Sweetening the Pill: a Theory of Waiting to Merge

Abstract

Merger policy is a permission-granting activity by government in which there may be disincentives to seek permission because of the benefit from having other firms merge. We set up a sequential merger game with endogenized antitrust policy to study one aspect of these disincentives. In particular, we delineate a pill-sweetening motive for waiting to merge: a small firm may choose to let other bigger firms move first, in order to get more mergers approved by government. We report the prevalence of pill sweetening to occur in equilibrium and find it to hinge on efficiency gains from a merger, differently sized firms, firms’ production technology, the presence of an antitrust authority, the alignment of interests between antitrust authorities and firms, and the number of firms in the industry.

Appendix

1.1 A.1 Some Notation

In Sections A2 and A3 of this Appendix, we provide the complete solution of the model. In order to do this, we introduce some notation that might seem a bit elaborate for this three-firm model, but it has been chosen in order to facilitate extensions to cases with more than three firms.³⁸

Recall that the set of possible outcomes of the merger game is

$$ {\Xi} :=\left\{ SQ\text{, }PO\text{,\ }PI\text{, }CM\right\} . $$

In order to ease notation, we will sometimes need to express an outcome by a single letter: Q = SQ; O = PO; I = PI; C = CM; and Ξ = {Q, O, I, C}. Furthermore, we denote the set of decision nodes in the merger game by N := {1, 1A,..., 4, 4A} (see Fig. 1 in the text).

The model has two exogenous parameters: a, which measures market size, and k, which measures firm asymmetry. As noted in Section 2, we restrict attention to those combinations (k, a) for which all existing firms produce positive quantities in all the four outcomes outlined above. We do this by, for every ξ ∈Ξ and every k ∈ (0, 1), restricting a such that $\phantom {\dot {i}\!}a\geq \underline {a}^{\xi }\left ( k\right )$, where, for each ξ ∈Ξ, $\phantom {\dot {i}\!}\underline {a}^{\xi }\left ( k\right )$ is described in the next Section. These outcome-wise restrictions can be summarized in the restriction

$$ a\geq \underline{a}\left( k\right) :=\max \left\{ \underline{a}^{SQ}\left( k\right) ,...,\underline{a}^{CM}\left( k\right) \right\} . $$

In the following, our attention is thus limited to parameter combinations $\left ( k,a\right ) \in Z:=\left \{ \left ( k,a\right ) \ |\ a\geq \underline {a} \left ( k\right ) \right \}$.

Our aim is, for each combination (k, a) ∈ Z of market size and firm asymmetry, to find the corresponding equilibrium outcome. We do this through backward induction by first solving the product-market game in each of the four situations. Thereafter, we proceed by looking at each node n ∈ N to determine, for each (k, a) ∈ Z, what the eventual outcome of the merger game is; i.e., we are looking for an outcome partitionΩⁿ of Z at each node, where ${\Omega }^{n}:=\left \{ Z_{\xi }^{n},Z_{\iota }^{n},...\right \}$ , and $Z_{\xi }^{n}$ consists of all (k, a) ∈ Z such that the outcome of the merger subgame starting at node n ∈ N is ξ ∈Ξ. The equilibrium outcome of the whole merger game then corresponds to Ω¹, the outcome partition at node 1.

Let M (n) be the entity that makes a decision at decision node n ∈ N. M (n) compares the possible outcomes that can follow each of its decisions. Let Γⁿ ⊆Ξ denote the set of outcomes that can occur after node n. For example, at node 4A in Fig. 1, Γ^4A = {PO, CM}. Denote by $\phantom {\dot {i}\!}V_{\xi \iota }^{n}\subset Z$ the relevant region of the parameter space at node n ∈ N for the comparison between outcomes ξ, ι ∈Γⁿ; that is, $V_{\xi \iota }^{n}$ is the set of combinations (k, a) such that taking one of the feasible actions at node n would lead to outcome ξ and taking another one would lead to outcome ι. Define $\phantom {\dot {i}\!}_{\xi }Y_{\iota }^{m}\subset Z$ as the set of combinations for which decision maker m prefers outcome ξ to outcome ι, where ξ, ι ∈Ξ. Let the decision maker at node n be denoted m = M (n). If n is a merger node, then m is a pair of firms choosing whether or not to propose a merger. If n is an AA node, then m is the AA. We express $\phantom {\dot {i}\!} Z_{\xi }^{n}$, introduced in the previous paragraph, as the collection of all parameter combinations for which outcome ξ is preferred by the decision maker M (n) at node n to another outcome in the relevant region of comparison between the two outcomes; to be precise:

$$ Z_{\xi }^{n}:=\cup_{\iota \in {\Gamma}^{n},\iota \neq \xi }\left( V_{\xi \iota }^{n}\cap _{\xi }Y_{\iota }^{M\left( n\right) }\right) ,\ n\in N,\ \xi \in {\Gamma}^{n}. $$

Let $\phantom {\dot {i}\!}\widetilde {N}$ denote the set of end nodes of the merger game. End nodes are not decision nodes, and outcome partitions at end nodes are degenerate: if the merger game ends in outcome ξ ∈Ξ at end node $\phantom {\dot {i}\!} \widetilde {n}\in \widetilde {N}$, then the outcome partition of that end node is $\phantom {\dot {i}\!}\left \{ Z_{\xi }^{\widetilde {n}}\right \} =\left \{ Z\right \}$. The relevant region at a decision node can thus be constructed recursively through the outcome partitions of the node’s immediate successors:

$$ V_{\xi \iota }^{n}:=\left[ \cup_{l\in I^{n}\cap {\Phi}_{\xi }}Z_{\xi }^{l} \right] \cap \left[ \cup_{l\in I^{n}\cap {\Phi}_{\iota }}Z_{\iota }^{h} \right] , $$

where Iⁿ is the set of immediate successor nodes of node n and

$$ {\Phi}_{\xi }:=\left\{ n\in N\cup \widetilde{N}\ |\ Z_{\xi }^{n}\neq \varnothing \right\} $$

is the set of nodes from which outcome ξ is a possible outcome. At every decision node in the present model, however, Iⁿ consists of two nodes, so that the expression simplifies to:

$$ V_{\xi \iota }^{n}=Z_{\xi }^{l}\cap Z_{\iota }^{h}, $$

where l, h ∈ Iⁿ, and l≠h, such that l ∈Φ_ξ and h ∈Φ_ι.³⁹

1.2 A.2 Product-Market Competition

The outcome of the quantity competition depends on which situation we are in. Below, we go through the four different situations that may occur in order to characterize the equilibrium in each of them.

Status Quo (SQ): {1,2,3}

In this situation, one firm of size k and two firms each of size $\phantom {\dot {i}\!}\frac {1-k }{2}$ compete. The first-order condition of firm 1 is: $\phantom {\dot {i}\!}a-X-x_{1}-\frac {1}{ k}=0$, while the first-order condition of firm s ∈ {2, 3} is: $\phantom {\dot {i}\!}a-X-x_{s}-\frac {2}{1-k}=0$. Imposing symmetry on the identical firms 2 and 3, we can write these conditions as: $2x_{1}+2x_{s}=a-\frac {1}{k}$, and $\phantom {\dot {i}\!}x_{1}+3x_{s}=a-\frac {2}{1-k}$. Solving this system, we have

$$ \begin{array}{@{}rcl@{}} x_{1}^{SQ} &=&\frac{1}{4}\left( a-\frac{3-7k}{k\left( 1-k\right) }\right) , \\ x_{2}^{SQ} &=&x_{3}^{SQ}=\frac{1}{4}\left( a-\frac{5k-1}{k\left( 1-k\right) } \right) , \end{array} $$

so that having non-negative quantities from all three firms requires $\underline {a}^{SQ}\left ( k\right ) :=\max \left \{ \frac {3-7k}{k\left ( 1-k\right ) },\frac {5k-1}{k\left ( 1-k\right ) }\right \}$, and total quantity is

$$ X^{SQ}=\frac{1}{4}\left( 3a-\frac{3k+1}{k\left( 1-k\right) }\right) . $$

Partial Out (PO): {1,23}

We have two firms: firm 1 of size k and firm 23 of size 1 − k. The first-order conditions of the firms are: $\phantom {\dot {i}\!}a-X-x_{1}-\frac {1}{k}=0;\$and $\phantom {\dot {i}\!} a-X-x_{23}-\frac {1}{1-k}=0$. Rewriting, we have: $\phantom {\dot {i}\!}2x_{1}+x_{23}=a-\frac {1}{k} ;\ x_{1}+2x_{23}=a-\frac {1}{1-k}$. Solving this system, we have

$$ \begin{array}{@{}rcl@{}} x_{1}^{PO} &=&\frac{1}{3}\left( a-\frac{2-3k}{k\left( 1-k\right) }\right) , \\ x_{23}^{PO} &=&\frac{1}{3}\left( a-\frac{3k-1}{k\left( 1-k\right) }\right) . \end{array} $$

Thus, $\underline {a}^{PO}\left ( k\right ) :=\max \left \{ \frac {2-3k}{k\left ( 1-k\right ) },\frac {3k-1}{k\left ( 1-k\right ) }\right \}$. Total quantity is

$$ X^{PO}=\frac{1}{3}\left( 2a-\frac{1}{k\left( 1-k\right) }\right) . $$

Partial In (PI): {12,3}

We have one big firm, 12, of size $\phantom {\dot {i}\!}k+\frac {1-k}{2}=\frac {1+k}{2}$ and one small firm, firm 3, of size $\phantom {\dot {i}\!}\frac {1-k}{2}$. The first-order condition of firm 12 is: $a-X-x_{12}-\frac {2}{1+k}=0$, while the first-order condition of firm 3 is: $\phantom {\dot {i}\!}a-X-x_{3}-\frac {2}{1-k}=0$. We rewrite to obtain: $2x_{12}+x_{3}=a-\frac {2}{1+k};\ x_{12}+2x_{3}=a-\frac {2}{1-k}$. Solving the system, we have

$$ \begin{array}{@{}rcl@{}} x_{12}^{PI} &=&\frac{1}{3}\left( a-\frac{2\left( 1-3k\right) }{1-k^{2}} \right) , \\ x_{3}^{PI} &=&\frac{1}{3}\left( a-\frac{2\left( 1+3k\right) }{1-k^{2}} \right) , \end{array} $$

so that non-negative quantities require $\phantom {\dot {i}\!}a\geq \underline {a}^{PI}\left ( k\right ) :=\max \left \{ \frac {2\left ( 1-3k\right ) }{1-k^{2}},\frac {2\left ( 1+3k\right ) }{1-k^{2}}\right \}$. Total quantity is

$$ X^{PI}=\frac{2}{3}\left( a-\frac{2}{1-k^{2}}\right) . $$

Complete Monopoly (CM): {123}

In complete monopoly, there is a single firm, 123, whose first-order condition is: a − 2x₁₂₃ − 1 = 0. In other words,

$$ X^{CM}=x_{123}^{CM}=\frac{a-1}{2}, $$

so that $\phantom {\dot {i}\!}\underline {a}^{CM}\left ( k\right ) :=1$.

Based on the above, we can now be specific about the function $\phantom {\dot {i}\!}\underline {a} \left ( k\right )$, which restricts the set Z of combinations (k, a) of interest and is given by the following piecewise relationship:

$$ \underline{a}\left( k\right) :=\left\{ \begin{array}{c} \underline{a}^{SQ}\left( k\right) =\frac{3-7k}{k\left( 1-k\right) }\text{, if }k\in \left( 0,\frac{1}{4}\right) \text{;} \\ \underline{a}^{PO}\left( k\right) =\frac{2-3k}{k\left( 1-k\right) }\text{, if }k\in \left[ \frac{1}{4},\frac{1}{3}\right) \text{;} \\ \underline{a}^{PI}\left( k\right) =\frac{2\left( 1+3k\right) }{1-k^{2}}\text{ , if }k\in \left[ \frac{1}{3},1\right) \text{.} \end{array} \right. $$

1.3 A.3 The Merger Game

In order to solve the game, we proceed by backward induction. Consider, therefore, node 4A in Fig. 1, where AA decides whether to approve a merger between firms 1 and 23. If AA says no to the merger, then the merger game stops in the PO situation, whereas a yes leads to CM; in other words, Γ^4A = {PO, CM}. The two immediate successors to node 4A are both end nodes, implying that $\phantom {\dot {i}\!}V_{CP}^{4A}=Z$. AA compares TW in the two outcomes and approves the merger if and only if⁴⁰

$$ \left( k,a\right) \in\ {\!}_{C}{Y_{O}^{A}}:=\left\{ \left( k,a\right) \in Z\ |\ a\leq a_{CO}^{A}\left( k\right) \right\} , $$

(A1)

where

$$ a_{CO}^{A}\left( k\right) :=\frac{27k^{2}-27k+16+6\sqrt{ 24k^{4}-48k^{3}+28k^{2}-4k+1}}{5k(1-k)} $$

(A2)

Intuitively, the merger is approved if the market is so small that there is no room for two firms in the market. Thus, the outcome partition at node 4A is ${\Omega }^{4A}=\left \{ Z_{CM}^{4A},Z_{PO}^{4A}\right \}$ , where $Z_{CM}^{4A}=V_{CO}^{4A}\cap \ {\!}_{C}{Y_{O}^{A}}=\_{C}{Y_{O}^{A}}$, and $\phantom {\dot {i}\!} Z_{PO}^{4A}=Z\backslash Z_{CM}^{4A}$.

At node 4, firms 1 and 23 decide whether or not to propose a merger. Possible outcomes are Γ⁴ = Γ^4A = {PO, CM}. The firms prefer to merge if

$$ \pi_{1}^{CM_{2}}\geq \pi_{1}^{PO}. $$

(A3)

Since

$$ \pi_{1}^{CM_{2}}=\frac{1}{2}\left( \pi_{123}^{CM}+\pi_{1}^{PO}-\pi_{23}^{PO}\right) , $$

(A4)

the condition in (A3) amounts to

$$ \pi_{123}^{CM}\geq \pi_{1}^{PO}+\pi_{23}^{PO}; $$

in other words, firms 1 and 23 prefer to merge exactly when the profit of the merged unit is larger than what the two firms can get separately. The condition holds for all (k, a) ∈ Z, so $\phantom {\dot {i}\!}_{C}{Y_{O}^{1}}=Z$. Thus, Ω⁴ = Ω^4A, and a merger is proposed at node 4 if and only if (A1) holds.

At node 3A, the AA decides whether to approve a merger between firms 2 and 3. Possible outcomes are Γ^3A = {CM, PO, SQ}. In particular, if the AA says no, then the merger game ends in an SQ outcome; if it says yes, then the game ends in CM if (A1) holds, in PO otherwise. Consider first the comparison between CM and SQ. The relevant region is $\phantom {\dot {i}\!}V_{CQ}^{3A}=Z_{CM}^{4}=Z_{CM}^{4A}$. The AA prefers CM to SQ if and only if $\left ( k,a\right ) \in {\!}_{C}{Y_{Q}^{A}}:=\left \{ \left ( k,a\right ) \in Z\ |\ a\leq a_{CQ}^{A}\left ( k\right ) \right \}$, where

$$ a_{CQ}^{A}\left( k\right) :=\frac{12k^{2}+3k+5+2\sqrt{45k^{4}-114k^{2}+96k-11 }}{3k(1-k)} $$

Consider next the comparison between PO and SQ. The relevant region is $\phantom {\dot {i}\!} V_{OQ}^{3A}=Z_{PO}^{4}=Z_{PO}^{4A}$. The AA prefers PO to SQ if $\phantom {\dot {i}\!}\left ( k,a\right ) \in \ _{O}{Y_{Q}^{A}}:=\left \{ \left ( k,a\right ) \in Z\ |\ a\leq a_{OQ}^{A}\left ( k\right ) \right \}$, where

$$ a_{OQ}^{A}\left( k\right) :=\frac{135k-19 + 12\sqrt{64k^{2}-12k+1}}{7k(1-k)} \text{.} $$

(A5)

Putting this together, we see that $\phantom {\dot {i}\!}{\Omega }^{3A}=\left \{ Z_{CM}^{3A},Z_{PO}^{3A},Z_{SQ}^{3A}\right \}$ (see Fig. 14).

Fig. 14

Outcomes at node 3A

Here, $\phantom {\dot {i}\!}Z_{CM}^{3A}=\left \{ \left ( k,a\right ) \in Z\ |\ a\leq \min \left \{ a_{CQ}^{A}\left ( k\right ) ,a_{CO}^{A}\left ( k\right ) \right \} \right \}$: when the market, measured by a, is small, both this merger and the next one (to be proposed at node 4) are accepted by the AA, and the merger game ends in a CM outcome; $Z_{PO}^{3A}=\left \{ \left ( k,a\right ) \in Z\ |\ a_{CO}^{A}\left ( k\right ) <a\leq a_{OQ}^{A}\left ( k\right ) \right \}$: when firm 1 is big (k is large), the AA prefers balancing it by accepting the merger between the two small firms 2 and 3 here at node 3A but will not allow a merger to CM later on at node 4A; and finally $Z_{SQ}^{3A}=\left \{ \left ( k,a\right ) \in Z\ |\ a>\max \left \{ a_{CQ}^{A}\left ( k\right ) ,a_{OQ}^{A}\left ( k\right ) \right \} \right \}$: when the market is large, there is no reason for the AA to allow any merger at all.

At node 3, no merger has taken place so far in the game when firms 2 and 3 consider whether or not to merge. We have Γ³ = {CM₂, PO, SQ}: In parallel to node 3A discussed above, we need to compare SQ with the outcomes CM and PO, but this time from the perspective of firms 2 and 3 rather than that of the AA; note that we now need to be explicit on which kind of complete monopoly is obtained. Consider first the comparison between CM₂ and SQ. The relevant region is $\phantom {\dot {i}\!}V_{CQ}^{3}=Z_{CM}^{3A}$. In order to find firm 2’s share of the profit in the completely monopolized industry, $\phantom {\dot {i}\!}\pi _{2}^{CM_{2}}$, we note that firms 2 and 3, if they merge, will eventually end up in the CM outcome. Thus, for firm 2 at node 3, merger is preferable to no merger if

$$ \pi_{2}^{CM_{2}}=\frac{1}{2}\left( \pi_{23}^{CM}+\pi_{2}^{SQ}-\pi_{3}^{SQ}\right) \geq \pi_{2}^{SQ} $$

(A6)

Since firms 2 and 3 are identical, we have $\phantom {\dot {i}\!}\pi _{2}^{SQ}=\pi _{3}^{SQ}$ . Using this and inserting from

$$ \pi_{23}^{CM}=\frac{1}{2}\left( \pi_{123}^{CM}+\pi_{23}^{PO}-\pi_{1}^{PO}\right) \text{,} $$

we can write (A6) as

$$ \frac{1}{4}\left( \pi_{123}^{CM}+\pi_{23}^{PO}-\pi_{1}^{PO}\right) \geq \pi_{2}^{SQ}. $$

This leads to the finding that firms 2 and 3, in the relevant region, always prefer CM₂ to SQ. In the comparison between PO and SQ, where $\phantom {\dot {i}\!}V_{OQ}^{3}=Z_{PO}^{3A}$, we find similarly that also PO is preferred to SQ for any $\phantom {\dot {i}\!}\left ( k,a\right ) \in Z_{PO}^{3A}$. The conclusion for node 3, therefore, is that a merger is proposed whenever it will be accepted at node 3A, i.e., Ω³ = Ω^3A.

Next, we move to node 2A, where the AA decides whether to approve a merger between firms 12 and 3. The choice is essentially between outcomes PI and CM, i.e., Γ^2A = {PI, CM}, and $\phantom {\dot {i}\!} V_{CB}^{2A}=Z$. We find that the AA prefers CM to PI if and only if $\left ( k,a\right ) \in \ _{C}{Y_{I}^{A}}:=\left \{ \left ( k,a\right ) \in Z\ |\ a\leq a_{CI}^{A}\left ( k\right ) \right \}$, where

$$ a_{CI}^{A}\left( k\right) :=\frac{27k^{2}+37 + 12\sqrt{6k^{4}-8k^{2}+6}}{ 5(k+1)(1-k)}\text{.} $$

This gives us $\phantom {\dot {i}\!}{\Omega }^{2A}=\left \{ Z_{CM}^{2A},Z_{PI}^{2A}\right \}$ (see Fig. 15).

Fig. 15

Outcomes at node 2A

Complete monopoly is fine with the AA if the market is small or if firm 1, and therefore even more so the merged entity 12, are anyway so big that the outside firm 3 does not make up any reasonable balance.

At node 2, firms 12 and 3 decide whether or not to join up to create a complete monopoly. The comparison is also here between CM and PI: Γ² = {PI, CM}, and $V_{CI}^{2}=Z$. We find that a merger is always preferable, and so a merger is proposed whenever it will be accepted: Ω² = Ω^2A.

At node 1A, the AA says yes or no to the merger between firms 1 and 2. If it says no, then the game moves to node 3 in Fig. 1. If it says yes, then the game moves to node 2. Thus, all outcomes are possible at this node: Γ^1A = {CM, PI, PO, SQ}. The two outcomes PO and SQ can only occur if the AA says no and moves the game to node 3. Therefore, there is no need to discuss the comparison between the two at node 1A. In the comparison between CM and PO, we note that $\phantom {\dot {i}\!} V_{CO}^{1A}=Z_{CM}^{2}\cap Z_{PO}^{3}$. Thus, $\phantom {\dot {i}\!}V_{CO}^{1A}\cap _{C}{Y_{O}^{A}}=\varnothing$; whenever the comparison between CM and PO is relevant at node 1A, the AA prefers PO. In the comparison between PO and PI, the relevant region is $\phantom {\dot {i}\!}V_{OI}^{1A}=Z_{PI}^{2}\cap Z_{PO}^{3}$. Note that, from the AA’s point of view, the two outcomes PO and PI are identical when $\phantom {\dot {i}\!}k=\frac {1}{3}$, in which case the industry consists of one firm of size $\phantom {\dot {i}\!}\frac {2}{3}$ (firm 12 in the case of PI and firm 23 in the case of PO) and one firm of size $\phantom {\dot {i}\!}\frac {1}{3}$ (firm 3 in the case of PI and firm 1 in the case of PO). With k going slightly below $\phantom {\dot {i}\!} \frac {1}{3}$, the big firm gets bigger in the case of PO and smaller in the case of PI. Thus, the AA prefers PI to PO whenever $\phantom {\dot {i}\!}k<\frac {1}{3}$: $\phantom {\dot {i}\!}_{I}{Y_{O}^{A}}:=\left \{ \left ( k,a\right ) \in Z\ |\ k<\frac {1}{3}\right \}$.

In the comparison between PI and SQ, the relevant region is $\phantom {\dot {i}\!} V_{IQ}^{1A}=Z_{SQ}^{3}$. We have that the AA prefers PI to SQ if and only if $\phantom {\dot {i}\!}\left ( k,a\right ) \in \ _{I}{Y_{Q}^{A}}:=\left \{ \left ( k,a\right ) \in Z\ |\ a\leq a_{IQ}^{A}\left ( k\right ) \right \}$, where

$$ a_{IQ}^{A}\left( k\right) :=\frac{135k^{2}-76k+45 + 24\sqrt{ 37k^{4}-68k^{3}+38k^{2}-4k+1}}{7k(k+1)(1-k)}\text{.} $$

(A7)

Finally, in the comparison between PI and CM, there is a possibility for the AA to obtain CM in stead of PI when $\phantom {\dot {i}\!}\left ( k,a\right ) \in V_{CI}^{1A}=Z_{PI}^{2}\cap Z_{CM}^{3}$. However, for any $\phantom {\dot {i}\!}\left ( k,a\right ) \in V_{CI}^{1A}$, the AA prefers PI to CM.

Our findings for node 1A are summarized in Fig. 16. As is evident from that figure, the AA says no to the merger proposal in order to obtain either SQ or PO. Saying no means moving the game over to node 3.

Fig. 16

Outcomes at node 1A

The AA prefers SQ when the market is large (high a) and the firms not very asymmetric (k not very small or very large): $\phantom {\dot {i}\!}Z_{SQ}^{1A}=Z_{SQ}^{3} \setminus _{I}{Y_{Q}^{A}}$. For intermediate market sizes or for a very small firm 1 (k small), the AA prefers PI: $Z_{PI}^{1A}=\left \{ \left ( k,a\right ) \in \left ( Z_{PI}^{2}\cap _{I}{Y_{Q}^{A}}\right ) |\ k<\frac {1}{3} \right \}$ . For intermediate market sizes or for a big firm 1 (k large), the AA prefers PO: $Z_{PO}^{1A}=\left \{ \left ( k,a\right ) \in Z_{PO}^{3}|\ k\geq \frac {1}{3}\right \}$ . Finally, the AA prefers CM when the market is small: $Z_{CM}^{1A}=Z_{CM}^{2}\cap Z_{CM}^{3}$.

At node 1, firms 1 and 2 decide whether or not to merge. A merger proposal would move the game to node 1A, where the AA decides whether or not to accept, whereas a decision not to merge would move the game to node 3, where firms 2 and 3 decides whether or not to merge. The first thing to note is that firm 1’s share of the monopolist’s profit in CM differs between CM₁ and CM₂. The crucial question is firm 1’s incentive to take part in the node-1 merger. Whereas firm 1’s share of the monopoly profit in CM₂ is

$$ \pi_{1}^{CM_{2}}=\frac{1}{2}\left( \pi_{123}^{CM}+\pi_{1}^{PO}-\pi_{23}^{PO}\right) , $$

as noted in (A4) above, its share in CM₁ is found by first finding 12’s share in the merger taking place at node 2,

$$ \pi_{12}^{CM_{1}}=\frac{1}{2}\left( \pi_{123}^{CM}+\pi_{12}^{PI}-\pi_{3}^{PI}\right) . $$

(A8)

At node 1, firm 1’s share in the merged unit’s profit, when the final outcome is complete monopoly, is

$$ \pi_{1}^{CM_{1}}=\frac{1}{2}\left( \pi_{12}^{CM_{1}}+\pi_{1}^{SQ}-\pi_{2}^{SQ}\right) . $$

(A9)

Now, comparing $\phantom {\dot {i}\!}\pi _{1}^{CM_{1}}$ in (A9) and $\phantom {\dot {i}\!}\pi _{1}^{CM_{2}}$ in (A4), we find that, when $\phantom {\dot {i}\!}\left ( k,a\right ) \in Z_{CM}^{1A}\cap Z_{CM}^{3}$, so that the final outcome is anyway CM, firm 1 prefers not to merge immediately if and only if $\phantom {\dot {i}\!}\pi _{1}^{CM_{2}}>\pi _{1}^{CM_{1}}$. After insertions from (A4), (A8), and (A9), this condition can be rewritten as

$$ \frac{1}{2}\pi_{123}^{CM}+\pi_{1}^{PO}+\pi_{2}^{SQ}+\frac{1}{2}\pi_{3}^{PI}>\pi_{1}^{SQ}+\pi_{23}^{PO}+\frac{1}{2}\pi_{12}^{PI}. $$

It follows that firm 1 prefers CM₂ to CM₁ when $\phantom {\dot {i}\!}\left ( k,a\right ) \in \ _{C_{2}}Y_{C_{1}}^{1}:=\{\left ( k,a\right ) \in Z\ | a>a_{C_{2}C_{1}}^{1}\left ( k\right ) \}$, where

$$ a_{C_{2}C_{1}}^{1}\left( k\right) :=\frac{-3k^{3}+18k^{2}+7k+2+2\sqrt{ -27k^{5}+102k^{4}+42k^{3}+10k^{2}-11k+4}}{3(k+1)(1-k)k}\text{.} $$

(A10)

Other comparisons at node 1 are more straightforward. We find, in the choice between PI and CM₂, that firm 1 always prefers CM₂ in the relevant region. Likewise, it always prefers, in the respective relevant regions, PI to SQ and PO to PI. See Fig. 4 in the text for details. Note in particular that the CM region is split in two by the (A10) curve.

1.4 A.4 The Four-Firm Case

In the main text, only the simplified picture of the merger game is provided, in which the AA nodes, where the AA makes decisions whether to accept proposed mergers, are subsumed. Here, in Fig. 17, we provide the full merger game. In two instances, nodes 2 and 6, a merger node is followed by two different AA nodes, since there are two different mergers available. The two AA nodes in each of the two cases are kept apart by denoting them 2A and 2B, respectively 6A and 6B. The detailed analysis of this game is in the Online Appendix.

Fig. 17

The four-firm merger game

1.5 A.5 Restricted AA

In this section, we provide details of the alternative model with a myopic AA discussed in Section 5.

At node 4A, there is no difference between the behavior of a myopic AA and that of a forward-looking one. Therefore, also node 4 is not affected by this new assumption.

At node 3A, the AA is now comparing PO with the status quo without considering that, for some parameters, the merger game leads to complete monopoly. This myopic AA will accept the merger proposal between firms 2 and 3 if and only if $\phantom {\dot {i}\!}a<a_{OQ}^{A}\left ( k\right )$ (see (A5)). Figure 18 presents the outcomes at node 3A. Comparing Fig. 18 with Fig. 14, one can see that the SQ region now is slightly larger.

Fig. 18

Restricted AA: outcomes at node 3A

At node 3, firm 2 will propose any merger that will be accepted at node 3A.

At node 2A, as at node 4A, there are no changes. Therefore, there are no changes at node 2 as well.

At node 1A, the myopic AA makes a comparison only between PI and SQ and accepts the merger proposal between firms 1 and 2 if and only if $\phantom {\dot {i}\!} a<a_{IQ}^{A}\left ( k\right )$ (see (A7)). The outcomes at node 1A are depicted in Fig. 19. This graph is dramatically different from Fig. 16. The myopic AA accepts this merger proposal more often than a forward-looking AA would do. In particular, there are now cases where firm 1 will have the merger accepted and eventually end up with complete monopoly, even when it is very big.

Fig. 19

Restricted AA: outcomes at node 1A

At node 1, equilibrium behavior is almost never affected by the assumption of the AA being myopic. The only difference occurs for a small parameter region where the firms no longer can obtain complete monopoly merger because the AA, at node 3A, no longer makes any comparison between CM and SQ. Instead, the firms at node 1 have to settle with the PI outcome in this case. This means that the parameter region giving rise to a decision not to merge because of pill sweetening has been slightly reduced.

Changes in the equilibrium outcome, as depicted in Fig. 13 in the text, are otherwise not attributable to changes in firms’ behavior at node 1 but rather to changes in the AA’s behavior at node 1A. In particular, we now have a large region of CM for high values of k. As seen in Fig. 13, there is also a thin slice of a PI region between CM and PO.