor’s payoffis the sum of the portion

originating from the new venture and the portion originating from the mature business; refer

to expression (2.1). The mature-business portion does not require any computation, so we

focus on new-venture payoffs next.

The new venture’s payoffs is an expected utility, that is, an integral with respected to the

unknown parameter ω of a quadratic function (expression 2.1). The integral of the quadratic

gives rise to a cubic form. Inside the cube lies a, the controlling agent’s strategy which, by

Proposition 1 Part 1, is a linear function of the cutoffs ωn−1, ωn. In turn, Part 3 says that ωn

is proportional to Un−1 (k) , so overall computating the expected payoffs requires evaluating

sums such as the following:N−1∑n=0

[Un (k)− Un−1 (k)]3 .

There are no off-the-shelf formulas for sums of powers of Chebyshev polynomial, so in Lemma

B.2 we derive the required formulas. Based on these formulas we are able to prove the

following proposition, which is the main technical contribution in this paper.

Proposition 2 (New-venture payoffs in most informative equilibrium). Fix a risk-adjusted

holdings ratio ρ. Equilibrium new-venture payoffs for expert and controlling investors in-

crease in the number of equilibrium partitions. Up to additive constants, equilibrium new-

venture payoffs have the following properties.

(1) If ρ ≥ 1, the supremum of the family of equilibrium new-venture payoffs is as follows.

For the controlling investor:

V C (ρ) =σ2X6

1− ρ(4ρ− 1)

.

9For part 5, note that hyperbolic sine is not periodic in the reals (it does have an imaginary period); becausehyperbolic sine is an increasing function it allows for arbitrarily many partitions.


For the expert investor:

V E (ρ) = −σ2X

6

ρ− 1

ρ (4ρ− 1)(4ρ− 3) .

(2) If 3/4 ≤ ρ ≤ 1, the family of equilibrium new-venture payoffs admits the following

upper bound. For the controlling investor:

V C (ρ) =σ2X6

(ρ− 1) (3ρ− 1)

(4ρ− 1).

For the expert investor:

V E (ρ) = −σ2X

3

1− ρ(4ρ− 1)

.

The exact equilibrium payoffs for the controlling and expert investors are given in the

appendix, equations B.3 and B.5, respectively. These bounds are tight in the sense

that for any ρ, there exists a value ρ ∈ (ρ, 1) such that V E (ρ) is attained.

(3) (communication breakdown) If 0 ≤ ρ < 3/4, no information can be transmitted.

In this case the controlling investor’s equilibrium new-venture payoff equals −σ2X/24

and the expert investor’s equilibrium new-venture payoff equals

− σ2X

2rEE(

1

2ρ− ω

)2.

Proof. See Propositions 8 and 9.

Proposition 2 provides surprisingly tractable expressions for (when ρ < 1, approximations

to) the controlling and the expert investors’new-venture payoffs in the best equilibrium.

Conveniently, equilibrium payoffs only depend on the holdings ratio ρ, not on holding lev-

els.10 Figure 1 depicts the controlling investor’s new-venture payoff for different values of

ρ. When ρ < 1 the actual payoff function is not smooth (orange line, analytical expression

provided in equation B.3) because as ρ increases the number of partitions contained in the

most informative equilibrium changes discretely; the function V j (ρ) (blue line, analytical

10To understand why levels do not matter except through ρ, consider two shareholding constellations,(θC , θE) and

(θ′C , θ

′E

), such that θC/θE = θ′C/θ

′E = ρ. Both investors hold more shares in one constel-

lation than in the other, and that should change their payoff. However, recall that investment level a isendogenous; when the controlling investor holds more shares, he will cut down on risk by choosing a lower a(Proposition 1 part 1). The controlling investor’s greater prudence has no effect on equilibrium communica-tion (equation 3.1 is unchanged), so the lower a exactly counteracts the effect of the higher θ in the payofffunction (2.1), and the equilibrium payoff is unchanged.


0.8 0.9 1.0 1.1 1.2 1.3ρ

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

Figure 1. Controlling investor’s new-venture payoff for different values of ρ.Note that for ρ ∈

(34, 1)proposition 2 gives the approximate payoff.

expression provided in Proposition 2) provides a smooth approximation to the exact payoff

function. From now on we will work with the approximate function.

4. Demand for Shares

In this section we compute each investor type’s inverse demand for shares. We break

inverse demand up into two components: the portion of demand that reflects the firm’s

mature business, and the portion that reflects the new venture.

4.1. New-Venture Related Demand for Shares

Inverse demand is defined as the marginal utility of increasing the investor’s equity hold-

ings. To compute new-venture related demand it is necessary to take a stand on which

equilibrium prevails among the many possible communication equilibria. Consistent with

standard practice in the literature on cheap talk, we focus on the most informative equilib-

rium. This equilibrium is focal because it is the equilibrium that is ex-ante preferred both

by the sender and receiver (refer to Proposition 2).

With this stipulation, the non-controlling investors’demand is linear in equity holdings

because they take the new venture’s value as given. For investors C, I the inverse demand

function is not linear because their equity holdings affect on communication and thus the

new venture’s value.


Definition 2. The inverse demand function for shares of investors j = C, I is the derivative

of V j (ρ) with respect to θj.

Note that inverse demand is defined based on the function V j (ρ), which, for ρ < 1, is only

an approximation of the actual utility experienced by investor j. Refer to the discussion of

approximation on page 14.

Proposition 3 (New—venture related demand for shares). The new—venture related inverse

demands for shares are as follows:

(1) for the controlling investor: σ2Xρ (2ρ− 1) /rEθE (4ρ− 1)2 for θC ∈(34rEθE, rEθE

),

which is increasing in θC ; zero for θC smaller than 34rEθE; and negative for θC larger

than rEθE;

(2) for the expert investor: σ2XrE (8ρ2 − 8ρ+ 1) /2θC (16ρ2 − 8ρ+ 1) > 0 for θE ∈(0, θC/rE) , which is decreasing in θE; and negative elsewhere;

(3) for i, a non-controlling investor: σ2X (θC − riθi)E[a∗ (Ω (ω))2

], where a∗ (Ω (ω)) de-

notes the equilibrium action taken by the controlling agent when the expert agent

employs reporting strategy Ω (ω) ; this is a decreasing function of θi.

Proof.

(1) The expression is the derivative of V C (ρ) from Proposition 2 with respect to θC .

Lemma C.1 provides the steps to the functional form, and Corollary C.2 shows that

demand is increasing when positive.

(2) The expression is the derivative of V E (ρ) from Proposition 2 with respect to θE.

Lemma C.1 provides the steps to the functional form, and Corollary C.4 shows that

demand is increasing when positive.

(3) Suppose the controlling investor takes optimal action a∗ (Ω (ω)) . Then the payoff

to noncontrolling investor i reads −σ2X2riE [riθia

∗ (Ω (ω))− ω]2 . Differentiating with

respect to θi and collecting terms yields the result. The steps are provided in Lemma

C.5.


The controlling investor’s new-venture related inverse demand is summarized in the top

right figure 2. The different regions are noteworthy. First, communication-related inverse

demand is 0 when θC < 34rEθE: this is because (1) there is no information transmission

in this region and (2) the controlling investor gets to freely choose the level of a and can

thus costlessly compensate for low shareholdings θC by choosing a high a. Demand is zero

because, at the margin, ownership is not valuable to an investor who has control. Second,

when θC ∈(34rEθE, rEθE

)information transmission becomes possible; thus, in addition to

effect (2) above, adding one more share improves alignment with the expert investor, which

means that the controlling investor will receive better information. Thus, her demand is

positive because of this communication motive. That this communication-related aspect of

demand is increasing reflects the increasing informational value of aligning interests with

the expert investor. The discontinuity in demand at rEθE is a measure of the informational

loss from marginal miscoordination when both investors are perfectly aligned. Finally, when

θC > rEθE, communication-related inverse demand is negative because every additional share

worsens the misalignment with the expert investor, thus worsening information transmission.


𝜃𝑖

𝑝

𝜓𝜎𝑋2𝜃𝐶

𝜃𝐶𝑟𝑖

𝜃𝐶

𝑝

𝑟𝐸𝜃𝐸

𝜎𝑋2

9𝑟𝐸𝜃𝐸

3

4𝑟𝐸𝜃𝐸

3𝜎𝑋2

32𝑟𝐸𝜃𝐸

0

−𝜎𝑋2

18𝑟𝐸𝜃𝐸

Panel A: Non-controlling Investor Panel B: Controlling Investor

𝜃𝐸

𝑝

𝑟𝐸𝜎𝑋2

4𝜃𝐶

𝑟𝐸𝜎𝑋2

18𝜃𝐶

4𝜃𝐶3𝑟𝐸

−3𝜎𝑋2

16𝜃𝐶

−𝜎𝑋2/9𝜃𝐶

𝜃𝐶𝑟𝐸

Panel C: Expert Investor

Figure 2. Investors’new-venture related demand for shares.

The expert investor’s demand is summarized in the bottom left panel in Figure 2. Because

this investor does not control a, he values ownership and has decreasing returns from holding

shares. This is why his demand for shares is decreasing. In addition, this investor also has

an informational motive for trading; as his holdings align more closely with the controlling

investor, he will be able to more credibly communicate information to her, so that her

actions will be more informed and also more closely aligned with his own preferences. This

communication motive explains the discontinuity at θC/rE : the discontinuity reflects the

cost of communication loss when moving away from the share allocation at which perfect

communication is possible.


A noncontrolling investor’s inverse demand is summarized in the top left panel in Figure

2. This investor takes both the action and the information structure as given, and therefore

he has a standard demand for a risky asset: linear and downward-sloping.

4.2. Overall Demand for Shares

The portion of demand that reflects the firm’s mature business is obtained by differen-

tiating with respect to θj the mature-business component of (2.1). That derivative equals

µY −Cov (Y, rjZj)− rjθjσ2Y . This is a standard downward-sloping demand for a risky assetand it is depicted in the two panels of Figure 3 by the dashed line. Adding together the

new-venture and mature-business components of demand yields total inverse demand for

shares.

𝜃𝐶

𝑝

𝑟𝐸𝜃𝐸3

4𝑟𝐸𝜃𝐸

𝜇𝑌 − 𝐶𝑜𝑣(𝑌, 𝑍𝐶)

0𝜃𝐸

𝑝

0𝜃𝐶𝑟𝐸

4

3

𝜃𝐶𝑟𝐸

𝜇𝑌 − 𝐶𝑜𝑣(𝑌, 𝑟𝐸𝑍𝐸)

Panel A: Controlling Investor Panel B: Expert Investor

Figure 3. Investors’overall demand (solid line) and mature business-related

demand (dashed line).


5. Applications

5.1. Shares Are Acquired On a Competitive Equity Market

This section shows that, if all investors (including notably the controlling agent) acquire

shares by trading in a competitive equity market, then in any competitive equilibrium in-

formation can be transmitted perfectly, and moreover there exists a competitive equilibrium

that achieves the effi cient risk allocation.

Proposition 4 (Competitive equilibrium is effi cient). Suppose Cov (Y, rjZj) is independent

of j and all investors acquire shares by trading in a competitive equity market. Then:

(1) at any competitive equilibrium ρ = 1 and information is perfectly transmitted;

(2) there exists a competitive equilibrium where risk is shared optimally among all in-

vestors.

Proof. The necessary condition for a competitive equilibrium reads:

µY − Cov (Y, rjZj)− rjθjσ2Y +∂V j (ρ∗)

∂θj= p for all j, (5.1)

where p represents the share price.

Part 1. Suppose by contradiction that there exists a competitive equilibrium with ρ∗ 6= 1.

There are two cases. In case ρ∗ < 1 we have ∂V E (ρ∗) /∂θE < 0 ≤ ∂V C (ρ∗) /∂θC (with

equality holding when ρ∗ < 3/4; refer to Panel B in Figure 2). In this case the competitive

equilibrium conditions imply:

− Cov (Y, rCZC)− θCσ2Y < −Cov (Y, rEZE)− rEθEσ2Y . (5.2)

Because Cov (Y, rjZj) is independent of j, this condition requires that θC > rEθE, which con-

tradicts the premise that ρ∗ < 1. In case ρ∗ > 1 we have ∂V E (ρ∗) /∂θE > 0 > ∂V C (ρ∗) /∂θC .

In this case the opposite inequality holds in (5.2), which implies θC < rEθE and thus a con-

tradiction of the premise that ρ∗ > 1.

Part 2. Consider an ancillary economy where we impose the restriction a ≡ 0. In this

economy the only tradeable assets is Y, and the first welfare theorem guarantees that the

competitive equilibrium allocates the θj’s effi ciently. The competitive equilibrium quantities


θ∗j and price p∗ in this economy solve:

µY − Cov (Y, rjZj)− rjθ∗jσ2Y = p∗ for all j. (5.3)

Because Cov (Y, rjZj) is independent of j, these equilibrium conditions yield that rjθ∗j is

independent of j. As a consequence ρ∗ = 1 in this ancillary economy. Therefore 0 ∈ ∂V j(ρ∗)

∂θj,

and so the ancillary-economy θ∗j’s and p∗ also solve (5.1). Therefore, the θ∗j’s and p

∗ also

constitute a competitive equilibrium for the economy where a is not restricted to equal zero.

At this equilibrium we know that the risk from the risk bundle Y + aX is shared effi ciently

because the conditions for effi cient risk sharing are that rjθ∗j is independent of j, regardless

of whether the risky asset being shares is Y (ancillary economy) or Y + aX (our main

model).

This proposition indicates that, absent heterogeneity in trading motives, which is ruled

out by the assumption that Cov (Y, rjZj) is independent of j, the competitive equilibrium

in the equity market produces an allocation where risk is shared optimally, information is

transmitted perfectly, and the allocation of control rights is irrelevant.

The proof of Proposition 4 indicates that the competitive equilibrium price p∗ does not

assign any value to the portion of risk coming from the new venture. This feature of the

equilibrium is due to the fact that in this model new-venuture risk is not scarce: it can be

costlessly increased by the controlling agent by increasing a. Therefore the marginal social

value of new-venture risk must be zero.

When Cov (Y, rjZj) varies with j investors have heterogeneous diversification motives,

creating a conflict between risk-sharing and communication. To demonstrate this, we have

worked out a a numerical example where Cov (Y, rEZE) > Cov (Y, ZC).11 The example

involves only an expert investor and a controlling investor; their demands for shares are

depicted in Figure 4. The market-clearing price leads to demands for shares θ∗E = 0.478,

θ∗C = 0.522, so that ρ = 1.091 and communication is not perfect in equilibrium. The fact

that ρ exceeds 1 means that in equilibrium the expert investor acquires too little “skin in

11The following parameter values are used rE = 1, σ2X = 1, σ2Y = 1, µY = 1, Cov (Y,ZC) = 0,Cov (Y, rEZE) =

14 . The equilibrium has the following features: p∗ = 0.386, θE = 0.478, θC = 0.522

(rounded to 3 decimal places). In this example ρ = 1.091 and thus, while a significant amount of informationis conveyed, communication is not perfect. The Mathematica code for the example can by downloaded atwww.kellogg.northwestern.edu/faculty/antic/research.html.

www.kellogg.northwestern.edu/faculty/antic/research.html


𝜃

𝑝

0

𝜇𝑌 − 𝐶𝑜𝑣(𝑌, 𝑍𝐸)

𝜃𝐶 4

3𝜃𝐶

𝜃𝐸3

4𝜃𝐸

𝜇𝑌 − 𝐶𝑜𝑣(𝑌, 𝑍𝐶)

𝑝∗

Figure 4. The controlling (green) and expert (blue) investor’s total demandfor shares in equilibrium, when rE = 1 and Cov (Y, ZC) < Cov (Y, ZE).

the game”relative to the controlling investor. The misalignement is due to heterogeneous

diversification motives: recall that the expert’s background portfolio is more correlated with

firm value than the controlling agent’s.

5.2. Firm Has an Entrenched Controlling Shareholder

Activist investors often seek to influence a business’strategy by transmitting information

to a controlling shareholder (management, board, controlling owner). In this setting investor

E can be interpreted as an activist investor, and investor C as the controlling shareholder.

In this section we assume that the controlling shareholder is entrenched, that is, she has

an exogenously given share θC of the enterprise. Aside from the controlling shareholder, all

other investors acquire shares in the competitive market.

The following proposition provides an ineffi ciency result in contrast to Proposition 4. To

highlight that the contrast is due to the non-tradability of θC and not to any heterogeneity

in background portfolios, we assume that there are no background portfolios.

Proposition 5 (Competitive equilibrium is ineffi cient when the firm has an entrenched

controlling investor). Suppose that Zj = 0 for all j and that the controlling investor’s share


of the enterprise is fixed at θC. All other investors acquire shares by trading in a competitive

equity market. In equilibrium:

(1) no investor ever holds too much “skin in the game”relative to the controlling share-

holder, that is, rjθ∗j ≤ θC for all j;

(2) there is a threshold θC such that perfect risk-sharing obtains in equilibrium if and

only if θC ≤ θC . This threshold is an increasing function of rE, ri, and a decreasing

function of the number of noncontrolling investors.

(3) there is a threshold θC > θC such that information is perfectly transmitted between

activist investor and controlling shareholder in any equilibrium if and only if θC ≤ θC .

This threshold is an increasing function of rE, ri and a decreasing function of the

number of noncontrolling investors.

Proof.

(1) The demand functions in Proposition 3 are negative for θE > θC/rE and θi > θC/ri.

Because the equilibrium price is nonnegative by assumption, neither investor E nor

investors i will want to hold more than θC/rE and θC/ri, respectively, in equilibrium.

(2) Perfect communication between controlling shareholder and informed activist takes

place if and only if rEθ∗E = θC . All agents share risk perfectly if and only if rjθ

∗j = θC

for j ∈ E, i (refer to condition 5.3). The demand functions in Proposition 3 indicatethat riθ

∗i = θC if and only if the share price is zero. The equilibrium price can be

zero only if aggregate demand at that price is below 1, that is, if

θC +θCrE

+∑i

θCri≤ 1.

θC is the solution where the condition holds with equality, whence

θC =1

1 + 1rE

+∑

i1ri

.

(3) Perfect communication between controlling shareholder and informed activist takes

place if and only if θ∗E = θC/rE; the informed activist is willing to hold this quantity

of shares as long as the price does not exceed rEσ2X/18θC (refer to Figure ??). At

this price, each noncontrolling-investor’s demand is given by equating these investors’


demand functions to the price (refer to Proposition 3):

σ2X (θC − riθi)1

3=

rEσ2X

18θC

θC − riθi =rE6θC

θi =θCri− 1

6θC.

A price below rEσ2X/18θC can be an equilibrium price only if aggregate demand at a

price of rEσ2X/18θC does not exceed 1, that is, if

θC +θCrE

+∑i

(θCri− 1

6θC

)≤ 1

θC

(1 +

1

rE+∑i

1

ri

)−∑i

1

6θC≤ 1.

θC is the solution where the condition holds with equality. Clearly, θC > θC . Also,

the LHS is a decreasing function of rE, ri and an increasing function of the number

of noncontrolling investors, which proves the last statement.

Proposition 5 part 1 shows that, no matter how much “skin in the game”the controlling

investor has, no other investor acquires more than that in equilibrium (note that “skin in the

game”is defined relative to one’s risk aversion). The reason is that the controlling investor’s

“skin in the game”determines her appetite for risk in choosing a, which in turn determines

the riskiness of the firm. Other investors anticipate this and buy just enough to achieve their

risk-return goals and, in the expert investor’s case, to communicate appropriately. Neither

of these motives lead investors to exceed the controlling investor’s “skin in the game.”

Part 2 is not surprising: there is a direct distorsion in risk allocation if any investor

(the controlling investor, in this case) is assigned too much risk. It is somewhat interesting

that, for some parameter values (between θC and θC) no additional distorsions (e.g., in

information transmission) are present. Put differently: in this setting, effi cient risk-sharing

fails more easily than full information transmission.


Part 3 is the most interesting part because of its contrast with the effi ciency result in

Proposition 4. If demand for risk is suffi ciently large, viz., when the controlling investor

owns a large share of the enterprise, or when there are many investors and they are not very

risk-averse, then the equilibrium share price is too high for the activist to hold exactly θC/rE.

Hence ρ > 1 and we cannot have full communication. The reason this problem arises is that

the controlling investor is not allowed to sell some of her θC . If she could, she would sell some

θC and simultaneously adopt a more aggressive investment strategy a. This processs would

bring her holdings down to the point where ρ = 1, thus recovering the effi cient allocation

(Proposition 4).

An intriguing implication of Proposition 5 part 3 is that, in the presence of entrenched

investors who hold shares partly to exercise control, the activist investor should be buying

more shares from a welfare perspective. This is because the price on the capital market

fails to reward the positive informational externality that the activist investor provides by

purchasing shares. This implication is intriguing because it would provide an argument for

subsidizing share purchases by activist investors.12 The next proposition makes the argument

formal by showing that allocating the activist investors more shares than he acquires in

equilibrium results in a welfare improvement.

Proposition 6 (Welfare gain from subsidizing the activist investor). Consider the equilib-

rium allocation when θC > θC. Taking one share away from a non-controlling investor i and

allocating it to the activist investor results in a welfare improvement.

Proof. In the case θC > θC the equilibrium price p∗ is such that the activist investor de-

mands θ∗E < θC/rE. Consider the effects of perturbing the equilibrium allocation by taking

one share away from a non-controlling investor i and allocating it to the activist investor.

The activist gains from receiving one more share is p∗ (this gain takes into account his

private inframarginal benefit from improving communication with the controlling investor).

The non-controlling investor’s marginal loss from giving up one share given the equilibrium

investment strategy is p∗; in addition, the non-controlling investor experiences an additional

12Current regulatory stance in the US leans the opposite way, i.e., toward restricting share purchases byactivist investors. For example, the recently introduced “Brokaw Act”bill aims to decrease the time investorshave to disclose a significant ownership stake in an equity security.


inframarginal benefit from a better investment strategy which results from improved com-

munication between the activist and the controlling investor (that this benefit is positive

is proved in Corollary C.7). Therefore, this perturbation results in a welfare gain for the

pair involved in the trade, as well as for all other non-controlling investors (from Corollary

C.7). Finally, the entrenched investor benefits from better communication. Hence, the new

allocation is welfare-improving.

The roots of the ineffi ciency featured in Proposition 6 lie in the mechanism that gov-

erns share allocation. In a capital market the noncontrolling investors are free-riding, in

equilibrium, on the expert investor’s communicative activity.

The ineffi ciency featured in Proposition 6 suggests that the firms should look for ways to

increase the expert investor’s equity ownership. One such mechanism could be to hire the

expert investor and put him to work as a manager. We discuss this in Section 5.3.

5.3. Shares Are Allocated As Compensation in a Principal-Agent Relation-

ship

In a principal—agent setting, the principal, not the market, sets the incentive scheme. We

consider an expert shareholder (a manager, say) whose compensation contract is designed

by the controlling shareholder (the principal). We now show that, if the space of contracts

includes a fixed salary plus shares in the enterprise, the optimal compensation scheme assigns

the manager too much equity for her to be willing to communicate truthfully with the

principal. Thus, in general the principal will face a trade-off between providing the agent

with incentives to exert optimal effort and incentives to communicate fully.

Our setting is as follows. An agent, who is indexed by E, has private information about

an investment and transmits this information to a principal. For simplicity we set Zj = 0

for both principal and agent, thus assuming away differences in diversification motives. The

agent’s payoff function is as that of the expert investor in the previous sections, with an

added moral hazard component: the agent exerts effort e at a private cost c (e) . Effort e

produces output e. If compensated with a salary S and shares θE, the agent’s payoff (net of

the portion due to Y which has no strategic effect) is:


uE (e; θE, S) = V E

(1− θErEθE

)︸︷︷︸old preferences

+ θEe− c (e)︸︷︷︸moral hazard part

+ S.

Output is owned by a principal, who is indexed by C. After communicating with the agent,

the principal chooses an investment strategy and has the payoff function of the controlling

investor in the previous sections; in addition, the principal owns the output e. The principal

chooses a compensation scheme for the agent comprised of a fixed salary S plus a share θE in

the enterprise. The principal’s payoff net of the portion due to Y is then:

uC (e; θE, S) = V C

(1− θErEθE

)︸︷︷︸old preferences

+ (1− θE) e︸︷︷︸worker’s output

− S.

Proposition 7 (Trade-off between moral hazard and imperfect communication). The prin-

cipal’s choice of compensation scheme (salary S plus share θE) is such that the agent receives

no fewer shares than θE = 1/1+rE, the amount that he would receive absent the moral hazard

problem.

Proof. The argument is standard in the principal—agent literature. The principal’s problem

is:

maxθE ,S

uC (e∗ (θE) ; θE, S)

subject toe∗ (θE) ∈ arg maxuE (e; θE, S) (incentive compatibility of effort)

uE (e∗ (θE) ; θE, S) ≥ 0 (individual rationality).

By standard arguments, the principal’s problem can be re-written as

maxθE

V C

(1− θErEθE

)+ V E

(1− θErEθE

)+ e∗ (θE)− c (e∗ (θE)) .

Thus, the principal’s problem coincides with welfare maximization under the constraint that

effort and reporting strategy are chosen by the agent.


Next, note that e∗ (θE)−c (e∗ (θE)) is an increasing function of θE. To see this, differentiate

∂

∂θE[e∗ (θE)− c (e∗ (θE))]

=∂

∂θE[(1− θE) e∗ (θE)] +

∂

∂θE[θEe

∗ (θE)− c (e∗ (θE))]

=∂

∂θE[(1− θE) e∗ (θE)] + e∗ (θE)

= −e∗ (θE) + (1− θE) e∗′ (θE) + e∗ (θE)

= (1− θE) e∗′ (θE) > 0.

Therefore, the principal’s problem is the sum of two functions that both peak when 1−θErEθE

= 1,

that is, when θE = 1/ (1 + rE), and a third function, which is increasing in θE.

The optimal contract trades off providing the agent with incentives to exert optimal effort

with providing incentives to communicate fully. As a result the agent is assigned too much

equity to effi ciently communicate with the principal, but less equity than would be optimal

if the only incentive issue was effort provision. In equilibrium the agent will feel that the

principal’s investment strategy is too aggressive and this will lead the agent to misreport his

signal.

5.4. Prudential Regulation

The premise of prudential regulation is that risk-taking by a single firm has externalities

on other stakeholders. Prudential regulation has received increasing attention since the

global financial crisis. These regulations can take two broad forms: (a) direct regulation

of risk-taking behavior (e.g., for financial institutions: capital requirements, stress tests,

“too big to fail”regulations, etc.), and (b) indirect regulation through checks on executive

compensation. In this section, we use the principal—agent model from Section 5.3 to evaluate

these two forms of regulation from a risk-taking perspective.

Let’s start with regulatory form (b). Bebchuk and Spamann (2009) make the case for

prudential regulation through checks on executive compensation. Such checks have been

implemented recently by British regulators. In a press release, the Bank of England explained

the goals of this policy as follows:


"The Prudential Regulation Authority (PRA) and Financial Conduct Author-

ity (FCA) are today publishing new remuneration rules... The new framework

aims to further align risk and individual reward in the banking sector, to dis-

courage irresponsible risk-taking and short-termism and to encourage more

effective risk management."13

The premise of the principal—agent model in Section 5.3 is that the principal, not the

agent, chooses the risky investment strategy a. This premise is realistic if large strategic bets

for a corporation require vetting by the board, i.e., the owners. If this premise is broadly

correct, then the executive can advise on and propose large risk-taking strategies, but does

not principally decide. The consequence of this observation is that checks on executive

compensation will affect risk-taking primarily through information transmission. Our model

allows us to examine this channel.

Section 5.3 shows that, within our model, optimal executive compensation, while second

best, falls short of the first best in that the executive (the agent) owns too many shares for

effi cient communication. The principal chooses this incentive scheme because he also wants

to incentivize the agent’s effort. Society (and the regulator), however, may care less about

whether the executive exerts effort, and may care more about the firm’s risk-taking. Thus,

the regulator may prefer the agent to have a less-powerful incentive scheme compared with

the one assigned to her by the principal. By marginally reducing the share-based amount

of compensation, the regulator can force a closer alignment between the principal’s and the

agent’s risk-taking incentives. This alignment does not result in a more cautious investment

strategy: indeed, the strategy is chosen by a controlling investor (owner) whose appetite for

risk-taking is unaffected by the regulation of executive compensation. Thus the benefit for

the regulator results from a more informed but not more cautious investment strategy.

We now turn to regulatory form (a): direct regulation of risk-taking behavior. Suppose

that the regulators discourage risk-taking by penalizing the controlling investor with κa2 for

any investment level a. The parameter κ > 0 captures the tightness of the regulation; the

quadratic specification is chosen purely for analytical convenience. The regulatory penalty

function κa2 captures in reduced form the regulator’s legal capacity to constrain risk-taking

13Quote from http://www.bankofengland.co.uk/publications/Documents/news/2015/ps1215.pdf .


behavior. The penalty is not intended as a monetary cost, and thus enterprise value is not

affected by it. Let us further assume, at least initially, that the agent’s compensation scheme

is fixed independently of κ.

Subtracting the regulatory penalty function from the controlling investor’s payoff(2.1) and

repeating the steps on page 36 shows that the controlling investor’s equilibrium investment

strategy takes the formωn+1 + ωn2θC + κ

,

which corresponds to the investment strategy of a controlling investor with shareholdings

θC + κ/2 (cf. Proposition 1 Part 1). Thus, if risk-taking behavior is directly regulated with

intensity κ, the amount of information transmitted in equilibrium is the same as that in the

unregulated cheap-talk game with a controlling investor with holdings θC + κ/2. Increasing

κ slightly above zero is beneficial in terms of information transmission because it increases

alignment between the principal’s and the agent’s payoffs. Moreover, there is also a direct

effect of making the principal (controlling agent) effectively less risk-tolerant and thus less

prone to risk-taking. In sum, direct regulation of risk-taking behavior improves matters

on two fronts: a direct effect (less risk-taking) and also an indirect effect (more accurate

investment strategy due to greater information transmission).

The previous paragraph assumes that the agent’s compensation scheme is fixed inde-

pendently of κ. But what if the agent’s compensation scheme is chosen by the principal

partly as a function of κ? Since increasing κ is functionally equivalent to having a more

risk-averse principal, we conjecture that the optimal compensation scheme from Section 5.3

would transfer more risk to the agent, that is, the compensation scheme would become more

stock-based. If this conjecture is true, this would be an intriguing indirect effect of the con-

ventional form of risk-taking regulation. But, to reemphasize, in our model giving the agent

a more high-powered incentive scheme has no direct effect on risk-taking behavior.

5.5. Expert Investor Acquires Information

Throughout the paper we have assumed that one of the investors, the expert investor,

is given knowledge of µX . In this section we explore a variant of the model where the

expert investor learns µX with probability P (θE) and learns nothing with complementary


probability. This specification allows information to be a correlated with the expert investor’s

holdings, echoing the premise in the literature on monitoring by large shareholders.14 We

explore this variant within the setting of Section 5.1, where all investors acquire shares by

trading on an equity market. For convenience, we assume that whether the expert investor

is informed or not becomes common knowledge before the cheap-talk stage.

Let’s start with the special case in which the probability P (θE) of becoming informed

is independent of θE. Then it would be an equilibrium for every investor j to ignore the

uncertainty associated with the realization of P (θE) and simply purchase shares in inverse

proportion to rj. This is because, regardless of whether the expert investor turns out to be

informed or not, the competitive equilibrium in either subgame has all investors j purchasing

shares such that θj/rj is independent of j, at a price given by equation (5.3). In other words,

the competitive equilibrium is independent of the probability of becoming informed.

If the probability P (θE) of becoming informed depends on θE the above is no longer an

equilibrium. This is because the expert investor’s decision to purchase shares has an effect

on his probability of becoming informed, and since the expert investor’s payoff is higher

in the subgame in which he becomes informed, he will have an incentive to over-purchase

shares. We conjecture that in this case the competitive equilibrium will feature the expert

investor acquiring too many shares for communication to be perfect, that is, we expect that

in equilibrium ρ > 1.

5.6. Microfounding the Controlling Shareholder based on Collective Decision-

making among Shareholders

Throughout the paper we have assumed that one of the investors is given the power

to choose the investment strategy. This assumption turned out to be innocuous in the

setting of Section 5.1, where no disagreement existed about the investment strategy after

shareholders traded on the share market. But, if there is disagreement among investors, who

is to be chosen as controlling investor? In this section we offer a model where the investment

action is chosen collectively among shareholders. To allow maximal scope for disagreement,

14In this literature, large shareholders are assumed to have access to better monitoring technology, and/orto have more incentive to monitor. This literature is discussed in the Introduction.


we assume that all the non-informed shareholders’ holdings are fixed exogenously. This

assumption could be relaxed to some degree (e.g., as in Section 5.3).

There is a continuum of shareholders with measure 1. Let i index all shareholders except

shareholder E, the expert shareholder. Shareholder i has risk-aversion parameter ri and is

exogenously assigned holdings θi. Denote hi = ri · θi. Both the cumulative distribution F (i)

of the index i and the distribution of hi vary smoothly with i.15 The investment decision

is determined collectively as follows: first the expert investor communicates information

publicly, then all shareholders simultaneously submit an investment recommendation, and

finally the median of the share-weighted distribution of recommendations is implemented.

In this setting it is a dominant strategy for each shareholder to make a sincere recom-

mendation, i.e., to recommend the investment action that he would choose if given the

opportunity to choose. These sincere recommendations are ordered by the index ri · θi. Tosee this, observe that all investors will recommend the action that maximizes their payoff

(expression 2.1) conditional on whatever posterior G (ω) the expert investor’s public com-

munication has generated in equilibrium. Thus, investor i recommends the investment level

a that solves

maxa−∫σ2X2ri

(hia− ω)2 dG (ω) .

The first-order conditions for this problem are

−∫

(hia− ω) dG (ω) = 0,

and it is immediate to see that shareholder i’s recommendation is monotonically decreasing

in hi. This monotonicity implies that the collectively chosen policy corresponds to the share-

weighted median of the distribution of the hi’s. This is a primitive of our model, which

is obtained by ordering investors according to the value of their hi and then picking the

share-weighted median of the distribution. The investor at that median is the controlling

shareholder, i.e., the shareholder indexed by C in the rest of the paper.

15In Section 5.1, the distribution of ri · θi is degenerate and thus does not vary smoothly with i.


6. Conclusion

This paper has provided a model of information transmission about a risky investment

prospect. The model is tailored to financial applications. Relative to the cheap-talk literature

the model innovates in two ways: first, in that the incentives to communicate are shaped by

equity ownership; second, in that equity ownership is determined endogenously. The analysis

of this model is innovative from a technical perspective, and thus should be considered an

additional contribution of this paper.

We have embedded this model into two settings. In the first setting, equity ownership is

determined by investors buying shares on a competitive equity market. The second setting

is a principal-agent relationship where the equity is granted as compensation by a principal

(board) to an agent (CEO). In both settings equity is acquired/granted anticipating the

communication stage that follows. We have asked whether, in either setting, the equilibrium

share allocation will feature full risk sharing and perfect communication.

We have provided conditions under which share-trading delivers perfect communication

and full risk-sharing. When frictions lead to a welfare-suboptimal outcome, we have identified

a novel source of ineffi ciency: a competitive market for shares fails to reward the positive

externality that an expert investor provides to other investors by purchasing shares.

Within a principal—agent setting, the model has highlighted a novel trade-off between

incentivizing effort provision and promoting information transmission. Also, the principal-

agent setting has been leveraged to obtain insights into the effects of prudential regulation.

Much of the literature on information transmission in organizations has focused on opti-

mizing the allocation of control, following Dessein’s (2002) pioneering insight. This paper

suggests that, in organizations where ownership can be easily adjusted, allowing the market

to allocate ownership can sometimes help solve the problem of information transmission.

References

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directors in corporate governance: A conceptual framework and survey.”Journal of Economic Literature

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[3] Admati, A. R., Pfleiderer, P., & Zechner, J. (1994). “Large shareholder activism, risk sharing, and

financial market equilibrium.”Journal of Political Economy, 1097-1130.

[4] Bebchuk, Lucian A., and Holger Spamann (2009). “Regulating Bankers’Pay.”Geo. LJ 98: 247.

[5] Crawford, Vincent and Sobel, Joel (1982) “Strategic Information Transmission,”Econometrica, 50, issue

6, p. 1431-51.

[6] Dessein, Wouter (2002) “Authority and Communication in Organizations.”Review of Economic Studies,

69, p. 811-38.

[7] Gillan, S., & Starks, L. T. (1998). “A survey of shareholder activism: Motivation and empirical evi-

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An Informational Rationale for Restrictive Amendment Procedures.” Journal of Law, Economics, &

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Economy, Vol. 94, 461-48.


Appendix A. Proofs

A.1. Lemma A.1

Lemma A.1. Up to µZ − r2σ2Z +

σ2X2rω2 (an additive constant that does not depend on a or

θ), the agents’payoffs can be written as:

−σ2X

2r(rθa− ω)2 + θ (µY − rCov (Y, Z))− r

2θ2σ2Y .

Proof. We have that V ar (θ (aX + Y ) + Z) = θ2a2V ar (X)+θ2V ar (Y )+V ar (Z)+2Cov (θY, Z),

since by assumption Cov (X, Y ) = Cov (X,Z) = 0. Thus

E (θ (aX + Y ) + Z)− r

2V ar (θ (aX + Y ) + Z)

= θaµX + θµY + µZ −r

2θ2a2σ2X −

r

2θ2σ2Y −

r

2σ2Z − rθCov (Y, Z)

= −σ2X(−θaµX

σ2X+r

2θ2a2

)+ θ (µY − rCov (Y, Z))− r

2θ2σ2Y + µZ −

r

2σ2Z

= −σ2X

2r

(r2θ2a2 − 2rθa

µ

σ2X+µ2

σ4X

)+ θ (µY − rCov (Y, Z))− r

2θ2σ2Y + µZ −

r

2σ2Z + σ2X

µ2

2rσ4X

= −σ2X

2r(rθa− ω)2 + θ (µY − rCov (Y, Z))− r

2θ2σ2Y + µZ −

r

2σ2Z +

σ2X2rω2.

Appendix B. Information Transmission with Fixed

Shareholdings

We start by providing a new, to our knowledge, characterization of the ratios of two

consecutive-ordered Chebyshev polynomial of the second kind.

Lemma B.1 (Ratios of Chebyshev polynomials of consecutive order). (1) Suppose |x| <1. Then Un−1 (x) /Un (x) < 1 if and only if n < 1

2

(π

arccosx− 1).

(2) Suppose x > 1. Then Un−1 (x) /Un (x) < 1 for all n, and limn→∞ Un−1 (x) /Un (x) =

exp (− arccoshx) .


Proof. (1) Denote φ = arccosx. Since |x| < 1,

sgn [Un (x)− Un−1 (x)]

= sgn [sin (n+ 1)φ− sinnφ] .

Now,

sin (n+ 1)φ− sinnφ

= sin

(nφ+

φ

2+φ

2

)− sin

(nφ+

φ

2− φ

2

)= 2 cos

(nφ+

φ

2

)sin

(φ

2

). (B.1)

where we have used the product-to-sum identity sin (x+ ρ)−sin (x− ρ) = 2 cos (x) sin (ρ) .

Let’s start by signing the second term; by the half-angle formula,

sin

(φ

2

)=

1− cos (φ)

2sgn

(2π − φ+ 4π

⌊φ

4π

⌋).

By definition, φ = arccos (x) ∈ [0, π] , and so the sgn operator returns +1. Therefore,

the whole right-hand side is positive, which shows that sin (φ/2) > 0. Thus, expression

(B.1) has the same sign as its first term. That term, cos (nφ− φ/2) , is positive as

long as:

nφ+φ

2<π

2.

Solve for n to get:

n <1

2

(π

φ− 1

).

(2) Un (x) > Un−1 (x) because arccoshx > 0 by definition. Denote φ = arccoshx. Since

x > 1,

Un−2Un−1

=sinh (φ (n− 1))

sinh (φn)

=exp (φ (n− 1))− exp (−φ (n− 1))

exp (φn)− exp (−φn),


where the second equality is De Moivre’s hyperbolic formula. Letting n → ∞ and

recalling that φ = arccoshx > 0 we get

limN→∞

Un−2Un−1

=exp (φ (n− 1))

exp (φn)

= exp (−φ) .

Lemma B.1 characterizes the family of Chebyshev polynomial of the second kind. This

family of polynomials is closely related to the equilibrium communication strategy because

both solve equation (3.1) .

B.1. Proof of Proposition 1

Proof. Part 1. In equilibrium, after receiving a message the controlling investor (receiver)

knows that the state ω is distributed uniformly over some interval (ωn, ωn+1). Then her

equilibrium action a∗C (ωn, ωn+1) is given by

a∗ (ωn, ωn+1|θC) = arg maxa−∫ ωn+1

ωn

(aθC − ω)2 dω.

The first-order condition w.r.t. a reads:

0 =

∫ ωn+1

ωn

(aθC − ω) dω

= aθC (ωn+1 − ωn)−ω2n+1 − ω2n+1

2

= aθC −ωn+1 + ωn

2.

Thus, the investor’s optimal action is:

a∗C (ωn, ωn+1) =ωn+1 + ωn

2θC.

Part 2. Let’s now turn to the expert investor’s strategy. At a cutpoint between different in-

tervals, the sender needs to be indifferent between inducing the equilibrium action associated


to either interval. Thus if the sender knows that ω = ωn it must be that:(ωn+1 + ωn

2θCθErE − ωn

)2=

(ωn + ωn−1

2θCθErE − ωn

)2.

Taking square roots on both sides of the equation while switching the sign of the right-hand

side yields:ωn+1 + ωn

2θCθErE − ωn = −ωn + ωn−1

2θCθErE + ωn

Rearranging we get the following difference equation:

ωn+1 = 2

(2θCrEθE

− 1

)ωn − ωn−1 (B.2)

which, when incremented, reads as in (3.1).

Part 3. For any kρ, all cheap talk equilibrium partitions satisfy equation (3.1) with initial

condition ω0 = 0. Chebyshev polynomials Un (kρ) are the unique solutions to (3.1) with

initial conditions U−1 (kρ) = 0, U0 (kρ) = 1. Thus if we denote ω1 = α then all cheap talk

equilibria must take the form ωn (kρ)n = αUn−1 (kρ)n . That α = UN−1 (kρ) follows from

the equilibrium requirement that ωN = 1.

Part 4. If ρ = 1 there is no conflict of interest. as discussed in Section 2.6.4, so perfect

communication is an equilibrium.

Part 5. ρ > 1 is equivalent to kρ > 1. In this case ωn (kρ) = αUn−1 (kρ) is monotonic in

n. Furthermore, for any N one can choose α such that αUN−1 (kρ) = 1, and thus construct

a communication equilibrium with exactly N partitions. But, although the number of par-

titions can be arbitrarily large, it does not follow that all partitions are arbitrarily small. To

see this, fix any equilibrium and compute the width of the topmost partition:

ωN − ωN−1

= 1− αUN−2 (kρ)

= 1− UN−2 (kρ)

UN−1 (kρ),


where the equalities make repeated use of the identity 1 = ωN = αUN−1. Letting N → ∞implies taking the limit across different equilibria. Using Lemma B.1 part 2 we have:

limN→∞

ωN − ωN−1 = 1− exp (− arccosh kρ) > 0.

The strict inequality holds because by definition arccosh kρ > 0. This shows that as we move

across equilibria with an increasing numbers of partitions N, even though N grows without

bound, the width of the topmost partition converges to a positive number and, in particular,

it does not vanish. Therefore, the extent of communication in this class of equilibria is

uniformly bounded away from full communication.

Part 6. ρ < 1 is equivalent to |kρ| < 1. In this case the Chebyshev polynomial Un (kρ) is

not monotonic in n, and so we need to worry about the monotonicity constraint ωn > ωn−1

which is required by the equilibrium definition. (Trying to construct an equilibrium with

too many partitions will run into the monotonicity constraint, and the maximal number

of partitions Nρ consistent with equilibrium is found by making the constraint bind.) The

monotonicity constraint requires that, for all n = 0, ..., N,

0 < sgn [ωn − ωn−1]

= sgn [Un−1 (kρ)− Un−2 (kρ)] .

In Lemma B.1 part 1, we saw that the above holds if n − 1 < 12

(π

arccos kρ− 1), or n <

12

(π

arccos kρ+ 1). Thus, the maximal integer Nρ that is consistent with this inequality is:

Nρ =

⌈1

2

(π

arccos kρ+ 1

)⌉− 1

=

⌈1

2

(π

arccos kρ− 1

)⌉.16

This Nρ is increasing in kρ because arccos (·) is a decreasing function over the interval[−1, 1] , and has a vertical asymptote at kρ = 1 because arccos (1) = 0. No information can

be communicated for kρ < 1/2 because in this range arccos (kρ) > π/3 and so Nρ = 1.

Part 7. If ρ < 0 then expert and controlling investors have opposing interests: if an action

increases the controlling investor’s payoff then it decreases the expert investor’s payoff. So

no communication can credibly take place in equilibrium.


B.2. Expected Utility Calculations

The order-n Chebyshev polynomial of the first kind, Tn (x) , is a polynomial function of x

defined as the unique solution to the following functional difference equation:

T0 (x) ≡ 0,

T1 (x) ≡ 1,

Tn (x)− 2xTn−1 (x) + Tn−2 (x) = 0.

The recursion that gives rise to the polynomials is the same as for polynomial of the second

kind, but the initial conditions are different. Nevertheless, the two classes of polynomials

are closely related. The relation we will use is the following:

Tn (x) = Un (x)− xUn−1 (x) .

We start by providing new, to our knowledge, explicit expressions for some useful sums of

series of Chebyshev polynomials.

Lemma B.2 (Sums Involving Cubes of Chebyshev Polynomials). We note the following

identities concerning Chebyshev polynomials:

(1)∑N−1

n=0 [Un (x)− Un−1 (x)]3 =

1−x2x+1

(2− 3 1+x

sin2(N arccosx)

)UN−1 (x)3 if |x| ≤ 1

1−x2x+1

(2 + 3 1+x

sinh2(N arccoshx)

)UN−1 (x)3 if x ≥ 1.

(2)∑N−1

n=0

[(Un (x)− Un−1 (x)) (Un (x) + Un−1 (x))2

]=

2+2x1+2x

(1− 1+x

2 sin2(N arccosx)

)UN−1 (x)3 if |x| ≤ 1

2+2x1+2x

(1 + 1+x

2 sinh2(N arccoshx)

)UN−1 (x)3 if x ≥ 1.

.

(3)∑N−1

n=0 Tn+1 (x)3+Tn (x)3 =

(x+1)2(x−1)

2x+1

(2x− 1− 3 x

sin2(N arccosx)

)UN−1 (x)3 if |x| ≤ 1

(x+1)2(x−1)2x+1

(2x− 1 + 3 x

sinh2(N arccoshx)

)UN−1 (x)3 if x ≥ 1.

Proof.

(1) We will rewrite Chebyshev polynomials in terms of exponentials. Note that by De

Moivre’s theorem:

sinnφ =enιφ − e−nιφ

2ιand sinhnφ =

enφ − e−nφ2

,


where we define φ = arccos kρ. Thus we may write:

Un−1 (kρ) =enφι

1[kρ<1] − e−nφι1[kρ<1]

eφι1[kρ<1] − e−φι

1[kρ<1]

=eξnφ − e−ξnφeξφ − e−ξφ ,

where ξ := ι1[kρ<1] . Now (dropping θ subscripts for simplicity):

N−1∑n=0

(Un−1 (k)− Un (k)) 3 =

N−1∑n=0

(eξnφ − e−ξnφeξφ − e−ξφ −

eξ(n+1)φ − e−ξ(n+1)φeξφ − e−ξφ

)3

=N−1∑n=0

(1

eξφ − e−ξφ

)3 (eξnφ − eξ(n+1)φ + e−ξ(n+1)φ − e−ξnφ

)3=

N−1∑n=0

(1

eξφ − e−ξφ

)3 (eξnφ − eξ(n+1)φ + e−ξ(n+1)φ − e−ξnφ

)3=

N−1∑n=0

(1

eξφ − e−ξφ

)3 (eξnφ

(1− eξφ

)+ e−ξ(n+1)φ

(1− eξφ

))3=

N−1∑n=0

(1− eξφ

eξφ − e−ξφ

)3 (eξnφ + e−ξ(n+1)φ

)3=

(1− eξφ

eξφ − e−ξφ

)3 N−1∑n=0

(eξnφ + e−ξ(n+1)φ

)3.

Let us work on computing the sum and we’ll add the coeffi cient later:

N−1∑n=0


)3=

N−1∑n=0

(e3ξnφ + 3eξ(n−1)φ + 3e−ξ(n+2)φ + e−3ξ(n+1)φ

)=

e3ξNφ − 1

e3ξφ − 1+ 3

eξ(N−1)φ − e−ξφeξφ − 1

+3e−ξ(N+2)φ − e−2ξφ

e−ξφ − 1+e−3ξ(N+1)φ − e−3ξφ

e−3ξφ − 1

=e3ξNφ − 1 + 1− e−3ξNφ

e3ξφ − 1+ 3

eξ(N−1)φ − e−ξφ + e−ξφ − e−ξ(N+1)φeξφ − 1

=e3ξNφ − e−3ξNφ

e3ξφ − 1+ 3

eξ(N−1)φ − e−ξ(N+1)φeξφ − 1

.


So that:

N−1∑n=0

(Un−1 (k)− Un (k)) 3 =

(1− eξφ

eξφ − e−ξφ

)3(e3ξNφ − e−3ξNφ

e3ξφ − 1+ 3

eξ(N−1)φ − e−ξ(N+1)φeξφ − 1

).

Now:(1− eξφ

eξφ − e−ξφ

)33eξ(N−1)φ − e−ξ(N+1)φ

eξφ − 1= −3

(1− eξφ

)2(eξφ − e−ξφ)3

(eξ(N−1)φ − e−ξ(N+1)φ

)= −3

(1− eξφ

)2e−ξφ

(eξφ − e−ξφ)2eξNφ − e−ξNφeξφ − e−ξφ

= −3

(eξφ + e−ξφ − 2

)(eξφ − e−ξφ)2

UN−1 (kρ)

= −6(kρ − 1)

(eξφ − e−ξφ)2UN−1 (kρ) ,

where the last line follows by eξφ + e−ξφ = 2kρ, since 2 cosφ = eιφ + e−ιφ and

2 coshφ = eφ + e−φ.

Similarly, we find that:(1− eξφ

eξφ − e−ξφ

)3e3ξNφ − e−3ξNφ

e3ξφ − 1

=1− 3eξφ + 3e2ξφ − e3ξφ

(e3ξφ − 1) (eξφ − e−ξφ)2

(eξNφ − e−ξNφ

) (e2ξNφ + e−2ξNφ + 1

)eξφ − e−ξφ

=1− e3ξφ − 3eξφ

(1− eξφ

)(e3ξφ − 1)

UN−1 (kρ)(e2ξNφ − 2 + e−2ξNφ + 3


=

(−1 +

3eξφ(1− eξφ

)(1− eξφ) (1 + eξφ + e2ξφ)

)UN−1 (kρ)


)2+ 3

(eξφ − e−ξφ)2

=

(−1− 2eξφ + e2ξφ

1 + eξφ + e2ξφ

)UN−1 (kρ)

(UN−1 (kρ)

2 +3


)=

(−e−ξφ − 2 + eξφ

e−ξφ + 1 + eξφ

)UN−1 (kρ)

(UN−1 (kρ)

2 +3


)=

2− 2kρ2kρ + 1

UN−1 (kρ)

(UN−1 (kρ)

2 +3


),

where we have used that 2kρ = eιφ + e−ιφ.


Putting everything together:

N−1∑n=0

(Un−1 (kρ)− Un (kρ))3

=2− 2kρ2kρ + 1

UN−1 (kρ)

(UN−1 (kρ)

2 +3


)− 6

(kρ − 1)

(eξφ − e−ξφ)2UN−1 (kρ)

= UN−1 (kρ)

(1− kρkρ + 1/2

(UN−1 (kρ)

2 +3


)+ 6

(1− kρ)(eξφ − e−ξφ)2

)

=1− kρkρ + 1/2

(1 +

3

UN−1 (kρ)2 (eξφ − e−ξφ)2

+ 6kρ + 1/2


)UN−1 (kρ)

3

=1− kρkρ + 1/2

(1 + 6

kρ + 1


)UN−1 (kρ)

3

=1− kρkρ + 1/2

(1 + 6

1 + kρ

(eξNφ − e−ξNφ)2

)UN−1 (kρ)

3

The result follows after noting that:


)2=

(2ι sinNφ)2 if |kρ| ≤ 1

(2 sinhNφ)2 if kρ ≥ 1

=

−4 sin2Nφ if |kρ| ≤ 1

4 sinh2Nφ if kρ ≥ 1.

(2) Note that we may write:

Un−1 (kρ) =enφι

1[kρ<1] − e−nφι1[kρ<1]

eφι1[kρ<1] − e−φι

1[kρ<1]

=eξnφ − e−ξnφeξφ − e−ξφ ,


where ξ := ι1[kρ<1] . Thus (dropping θ subscripts for simplicity):

N−1∑n=0

(Un (x)− Un−1 (x)) (Un (x) + Un−1 (x))2

=

N−1∑n=0

(eξ(n+1)φ − e−ξ(n+1)φ

eξφ − e−ξφ − eξnφ − e−ξnφeξφ − e−ξφ

)(eξ(n+1)φ − e−ξ(n+1)φ

eξφ − e−ξφ +eξnφ − e−ξnφeξφ − e−ξφ

)2

=N−1∑n=0

(1

eξφ − e−ξφ

)3 (eξ(n+1)φ − eξnφ + e−ξnφ − e−ξ(n+1)φ

) (eξ(n+1)φ + eξnφ − e−ξnφ − e−ξ(n+1)φ

)2=

N−1∑n=0

(eξφ − 1

) (eξφ + 1



) (eξnφ − e−ξ(n+1)φ

)2=

(eξφ − 1

) (eξφ + 1


N−1∑n=0



)2Now, we have that:

N−1∑n=0



)2=

N−1∑n=0

(e2ξnφ − e−2ξ(n+1)φ


)=

N−1∑n=0

(e3ξnφ − e2ξnφe−ξ(n+1)φ − eξnφe−2ξ(n+1)φ + e−3ξ(n+1)φ

)=

e3ξNφ − 1

e3ξφ − 1− e−2ξφ − eξ(N−2)φ

e−ξφ − 1− e−ξ(N+2)φ − e−2ξφ

e−ξφ − 1+

1− e−3ξNφe3ξφ − 1


e3ξφ − 1− e−2ξφ − eξ(N−2)φ + e−ξ(N+2)φ − e−2ξφ

e−ξφ − 1


e3ξφ − 1− e−ξ(N+2)φ − eξ(N−2)φ

e−ξφ − 1.

Thus:

N−1∑n=0

(Un (x)− Un−1 (x)) (Un (x) + Un−1 (x))2

=

(eξφ − 1

) (eξφ + 1


(e3ξNφ − e−3ξNφ

e3ξφ − 1− e−ξ(N+2)φ − eξ(N−2)φ

e−ξφ − 1

).


We will work on computing the two parts in turn:(eξφ − 1

) (eξφ + 1


e3ξNφ − e−3ξNφe3ξφ − 1

=e2ξφ + 2eξφ + 1

e2ξφ + eξφ + 1


) (e2ξNφ + 1 + e−2ξNφ


=eξφ + 2 + e−ξφ

eξφ + 1 + e−ξφUN−1 (kρ)

(e2ξNφ − 2 + e−2ξNφ

)+ 3


=2 + 2kρ1 + 2kρ

UN−1 (kρ)

(UN−1 (kρ)

2 +3


)=

2 + 2kρ1 + 2kρ

UN−1 (kρ)

(UN−1 (kρ)

2 +3


)The second part is:(

eξφ − 1) (eξφ + 1


e−ξ(N+2)φ − eξ(N−2)φe−ξφ − 1

= −(eξφ − 1

) (eξφ + 1

)2e−2ξφ

(eξφ − e−ξφ)3eξNφ − e−ξNφe−ξφ − 1

=

(1− eξφ

) (eξφ + 1

)2e−2ξφ

(eξφ − e−ξφ)2 (e−ξφ − 1)UN−1 (kρ)

=

(eξφ + 1

)2e−ξφ


=2kρ + 2


where we have used eξφ + e−ξφ = 2kρ in the last step.

Putting this together, we have that:

N−1∑n=0

(Un (x)− Un−1 (x)) (Un (x) + Un−1 (x))2

=2 + 2kρ1 + 2kρ

UN−1 (kρ)

(UN−1 (kρ)

2 +3


)+

2kρ + 2


=2 + 2kρ1 + 2kρ

UN−1 (kρ)

(UN−1 (kρ)

2 +3

(eξφ − e−ξφ)2+

1 + 2kρ


)

=2 + 2kρ1 + 2kρ

UN−1 (kρ)3

(1 + 2

1 + kρ


)

=2 + 2kρ1 + 2kρ

UN−1 (kρ)3

(1 + 2

1 + kρ

(eξNφ − e−ξNφ)2

),


where the result follows after noting that:


)2=

(2ι sinNφ)2 if |kρ| ≤ 1

(2 sinhNφ)2 if kρ ≥ 1

=

−4 sin2Nφ if |kρ| ≤ 1

4 sinh2Nφ if kρ ≥ 1.

(3) Note that by the product-to-sum identity for Chebyshev polynomials 2TI (kρ)Tn (kρ) =

TI+n − T|I−n| and hence:

Tn (kρ)3 = Tn (kρ)Tn (kρ)

2

= Tn (kρ)

(T2n (kρ) + 1

2

)=

1

2T2n (kρ)Tn (kρ) +

1

2Tn (kρ)

=1

4(T3n (kρ) + Tn (kρ)) +

1

2Tn (kρ)

=1

4T3n (kρ) +

3

4Tn (kρ) .

Thus we want to evaluate:

Nρ−1∑n=0

Tn+1 (kρ)3 + Tn (kρ)

3 =1

4

Nρ−1∑n=0

T3n+3 (kρ) + 3Tn+1 (kρ) + T3n (kρ) + 3Tn (kρ) .

Now we can again rewrite the Chebyshev polynomials in terms of exponentials, i.e.,

De Moivre’s theorem implies:

cosnφ =eιnφ + e−ιnφ

2and coshnφ =

enφ + e−nφ

2,

where we define φ = arccos kρ. This means that Chebyshev polynomials of the first

kind may be written as:

Tn (kρ) =eι1[kρ<1]nφ + e−ι

1[kρ<1]nφ

2

=eξnφ + e−ξnφ

2,


where ξ := ι1[kρ<1] . Thus:

Nρ−1∑n=0


3

=1

8

Nρ−1∑n=0

eξ(3n+3)φ + e−ξ(3n+3)φ + 3eξ(n+1)φ + 3e−ξ(n+1)φ

+eξ3nφ + e−ξ3nφ + 3eξnφ + 3e−ξnφ

=

1

8

Nρ−1∑n=0

eξ3nφ(eξ3φ + 1

)+ e−ξ3nφ

(e−ξ3φ + 1

)+3eξnφ

(eξφ + 1

)+ 3e−ξnφ

(e−ξφ + 1

)

=1

8

eξ3φNρ−1eξ3φ−1

(eξ3φ + 1

)+ e−ξ3φNρ−1

e−ξ3φ−1(e−ξ3φ + 1

)+3 e

ξφNρ−1eξφ−1

(eξφ + 1

)+ 3 e

−ξφNρ−1e−ξφ−1

(e−ξφ + 1

) .

Now, we’ll look at these in turn:

3eξφNρ − 1

eξφ − 1

(eξφ + 1

)+ 3

e−ξφNρ − 1

e−ξφ − 1

(e−ξφ + 1

)= −3

(eξφNρ − 1

) (eξφ − e−ξφ

)(eξφ − 1) (e−ξφ − 1)

+ 3

(e−ξφNρ − 1

) (eξφ − e−ξφ

)(e−ξφ − 1) (eξφ − 1)

= −3

(eξφ − e−ξφ

) (eξφNρ − 1− e−ξφNρ + 1

)2− eξφ − e−ξφ

= 3

(eξφ/2 − e−ξφ/2

) (eξφ/2 + e−ξφ/2

) (eξφNρ − e−ξφNρ

)eξφ − 2 + e−ξφ

= 3

(eξφ/2 + e−ξφ/2

)2 (eξφNρ − e−ξφNρ

)eξφ − e−ξφ

= 3(eξφ + e−ξφ + 2

)UN−1 (kρ)

= 6 (2kρ + 2)UN−1 (kρ) ,

where we have used eξφ + e−ξφ = 2kρ.


The other part is a little more complicated:

eξ3φNρ − 1

eξ3φ − 1

(eξ3φ + 1

)+e−ξ3φNρ − 1

e−ξ3φ − 1

(e−ξ3φ + 1

)= −

(eξ3φNρ − 1

) (eξ3φ − e−ξ3φ

)(eξ3φ − 1) (e−ξ3φ − 1)

+

(e−ξ3φNρ − 1

) (eξ3φ − e−ξ3φ

)(e−ξ3φ − 1) (eξ3φ − 1)

= −(eξ3φ − e−ξ3φ

) (eξ3φNρ − 1− e−ξ3φNρ + 1

)2− eξ3φ − e−ξ3φ

= −(eξ3φ/2 − e−ξ3φ/2

) (eξ3φ/2 + e−ξ3φ/2

) (eξ3φNρ − e−ξ3φNρ

)2− eξ3φ − e−ξ3φ

=

(eξ3φ/2 − e−ξ3φ/2

) (eξ3φ/2 + e−ξ3φ/2


) (e2ξφNρ + e−2ξφNρ + 1

)(eξ3φ/2 − e−ξ3φ/2)2

=

(eξ3φ/2 + e−ξ3φ/2

)(eξ3φ/2 − e−ξ3φ/2)

(eξφNρ − e−ξφNρ

) (e2ξNρφ − 2 + e−2ξNρφ + 3

)=

(eξ3φ + e−ξ3φ + 2


) ((eξφ − e−ξφ

)2UN−1 (kρ)

2 + 3)

eξ3φ − e−ξ3φ

=

((eξφ + e−ξφ

)3 − 3(eξφ + e−ξφ

)+ 2)UN−1 (kρ)

((eξφ − e−ξφ

)2UN−1 (kρ)

2 + 3)

((eξφ + e−ξφ)2 − 1

)=

((2kρ)

3 − 3 (2kρ) + 2)UN−1 (kρ)


)2UN−1 (kρ)

2 + 3)

(2kρ)2 − 1

= 2(kρ + 1) (2kρ − 1)2 UN−1 (kρ)


)2UN−1 (kρ)

2 + 3)

(2kρ − 1) (2kρ + 1)

= 2(kρ + 1) (2kρ − 1)UN−1 (kρ)


)2UN−1 (kρ)

2 + 3)

2kρ + 1


Putting this together, we have:

Nρ−1∑n=0


3

=1

8

6 (kρ + 1)UN−1 (kρ) + 2(kρ + 1) (2kρ − 1)UN−1 (kρ)


)2UN−1 (kρ)

2 + 3)

2kρ + 1

=

1

4(kρ + 1)UN−1 (kρ)

32kρ + 1

2kρ + 1+

(2kρ − 1)((eξφ − e−ξφ

)2UN−1 (kρ)

2 + 3)

2kρ + 1

=

1

4

kρ + 1

2kρ + 1

(3 (2kρ + 1)

UN−1 (kρ)2 + (2kρ − 1)

(eξφ − e−ξφ

)2+

3 (2kρ − 1)

UN−1 (kρ)2

)

=1

4

kρ + 1

2kρ + 1

(12kρ


+ 2kρ − 1

)UN−1 (kρ)

3

=1

4

kρ + 1

2kρ + 1

(eξφ − e−ξφ

)2( 12kρ


+ 2kρ − 1

)UN−1 (kρ)

3

=(kρ + 1)2 (kρ − 1)

2kρ + 1

(2kρ − 1 +

12kρ


)UN−1 (kρ)

3

As in the proof of part 1, the result follows since:


)2=

(2ι sinnφ)2 if |kρ| ≤ 1

(2 sinhnφ)2 if kρ ≥ 1

=

−4 sin2 nφ if |kρ| ≤ 1

4 sinh2 nφ if kρ ≥ 1.

Regardless of whether the series being summed involves polynomials of the first or the

second kind, their sums are expressed as a function of UN−1 (x) , a single polynomial of the

second kind. This polynomial happens to be the scaling factor α identified in Proposition 1

part 3. We now leverage Lemma B.2 to compute the investor’s equilibrium payoff.

Proposition 8 (Closed-form solutions for controlling investor’s equilibrium payoff). Fix ρ.

Up to the additive constant σ2X2E (ω2), the controlling investor’s equilibrium payoffs are as

follows:


(1) If ρ < 3/4, the controlling investor’s equilibrium payoff equals −σ2X/24.

(2) If 3/4 ≤ ρ ≤ 1, the controlling investor’s payoff in an equilibrium with N > 1

partitions is

VC (ρ,N) =σ2X6

ρ− 1

(4ρ− 1)

(3ρ

sin2 (N arccos kρ)− 1

)(B.3)

≤ V C (ρ) =σ2X6

(ρ− 1) (3ρ− 1)

(4ρ− 1).

The payoff VC (ρ,N) is increasing in N. For any ρ, there exists a value ρ ∈ [ρ, 1)

such that VC (ρ,Nρ) attains the bound V C (ρ).

(3) If ρ ≥ 1, the controlling investor’s payoff in an equilibrium with N > 1 partitions is

VC (ρ,N) =σ2X6

1− ρ4ρ− 1

(3ρ

sinh2 (N arccosh kρ)+ 1

)≤ V C (ρ) =

σ2X6

1− ρ(4ρ− 1)

.

The payoff VC (ρ,N) is increasing in N and limN→∞ VC (ρ,N) = V C (ρ) .

Proof. In light of Lemma A.1, the investor’s expected utility in an equilibrium when N

intervals are being communicated is:

VC (ρ,N) = −σ2X

2

N−1∑n=0

∫ ωn+1

ωn

(ω − a∗C (ωn, ωn+1|θC) θC)2 dω

= −σ2X

6

N−1∑n=0

[(ωn+1 − a∗C (ωn, ωn+1|θC) θC)3 − (ωn − a∗C (ωn, ωn+1|θC) θC)3

]= −σ

2X

6

N−1∑n=0

[(ωn+1 −

ωn+1 + ωn2

)3−(ωn −

ωn+1 + ωn2

)3]

= −σ2X

6

N−1∑n=0

[1

4(ωn+1 − ωn)3

]

= −σ2X

24

N−1∑n=0

[(Un (kρ)

UN−1 (kρ)− Un−1 (kρ)

UN−1 (kρ)

)3]

= −σ2X

24

1

[UN−1 (kρ)]3

N−1∑n=0

[Un (kρ)− Un−1 (kρ)]3 , (B.4)

where to get the second equality we have substituted for a∗ (ωn, ωn+1|θC) using Lemma 1.


(1) If ρ < 3/4, no information can be communicated. In this case the investor would

choose action 1/2θC (see Lemma 1) giving rise to an expected payoffwhich, by Lemma

A.1, equals

−σ2X

2E(

1

2− ω

)2= −σ

2X

24.

(2) In the case 3/4 ≤ ρ ≤ 1, we have |kρ| ≤ 1 and substituting the relevant expression

from Lemma B.2 into (B.4) we get

VC (ρ,N) =σ2X24

1− kρ2kρ + 1

(2− 3

1 + kρsin2 (N arccos kρ)

)=

σ2X6

ρ− 1

4ρ− 1

(3

ρ


),

where the last equality follows from substituting 2ρ− 1 for kρ and collecting terms.

Note that the payoff VC (ρ,N) is increasing in N , since:

VC (ρ,N)− VC (ρ,N − 1) =σ2X6

ρ− 1

4ρ− 13ρ

(1


sin2 ((N − 1) arccos kρ)

)=

σ2X6

ρ− 1

4ρ− 13ρ

(sin2 ((N − 1) arccos kρ)− sin2 (N arccos kρ)

sin2 (N arccos kρ) sin2 ((N − 1) arccos kρ)

)and since in an equilibrium where N > 1 it must be ρ > 3/4 (Proposition 1 part 6), so

the factor multiplying the parenthesis is negative and thus VC (ρ,N) ≥ VC (ρ,N − 1)

if and only if

sin2 (N arccos kρ) ≥ sin2 ((N − 1) arccos kρ) , or

sin (N arccos kρ) ≥ sin ((N − 1) arccos kρ) ,

where the second line follows since sin is positive on the relevant range. Let φ =

arccos kρ and note that:

sin (Nφ)− sin ((N − 1)φ) = sin

(Nφ− φ

2+φ

2

)− sin

(Nφ− φ

2− φ

2

)= 2 cos

(Nφ− φ

2

)sin

(φ

2

),


and since sin (φ/2) > 0 we have that sin (Nφ) ≥ sin ((N − 1)φ) if and only if

cos (Nφ− φ/2) , is positive, i.e., if and only if

Nφ− φ

2≤ π

2,

or after solving for N , if

N ≤ 1

2

(π

arccos kρ− 1

)≤

⌈1

2

(π

arccos kρ− 1

)⌉= Nρ.

To establish the tightness of the upper bound, note that

VC (ρ,Nρ) =σ2X6

ρ− 1

(4ρ− 1)

(3ρ

sin2 (Nρ arccos kρ)− 1

).

Now, the approximation is tight whenever⌈1

2

(π

arccos kρ− 1

)⌉arccos kρ =

π

2,

or ⌈1

2

π

arccos kρ− 1

2

⌉=

1

2

π

arccos kρ,

which holds whenever πarccos kρ

∈ Z and this occurs arbitrarily often as limρ→1 arccos kρ =

0.

(3) In the case ρ ≥ 1 we have |kρ| ≥ 1, and substituting the relevant expression from

Lemma B.2 into (B.4) we get

VC (ρ,N) =σ2X24

1− kρ2kρ + 1

(2 + 3

1 + kρ

sinh2 (N arccosh kρ)

)=

σ2X6

1− ρ4ρ− 1

(1 + 3

ρ

sinh2 (N arccosh kρ)

).

This expression is negative and is increasing in N because the function sinh (·) ismonotonically increasing. As N →∞ this converges to σ2X

61−ρ(4ρ−1) .


Proposition 8 provides a convenient expression for the investor’s equilibrium payoff in any

equilibrium, and a tight upper bound V C (ρ) for the payoff in the best equilibrium for the

investor. The best equilibrium is achieved at the maximal number of partitions compatible

with equilibrium.

Proposition 9 (Closed-form solutions for expert investor’s equilibrium payoff). Fix ρ. Up

to the additive constant σ2X2rEE (ω2), the expert investor’s equilibrium payoffs are as follows:

(1) If ρ < 3/4, no information can be communicated. In this case the expert investor’s

equilibrium payoff equals

− σ2X

2rEE(

1

2ρ− ω

)2.

(2) If 3/4 ≤ ρ ≤ 1, the expert investor’s payoff in an equilibrium with N > 1 partitions

is:

VE (ρ,N) = −σ2X

6

1− ρρ (4ρ− 1)

(3

2ρ− 1

sin2 (N arccos kρ)+ 3− 4ρ

)(B.5)

≤ V E (ρ) = −σ2X

3

1− ρ(4ρ− 1)

The payoff VE (ρ,N) is increasing in N. For any ρ, there exists a value ρ ∈ [ρ, 1)

such that VE (ρ,Nρ) attains the bound V E (ρ).

(3) If ρ ≥ 1, the expert investor’s payoff in an equilibrium with N > 1 partitions is:

VE (ρ,N) = −σ2X

6ρ

(ρ− 1)

4ρ− 1

(3

2ρ− 1

sinh 2 (N arccosh kρ)+ 4ρ− 3

)≤ V E (ρ) = −σ

2X

6

ρ− 1

ρ (4ρ− 1)(4ρ− 3) ,

The payoff VE (ρ,N) is increasing in N and limN→∞ VE (ρ,N) = V E (ρ) .


Proof. In light of Lemma A.1, the expert investor’s expected utility in an equilibrium when

N intervals are being communicated is

VE (ρ,N) = −σ2X

2

N−1∑n=0

∫ ωn+1

ωn

(ω − θErEa∗ (ωn, ωn+1|θC))2 dω

= −σ2X

6

N−1∑n=0

[(ωn+1 − θErEa∗ (ωn, ωn+1|θC))3 − (ωn − θErEa∗ (ωn, ωn+1|θC))3

]= −σ

2X

6

N−1∑n=0

[(ωn+1 − θErE

ωn+1 + ωn2θC

)3−(ωn − θErE

ωn+1 + ωn2θC

)3]

= −σ2X

6

N−1∑n=0

[(Un (kρ)

UN−1 (kρ)− Un (kρ) + Un−1 (kρ)

(kρ + 1)UN−1 (kρ)

)3−(Un−1 (kρ)

UN−1 (kρ)− Un (kρ) + Un−1 (kρ)

(kρ + 1)UN−1 (kρ)

)3]

= − σ2X6 (kρ + 1)3 UN−1 (kρ)

3

N−1∑n=0

[(kρUn (kρ)− Un−1 (kρ))

3 − (kρUn−1 (kρ)− Un (kρ))3]

=−σ2X

6 (kρ + 1)3 UN−1 (kρ)3

N−1∑n=0

[Tn+1 (kρ)

3 + Tn (kρ)3] , (B.6)

where the last line follows by an identity linking the Chebyshev polynomials of the first and

second kinds, Tn (kρ) = Un (kρ)−kρUn−1 (kρ), and by incrementing this identity and applying

the defining linear difference equation

Tn+1 (kρ) = Un+1 (kρ)− kρUn (kρ)

= 2kρUn (kρ)− Un−1 (kρ)− kρUn (kρ)

= kρUn (kρ)− Un−1 (kρ) .

(1) If ρ < 3/4, no information can be communicated. In this case the controlling investor

would choose action 1/2θC (see Lemma 1) giving rise to an expected payoff which,

by Lemma A.1, equals

− σ2X

2rEE(rEθE2θC

− ω)2

= − σ2X

2rEE(

1

2ρ− ω

)2.


(2) In the case 3/4 ≤ ρ ≤ 1, we have |kρ| ≤ 1 and substituting the relevant expression

from Lemma B.2 into (B.6) we get

VE (ρ,N) = − σ2X6 (kρ + 1)

(kρ − 1)

2kρ + 1

(2kρ − 1− 3

kρsin2 (N arccos kρ)

)= −σ

2X

6

1− ρρ (4ρ− 1)

(3

2ρ− 1

sin2 (N arccos kρ)+ 3− 4ρ

),

where the last equality follows from substituting 2ρ − 1 for kρ and collecting terms.

When sin2 = 1 the expression reduces to V E (ρ) . Monotonicity in N and tightness

of the bound are proved, mutatis mutandis, as in Proposition 8 part 1.

(3) In the case ρ ≥ 1 we have |kρ| ≥ 1, and substituting the relevant expression from

Lemma B.2 into (B.6) we get

VE (ρ,N) = − σ2X6 (kρ + 1)

(kρ − 1)

2kρ + 1

(2kρ − 1 + 3

kρsinh 2 (N arccosh kρ)

)= −σ

2X

6ρ

(ρ− 1)

4ρ− 1

(3

2ρ− 1

sinh 2 (N arccosh kρ)− 3 + 4ρ

),

where the last equality follows from substituting 2ρ − 1 for kρ and collecting terms.

The upper bound V E (ρ) is approached for N →∞.

Appendix C. Communication-related Demand for Shares

Lemma C.1 (Controlling Investor’s Communication-related Demand). Assume the action

a∗C (Ω) is optimally chosen by controlling investor C under the advice of a expert investor

with rE, θE. Let DC (θC |rC , a∗) denote the controlling investor C’s communication-relateddemand function. Then DC (θC |rC , a∗C) =

σ2XrEθE

ρ(2ρ−1)(4ρ−1)2 for θC ∈

(34rEθE, rEθE

). For levels of

θC smaller than 34rEθE demand equals zero, and it is negative for levels of θC larger than

rEθE.


Proof. From Proposition 8, the investor’s equilibrium highest payoff V C (ρ) equals approxi-

mately:

V C (ρ) '

−σ2X

24if 0 ≤ ρ ≤ 3

4

σ2X6(ρ−1)(3ρ−1)(4ρ−1) if 3

4≤ ρ ≤ 1

σ2X6

1−ρ(4ρ−1) if 1 ≤ ρ

Differentiating the above, we have that:

∂V C (ρ)

∂ρ'

0 if 0 ≤ ρ ≤ 3

4

σ2Xρ(2ρ−1)(4ρ−1)2 if 3

4≤ ρ ≤ 1

−σ2X2

1(4ρ−1)2 if 1 ≤ ρ

.

Since DC (θC |rC , a∗C) = ∂V C(ρ)∂ρ· ∂ρ∂θC, demand of the controlling investor is given by:

DC (θC |rC , a∗C) '

0 if 0 ≤ θC ≤ 3

4rEθE

σ2XrEθE

ρ(2ρ−1)(4ρ−1)2 if 3

4rEθE ≤ θC ≤ rEθE

− σ2X2rEθE

1(4ρ−1)2 if rEθE ≤ θC

Corollary C.2 (Features of Controlling Investor’s Communication-related Demand). The

controlling investor’s communication-related demand is zero on(0, 3

4rEθE

), it is increasing

on(34rEθE, rEθE

), and it is negative on (rEθE,∞) . The controlling investor purchases a

positive amount of shares only if prices are in[0,

5σ2X192rEθE

].

Proof. DC (θC |rC , a∗C) is constant on(0, 3

4rEθE

). DC (θC |rC , a∗C) is increasing on

(34rEθE, rEθE

)since:

∂

∂ρDC (θC |rC , a∗C) ∝ 2

ρ

(4ρ− 1)2+

2ρ− 1

(4ρ− 1)2− 8ρ

2ρ− 1

(4ρ− 1)3

=1

(4ρ− 1)2

[2ρ+ 2ρ− 1− 8ρ

2ρ− 1

(4ρ− 1)

]=

1

(4ρ− 1)3[(4ρ− 1)2 − 8ρ (2ρ− 1)

]=

1

(4ρ− 1)3> 0.


The perfect-communication conclusion reflects perfect alignment of preferences between con-

trolling investor and expert investor. Clearly there can be no equilibrium at if the controlling

investor owns an amount different to rEθE as his demand is weakly increasing up to this

point and negative after it.

To calculate the maximum willingness to pay p we need to ensure that:

0 ≤∫ rEθE

0

(DC (θC |rC , a∗C)− p) dθC

=

∫ rEθE

34rEθE

σ2XrEθE

ρ (2ρ− 1)

(4ρ− 1)2dθC −

∫ rEθE

0

p dθC

=

∫ rEθE

34rEθE

σ2XrEθE

θCrEθE

(2 θCrEθE

− 1)

(4 θCrEθE

− 1)2 dθC − prEθE

=

∫ rEθE

34rEθE

σ2XrEθE

θC2θC − rEθE

(4θC − rEθE)2dθC − prEθE

=5σ2X192− prEθE,

or if p ≤ 5σ2X192rEθE

.

Lemma C.3 (Expert Investor’s Communication-related Demand). Assume the action a∗C (ωn)

is optimally chosen by controlling investor C under the advice of an expert investor with risk

aversion parameter rE and holdings θE. Let DE (θE|rE, a∗C) denote the expert investor’s

demand function. For θE ∈ (0, θC/rE) we have DE (θE|rE, a∗C) =σ2X2rEθC

[8ρ2−8ρ+116ρ2−8ρ+1

]> 0 . De-

mand is negative for θE ∈ (θC/rE, 4θC/3rE) . Demand is also negative for θE ∈ (4θC/3rE, 1) .

Proof. From Proposition 8, the expert investor’s equilibrium highest payoff V E (ρ) equals

approximately:

V E (ρ) '

− σ2X2rEE(12ρ− ω

)2if 0 ≤ ρ ≤ 3

4

−σ2X3

1−ρ(4ρ−1) if 3

4≤ ρ ≤ 1

−σ2X6

ρ−1ρ(4ρ−1) (4ρ− 3) if 1 ≤ ρ

where ρ = θC/rEθE.


Case ρ < 3/4. This case corresponds to θE > 4θC/3rE.

DE (θE|rE, a∗C) =∂V E (ρ)

∂θE

= − σ2X

2rE

∂

∂θEE(rEθE2θC

− ω)2

= − σ2X

2θCE(rEθE2θC

− ω)

= − σ2X

2θC

(1

2ρ− 1

2

)< 0.

Case 34≤ ρ ≤ 1. Since ∂

∂ρ

(−σ2X3

1−ρ(4ρ−1)

)=

σ2X(4ρ−1)2 by what we found in the controlling-

investor ρ ≥ 1 case, we have that:


∂ρ· ∂ρ∂θE

=σ2X

(4ρ− 1)2

(−ρ 1

θE

)=

−σ2XρθE (4ρ− 1)2

.

Case 1 ≤ ρ. This case corresponds to θE ≤ θC/rE. On the range 1 ≤ ρ we have:

∂V E (ρ)

∂ρ= −σ

2X

6

(∂

∂ρ

ρ− 1

ρ (4ρ− 1)(4ρ− 3)

)= −σ

2X

6

(3 (8ρ2 − 8ρ+ 1)

ρ2 (4ρ− 1)2

)Now:


∂ρ· ∂ρ∂θE

=∂V E (ρ)

∂ρ·(− θC

(rEθE)2rE

)=

∂V E (ρ)

∂ρ·(−ρ 1

θE

)


So

DE (θE|rE, a∗C) =σ2X

2ρθE

(8ρ2 − 8ρ+ 1

(4ρ− 1)2

)=

σ2X2

rEθC

(8ρ2 − 8ρ+ 1

16ρ2 − 8ρ+ 1

).

Which means that:

DE (θE|rE, a∗C) '

0 if 0 ≤ ρ ≤ 3

4

−σ2XρθE(4ρ−1)2

if 34≤ ρ ≤ 1

σ2X2rEθC

(8ρ2−8ρ+1(4ρ−1)2

)> 0 if 1 ≤ ρ

.

Corollary C.4 (Features of Expert Investor’s Communication-related Demand). The expert

investor’s communication-related demand is positive and decreasing in θE over the interval

(0, θC/rE) . DE (0|rE, a∗C) =σ2X4rEθCand DE

(θCrE|rE, a∗C

)=

rEσ2X

18θC. Since expert investor’s de-

mand is negative or zero elsewhere, if the expert investor purchases any positive amount

of shares in equilibrium he purchases within the interval (0, θC/rE) . The expert investor

purchases θC/rE only if prices are in[0,

σ2X18

rEθC

].

Proof. The function 8ρ2−8ρ+116ρ2−8ρ+1 is positive if ρ > 1; it is also increasing in ρ since d

dρ

(8ρ2−8ρ+116ρ2−8ρ+1

)=

16ρ

(4ρ−1)3 > 0 since ρ > 1/4. The function has value 1/9 at ρ = 0 and it asymptotes

horizontally to 1/2 as ρ → ∞. Therefore, since ρ is a decreasing function of θE, we

have that DE (θE|rE, a∗C) is decreasing in θE on (0, θC/rE) . DE (0|rE, a∗C) =σ2X4rEθCand

DE

(θCrE|rE, a∗C

)=

σ2X18

rEθC.

Lemma C.5 (Non-controlling investor’s demand). Assume the action a∗ (Ω) is optimally

chosen by the controlling agent under information set Ω . Let Di (θi|ri, a∗) denote the de-mand function of a non-controlling investor with risk aversion ri. Then Di (θi|ri, a∗) =

σ2X (rCθC − riθi)E[a∗ (Ω)2

].


Proof.

Di (θi|ri, a∗) = − ∂

∂θi

σ2X2riE [riθia

∗ (ω)− ω]2

= −σ2XE [a∗ (ω) (riθia∗ (ω)− ω)]

= σ2XE [E [a∗ (ω) (ω − riθia∗ (Ω)) |Ω]]

= σ2XE [a∗ (Ω)E [ω − rCθCa∗ (Ω) + rCθCa∗ (Ω)− riθia∗ (Ω) |Ω]] . (C.1)

Now, a∗ (Ω) must solve the following maximization problem:

maxa− σ

2X

2rCE[(rCθCa− ω)2 |Ω

],

and the first order conditions w.r.t. a read E [ω − rCθCa∗ (Ω) |Ω] = 0. Use this expression to

simplify (C.1), then isolate a∗ (Ω) to get the desired expression.

Lemma C.6 (Controlling investor’s optimal action). Let a∗ (ρ,Ω) denote the action cho-

sen by the controlling agent if players are playing the cheap talk equilibrium with the finest

partition given ρ and the controlling investor knows information set Ω. We have that

E[a∗ (ρ,Ω)2

]=

θ2C+θErEθC

16θCθ3ErE−4θ4Er2E

if θC ≤ rEθE

θC4θCθ

2E−θ3ErE

if rEθE ≤ θC.

Proof. Note that:

ωn =Un−1 (kρ)

UN−1 (kρ)=

enφi1[kθ<1] − e−nφi

1[kθ<1]

eNφi1[kθ<1] − e−Nφi

1[kθ<1]=eξnφ − e−ξnφeξNφ − e−ξNφ

where ξ := ι1[kρ<1] and that:

a∗ (ρ, [ωn, ωn+1]) =ωn + ωn+1

2θE.

Now since:

E[a∗ (ρ,Ω)2

]=

N−1∑n=0

(ωn+1 − ωn)

(ωn + ωn+1

2θE

)2

=1

4θ2E

N−1∑n=0

(ωn+1 − ωn) (ωn + ωn+1)2 ,


we need to evaluate (at the highest possible value of N , more on this later):

N−1∑n=0

(ωn+1 − ωn) (ωn + ωn+1)2

=1

UN−1 (kρ)3

N−1∑n=0

(Un (kρ)− Un−1 (kρ)) (Un (kρ) + Un−1 (kρ))2 .

Note that lemma B.2 part 2 gives exactly the sum part of this expression. Dividing into two

regimes we have that for ρ > 1 (where N →∞):

E[a∗ (ρ,Ω)2

]=

1θ2E

ρ4ρ−1

(1 + ρ

sin 2(Nρ arccos 2ρ)

)if 0 ≤ ρ ≤ 1

1θ2E

ρ4ρ−1 if 1 ≤ ρ

≤

14θ2E

ρ(1+ρ)4ρ−1 if 0 ≤ ρ ≤ 1

1θ2E

ρ4ρ−1 if 1 ≤ ρ

=

θ2C+θErEθC

16θCθ3ErE−4θ4Er2E

if θC ≤ rEθE

θC4θCθ

2E−θ3ErE

if rEθE ≤ θC.

Corollary C.7. The demand function of a non-controlling investor is increasing if the con-

trolling investor’s preferences are more aligned with the expert investor’s preferences, i.e., if

ρ→ 1, or θC gets closer to rEθE.

Proof. By lemma C.5 the demand of the non-controlling investor is increasing in E[a∗ (ρ,Ω)2

].

From lemma C.6 we have that:

d

dθCE[a∗ (ρ,Ω)2

]∣∣∣∣rEθE≤θC

= −rEθE

1

(4θC − θErE)2< 0.

andd

dθCE[a∗ (ρ,Ω)2

]∣∣∣∣θC≤rEθE

=4θ2C − 2θCθErE − θ2Er2E4θ3ErE (4θC − θErE)2

,

which is positive iff θC > 14θErE

(√5 + 1

)or if ρ > 1

4

(√5 + 1

)= 1

2ϕ. The lower-bound

on ρ above can be significantly tightened by not using the approximate actions, since these

approximations are only very accurate when ρ is close to 1.