Strategic Merger Approvals Under Incomplete Information

Abstract

We examine a signaling game where the merging entity privately observes the cost-reduction effect from the merger, but the competition authority does not. The latter, however, observes the firm’s submission costs in the merger request, using them to infer its type. We identify pooling equilibria where all firm types, even those with small efficiencies, submit a merger request, which is approved by the regulator. This merger profile cannot be supported under complete information, thus leading to inefficiencies. We investigate under which parameter conditions inefficient mergers are less likely to arise in equilibrium, and which policies hinder them, ultimately improving information transmission from firms to the competition authority.

Appendix

1.1 Notation table

$\theta$	Cost-reduction effect of the merger
$\overline{\theta }$	Cutoff for a merger to increase consumer surplus
$\overline{\theta }_{W}$	Cutoff for a merger to be welfare enhancing
$\widehat{\theta }$	Cutoff for a merger to be profit enhancing
f	Administrative fee
$R_{H}^{SE}$	Investment of the high-type merging entity in a separating equilibrium
$R_{L}^{SE}$	Investment of the low-type merging entity in a separating equilibrium
$R^{PE}$	Investment of every type of merging entity in a pooling equilibrium
$\widehat{\theta }(R^{PE})$	Cutoff for a merger to be sustained in the pooling equilibrium
$\widetilde{R}$	Fees below which mergers are profitable regardless of $\theta$
$\widehat{R}$	Fees below which mergers satisfy $\overline{\theta }>\widehat{\theta }$

1.2 Proof of Lemma 1

In a case of no mergers, every firm solves,

$$\begin{aligned} \max _{q_{i}\ge 0}\ (1-q_{i}-q_{j})q_{i}-cq_{i} \end{aligned}$$

Differentiating with respect to $q_{i}$, we find firm i’s best response function $q(q_{j})=\frac{1-c}{2}-\frac{1}{2}q_{j}$ In a symmetric equilibrium, $q_{i}=q_{j}=q$. Therefore, the equilibrium output in this setting is

$$\begin{aligned} q_{i}^{NM}=\frac{1-c}{3} \end{aligned}$$

and equilibrium profits become $\pi _{i}^{NM}=\left( \frac{1-c}{3}\right) ^{2}=\left( q_{i}^{NM}\right) ^{2}$.

Now consider a merger is approved, then, firms solve

$$\begin{aligned} \max _{q\ge 0}\ (1-q)q-(c-x)q \end{aligned}$$

Differentiating with respect to q, yields $q^{M}=\frac{1-\left( c-x\right) }{2}$, entailing merger profits

$$\begin{aligned} \pi ^{M}=\left( \frac{1-\left( c-x\right) }{2}\right) ^{2}=\left( q^{M}\right) ^{2}. \end{aligned}$$

1.3 Proof of Lemma 2

In this setting, an increase in consumer surplus is equivalent to an increase in output. In particular, $q^{M}\ge 2q_{i}^{NM}$ holds if and only if

$$\begin{aligned} \frac{1-c+x}{2}\ge 2\frac{1-c}{3}. \end{aligned}$$

Rearranging, and solving for x yields $x\ge \frac{1-c}{3}$ or, alternatively, $\theta \equiv \frac{x}{1-c}\ge \frac{1}{3}\equiv \overline{ \theta }$. If the CA considers, instead, total welfare, it approves the merger if and only if

$$\begin{aligned}{} & {} \frac{1}{2}\left( \frac{1-c+x}{2}\right) ^{2}+\left( \frac{1-c+x}{2} \right) ^{2} \\{} & {} \quad \ge \frac{1}{2}\left( \frac{2\left( 1-c\right) }{3}\right) ^{2}+2\left( \frac{1-c+x}{3}\right) ^{2} \end{aligned}$$

where the left (right) side represents the sum of consumer and producer surplus when the merger is approved (blocked, respectively). Rearranging, and solving for x yields $x\ge \frac{1-c}{3}$ or, alternatively,

$$\begin{aligned} \theta \equiv \frac{x}{1-c}\ge \frac{4\sqrt{6}-9}{9}\simeq 0.089\equiv \overline{\theta }_{W} \end{aligned}$$

1.4 Proof of Proposition 1

A merger is profitable if $\pi ^{M}-R\ge 2\pi _{i}^{NM}$, which holds if

$$\begin{aligned} \left( \frac{1-\left( c-x\right) }{2}\right) ^{2}-R\ge 2\left( \frac{1-c}{3} \right) ^{2} \end{aligned}$$

and rearranging, we find

$$\begin{aligned} \theta \equiv \frac{x}{1-c}\ge \frac{2}{1-c}\sqrt{2\left( \frac{1-c}{3} \right) ^{2}+R}-1\equiv \widehat{\theta } \end{aligned}$$

In the absence of submission costs, $R=0$, this cutoff simplifies to $\theta \ge \frac{2\sqrt{2}}{3}-1\simeq -0.06$, which is always satisfied. In addition, $\widehat{\theta }>\overline{\theta }$ holds if and only if

$$\begin{aligned} \frac{2}{1-c}\sqrt{2\left( \frac{1-c}{3}\right) ^{2}+R}-1>\frac{1}{3} \end{aligned}$$

or, rearranging, and solving for R, $R>2\left( \frac{1-c}{3}\right) ^{2}\equiv \widehat{R}$. Therefore, $\widehat{\theta }>\overline{\theta }$ holds when $R>\widehat{R}$; otherwise, $\widehat{\theta }\le \overline{ \theta }$ holds. Finally, it is easy to check that cutoff $\widehat{\theta }$ increases in R, as it enters positively, and in c since

$$\begin{aligned} \frac{\partial \widehat{\theta }}{\partial c}=\frac{6R}{ (1-c)^{2}\sqrt{2\left( 1-c\right) ^{2}+9R}}>0. \end{aligned}$$

1.5 Proof of Proposition 2

SE where only the high-type submits a merger request. Updated beliefs. In this separating strategy profile, the CA updates its beliefs according to Bayes’ rule, obtaining $\mu \left( \theta _{H}|M\right) =1$ and $\mu \left( \theta _{L}|M\right) =0$.

Receiver’s response. Given the above beliefs, the CA is convinced of facing a high-type firm, and responds approving a merger upon observing one since $\theta _{H}>\overline{\theta }$ by definition.

Sender’s messages. Anticipating these beliefs, the $\theta _{H}$ -type entity submits a merger approval, as prescribed in this separating strategy profile, if and only if $\pi ^{M,H}-f\ge 2\pi _{i}^{NM}$, since it anticipates that the request will be approved by the CA. As shown in Proposition 1, this inequality entails that

$$\begin{aligned} \theta _{H}\equiv \frac{x_{H}}{1-c}\ge \frac{2}{1-c}\sqrt{2\left( \frac{1-c }{3}\right) ^{2}+f}-1\equiv \widehat{\theta }. \end{aligned}$$

This condition, however, holds by definition since $\theta _{H}>\overline{ \theta }$ and $\overline{\theta }>\widehat{\theta }$, implying that $\theta _{H}>\widehat{\theta }$ as well. In contrast, the $\theta _{L}$-type entity does not submit a merger request, as required in this separating strategy profile, if and only if $\pi ^{M,L}-f\le 2\pi _{i}^{NM}$. Rearranging this inequality, yields $\theta _{L}\le \widehat{\theta }$.

PE where both firm types submit a merger request. Updated beliefs. In this pooling strategy profile, the CA cannot update its beliefs according to Bayes’ rule, keeping its prior probabilities unaffected, $\mu \left( \theta _{H}|M\right) =p$ and $\mu \left( \theta _{L}|M\right) =1-p$.

Receiver’s response. Given the above beliefs, the CA responds approving a merger upon observing a request if and only if

$$\begin{aligned} p\frac{1-c+x_{H}}{2}+(1-p)\frac{1-c+x_{L}}{2}\ge 2\frac{1-c}{3}. \end{aligned}$$

Rearranging, yields $px_{H}+(1-p)x_{L}\ge \frac{1-c}{3}$ or, alternatively,

$$\begin{aligned} \frac{px_{H}+(1-p)x_{L}}{1-c}\ge \frac{1}{3}\equiv \overline{\theta } \end{aligned}$$

which we can also express as

$$\begin{aligned} E[\theta ]\equiv p\frac{x_{H}}{1-c}+(1-p)\frac{x_{L}}{1-c}=p\theta _{H}+(1-p)\theta _{L}\ge \overline{\theta }. \end{aligned}$$

or, after solving for p, we obtain $p>\frac{\overline{\theta }-\theta _{L} }{\theta _{H}-\theta _{L}}\equiv \widehat{p}$, which holds by assumption.

Sender’s messages. The $\theta _{H}$-type entity submits a merger approval, as prescribed in this pooling strategy profile, if and only if $\pi ^{M,H}-f\ge 2\pi _{i}^{NM}$, since it anticipates that the request will be approved by the CA. This inequality entails $\theta _{H}\ge \widehat{ \theta }$. Similarly, the $\theta _{L}$-type entity submits a merger approval, as required in this pooling strategy profile, if and only if $\pi ^{M,L}-R\ge 2\pi _{i}^{NM}$, which yields $\theta _{L}\ge \widehat{\theta }$.

PE where both firm types do not submit a merger request. Updated beliefs. In this pooling strategy profile, the CA’s information set is not reached since no merger approval request is submitted. Then, the CA holds off-the-equilibrium beliefs $\mu \left( \theta _{H}|M\right) =\mu$ and $\mu \left( \theta _{L}|M\right) =1-\mu$.

Receiver’s response. Given the above beliefs, the CA responds approving a merger upon observing one (which can only happen off-the-equilibrium path) if and only if $\mu \theta _{H}+(1-\mu )\theta _{L}>\overline{\theta }$ or, alternatively, $\mu >\frac{\overline{\theta } -\theta _{L}}{\theta _{H}-\theta _{L}}\equiv \widehat{p}$. We next separately analyze the case in which $\mu >\widehat{p}$ (and the CA responds approving merger, if one is submitted off the equilibrium) and that in which $\mu \le \widehat{p}$ (and the CA blocks the merger).

Sender’s messages. If $\mu >\widehat{p}$, the $\theta _{H}$-type entity, does not submit a merger approval, as prescribed in this pooling strategy profile, if and only if $\pi ^{M,H}-f<2\pi _{i}^{NM}$, since it anticipates that a deviation towards sending a request will be approved by the CA (given its off-the-equilibrium beliefs). This inequality entails $\theta _{H}<\widehat{\theta }$. Similarly, the $\theta _{L}$-type entity, does not submit a merger approval, as required in this pooling strategy profile, if and only if $\pi ^{M,L}-f<2\pi _{i}^{NM}$. This inequality yields $\theta _{L}<\widehat{\theta }$. However, condition $\theta _{i}< \widehat{\theta }$ for all types $i=\{H,L\}$ is not compatible with the initial condition that $\theta _{H}>\overline{\theta }>\theta _{L}$, implying that this pooling strategy profile cannot be sustained as a PE.

If, instead, $\mu \le \widehat{p}$ holds, both entity types anticipate that the CA will respond blocking merger approval requests (off-the-equilibrium path). In this context, the $\theta _{H}$-type entity does not submit a request, as prescribed in this pooling strategy profile, if and only if $\pi ^{NM}-f<2\pi _{i}^{NM}$, which simplifies to $-f<\pi _{i}^{NM}$, which holds by definition. A similar argument applies to the $\theta _{L}$-type entity. As in the previous case, since condition $\theta _{i}<\widehat{\theta }$ for all $i=\{H,L\}$ is not compatible with the initial assumption $\theta _{H}> \overline{\theta }>\theta _{L}$, this pooling strategy profile cannot be sustained as a PE.

In summary, the pooling strategy profile where no merging entity types submits a merger request cannot be sustained as a PE.

Applying the Cho and Kreps’ Intuitive Criterion. We need to consider only the PE where both firm types submit a merger request. In this case, if any firm i deviates towards not submitting a merger request (off-the-equilibrium path), the CA is not called on to move, which implies that this player does not hold off-the-equilibrium beliefs. As a consequence, we cannot further restrict the set of types that could have sent such an off-the-equilibrium message, which ultimately implies that we cannot restrict the CA’s beliefs either. As a consequence, this pooling PE survives the Intuitive Criterion.

1.6 Proof of Corollary 1

Cost-reduction effects $\theta _{H}$ and $\theta _{L}$, satisfy $1>\theta _{H}>\overline{\theta }>\theta _{L}>0$. If cutoffs $\widehat{\theta }$ and $\overline{\theta }$ satisfy $\widehat{\theta }\le \overline{\theta }$, then, we obtain that $1>\theta _{H}>\overline{\theta }$, while $\theta _{L}$ satisfies $\overline{\theta }>\theta _{L}>\widehat{\theta }$, which entails that the PE of Proposition 3 can be supported in the region $(1-\overline{ \theta })(\overline{\theta }-\widehat{\theta })$.

If, instead, cutoffs $\widehat{\theta }$ and $\overline{\theta }$ satisfy $\widehat{\theta }>\overline{\theta }$, then, the PE of Proposition 3 cannot be sustained.

1.7 Proof of Corollary 2

The region where a PE can be sustained is given by $PE=\left( 1-\overline{ \theta }\right) (\overline{\theta }-\widehat{\theta })$. Differentiating with respect to $\widehat{\theta }$, yields $\frac{\partial PE}{\partial \widehat{\theta }}=-\left( 1-\widehat{\theta }\right)$, which is negative since cutoff $\widehat{\theta }$ satisfies $\widehat{\theta }<1$ by definition. Therefore, PE unambiguously shrinks as cutoff $\widehat{\theta }$ increases.

The region where the SE can be supported is $SE=\left( 1-\overline{\theta } \right) \widehat{\theta }$. This region satisfies $\frac{\partial SE}{ \partial \widehat{\theta }}=1-\overline{\theta }$, which is positive given that $\overline{\theta }<1$ by definition.

1.8 Proof of Proposition 3

SE where $R_{H}\ne R_{L}$. Updated beliefs. In this separating strategy profile, the CA updates its beliefs according to Bayes’ rule, obtaining $\mu \left( \theta _{H}|R_{H}\right) =1$ and $\mu \left( \theta _{H}|R_{L}\right) =0$. For simplicity, we consider that, upon observing an off-the-equilibrium message $R\ne R_{H}\ne R_{L}$, the CA’s off-the-equilibrium beliefs satisfy $\mu \left( \theta _{H}|R\right) =0$.

Receiver’s response. Given the above beliefs, the CA is convinced that it faces a high-type firm upon observing $R_{H}$, and responds by approving a merger since $\theta _{H}>\overline{\theta }$ by definition. In contrast, upon observing $R_{L}$, the CA is convinced that it faces a low-type firm, and it thus blocks the merger since $\theta _{L}<\overline{ \theta }$ by assumption. A similar argument applies upon observing any other $R\ne R_{H}$, leading to a merger blocking decision.

Sender’s messages. Anticipating these responses, the $\theta _{H}$ -type entity invests $R_{H}$, as prescribed in this separating strategy profile, if $\pi ^{M,H}-R_{H}\ge 2\pi _{i}^{NM}$, where the right-hand side assumes that the high-type deviates to zero (no merger request). This inequality simplifies to $\pi ^{M,H}-2\pi _{i}^{NM}\ge R_{H}$ or, solving for $\theta _{H}$, we know from Proposition 2 that this inequality yields $\theta _{H}>\widehat{\theta }(R_{H})$, where cutoff $\widehat{\theta }(\cdot )$ is evaluated at $R=R_{H}$. Alternatively, the high-type firm could deviate to $R_{L}$, but doing so would yield even a lower payoff on the right-hand side of the above inequality, $\pi ^{M,H}-R_{H}\ge 2\pi _{i}^{NM}-R_{L}$, which implies that $\theta _{H}>\widehat{\theta }(R_{H})$ is a sufficient condition for $\pi ^{M,H}-R_{H}\ge 2\pi _{i}^{NM}-R_{L}$. Condition $\theta _{H}>\widehat{\theta }(R_{H})$, however, holds by definition since $\theta _{H}>\overline{\theta }$ and $\overline{\theta }> \widehat{\theta }(R)$ for all admissible values of R, including $\widehat{ \theta }(R_{H})$, which implies that $\theta _{H}>\widehat{\theta }(R_{H})$ as well.

In contrast, the $\theta _{L}$-type entity chooses $R_{L}$, instead of deviating to $R_{H}$, if and only if $2\pi _{i}^{NM}-R_{L}\ge \pi ^{M,L}-R_{H}$, as the CA blocks the merger upon observing $R_{L}$ but approves it upon observing $R_{H}$. Among all of the values of $R_{L}$ that lead to the CA’s blocking of a merger, the most profitable is, of course, $R_{L}=0$ (minimizing submission costs), so the above inequality becomes $2\pi _{i}^{NM}\ge \pi ^{M,L}-R_{H}$. Solving for $\theta _{L}$, yields $\theta _{L}\le \widehat{\theta }(R_{H})$ which, solving for $R_{H}$, is equivalent to $R_{H}\ge \pi ^{M,L}-2\pi _{i}^{NM}$. Combining the inequalities we found from the high- and low-type firms, we obtain that a SE can be sustained if $\theta _{H}>\widehat{\theta }(R_{H})>\theta _{L}$, where the high-type firm invests $R_{H}\ge \pi ^{M,L}-2\pi _{i}^{NM}$.

Among all these SEs, however, only the least-costly SE, where $R_{H}^{SE}=\pi ^{M,L}-2\pi _{i}^{NM}$, survives Cho and Kreps’ Intuitive Criterion. However, cutoff $\widehat{\theta }(R_{H}^{SE})$ evaluated at $R_{H}^{SE}=\pi ^{M,L}-2\pi _{i}^{NM}$ simplifies to $\theta _{L}$, which implies that inequality $\theta _{L}\le \widehat{\theta }(R_{H}^{SE})$ holds for all parameter values. (The above argument assumes that f satisfies $f<\pi ^{M,L}-2\pi _{i}^{NM}$. Otherwise, the only SE entails $R_{H}=f$.)

PE where both firm types invest a positive amount. Updated beliefs. In this pooling strategy profile, the CA cannot update its beliefs according to Bayes’ rule. Therefore, upon observing R, where $R\ge f$, its beliefs are $\mu \left( \theta _{H}|R\right) =p$ and $\mu \left( \theta _{L}|R\right) =1-p$, whereas upon receiving any off-the-equilibrium message $R^{\prime }\ne R$, where $R^{\prime }\ge f$, its off-the-equilibrium beliefs are $\mu \left( \theta _{H}|R^{\prime }\right) =0$.

Receiver’s response. Given the above beliefs, upon observing R, in equilibrium, the CA responds by approving a merger upon observing a request if and only if

$$\begin{aligned} p\frac{1-c+x_{H}}{2}+(1-p)\frac{1-c+x_{L}}{2}\ge k\frac{1}{3}. \end{aligned}$$

Rearranging, yields $px_{H}+(1-p)x_{L}\ge \frac{1-c}{3}$ or, alternatively,

$$\begin{aligned} E[\theta ]\equiv p\frac{x_{H}}{1-c}+(1-p)\frac{x_{L}}{1-c}=p\theta _{H}+(1-p)\theta _{L}\ge \overline{\theta }. \end{aligned}$$

or, after solving for p, we obtain $p>\frac{\overline{\theta }-\theta _{L} }{\theta _{H}-\theta _{L}}\equiv \widehat{p}$, which holds by assumption. In contrast, upon observing the off-the-equilibrium message $R^{\prime }$, the CA responds blocking the merger since $\mu \left( \theta _{H}|R^{\prime }\right) =0$ and $\theta _{L}<\overline{\theta }$ by assumption.

Sender’s messages. Anticipating these responses, the $\theta _{H}$ -type entity invests R, as prescribed in this pooling strategy profile, if $\pi ^{M}-R\ge 2\pi _{i}^{NM}$, where the right-hand side assumes that the high-type deviates to zero investment (no merger request) because any deviation to $R^{\prime }\ne R$ guarantees that a merger will be blocked and $R^{\prime }=0$ minimizes the firm’s submission cost. This inequality simplifies to $\pi ^{M,H}-2\pi _{i}^{NM}\ge R_{H}$ or, solving for $\theta _{H}$, we know from Proposition 2 that this inequality yields $\theta _{H}> \widehat{\theta }(R)$.

Similarly, the $\theta _{L}$-type entity chooses R, instead of deviating to any other $R^{\prime }\ne R$, which guarantees that a merger will be blocked, if and only if $\pi ^{M,L}-R\ge 2\pi _{i}^{NM}$. (The right-hand side of this inequality follows a similar argument as for the high-type firm.). This inequality simplifies to $\pi ^{M,L}-2\pi _{i}^{NM}\ge R$ or, solving for $\theta _{L}$, we know from Proposition 2 that this inequality yields $\theta _{L}>\widehat{\theta }(R)$.

Combining the inequalities we found from the high- and low-type firms, we obtain that a PE can be sustained if $\theta _{H}>\widehat{\theta }(R)$ and $\theta _{L}>\widehat{\theta }(R)$, but since $\theta _{H}>\theta _{L}$ by definition, a sufficient condition for both inequalities to hold is $\theta _{L}>\widehat{\theta }(R)$, which is equivalent to $\pi ^{M,L}-2\pi _{i}^{NM}\ge R$.

Applying the Cho and Kreps’ Intuitive Criterion does not have a bite in this case. Specifically, both firm types have incentives to deviate from R to $R^{\prime }$, where $R>R^{\prime }\ge f$, if the merger request is still approved. As a consequence, the CA cannot update its off-the-equilibrium beliefs.

PE where both firm types invest zero. Updated beliefs. In this pooling strategy profile, where $R=0$ for both firm types, the CA’s information set is not reached since no merger approval request is submitted. Then, the CA holds off-the-equilibrium beliefs $\mu \left( \theta _{H}|M\right) =\mu$ and $\mu \left( \theta _{L}|M\right) =1-\mu$.

Receiver’s response. Given the above beliefs, the CA responds by approving a merger upon observing one (which can happen only off the equilibrium path) if and only if $\mu \theta _{H}+(1-\mu )\theta _{L}> \overline{\theta }$ or, alternatively, $\mu >\frac{\overline{\theta }-\theta _{L}}{\theta _{H}-\theta _{L}}\equiv \widehat{p}$. We next separately analyze the case in which $\mu >\widehat{p}$ (and the CA responds by approving merger, if one is submitted off the equilibrium) and that in which $\mu \le \widehat{p}$ (and the CA blocks the merger).

Sender’s messages. If $\mu >\widehat{p}$, the $\theta _{H}$-type entity does not invest $R=0$, as prescribed in this pooling strategy profile, if and only if $\pi ^{M}-R^{\prime }<2\pi _{i}^{NM}$, where $R^{\prime }\ge f$, since it anticipates that a deviation towards sending a request will be approved by the CA (given its off-the-equilibrium beliefs). This inequality entails $\theta _{H}<\widehat{\theta }(R^{\prime })$. Similarly, the $\theta _{L}$-type entity, does not invest, as required in this pooling strategy profile, if and only if $\pi ^{M}-R^{\prime }<2\pi _{i}^{NM}$, which yields $\theta _{L}<\widehat{\theta }(R^{\prime })$. However, condition $\theta _{i}<\widehat{\theta }(R^{\prime })$ for all $i=\{H,L\}$ is not compatible with the initial assumption $\theta _{H}> \overline{\theta }>\theta _{L}$, since $\widehat{\theta }(R)<\overline{ \theta }$ for all R, which implies that this pooling strategy profile cannot be sustained as a PBE.

Summary. Cutoff $\widehat{\theta }(R^{PE})\$lies below $\widehat{ \theta }(R_{H}^{SE})$ since $R^{PE}$ satisfies $f\le R^{PE}\le R_{H}^{SE}$, thus giving rise to three regions: (i) if $\theta _{L}\le \widehat{ \theta }(R^{PE})$, only a SE can be sustained; (ii) if $\widehat{\theta } (R^{PE})<\theta _{L}\le \widehat{\theta }(R_{H}^{SE})$, both a SE and PE coexist; and (iii) otherwise, only a PE can be supported.