Investor’s Information Sharing with Firms in Oligopoly∗ Myeonghwan Cho† University of Seoul July 28, 2017

Abstract We study the incentives for an investor to transmit information to its invested firms in oligopoly. The investor has more information on market conditions than the firms and reveals it publicly or privately before the firms establish their production. When the investor uses a public channel to transmit information, the investor does not reveal any of its information to the firms. When the investor uses a private channel to transmit information, it is possible that the investor partially reveals its private information to a firm. Indeed, this is possible only when the investor owns a relatively larger share of a firm than that of the other firm. JEL Classification: C72, D21, D82, D83, L13. Keywords: Oligopoly, Cournot competition, Information asymmetry, Information transmission.

∗

This work was supported by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea (NRF-2017S1A5A2A01023563). We would like to thank Sangwon Park, Hwan-sik Choi, and participants at the 2016 KAEA Annual Meeting and 2016 KIEA Annual Meeting for their helpful comments. † Department of Economics, University of Seoul, Seoulsiripdaero 163, Dongdaemun-gu, Seoul, Republic of Korea. Tel: +82 2 6490 2067; Fax: +82 2 6490 2054; E-mail: [email protected]

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1

Introduction

Recently, institutional investors have played an important role in the economy. The main role of the institutional investors is to make investments to several firms. In addition, they generate and provide various information on the market. Of course, the firms are usually more knowledgeable about the market for their products than their investors. But, we frequently observe that the investors also have some information that the firms do not have and provide it in order to improve the reliability from the market and the performance of their invested firms. For example, many investment banks such as J.P. Morgan, Goldman Sachs, and Morgan Stanley operate economic research teams that provide analysis of economic environments and market trends. Venture capitals not only provide funding for early-stage firms, but also provide a variety of consulting services for their invested firms. These consulting services include providing information about market to develop business strategies. Also, institutional investors with a global network may be better aware of the situation in the overseas market than domestic companies. In this paper, we are interested in the investor’s incentive in providing information to its invested firms in an oligopoly market. While an investor making investments to several firms is interested in the performance of its portfolio, which is the (weighted) joint profit of its invested firms, an individual firm is interested only in its own profit maximization. Thus, the investors may have an incentive to behave strategically in providing information to the firms in order to induce the profit maximizing firms to make a decision improving the joint profit. When the investor provides information to its invested firms, it may make use of a public or private channel. When an investor provides information utilizing a public channel, the same information is provided to all firms. For example, the investment banks produce and publicly disclose a report about economic or market trends. When an investor provides information utilizing a private channel, only a specific firm receives the information. For example, an institutional investor provides a consultancy service for a specific firm. The aim of this paper is to investigate how the investor provides its private 2

information to firms when it utilizes a public channel or a private channel. To this end, we consider a model with two firms producing homogeneous goods and one investor owning shares of these firms. The investor possesses information about the market condition that the firms do not have and decides how to provide the information to the firms, utilizing either a public channel or a private channel. To focus on the investor’s incentive to transmit its private information, it is assumed that the investor is more informed about the market conditions than the firms. This assumption tries to capture that the investor has some information that the firms cannot access, not that the investor has additional information including information the firms have. The information provided by the investor cannot be verified, and so the investor can provide inaccurate information to the firms. After the investor delivers information about market conditions, the firms engage in Cournot competition and choose output to maximize profit based on the information provided by the investor. The profit of each firm is determined by the outputs chosen by the firms and shared by the investor and the relevant firm in accordance with their share ratio. We first find that, when the investor utilizes a public channel, no information is provided to the firms in any equilibrium. Since each firm’s output that is optimum to the investor is different from the output that is optimum to the firm, the investor may be reluctant to provide its private information to the firms. Meanwhile, when the investor provides information to a specific firm utilizing a private channel, it is possible in an equilibrium for the investor to partially reveal information about market conditions to the relevant firm. Indeed, when the investor owns a relatively larger share of a specific firm than that of the other firm, the investor’s concern for the relevant firm’s profit is high enough for the investor to partially provide its information to the firm in order to improve its performance. There are a number of studies that investigate information sharing in an oligopoly. Examples are Novshek and Sonnenschein (1982), Clarke (1983), Vives (1984), Fried (1984),

3

Li (1985), Gal-Or (1985, 1986), Shapiro (1986), Kirby (1988), Ziv (1993), Raith (1996), and Jansen (2008). Most of these studies focus on information sharing between the firms in an oligopoly by considering the following two stage decision process. In the first stage, the firms make a decision whether or how to share their private information. In the second stage, the firms play a Bayesian game of market based competition under incomplete information. A variety of results are provided in these studies depending on the particular specifications of the model. Indeed, the results depend on the type of market based competition (Cournot or Bertrand competition), the source of uncertainty (demand or cost side), the type of uncertainty (common or private value), and the verifiability of revealed information. Our model departs from these studies by introducing the investor and focusing on the investor’s incentive to transmit private information to the oligopoly firms. Eliaz and Forges (2015) studies the optimal information transmission of a planner to the firms in Cournot oligopoly. Their model shares with ours the features that the planner concerning the joint profit of the firms is more informed on the market condition and transmits its information to the firms. But, in Eliaz and Forges (2015), the information transmitted by the planner is verifiable although the planner can manipulate the accuracy of information. This paper is also related to the studies on strategic transmission of private information. Many of those studies consider the model in which individuals play a role of either a sender who has private information and makes a decision how to reveal it or a receiver who chooses an action that affects her own and the sender’s payoffs. Milgrom (1981) and Milgrom and Roberts (1986) consider the situation in which a sender may withhold information but may not lie because information is verifiable. They find that a complete revelation of information can be achieved as an equilibrium. Okuno-Fujiwara et al. (1990) also adopt this approach to find sufficient conditions for complete revelation of private information. Crawford and Sobel (1982) are interested in the possibility of partial revelation of private information under a model in which there are one sender and one receiver

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and the sender’s revealed information is not verifiable. Melumad and Shibano (1991) borrow this approach to compare the equilibria in models with and without commitment on decision rule. Alonso et al. (2008) extend the model of Crawford and Sobel (1982) to the situation with multiple senders and compare the centralized and decentralized decision making processes. Farrell and Gibbons (1989) and Goltsman and Pavlov (2011) also extend the model of Crawford and Sobel (1982) by considering multiple receivers whose payoffs do not depend on each other’s actions. Compared with these studies, we also consider a model with one sender and multiple receivers, in which the sender’s revealed information is not verifiable. But the receivers in our model are strategically interdependent through Cournot competition. Eliaz and Serrano (2014) is another work that considers a model with one sender and multiple receivers. In their model, the receivers are strategically dependent but the sender’s revealed information is verifiable. The remainder of this paper is organized as follows. Section 2 explains the model. In Section 3, we characterize an equilibrium when the investor uses a public channel to transmit its private information. In Section 4, we characterize an equilibrium when the investor uses a private channel to transmit its private information. We provide some discussion in Section 5 and conclusions in Section 6.

2

Model

There are one investor and two firms indexed by i = 1, 2. The investor owns a share of each firm. Let αi ∈ (0, 1) be the investor’s share of firm i. The firms produce a homogeneous good with a marginal cost c > 0 and are involved in Cournot competition. Let qi be the output of firm i. The market demand for the good is linear and given by

p = a + θ − q1 − q2 ,

5

(1)

where a > 0 is constant and θ is randomly drawn from a distribution. A realization of θ is private information to the investor, which can be interpreted as the investor being more informed on the market condition than the firms.1 The distribution of θ is public ¯ that is, supp(f ) = information and has a continuous density f with a support of [0, θ]; ¯ For notational simplicity, let Eθ = E[θ] be the expectation of θ. We assume that a [0, θ]. is large enough to satisfy ¯ a − c > 3θ.

(2)

This can also be interpreted as relatively small uncertainty in the market condition. Given an output (q1 , q2 ) of the firms, each firm i’s (ex-post) profit is

πi (qi , qj ; θ) = (a − c + θ − qi − qj )qi ,

(3)

and it is shared with the investor in accordance with their share ratio. Thus, given firm i’s profit πi (qi , qj ; θ), firm i’s (ex-post) payoff is

ui (qi , qj ; θ) = (1 − αi )πi (qi , qj ; θ)

(4)

= (1 − αi )(a − c + θ − qi − qj )qi

and the investor’s (ex-post) payoff is

v(qi , qj ; θ) = αi πi (qi , qi ; θ) + αj πj (qi , qj ; θ)

(5)

= (a − c + θ − qi − qj )(αi qi + αj qj ).

In this environment, we are interested in the investor’s incentive to transmit its private information to the firms. Before the firms choose their output, the investor transmits its 1

With this assumption, we do not insist that the investor is more knowledgeable about the market conditions than the firms in general. This assumption tries to capture that the investor has some information that the firms cannot access, although the firms may have a better sense for the market demand. This allows us to focus on the investor’s incentives to transmit its information.

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information using a public channel or a private channel. When the investor uses a public ¯ about θ to both firms. When the investor uses a channel, it sends a message m ∈ [0, θ] ¯ to only one firm, say firm i. We assume private channel, it sends a message m ∈ [0, θ] that the firms cannot verify the investor’s messages. So, the investor may be able to hide or partially reveal its information by sending a message m different from θ. After the investor sends a message, each firm forms a belief on θ and chooses its output to maximize its expected payoff. In summary, the decision process consists of two stages. In Stage 1, the investor ¯ to the firms (or to observes a realization of θ and sends a (random) message m ∈ [0, θ] firm i). In Stage 2, each firm i decides its output qi ∈ R+ . Because the investor can send a random message, the investor’s strategy is represented ¯ → ∆([0, θ]), ¯ where ∆([0, θ]) ¯ is the set of distribution on [0, θ]. ¯ Here, as a function µ : [0, θ] we allow the investor to use a mixed strategy (or to send a random message).2 We refer to µ(θ) as a random message whose distribution is generated by the investor’s strategy µ given that θ is realized. If the investor uses a pure strategy µ, its strategy can be expressed ¯ → [0, θ], ¯ where µ(θ) is a message that the investor sends when it as a function µ : [0, θ] observes θ. The truth-telling strategy µT is a strategy in which the investor truthfully ¯ reveals its private information on θ. That is, µT (θ) = θ for each θ ∈ [0, θ]. When the investor sends a message to firm i (using a public or a private channel), ¯ → R+ . Here, σi (m) represents firm i’s strategy can be represented as a function σi : [0, θ] the output that firm i chooses when it receives a message m from the investor. When the investor does not send a message to firm i (using a private channel), firm i’s strategy can be represented as σi ∈ R+ . Of course, the firms can use a mixed strategy. However, given a message m, the firm’s optimal decision on outputs is uniquely determined, and so the firms will not use a mixed strategy in any equilibrium. Thus, we can restrict our attention to pure strategies for the firms. ¯ → ∆([0, θ]) ¯ is a behavioral strategy rather than a mixed strategy. It is well known Formally, µ : [0, θ] that, if a condition known as perfect recall is satisfied, the mixed and behavioral strategies are equivalent. 2

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Given the investor’s strategy µ, message m is observable under µ if m belongs to the support of µ(θ) for some θ. That is, m ∈ supp(µ(θ)) for some θ. In other words, message m is observable under µ if it can be sent by the investor who plays strategy µ. For the investor’s strategy µ, let M(µ) be the set of messages that are observable under µ. That S is, M(µ) = θ∈[0,θ]¯ supp(µ(θ)). In the analysis, we use a perfect Bayesian equilibrium as a solution concept. A perfect Bayesian equilibrium (µ∗ , σ1∗ , σ2∗ ) has to satisfy two conditions, called consistency and sequential rationality. In general, consistency requires that each firm i forms a belief using Bayes’ rule from µ∗ and σj∗ as long as it observes a message m that is observable under µ∗ . Sequential rationality requires that the investor observing θ maximizes its payoff given that the firms play (σ1∗ , σ2∗ ), and that each firm i maximizes its expected payoff under its belief.3 If firm i has a belief that is consistent with µ, for a message m ∈ M(µ), firm i has to form an expectation on θ as E[θ|µ(θ) = m]. For m ∈ M(µ), let

e(m; µ) = E[θ|µ(θ) = m].

(6)

We say that e(m; µ) is an expectation induced by m under µ. For notational simplicity, we write e(m) = e(m; µ) for the investor’s strategy µ if there is no possibility of confusion. We also say that an expectation e is induced under µ if e = e(m) holds for some m ∈ M(µ). For the investor’s strategy µ, let E(µ) be the set of expectations e that are induced under µ. That is, E(µ) = {e : e = e(m; µ) for some m ∈ M(µ)}.

(7)

¯ e(m; µT ) = m for each Note that, for a truth-telling strategy µT , M(µT ) = [0, θ], ¯ and E(µT ) = [0, θ] ¯ hold. For a strategy µU in which the investor always m ∈ [0, θ], 3 Formally, a perfect Bayesian equilibrium consists of a belief system and a strategy profile that satisfy consistency and sequential rationality. For a formal definition of perfect Bayesian equilibrium, see Osborne and Rubinstein (1994).

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¯ whatever θ is realized, sends a random message µU (θ) uniformly distributed on [0, θ] ¯ e(m; µU ) = Eθ for each m ∈ [0, θ], ¯ and E(µU ) = {Eθ } hold. The strategy M(µU ) = [0, θ], µU is said to be uninformative because the messages that are observable under µU do not change the firm’s prior belief on θ.

3

Public information transmission channel

In this section, we consider the situation where the investor uses a public channel to transmit its private information. Here, the investor has to send the same message to both firms, and so the firms have the same expectation on θ. To find a perfect Bayesian equilibrium (µ† , σ1† , σ2† ), we use backward induction. Consider firm i’s problem in Stage 2 where the investor plays an equilibrium strategy µ† and the firms observe message m ∈ M(µ† ). Because the firms have a belief consistent with µ† , each firm i’s choice σi† (m) in the equilibrium has to maximize E[ui (qi , σj† (m); θ)|µ† (θ) = m] = E[(1 − αi )(a − c + θ − qi − σj† (m))qi |µ† (θ) = m]

(8)

with respect to qi ∈ R+ .4 From the first order necessary condition, we have 1 1 1 σi† (m) = (a − c) + E[θ|µ† (θ) = m] − E[σj† (m)|µ† (θ) = m]. 2 2 2

(9)

Taking-expectation conditional on µ† (θ) = m to both sides of (9), we have 1 1 1 E[σi† (m)|µ† (θ) = m] = (a − c) + E[θ|µ† (θ) = m] − E[σj† (m)|µ† (θ) = m]. 2 2 2

(10)

Notice that ui (qi , σj† (m); θ) is concave in qi . Thus, the first order necessary conditions are sufficient for the solution of the problem of maximizing ui (qi , σj† (m); θ). This is true for all other maximization problems considered in this paper. 4

9

Solving the equations in (10) for i and j, we obtain 1 1 E[σi† (m)|µ† (θ) = m] = E[σj† (m)|µ† (θ) = m] = (a − c) + e(m) 3 3

(11)

where e(m) = E[θ|µ† (θ) = m]. Plugging (11) into (9), each firm i’s equilibrium strategy σi† should satisfy, for each m ∈ M(µ† ), 1 1 σi† (m) = (a − c) + e(m). 3 3

(12)

We next move to Stage 1 in which the investor chooses a message to maximize its ¯ is realized. Given payoff given that each firm i plays σi† in (12). Suppose that θ ∈ [0, θ] that each firm i observes a message m ∈ M(µ† ) and plays an equilibrium strategy σi† in (12), firm i’s (ex-post) profit is5 Π†i,m = (a − c + θ − σi† (m) − σj† (m))σi† (m)

(13)

1 = (a − c + e(m))(a − c + 3θ − 2e(m)) 9 and firm i’s (ex-post) payoff is † Ui,m = (1 − αi )Π†i,m

(14)

1 = (1 − αi )(a − c + e(m))(a − c + 3θ − 2e(m)). 9 In addition, given that each firm plays σi† in (12), the investor can receive a (ex-post) payoff Vm† = αi Π†i,m + αj Π†j,m

(15)

1 = (αi + αj )(a − c + e(m))(a − c + 3θ − 2e(m)) 9 5

† Note that the condition in (2) ensures Π†i,m > 0, and so Ui,m > 0 and Vm† > 0.

10

by sending a message m ∈ M(µ† ). Thus, for µ† to be an equilibrium strategy for the ¯ if m ∈ supp(µ† (θ)) and m0 ∈ M(µ† ), investor, it has to be satisfied that, for any θ ∈ [0, θ], then Vm† ≥ Vm† 0 holds. If Vm† < Vm† 0 holds for m ∈ supp(µ† (θ)) and m0 ∈ M(µ† ), the investor observing θ has an incentive to deviate from µ† by sending a message m0 . Note that, for any m ∈ supp(µ† (θ)), Vm† in (15) has the same value. Lemma 1 states that the firms have the same expectation on θ regardless of messages along the equilibrium path. Lemma 1 Let (µ† , σ1† , σ2† ) be an equilibrium. Then E(µ† ) is a singleton with an element Eθ . Proof. Suppose that there are e ∈ E(µ† ) and e0 ∈ E(µ† ) with e > e0 . Then, there exist m ∈ M(µ† ) and m0 ∈ M(µ† ) such that e(m) = e and e(m0 ) = e0 . Consider the investor who observes θ such that m ∈ supp(µ† (θ)). For m and m0 , define the investor’s payoffs Vm† and Vm† 0 as in (15), respectively. Then, (2) implies that 1 Vm† − Vm† 0 = (αi + αj )(e(m0 ) − e(m))(2e(m0 ) + 2e(m) + a − c − 3θ) < 0. 9

(16)

This means that the investor observing θ has an incentive to send message m0 , which contradicts that µ† is an equilibrium strategy. Thus, E(µ† ) has at most one element ¯ Since E[θ|µ† (θ) = m] = e† for all m ∈ M(µ† ), it is satisfied that e† ∈ [0, θ]. Eθ = E[E[θ|µ† (θ) = m]|m ∈ M(µ† )] = E[e† |m ∈ M(µ† )] = e† .

(17)

This completes the proof. Lemma 1 implies that the investor always sends an uninformative message in any equilibrium. This also implies the non-existence of equilibrium in which the investor uses a truth-telling strategy. Note that the investor cares about the (weighted) joint profit of two firms, while each firm cares about its own profit. Because the firms have the 11

same marginal cost which is constant, the investor wants the firm i with a smaller share (αi < αj ) to produce nothing and the firm j with a larger share to produce all goods in the market. This is, of course, not desirable to firm i. This can be interpreted as the discrepancy in the optimum level of outputs between the sender (investor) and the receivers (the firms) is high enough. Thus, the investor is not willing to provide its private information to the firms. This result is consistent with an insight from Crawford and Sobel (1982). We note that the assumption in (2) used in the proof of Lemma 1 is stronger than required. Indeed, the condition for Lemma 1 to hold is that there does not exist y = (y1 , . . . , yK−1 ) ∈ RK−1 satisfying 0 ≡ y0 < y1 < . . . < yK−1 ≤ yK ≡ θ¯ and, for each + k = 1, . . . , K − 1, yk =

2(ek + ek+1 ) a − c + , 3 3

(18)

where ek = E[θ|yk−1 ≤ θ ≤ yk ]. Similar arguments in Section 4 can be applied to verify this condition. Notice that (2) implies the non-existence of y ∈ RK−1 satisfying (18). + ¯ including the uniform distribution, satisfy the In addition, many distributions on [0, θ], condition. We add the assumption in (2) for the simplicity of analysis. Proposition 1 shows the existence of an equilibrium that has the property in Lemma 1 and the uniqueness of such an equilibrium in terms of the payoffs. Proposition 1 There exists an equilibrium (µ† , σ1† , σ2† ) such that, for any m ∈ M(µ† ), e(m; µ† ) = Eθ , and

(19)

1 1 σ1† (m) = σ2† (m) = (a − c) + Eθ . 3 3

(20)

Indeed, every equilibrium (µ† , σ1† , σ2† ) satisfies (19) and (20). Proof. Consider the investor’s strategy µ† such that, for all θ, µ† (θ) is uniformly distributed ¯ Note that M(µ† ) = [0, θ] ¯ holds. Since, for all m ∈ [0, θ], ¯ e(m; µ† ) = Eθ holds, the on [0, θ]. investor does not have an incentive to deviate from µ† . In addition, since e(m; µ† ) = Eθ 12

¯ (12) implies that (20) is an equilibrium between the firms after they for all m ∈ [0, θ], ¯ This proves the first assertion. Let (µ† , σ † , σ † ) be an observe any message m ∈ [0, θ]. 1 2 equilibrium. Because Lemma 1 implies E(µ† ) = {Eθ }, (19) and (20) should be satisfied for any m ∈ M(µ† ). This implies the second assertion. Note that the equilibrium payoffs depend on the firms’ chosen outputs not on the messages of the investor. Thus, if the output is uniquely determined in equilibria, the equilibrium payoffs are uniquely determined. In the proof of Proposition 1, we construct an investor’s equilibrium strategy µ† that the investor always sends a random message ¯ We note that the strategy µ† is not the only equilibrium uniformly distributed on [0, θ]. strategy for the investor. For example, one can show that a strategy µo in which the investor always sends the same message regardless of the realization of θ constitutes an equilibrium. Notice that the strategies µ† and µo do not provide any of the investor’s private information on θ, and the firms do not change their prior belief on θ after receiving a message from the investor. Thus, Proposition 1 can be interpreted as, when the investor uses a public channel of information transmission, it does not reveal any of its private information with the firms. From Proposition 1, we can obtain each firm i’s (ex-post) equilibrium payoff as 1 Ui† = (1 − αi )(a − c + Eθ )(a − c + 3θ − 2Eθ ) 9

(21)

and the investor’s (ex-post) equilibrium payoff as 1 V † = (αi + αj )(a − c + Eθ )(a − c + 3θ − 2Eθ ). 9

(22)

From (21) and (22), we also obtain each firm i’s ex-ante equilibrium payoff as 1 E[Ui† ] = (1 − αi )(a − c + Eθ )2 9

13

(23)

and the investor’s ex-ante equilibrium payoff as 1 E[V † ] = (αi + αj )(a − c + Eθ )2 . 9

4

(24)

Private information transmission channel

In this section, we consider the situation where the investor uses a private channel to transmit its private information. Private channel means that the investor sends a message to only one firm, say firm i. Because only firm i receives the investor’s message m, firm i’s ¯ → R+ , and firm j’s strategy is a scalar σj ∈ R+ . Again, strategy is a function µi : [0, θ] backward induction is used to find a perfect Bayesian equilibrium (µ‡ , σ1‡ , σ2‡ ). Consider firm i’s problem in Stage 2 where the investor plays an equilibrium strategy µ‡ and firm i observes message m ∈ M(µ‡ ). Because firm i has a belief consistent with µ‡ , firm i’s equilibrium strategy σi‡ (m) has to maximize E[ui (qi , σj‡ ; θ)|µ‡ (θ) = m] = E[(1 − αi )(a − c + θ − qi − σj‡ )qi |µ‡ (θ) = m]

(25)

with respect to qi ∈ R+ . In addition, because firm j does not receive any message from the investor, firm j’s equilibrium strategy σj‡ has to maximize E[uj (σi‡ (m), qj ; θ)] = E[(1 − αj )(a − c + θ − σi‡ (m) − qj )qj ]

(26)

with respect to qj ∈ R+ . The first order necessary conditions for problems (25) and (26) imply that 1 σi‡ (m) = (a − c) + 2 1 σj‡ = (a − c) + 2

1 1 E[θ|µ‡ (θ) = m] − E[σj‡ |µ‡ (θ) = m] 2 2 1 1 E[θ] − E[σi‡ (m)]. 2 2

14

(27) (28)

Taking an expectation on both sides of (27) and (28), we have 1 E[σi‡ (m)] = (a − c) + 2 1 E[σj‡ ] = (a − c) + 2

1 Eθ − 2 1 Eθ − 2

1 E[σj‡ ] 2 1 E[σi‡ (m)], 2

(29) (30)

and so E[σi‡ (m)] = (1/3)(a − c) + (1/3)Eθ . Plugging this into (28), we have 1 1 σj‡ = (a − c) + Eθ . 3 3

(31)

In addition, since E[σj‡ |µ‡ (θ) = m] = E[σj‡ ] = (1/3)(a − c) + (1/3)Eθ holds, we obtain from (27) that, for each m ∈ M(µ‡ ), 1 1 1 σi‡ (m) = (a − c) + e(m) − Eθ , 3 2 6

(32)

where e(m) = E[θ|µ‡ (θ) = m]. For the investor’s optimal choice in Stage 1, suppose that firm i plays σi‡ in (32), and firm j plays σj‡ in (31). Given that the investor sends a message m ∈ M(µ‡ ), firm i’s and firm j’s (ex-post) profits can be calculated as Π‡i,m = (a − c + θ − σi‡ (m) − σj‡ )σi‡ (m) =

1 (2(a − c) + 6θ − Eθ − 3e(m))(2(a − c) − Eθ + 3e(m)) 36

Π‡j,m = (a − c + θ − σi‡ (m) − σj‡ )σj‡ =

(33)

(34)

1 (2(a − c) + 6θ − Eθ − 3e(m))(2(a − c) + 2Eθ ), 36

and so firm i’s and firm j’s (ex-post) payoffs are ‡ Ui,m = (1 − αi )Π‡i,m

=

(35)

1 (1 − αi )(2(a − c) + 6θ − Eθ − 3e(m))(2(a − c) − Eθ + 3e(m)) 36 15

‡ Uj,m = (1 − αj )Π‡j,m

=

(36)

1 (1 − αj )(2(a − c) + 6θ − Eθ − 3e(m))(2(a − c) + 2Eθ ). 36

In addition, the investor observing θ receives the (ex-post) payoff of Vm‡ = αi Π‡i,m + αj Π‡j,m =

(37)

1 (2(a − c) + 6θ − Eθ − 3e(m)) 36 × (αi (2(a − c) − Eθ + 3e(m)) + αj (2(a − c) + 2Eθ ))

by sending a message m ∈ M(µ‡ ). Note that, for the investor’s strategy µ‡ to constitute an ¯ and for messages m ∈ supp(µ‡ (θ)) equilibrium, it has to be satisfied that, for each θ ∈ [0, θ] and m0 ∈ M(µ‡ ), Vm‡ and Vm‡ 0 defined as in (37), respectively, satisfy Vm‡ ≥ Vm‡ 0 . Lemma 2 For the investor’s equilibrium strategy µ‡ , E(µ‡ ) has a finite number of elements. Proof. Let the investor’s strategy µ‡ constitute an equilibrium (µ‡ , σ1‡ , σ2‡ ). Suppose that e ∈ E(µ‡ ) and e0 ∈ E(µ‡ ) satisfy e0 < e. Then, there exist messages m ∈ M(µ‡ ) and ¯ such m0 ∈ M(µ‡ ) such that e = e(m) and e0 = e(m0 ). In addition, there exists θ ∈ [0, θ] that m ∈ supp(µ‡ (θ)) and θ ≤ e(m). Then, because (µ‡ , σ1‡ , σ2‡ ) is an equilibrium, we can see from (37) that

Vm‡

−

Vm‡ 0

αi = (e(m0 ) − e(m)) 2



e(m0 ) + e(m) αj + 2 αi



a − c + Eθ 3



 −θ

≥0

(38)

should hold. Since e(m0 ) < e(m) and θ ≤ e(m), (38) implies that 2αj (a − c + Eθ ) ≤ e(m) − e(m0 ) = e − e0 . 3αi

(39)

¯ (39) implies the result. Since e and e0 are arbitrary in E(µ‡ ) and satisfy 0 ≤ e0 < e ≤ θ, 16

Lemma 2 shows that firm i’s expectations on θ that can be formed under the belief consistent with the equilibrium strategy µ‡ are finite. This implies that there is no equilibrium in which the investor plays a truth-telling strategy. Note that, if the investor ¯ and E(µT ) = [0, θ] ¯ hold. employs the truth-telling strategy µT , e(m) = m for all m ∈ [0, θ] In addition, under the belief that a truth-telling strategy is played, the investor observing θ > 0 has an incentive to send a message that is slightly lower than θ, which induces firm i to reduce its output.6 Although the investor does not completely reveal its private information in an equilibrium, it might be possible that the investor partially reveals its private information to firm i. We next focus on the equilibria having this property. Proposition 2 characterizes such equilibria. Due to Lemma 2, E(µ‡ ) for an equilibrium strategy µ‡ can be represented as E(µ‡ ) = {e1 , . . . , eK } for some K ∈ N. Proposition 2 Let (µ‡ , σi‡ , σj‡ ) be an equilibrium with E(µ‡ ) = {e1 , . . . , eK } for some K ∈ N. Without loss of generality, let e1 < · · · < eK . Then, the following holds.7 (a) There exists y = (y1 , . . . , yK−1 ) ∈ RK−1 such that + 0 ≡ y0 < y1 < . . . < yK−1 ≤ yK ≡ θ¯

(40)

and, for each k = 1, . . . , K − 1, ek + ek+1 αj + yk = 2 αi



a − c + Eθ 3

 .

(41)

¯ k = 1, . . . , K, such that θ ∈ (yk−1 , yk ) (b) There is a collection of disjoint sets Mk ⊂ [0, θ], implies supp(µ‡ (θ)) ⊂ Mk . (c) When firm i observes a message m ∈ Mk , firm i forms an expectation on θ as

ek = E[θ|yk−1 ≤ θ ≤ yk ]. 6

(42)

If αi = 1, it is possible that the investor uses a truth-telling strategy in an equilibrium although firm i’s profit is always zero. 7 If K = 1 and so E(µ) is a singleton, (a) and (b) in Proposition 2 are redundant.

17

(d) For each θ ∈ (yk−1 , yk ) and m ∈ supp(µ‡ (θ)), 1 σi‡ (m) = (a − c) − 3 1 σj‡ = (a − c) + 3

1 1 Eθ + ek 6 2 1 Eθ 3

(43) (44)

Proof. See Appendix. The proofs omitted in the text are found in the Appendix. In Proposition 2, (a) and (b) provide the characteristics of the investor’s equilibrium strategies. In any equilibrium, ¯ with threshold y = (y1 , . . . , yK−1 ) satisfying (40) the investor makes a partition of [0, θ] and (41) and determines disjoint sets Mk , k = 1, . . . K, of messages. Then, if the investor observes θ ∈ (yk−1 , yk ), it sends a (random) message m ∈ Mk .8 (c) provides a property of firm i’s belief at the reachable information sets along the equilibrium path. Note that the belief in (c) is consistent with the investor’s strategy µ‡ . (d) describes the equilibrium strategies for the firms. From (d) in Proposition 2, we see that, when θ ∈ (yk−1 , yk ) is realized, the market output at an equilibrium (µ‡ , σi‡ , σj‡ ) is determined by 1 1 2 Q‡ = σi‡ (m) + σj‡ = (a − c) + Eθ + ek , 3 6 2

(45)

where ek = E[θ|yk−1 ≤ θ ≤ yk ]. This is an ex-post market output, and it varies with equilibria and the realizations of θ. The ex-ante equilibrium market output is uniquely determined as 2 2 E[Q‡ ] = (a − c) + Eθ , 3 3

(46)

which is invariant to equilibria. Proposition 2 also implies that the equilibria with the same y = (y1 , . . . , yK−1 ) are 8

It is possible that, in an equilibrium, the investor observing θ = yk sends (random) messages m ∈ Mk ∪ Mk+1 .

18

outcome equivalent in that they yield the same outcome in the game. To see this, note that the distribution of outcomes in an equilibrium depends on the distribution of ek . From (c) in Proposition 2, we see that, given the distribution of θ, the distribution of ek depends only on y and so does the distribution of the outcomes. An equilibrium (µ‡ , σi‡ , σj‡ ) that satisfies the properties in Proposition 2 is referred to as a K-partition equilibrium. Here, K represents the number of elements in the partition of ¯ that is generated by µ‡ . With a slight abuse of notation, we denote a partition of [0, θ] ¯ [0, θ] by y = (y1 , . . . , yK−1 ) ∈ RK−1 satisfying (40) A partition y ∈ RK−1 means a partition + + ¯ such that (yk−1 , yk ) ⊂ Θk ⊂ [yk−1 , yk ] for each k = 1, . . . , K. {Θ1 , . . . , ΘK } of [0, θ] Note that a 1-partition equilibrium is an uninformative equilibrium in the sense that the investor’s message does not change the firm’s prior belief on θ. In addition, we refer to a K-partition equilibrium with K ≥ 2 as an informative equilibrium because the investor’s ¯ message provides information that θ belongs to a proper subset of [0, θ]. Note that Proposition 2 does not claim the existence of an equilibrium. Indeed, there is a 1-partition equilibrium that yields the same outcome as the equilibrium when the investor use a public channel to transmit information. For instance, a strategy such that ¯ the investor always sends to firm i a random message that is uniformly distributed on [0, θ] whatever θ is realized and each firm always chooses an output as in (20) is an equilibrium. This equilibrium is uninformative in that the investor does not reveal any of its private information. As mentioned earlier, there can be other equilibria in which the investor partially reveals its private information to firm i. The conditions for the existence of such equilibria are provided in Lemma 3 and Proposition 3. Lemma 3 If there exists y = (y1 , . . . , yK−1 ) ∈ RK−1 for K ≥ 2 satisfying (40), (41) and + (42), there exists a K-partition equilibrium (µ‡ , σi‡ , σj‡ ). ¯ such that Θ1 = [y0 , y1 ] and Θk = (yk−1 , yk ] Proof. Let {Θ1 , . . . , ΘK } be a partition of [0, θ] for each k = 2, . . . , K. Let µ‡ be the investor’s strategy such that, when it observes θ ∈ Θk ,

19

it sends a random message µ‡ (θ) that is uniformly distributed on [yk−1 , yk ]. Note that ¯ M(µ‡ ) = [0, θ]. The firm i forms a belief as follows. When it observes the investor’s message m ∈ Θk , it believes that θ is distributed with a conditional density f (θ|θ ∈ Θk ), and so it has an expectation on θ as ek = E[θ|yk−1 ≤ θ ≤ yk ]. This belief is consistent with µ‡ in every reachable information set. Let (σi‡ , σj‡ ) be the firms’ strategies as in (43) and (44). The arguments used to obtain (32) and (31) show that (σi‡ , σj‡ ) in (43) and (44) maximizes each firm i’s (resp. firm j’s) expected payoff given that the other firm j plays σj‡ (resp. the other firm i plays σi‡ ). Similar to the proof of Proposition 2, it can be shown that, for any θ ∈ Θk , for any m ∈ Θk , and for any m0 ∈ Θk0 , Vm‡ − Vm‡ 0 ≥ 0 is satisfied, where Vm‡ and Vm‡ 0 are determined as in (37) for messages m and m0 , respectively. This means that the investor does not have an incentive to deviate from µ‡ . Lemma 3 implies that the existence of a K-partition equilibrium is equivalent to the existence of y = (y1 , . . . , yK−1 ) ∈ RK−1 satisfying (40), (41), and (42). Thus, in order to + find a K-partition equilibrium, it is enough to find y = (y1 , . . . , yK−1 ) ∈ RK−1 satisfying + (40), (41), and (42). In the proof of Lemma 3, we construct an equilibrium in which the investor sends a random message m that is uniformly distributed on [yk−1 , yk ] when it observes θ ∈ (yk−1 , yk ). Of course, this is not a unique K-partition equilibrium with information partition y = (y1 , . . . , yK−1 ) ∈ RK−1 . For example, we can also construct + another K-partition equilibrium in which the investor sends a message yk when it observes θ ∈ (yk−1 , yk ). From Proposition 2, we see that all K-partition equilibria are equivalent in terms of the outcome of the game as long as the investor’s strategy generates the same information partition y = (y1 , . . . , yK−1 ) ∈ RK−1 . + We note that Lemma 3 does not rule out the existence of multiple K-partition equi. However, it can be show that, if libria with different information partitions y ∈ RK−1 + the distribution of θ satisfies the relevant condition, every K-partition equilibrium yields the unique information partition and so the same outcome. In other words, for any equi-

20

librium strategy µ‡ for which the information partition has K elements, the partition y ∈ RK−1 generated by µ‡ is uniquely determined. The condition for the uniqueness + of an equilibrium in terms of information partition is that the change in the conditional expectation of types given an interval of types is smaller than the change in the boundary of the interval.9 An example of the distributions satisfying this conditions is a uniform distribution. Proposition 3 Given (αi , αj ), there exists an integer K(αi , αj ) ∈ N such that, for each K with 1 ≤ K ≤ K(αi , αj ), there exists a K-partition equilibrium (µ‡ , σ1‡ , σ2‡ ). In addition, ¯ ¯ if (3/2)(θ/(a−c+E θ )) < (αj /αi ), then K(αi , αj ) = 1, and if (αj /αi ) < (3/2)((θ−Eθ )/(a− c + Eθ )), then K(αi , αj ) ≥ 2. Proof. See Appendix. Proposition 3 provides a condition on αi and αj for the existence of an informative K-partition equilibrium in which the investor partially reveals its private information to firm i. The condition is that the investor’s share of firm i is larger than its share of firm j. Indeed, the relative ratio of αi and αj matters for the existence of informative equilibrium. A decrease in αj /αi can be interpreted as firm i’s profit becomes more important to the investor’s portfolio, and so the investor’s interest becomes more similar to that of firm i. Thus, as αj /αi decreases, the investor may become more willing to share its information with firm i. In addition, if there is a K-partition equilibrium for K ≥ 2, it is also possible to construct a K 0 -equilibrium with K 0 < K. In general, this means that, if there is an equilibrium in which the investor partially reveals its private information, there is also an equilibrium in which the investor reveals less information. Proposition 3 also implies the existence of an uninformative equilibrium in which the investor does not reveal its private information at all regardless of the share of the firms. ¯ with x 6= x0 and y 6= y 0 , |E[θ|x ≤ θ ≤ y 0 ] − E[θ|x ≤ θ ≤ y]| < Formally, for each x, x0 , y, y 0 ∈ [0, θ] 0 |y − y| and |E[θ|x ≤ θ ≤ y] − E[θ|x ≤ θ ≤ y]| < |x0 − x|. Indeed, this condition is stronger than required. 9

0

21

The conditions in Proposition 3 imply the following proposition: given a fixed investor’s share of the firms, if the investor reveals its information to firm i in an equilibrium, it does not reveal its information to firm j in any equilibrium. Proposition 4 Let αi and αj be fixed. If there exists an informative equilibrium in which the investor sends a message only to firm i, then there does not exist an informative equilibrium in which the investor sends a message only to firm j. Proof. If there exists a K-partition equilibrium with K ≥ 2 when the investor sends a message only to firm i, Proposition 3 together with (2) implies that αi and αj satisfy 2 αj ≤ αi 3



 θ¯ 1 αi < < . a − c + Eθ 2 αj

(47)

Then, Proposition 3 implies the result. Consider a K-partition equilibrium (µ‡ , σ1‡ , σ2‡ ) with an information partition y = (y1 , . . . , yK−1 ) ∈ RK−1 . From (35), (36), and (37), we can that, when the investor observes + θ ∈ (yk−1 , yk ), firm i’s, firm j’s, and the investor’s (ex-post) payoffs are respectively 1 (1 − αi )(2(a − c) + 6θ − Eθ − 3ek )(2(a − c) − Eθ + 3ek ) 36 1 (1 − αj )(2(a − c) + 6θ − Eθ − 3ek )(2(a − c) + 2Eθ ) Uj‡ = 36 1 V‡ = (2(a − c) + 6θ − Eθ − 3ek ) 36 Ui‡ =

(48) (49) (50)

× (αi (2(a − c) − Eθ + 3ek ) + αj (2(a − c) + 2Eθ )),

where ek = E[θ|yk−1 ≤ θ ≤ yk ]. Taking expectations on (48), (49), and (50), we derive firm i’s, firm j’s, and the investor’s ex-ante payoffs as follows.10 1 1 E[Ui‡ ] = (1 − αi )(a − c + Eθ )2 + (1 − αi )(E[e2k ] − Eθ2 ) 9 4 10

In deriving (51), (52), and (53), notice that E[ek θ] = E[e2k ], where ek = E[θ|yk−1 ≤ θ ≤ yk ].

22

(51)

1 E[Uj‡ ] = (1 − αj )(a − c + Eθ )2 9 1 1 E[V ‡ ] = (αi + αj )(a − c + Eθ )2 + αi (E[e2k ] − Eθ2 ). 9 4

(52) (53)

Then, we can easily compare the ex-ante payoffs between an uninformative equilibrium and an informative equilibrium in which the investor partially reveals its private information. Proposition 5 Consider the situation in which the investor sends a message only to firm i. Let (µ‡ , σi‡ , σj‡ ) be a K-partition equilibrium for some K ≥ 2. Let Ui‡ , Uj‡ , and V ‡ be firm i’s, firm j’s, and the investor’s payoffs at (µ‡ , σi‡ , σj‡ ), respectively. In addition, let (µ0‡ , σi0‡ , σj0‡ ) be an uninformative equilibrium (that is, a 1-partition equilibrium). Let Ui0‡ , Uj0‡ , and V 0‡ be firm i’s, firm j’s, and the investor’s payoffs at (µ0‡ , σi0‡ , σj0‡ ), respectively. Then, E[Ui‡ ] > E[Ui0‡ ], E[Uj‡ ] = E[Uj0‡ ], and E[V ‡ ] > E[V 0‡ ] are satisfied. Proof. For an uninformative equilibrium (µ0‡ , σi0‡ , σj0‡ ), it is satisfied that E[e2k ] − Eθ2 = E[Eθ2 ] − Eθ2 = 0. For a K-partition equilibrium for some K ≥ 2, it is satisfied that E[e2k ] − Eθ2 = E[(ek − Eθ )2 ] > 0. Then, the results trivially follow from (51), (52), and (53). Proposition 5 means that, before the realization of θ, the investor and firm i prefer an informative equilibrium to an uninformative equilibrium. For intuition underlying this result, note that firm i’s ex-post profit at (µ‡ , σi‡ , σj‡ ) can be represented as Π‡i =

1 1 (2(a − c) + 3θ − Eθ )2 − (θ − ek )2 , 36 4

(54)

which depends on θ − ek . In addition, firm i’s ex-post profit at (µ0‡ , σj0‡ , σj0‡ ) can be represented as Π0‡ i =

1 1 (2(a − c) + 3θ − Eθ )2 − (θ − Eθ )2 , 36 4

(55)

which depends on θ − Eθ . Since Π‡i and Π0‡ i are concave in θ − ek and θ − Eθ , respectively, 23

and the uncertainty of θ −ek is smaller than that of θ −Eθ , the investor’s partial revelation of private information improves firm i’s profit and so improves firm i’s (ex-ante) payoff and the investor’s payoff. In Section 3, it is shown that, when the investor sends messages to both firms using a public channel of information transmission, the unique equilibrium (µ† , σ1† , σ2† ) is uninformative. In this equilibrium, both firms form an expectation on θ as Eθ in deciding their outputs. Thus, the outcome of the equilibrium (µ† , σ1† , σ2† ) is the same as the outcome of the uninformative equilibrium (µ0‡ , σi0‡ , σj0‡ ) in Proposition 5. This directly leads us to Corollary 1. Corollary 1 Let (µ† , σ1† , σ2† ) be an equilibrium in the situation that the investor sends a message to both firms using a public channel. Let Ui† , Uj† , and V † be the payoff of firm i, firm j, and the investor at (µ† , σ1† , σ2† ), respectively. Let (µ‡ , σi‡ , σj‡ ) be a K-partition equilibrium for some K ≥ 2 in the situation that the investor sends a message only to firm i. Let Ui‡ , Uj‡ , and V ‡ be the payoff of firm i, firm j, and the investor at (µ‡ , σi‡ , σj‡ ), respectively. Then, E[Ui‡ ] > E[Ui† ], E[Uj‡ ] = E[Uj† ], and E[V ‡ ] > E[V † ] are satisfied. Proof. Since the outcome of the equilibrium (µ† , σ1† , σ2† ) is the same as the outcome of the uninformative equilibrium (µ0‡ , σi0‡ , σj0‡ ) in Proposition 5, Proposition 5 directly implies the result. Corollary 1 implies that the investor will not be worse off using the private channel compared with the public channel of information transmission. Thus, the investor (weakly) prefers the private channel to the public channel of information transmission and will choose to use a private channel if afforded the option. In addition, firm i also prefers the private channel to the public channel of information transmission. Thus, the investor can easily obtain firm i’s agreement on sharing private information between themselves. Note that, although firm j is indifferent between the private and the public channel of information transmission, use of a private channel will improve the Pareto efficiency 24

(among the investor and the firms) in the view of ex-ante payoffs.11 We next discuss the comparative statics. In particular, we are interested in the effects of αi and αj on the investor’s equilibrium payoff. Recall that αi and αj are the investor’s share of firm i and firm j, respectively. From (41), the information partition y = (y1 , . . . , yK−1 ) ∈ RK−1 in a K-partition equilibrium with K ≥ 2 is not affected by αi + or αj as long as αj /αi is constant. This implies that E[e2k ] in (53) is constant as long as αj /αi is constant. Then, it is easy to see that the investor’s ex-ante equilibrium payoff E[V ‡ ] increases as αi and αj increase at the same rate. Intuitively, this is clear because the investor’s large share of both firms yields a large proportion of the profits and so a high payoff. A more interesting question is how the investor’s payoff is affected by the change in relative share of the firms. To answer this question, we restrict αi and αj to satisfy αi + αj = α ¯ for some α ¯ > 0. Because the firms are ex-ante identical, this restriction can be interpreted as a fixed total amount of investment in the firms. Under this assumption, increasing αi can be interpreted as the investor making a larger investment in firm i than in firm j. Because the investor’s optimal decisions are invariant with α ¯ , we set α ¯ to 1 without loss of generality. Thus, in what follows, we assume that

αi + αj = 1.

(56)

Consider a K-partition equilibrium (µ‡ , σi‡ , σj‡ ) in which the information partition is for K ≥ 2. Under the condition in (56), the investor’s ex-ante y = (y1 , . . . , yK−1 ) ∈ RK−1 + payoff at (µ‡ , σi‡ , σj‡ ) becomes 1 1 E[V ‡ ] = (a − c + Eθ )2 + αi E[(ek − Eθ )2 ]. 9 4

(57)

Note that E[(ek − Eθ )2 ] in (57) is the variance of the conditional expectation ek . 11

In Section 5, we also discuss that the use of private channel of information transmission improves the consumer surplus.

25

Proposition 6 shows how E[V ‡ ] changes as αi increases. Proposition 6 Suppose that, for K ≥ 2, there exists a K-partition equilibrium for some α ¯ i ∈ (0, 1). Then, there exists a K-partition equilibrium for any αi ∈ [¯ αi , 1). In addition, the investor’s ex-ante payoff increases as αi increases from α ¯i. Proof. See the Appendix. Proposition 6 says that, if there is an equilibrium with a number of elements in the information partition, there also exists an equilibrium with the same number of elements in the information partition when the investor increases its share of firm i. Thus, K(αi , 1− αi ), the maximum number of elements in the information partitions in equilibria, does not decrease as αi increases. Because the larger number of elements in the information partition can be interpreted as the investor revealing more of its private information, this observation coincides with the intuition: when the investor has a larger share of firm i, the optimal choices between the investor and firm i become more similar and the investor’s incentive to conceal its private information from firm i is reduced. Proposition 6 also says that, given a number of elements in an information partition, the investor’s ex-ante payoff increases as it invests more in firm i. Indeed, the investor’s payoff depends on the variance of the conditional expectation on the market conditions multiplied by the investor’s share of firm i. Note that an increase in the investor’s share of firm i reduces the difference in the preferences between the investor and firm i. Thus, the investor with a greater share of firm i is willing to reveal its private information more accurately to firm i. This reduces the variance of the conditional expectation on the market conditions and increases the ex-ante payoff to firm i and the investor.12 12

One might be interested in the effect of αi on firm i’s and firm j’s ex-ante payoffs. From (51) and (52), we can easily see that firm j’s ex-ante payoff increases as αi increases, but it is not clear whether an increase in αi improves firm i’s ex-ante payoff.

26

5

Discussions

In the previous sections, we focus on the investor’s incentive to share its private information with firms and the effect of the investor’s information sharing on the payoffs. One might be interested in how the consumer surplus and social welfare are affected by the investor’s information sharing with firms. Given a market output Q in an equilibrium, the price p is determined by the market demand in (1) and so the consumer surplus can be defined by13 Z CS = 0

Q

1 (a + θ − Q − p)dQ = Q2 . 2

(58)

From Propositions 1 and 2, given a realization of θ ∈ (yk−1 , yk ), the market output Q† at the uninformative equilibrium (µ† , σ1† , σ2† ) under a public channel and the market output Q‡ at an informative equilibrium (µ‡ , σ1‡ , σ2‡ ) under a private channel are, respectively, 2 Q† = (a − c) + 3 2 Q‡ = (a − c) + 3

2 Eθ 3 1 1 Eθ + ek , 6 2

(59) (60)

where ek = E[θ|yk−1 ≤ θ ≤ yk ]. Thus, the ex-ante consumer surplus at each equilibrium is respectively determined by  1 † 2 E[CS ] = E (Q ) = 2   1 ‡ 2 ‡ E[CS ] = E (Q ) = 2 †



2 (a − c − Eθ )2 9 1 2 (a − c − Eθ )2 + E[(ek − Eθ )2 ]. 9 8

(61) (62)

Since E[CS † ] < E[CS ‡ ], the informative equilibrium is beneficial to the demand side (consumers) as well as the supply side (the firms and the investor). In addition, the 13

Indeed, the market demand in (1) can be derived from a utility maximization problem of the representative consumer whose utility function is W (Q, C) = (a+θ)Q−(1/2)Q2 +C and budget constraint is pQ + C = M , where C is the consumption on other goods and M is the consumer’s income. Then, it is straightforward that the consumer surplus in (58) is the consumer’s additional benefit from consuming the goods produced by the firms. That is, CS = W (Q∗ , C ∗ ) − W (0, M ) where (Q∗ , C ∗ ) is the solution to the consumer’s utility maximization problem given that p = a + θ − Q∗ .

27

investor’s sharing information with a firm improves the social welfare that consists of surplus on the demand and the supply sides. In the paper, we consider two ways for the investor to transmit information. The investor using a public channel sends the same message to both firms. The investor using a private channel sends a message only to one firm (firm i). However, it is also ¯ to firm 1 and another message possible that the investor sends a message m1 ∈ [0, θ] ¯ to firm 2. In this case, the investor’s strategy can be represented as a function m2 ∈ [0, θ] ¯ → ∆([0, θ] ¯ 2 ). Here, µ(θ) = (µ1 (θ), µ2 (θ)) for each θ, and µi (θ) can be interpreted µ : [0, θ] as a (random) message that the investor observing θ sends to firm i. After the investor sends messages to the firms, each firm simultaneously decides its output. Of course, there is also an equilibrium in this situation, but we have some difficulties in characterizing the equilibrium. For example, let (µo , σ1o , σ2o ) be an equilibrium and consider the situation that each firm receives a message from the investor and has to choose the output. Suppose that ¯ for each firm. the investor’s strategy µo = (µo1 , µo2 ) generates different partitions on [0, θ] ¯ generated by µo . For each θ, let Θ ¯ i (θ) be an ¯ i be the information partition on [0, θ] Let Θ i ¯ 1 and Θ ¯ 2 is the partition containing element of Θi to which θ belongs. If the meet of Θ ¯ the firms fail to form a common belief on θ except that θ belongs to [0, θ]. ¯ 14 For only [0, θ], ¯ i . With a message mi ∈ supp(µi (θ)), firm i forms a belief that θ belongs to some Θik ∈ Θ ¯ i to slight abuse of notation, for message mi ∈ supp(µi (θ)), let Θi (mi ) be an element of Θ which θ belongs under the belief of firm i receiving mi from the investor. We can find an equilibrium strategy (σ1o , σ2o ) for the firms as follows. Given a message mi ∈ supp(µi (θ)), for each t, let Kit (mi ) be firm i’s t-order conditional expectation on θ, which is defined sequentially as

Ki1 (mi ) = E[θ|Θi (mi )], 14

¯ 2. Θ

(63)

¯ the meet of Θ ¯ 1 and Θ ¯ 2 of [0, θ], ¯ 1 and Θ ¯ 2 is the finest common coarsening of Θ ¯ 1 and For partitions Θ

28

¯ j ]|Θi (mi )], Ki2 (mi ) = E[E[θ|Θ ¯ i ]|Θ ¯ j ]|Θi (mi )], Ki3 (mi ) = E[E[E[θ|Θ ¯ j ]|Θ ¯ i ]|Θ ¯ j ]|Θi (mi )], · · · . Ki4 (mi ) = E[E[E[E[θ|Θ Then, firm i’s equilibrium strategy σio has to satisfy that, for each m ∈ supp(µoi (θ)), σio (mi )

t−1 ∞  1 1X 1 − = (a − c) + Kit (mi ). 3 2 t=1 2

(64)

¯ it is possible that ¯ 1 and Θ ¯ 2 is the partition containing only [0, θ], Note that, if the meet of Θ Kit (mi ), t = 1, 2, . . ., have different values. In addition, the investor’s payoff by sending a message mi ∈ supp(µoi (θ)) also depends on Kit (mi ), t = 1, 2, . . .. To find the condition for the investor not to deviate from µoi , we may have to explicitly figure out how Kit (mi ) depends on messages mi . But this seems not an easy task. Although we cannot fully characterize the equilibria, we can find some properties of equilibria for the situation that the investor sends separate messages to each firm. First, if the investor’s strategy µo = (µo1 , µo2 ) generates the same information partition y = (y1 , . . . , yK−1 ) with K ≥ 2 for both firms, (µo , σ1o , σ2o ) cannot be an equilibrium. In other words, there is no equilibrium (µo , σ1o , σ2o ) in which the investor’s strategy is informative and generates the same information partition for the firms. If σio and σjo ¯ the firms with a belief consistent with generate the same information partition on [0, θ], µo have a common belief on the distribution of θ. Then, arguments similar to those proving Lemma 2 can be applied to verify the assertion. Second, there is an equilibrium (µo , σ1o , σ2o ) in which, for some i, µoi is informative and µoj is uninformative. For µoi to be informative, the condition (αj /αi ) < (3/2)((θ¯ − Eθ )/(a − c + Eθ )) in Proposition 3 should hold. Then, one can show that the investor’s strategy µo = (µoi , µoj ) such that µoi ¯ is the same as µ‡ in Proposition 2 and, for each θ, µoj (θ) is uniformly distributed on [0, θ] constitutes an equilibrium. Here, firm i’s strategy σio is the same as σi‡ in (43), and firm

29

j’s strategy σjo is σjo (mj ) = (1/3)(a − c) + (1/3)Eθ for any mj . Arguments similar to those used in Section 4 can be applied to prove this assertion. We also note that there is always an uninformative equilibrium (µo , σ1o , σ2o ) in which the investor always send random ¯ messages µoi (θ) and µoj (θ) that are independently and uniformly distributed on [0, θ]. The results in this paper can be extended to the situation with more than two firms. Suppose that there are n firms in the market and the market demand is given by p = P a + θ − nk=1 qk , where θ is private information to the investor. We also assume that a − c is large enough compared with the uncertainty in the market condition. Each firm i’s payoff and the investor’s payoff are defined similarly to (4) and (5). The investor sends a common message to a subset of the firms, which can be the set containing all firms or a set containing only one firm. It is assumed that the firms know who receives the message from the investor. In this environment, we can also characterize the equilibrium. Indeed, similar arguments in Section 3 show that, when the investor sends a message to more than one firm, there is an equilibrium in which the investor sends an uninformative message, and the equilibrium is unique in terms of the outcome of the game. In addition, when the investor sends a message only to one firm i and the investor’s share of firm i is large enough compared with its shares of the other firms, there exist equilibria in which the investor sends an informative message. In these equilibria, the investor’s strategy ¯ that has a finite number of elements. The properties of generates a partition on [0, θ] these equilibria are similar to those of the equilibria in Section 4.

6

Conclusion

This paper studies the incentives of an investor to disclosure its private information on market conditions to the firms in an oligopoly. Because there is a discrepancy in the optimum level of outputs between the investor and the firms, the investor has an incentive to use its private information strategically. In particular, we consider two cases of investor’s information transmission to the firms. In the first case, the investor publicly provides its 30

information to the firms. In this case, the investor does not reveal any of its information to the firms in any equilibrium. This result comes from that the discrepancy in the optimum level of outputs between the investor and the firms is so large that the investor cannot increase its payoff by manipulating the transmission of its private information. In the second case, the investor privately transmits its information to a specific firm. In this case, there can be an equilibrium in which the investor partially reveals its information to the firm. Indeed, such an equilibrium exists when the investor owns a relatively larger shares of a firm than that of the other firm. In addition, the payoffs of the investor and the firm receiving information from the investor are improved in an equilibrium with the investor’s partial revelation of information compared to an equilibrium in which the investor does not reveal any of its information. In this paper, we consider the model in which the investor’s shares of the firms are exogenously given. One might be interested in the situation in which the investor and the firms decide the investor’s share of the firms through a bargaining procedure. As discussed in Section 4, when the investor transmits its information privately to firm i, an increase in the investor’s share of firm i improves the investor’s (ex-ante) payoff given that the total amount of investment is fixed. But, an increase in the investor’s share of firm i can decrease firm i’s payoff though firm i can take advantage of the investor’s private information. Thus, some bargaining procedure can play a role in determining the investor’s share of firm i. Our model also assumes that the investor transmits its information either publicly or privately. However, it sounds more realistic that the investor provides a different quality of information to the firms using both public and private channels. For example, the investor makes a public report on market conditions and provides private consultancy services to its invested firms. Study of the behavior of the investor and the firms in such environments are left for future research.

31

7

Appendix

Proof of Proposition 2. For each k = 1, . . . K, let

Mk = {m ∈ M(µ‡ ) : E[θ|µ‡ (θ) = m] = ek }

(65)

be a set of messages that induce expectation ek under µ‡ , and let Θk = {θ : supp(µ‡ (θ)) has an element m ∈ Mk }

(66)

be a set of the investor’s observations θ that can induce expectation ek under µ‡ . The S ¯ collection of sets Mk , k = 1, . . . , K, is a partition of M(µ‡ ). In addition, K k=1 Θk = [0, θ] holds. Consider messages m ∈ Mk , and m0 ∈ Mk0 for some k and k 0 with k 6= k 0 . Suppose that θ ∈ Θk ∩ Θk0 . Since (µ‡ , σi‡ , σj‡ ) is an equilibrium, the investor observing θ always sends messages that yield the same expected payoff under µ‡ . This means that, for θ ∈ Θk ∩Θk0 ,

Vm‡

−

Vm‡ 0

αi = (ek0 − ek ) 2



ek0 + ek αj + 2 αi



a − c + Eθ 3



 −θ

=0

(67)

holds. Thus, we have ek0 + ek αj θ= + 2 αi



a − c + Eθ 3

 .

(68)

This implies that Θk ∩ Θk0 has at most one element. For the investor not to deviate from µ‡ , it has to be satisfied that, for any θ ∈ Θk and for any m ∈ Mk and m0 ∈ Mk0 with k 6= k 0 , Vm‡

−

Vm‡ 0

αi = (ek0 − ek ) 2



ek0 + ek αj + 2 αi

32



a − c + Eθ 3



 −θ

≥ 0.

(69)

Since e1 < · · · < eK holds, (69) implies that, for each k = 1, . . . K − 1, ek−1 + ek αj + 2 αi



a − c + Eθ 3



ek + ek+1 αj ≤ inf Θk ≤ sup Θk ≤ + 2 αi



a − c + Eθ 3

 . (70)

This is equivalent to ek + ek+1 αj sup Θk ≤ + 2 αi Because

SK

k=1



a − c + Eθ 3

 ≤ inf Θk+1 .

(71)

¯ holds and Θk ∩ Θk+1 has at most one element, (71) implies that, Θk = [0, θ]

for each k = 1, . . . , K − 1, there exists yk such that

yk = sup Θk = inf Θk+1

ek + ek+1 αj = + 2 αi



a − c + Eθ 3

 .

(72)

Since e1 < · · · < eK , (72) implies that yk−1 < yk for each k = 1, . . . , K − 1. This proves (a). Notice that (yk−1 , yk ) ⊂ Θk ⊂ [yk−1 , yk ]. Consider θ ∈ (yk−1 , yk ) and message m ∈ supp(µ‡ (θ)). Suppose that m ∈ / Mk . Since the collection of Mk forms a partition of M(µ‡ ), m ∈ Mk0 for some k 0 6= k should hold. This implies θ ∈ Θk0 , which is a contradiction. Thus, m ∈ Mk should be satisfied and (b) is proved. For (c), note that {θ : supp(µ‡ (θ)) ⊂ Mk } = {θ : yk−1 ≤ θ ≤ yk } holds almost surely. This implies ek = E[θ|supp(µ‡ (θ)) ⊂ Mk ] = E[θ|yk−1 ≤ θ ≤ yk ].

(73)

Finally, (d) follows from the same argument used to obtain (31) and (32). as Proof of Proposition 3. For each K ≥ 2, define a set YK−1 ⊂ RK−1 +  YK−1 = y = (y1 , . . . , yK−1 ) : 0 ≡ y0 ≤ y1 ≤ · · · ≤ yK−1 ≤ yK ≡ θ¯ .

(74)

Note that YK−1 is compact and convex. For each K ≥ 2, define a function HK−1 : YK−1 → 33

RK−1 by, for each k = 1, . . . , K − 1, ek + ek+1 αj HK−1,k (y) = + 2 αi



a − c + Eθ 3

 (75)

where ek = E[θ|yk−1 ≤ θ ≤ yk ]. If HK−1 has a fixed point in YK−1 for some K ≥ 2, let K(αi , αj ) = max{K : HK−1 has a fixed point in YK−1 }. Otherwise, let K(αi , αj ) = 1. If y = (y1 , . . . , yK−1 ) ∈ YK−1 is a fixed point of HK−1 , it should satisfy that, for each k, αj yk−1 + αi



a − c + Eθ 3



ek + ek+1 αj ≤ + 2 αi



a − c + Eθ 3

 = yk .

(76)

This implies that αi K≤ αj



3 a − c + Eθ



¯ θ.

(77)

Thus K (αi , αj ) exists. The existence of a K-partition equilibrium (µ‡ , σ1‡ , σ2‡ ) follows from Lemma 3. ∗ Let K 0 = K −1 ≥ 2. Suppose that HK−1 has a fixed point y∗ = (y1∗ , . . . , yK−1 ) ∈ YK−1 . 0

−1 Define a set Y K 0 −1 ⊂ RK as +

Y K 0 −1 =

  

y = (y1 , . . . , yK 0 −1 ) :

 

  

¯ and 0 ≡ y0 ≤ y1 ≤ · · · ≤ yK 0 ≡ θ, yk∗

≤ yk ≤

∗ yk+1

.

(78)

0

 for each k = 1, . . . , K − 1 

Y K 0 −1 is compact and convex. In addition, y = (y1 , . . . , yK 0 −1 ) ∈ Y K 0 −1 implies that, for each k = 1, . . . , K 0 − 1,

yk∗

  ∗ ∗ E[θ|yk−1 ≤ θi ≤ yk∗ ] + E[θ|yk∗ ≤ θi ≤ yk+1 ] αj a − c + E θ = + 2 αi 3   E[θ|yk−1 ≤ θi ≤ yk ] + E[θ|yk ≤ θ ≤ yk+1 ] αj a − c + Eθ ≤ + 2 αi 3   ∗ ∗ ∗ ∗ E[θ|yk ≤ θ ≤ yk+1 ] + E[θ|yk+1 ≤ θ ≤ yk+2 ] αj a − c + Eθ ∗ ≤ + = yk+1 . 2 αi 3

(79)

This means that, for each y ∈ Y K 0 −1 , HK 0 −1 (y) ∈ Y K 0 −1 holds. Because HK 0 −1 is 34

continuous on Y K 0 −1 , HK 0 −1 has a fixed point y∗∗ ∈ Y K 0 −1 . Then, Lemma 3 establishes the existence of a K-partition equilibrium (µ‡ , σ1‡ , σ2‡ ) for K < K(αi , αj ). ¯ − c + Eθ )) < (αj /αi ), K < 2 holds. Thus we have Notice from (77) that, if (3/2)(θ/(a ¯ → R+ as K(αi , αj ) = 1. To find a condition for K(αi , αj ) ≥ 2, define a function Φ : [0, θ]   ¯ αj a − c + Eθ E[θ|0 ≤ θ ≤ y1 ] + E[θ|y1 ≤ θ ≤ θ] + − y. Φ(y1 ) = 2 αi 3

(80)

Notice that (αj /αi ) < (3/2)((θ¯ − Eθ )/(a − c + Eθ )) implies ¯ − Eθ αj  a − c + Eθ  θ ¯ =− Φ(θ) + < 0. 2 αi 3

(81)

Since Φ is continuous and satisfies Eθ αj Φ(0) = + 2 αi



a − c + Eθ 3

 > 0,

(82)

¯ such that Φ(y1 ) = 0. Then, Lemma 3 implies the existence of a there exists y1 ∈ [0, θ] 2-partition equilibrium (µ‡ , σ1‡ , σ2‡ ). To prove Proposition 6, Lemma 4 is useful. Lemma 4 Suppose that, for αi0 ∈ (0, 1) there exists a K-partition equilibrium (µ0‡ , σ10‡ , σ20‡ ) 0 ) ∈ RK−1 . Then, for any αi00 ∈ (0, 1) with with an information partition y0 = (y10 , . . . , yK−1 +

αi00 > αi0 , there exists a K-partition equilibrium (µ00‡ , σ100‡ , σ200‡ ) with an information partition 00 y00 = (y100 , . . . , yK−1 ) ∈ RK−1 satisfying yk00 ≤ yk0 for each k = 1, . . . , K − 1. +

Proof. Let αi0 ∈ (0, 1) and αi00 ∈ (0, 1) satisfy αi0 < αi00 . For αi0 ∈ (0, 1), let (µ0‡ , σ10‡ , σ20‡ ) be 0 an equilibrium in which µ‡ generates an information partition y0 = (y10 , . . . , yK−1 ) ∈ RK−1 +

¯ Define a set YeK−1 ⊂ RK−1 on [0, θ]. as +

YeK−1

  

  ¯ 0 ≡ y0 ≤ y1 ≤ · · · ≤ yK ≡ θ and  = y = (y1 , . . . , yK−1 ) : .    yk ≤ yk0 for each k = 1, . . . , K − 1  35

(83)

00 YeK−1 is compact and convex. Define a function HK−1 : YeK−1 → RKi −1 as, for each

k = 1, . . . , Ki − 1, 00 HK−1,k (y)

ek + ek+1 1 − αi00 + = 2 αi00



a − c + Eθ 3

 ,

(84)

where ek = E[θ|yk−1 ≤ θ ≤ yk ]. Then, for each y ∈ YeK−1 , it is satisfied that, for each k = 1, . . . , K − 1,

00 HK−1,k (y)

  ek + ek+1 1 − αi00 a − c + Eθ = + 2 αi00 3   0 ek + ek+1 1 − αi a − c + Eθ ≤ + = yk0 , 2 αi0 3

(85)

0 where ek = E[θ|yk−1 ≤ θ ≤ yk ] and e0k = E[θ|yk−1 ≤ θ ≤ yk0 ]. This means that, for each 00 0 00 y ∈ YeK−1 , HK−1 (y) ∈ YeK−1 holds. Since HK−1 is continuous on YeK−1 , HK−1 has a fixed

point y00 ∈ YeK−1 . Then, Lemma 3 implies the result. Proof of Proposition 6. The existence of a K-partition equilibrium follows from Lemma 4. For each αi ∈ [¯ αi , 1), let y(αi ) = (y1 (αi ), . . . , yK−1 (αi )) ∈ RK−1 be an information + partition generated by K-partition equilibrium. For each k = 1, . . . , K, let ek (αi ) = ¯ Let V ‡ (αi ) be the investor’s E[θ|yk−1 (αi ) ≤ θ ≤ yk (αi )], where y0 (αi ) = 0 and yK (αi ) = θ. (ex-post) payoff at the K-partition equilibrium for αi . Notice that 1 E[V ‡ (αi )] = (a − c + Eθ )2 + 9 1 = (a − c + Eθ )2 + 9

1 αi E[(ek (αi ) − Eθ )2 ] 4 1 αi (E[ek (αi )2 ] − Eθ2 ), 4

(86)

and so dE[V ‡ (αi )] 1 = dαi 4

  d(E[ek (αi )2 ]) 2 E[(ek (αi ) − Eθ ) ] + αi . dαi

(87)

Thus, if we show that dE[ek (αi )2 ]/dαi > 0, we have dE[V ‡ (αi )]/dαi > 0 and complete the proof.

36

Note that15 dE [ek (αi )2 ] d = dαi dαi =

K X k=1

−

K Z X k=1

=

k=1

−

=

K X

(88)


dyk (αi ) ek (αi )2 f (yk (αi )) − 2ek (αi )yk (αi )f (yk (αi )) dαi


dyk (αi ) ek+1 (αi )2 f (yk (αi )) − 2ek+1 (αi )yk (αi )f (yk (αi )) dαi K−1 X

k=1  K−1 X k=1

ek (αi )2 f (θ)dθ

yk−1 (α)


dyk−1 (αi ) yk (αi )2 f (yk−1 (αi )) − 2ek (αi )yk−1 (αi )f (yk−1 (αi )) dαi

k=1 K−1 X

!

yk (α)


dyk (αi ) ek (αi )2 f (yk (αi )) − 2ek (αi )yk (αi )f (yk (αi )) dαi

dyk (αi ) (ek+1 (αi ) − ek (αi ))((ek+1 (αi ) + ek (αi )) − 2yk (αi ))f (yk (αi )) dαi

 .

The third equality in (88) comes from (dy0 (αi )/dαi )(αi ) = 0 and (dyK (αi )/dαi )(αi ) = 0. For each k = 1, . . . , K − 1, ek+1 (αi ) − ek (αi ) > 0 and (ek+1 (αi ) + ek (αi )) − 2yk (αi ) < 0 hold. In addition, Lemma 4 implies that, for each k = 1, . . . K − 1, (dyk /dαi )(αi ) < 0 holds for each αi ∈ (¯ αi , 1). Therefore, dE[ek (αi )2 ]/dαi > 0 is satisfied. This completes the proof.

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37

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