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Sweetening the Pill: a Theory of Waiting to Merge


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.

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  1. 1.

    This disincentive for merger has been discussed extensively in previous work—notably by Stigler (1950) and Salant et al. (1983)—but it has not previously been spelled out in a setting where antitrust policy is endogenized. The literature makes at most the simple assumption that only one merger can take place, because of some unmodeled antitrust authority.

  2. 2.

    See, e.g., Bloch (1996), Fridolfsson and Stennek (2005a), and Rodrigues (2014) for analyses where this is played out in various ways.

  3. 3.

    In Section 5 we discuss the alternative, to let one of the other firms, equal sized by assumption, move first and find this to be less interesting.

  4. 4.

    The notion that the antitrust authority is forward-looking is common in the literature (see, e.g., Mermelstein et al. 2018). Moreover, Hovenkamp (2019) discusses how merger policy can and should look ahead and accommodate the effects a merger will have on the future of the industry involved. While he does not mention future horizontal mergers, his arguments might be applicable also to such cases. Still, it appears to be a widely held view that it is illegal, in both the EU and the US, for an antitrust authority to base current merger decisions on what they will imply for future merger decisions. The implications for our analysis of the alternative position that it is impossible for the antitrust authority to be forward-looking when deciding on merger proposals are discussed in Section 5.

  5. 5.

    The importance of firm asymmetry for outcomes of merger games is also stressed by others. Tombak (2002) finds that introducing firm asymmetry into the Kamien and Zang (1993) analysis increases the scope for merger to monopoly. Qiu and Zhou (2007) report that firm heterogeneity is crucial for the creation of a merger wave in their model. Fridolfsson (2007) finds that, with asymmetry, big firms will merge—a result resembling ours on the free-riding motive. Cunha and Vasconcelos (2018) discuss sequential mergers in an industry with both Stackelberg leaders and followers. Barros (1998) uses a different cost structure than ours, and thus a different kind of firm asymmetry, that we discuss in more detail in Section 5 below. Gowrisankaran (1999), in his analysis, simply lets bigger firms merge first, an assumption that fits well with our prediction that indeed big firms merge before small ones.

  6. 6.

    See, e.g., Farrell and Shapiro (1990, in particular Sec. III.D) on how the external effect of a merger increases in the concentration of the non-merging firms in the presence of efficiency gains from a merger. This implies that a merger is more benign, in terms of its external effect, if other firms have merged before it.

  7. 7.

    Current policy in both the EU and the USA is strongly consumer biased (see, e.g., Whinston 2007). Outside the EU and the USA, the picture is mixed. Ross and Winter (2005) argue that Canadian merger policy is close to the total welfare standard. The International Competition Network (2011) also lists Australia and New Zealand, as well as some emerging economies, among jurisdictions close to a total welfare standard. It should be noted that Ashenfelter et al. (2014) and Kwoka (2015) find that actual US merger policy has been closer to the total welfare standard than the statutes indicate. And scholars like Carlton (2007), Blair and Sokol (2012), and Kaplow (2012) argue in favor of moving US antitrust law towards a total welfare standard. Moreover, Glick (2018) and other so-called neo-Brandeisian scholars argue that the consumer-versus-total welfare discussion misses the point and that more emphasis should be put on market structure. As discussed in the text, the scope for pill sweetening is greatest in countries close to the total welfare standard.

  8. 8.

    A protocol is a sequence of proposers and, for each proposer, a sequence of respondents. Our protocol is simple: with pairwise mergers, there is, for each proposer, a single respondent.

  9. 9.

    In addition to Lyons (2003), see, e.g., Armstrong and Vickers (2010), Nocke and Whinston (2013), and Burguet and Caminal (2015).

  10. 10.

    Related to the work of Motta and Vasconcelos (2005) is that of Fumagalli and Vasconcelos (2009), who discuss a model of sequential mergers with multiple antitrust authorities, two national ones and one supranational, and the effect of varying the antitrust authorities’ objectives.

  11. 11.

    But see Mermelstein et al. (2018), who allow for efficiency gains from mergers and also allow for firms to grow organically, i.e., through investments, in addition to through mergers.

  12. 12.

    But see Jeziorski (2015), who, in his study of the US radio broadcasting industry, extends the analysis of Nocke and Whinston (2010) to allow for overlapping mergers. Also Nocke and Whinston (2013) allow overlapping sets of (potentially) merging firms, but they do not discuss dynamics.

  13. 13.

    When we, in our analysis, disregard the possibility of a threewise merger straight to monopoly, we do it without loss of generality. Clearly, such a merger would only be allowed, and therefore only happen, when it would be socially optimum, and so allowing it would not affect the prevalence of pill sweetening.

  14. 14.

    The market demand can, in other words, be written: D (p) = ap. It is, of course, only because D(p) = − 1—and, correspondingly, p(X) = − 1—that we this simply can interpret a as a demand shifter and thus as a measure of market size.

  15. 15.

    To be exact, the function in (1) should be viewed as a short-run cost function where one production factor, ki, is fixed. This formulation is similar to other cost functions used in the merger literature (see, e.g., Horn and Persson 2001b, Motta and Vasconcelos 2005, Vasconcelos 2005, Fumagalli and Vasconcelos 2009, and Cunha and Vasconcelos 2018). The introduction of efficiency effects from mergers in this manner originates with Perry and Porter (1985). Their focus is on cases where the long-run cost function C (xi, ki) is homogeneous of degree one, which implies that the short-run one features decreasing returns to scale. In Section 5, we discuss equilibrium outcomes of our model when the short-run cost function exhibits non-constant returns to scale.

  16. 16.

    By the convention we adopt here, {1, 23} denotes a two-firm industry consisting of firm 1 and the entity stemming from the merger between firms 2 and 3. With this notation, the set of non-empty subsets of S is {1, 2, 3, 12, 13, 23, 123}.

  17. 17.

    Since firms 2 and 3 are identical, letting ΘPI = {13, 2} would be the same.

  18. 18.

    See for example Motta (2004, sec.; conditions ensuring positive quantities are discussed in the Appendix. Note that we this way disregard cases where a merger entails exit by the non-merging firm (or firms, in the four-firm analysis of Section 4), an aspect of our analysis that contrasts it to that of Motta and Vasconcelos (2005).

  19. 19.

    In Section 5 below, we discuss an alternative assumption, letting firms be forward-looking while the AA is myopic.

  20. 20.

    But see Section 5 for a brief discussion of the effects of having other ways of splitting the gains from merger than equally.

  21. 21.

    In a game-theoretic sense, the number of decision nodes is greater than 8. As will become clear in the text, some of our 8 nodes can be reached by different routes through the graph in Fig. 1.

  22. 22.

    Having the differently sized firm 1 move first is the only interesting case. In Section 5 below, we discuss why having one of the other firms move first is of less interest. Giving firm 1 the decision power on merging but not the full bargaining power in the negotiations with the merging partner is of little significance for our results, as we discuss in Section 5.

  23. 23.

    In these figures, the vertical axis is truncated at 4.5 for convenience. The reason is that, for a < 4.5, not even monopoly is profitable, whatever k is.

  24. 24.

    The procedure to determine the socially optimum market structure is: for each (k, a) ∈ Z, identify the market structure that maximizes TW, i.e., that picks the ξ ∈Ξ with the highest TWξ, given in (2).

  25. 25.

    The only way the pill-sweetening motive can occur in our model is by firm 1 in the future joining a unit that has merged in the meantime, i.e., the merged entity 23. With more firms in the industry at the outset, there is a scope for the pill-sweetening motive to occur also through firm 1 in the future merging with another firm than the unit that has merged in the meantime, in order to restore some of the imbalance in the industry created by the first merger (see our discussion in Section 4).

  26. 26.

    For a complete description and a detailed analysis of the four-firm merger game, see our Online Appendix.

  27. 27.

    The bargaining-power and free-riding motives show up in Fig. 8 in much the same way as we saw in the analysis of the three-firm case. In addition, we observe a new motive for not merging that did not occur earlier: there are cases where the AA would allow one merger by firms 1 and 2, so that we would end up in the PI outcome, but where the merger is not profitable for the firms involved simply because of the output contraction involved; this happens for (k, a) combinations in the “Contraction SQ” region of Fig. 8. We call this the contraction motive for not merging. Although this is a rather prosaic reason for abstaining from merger, we note that it does occur in our model solely because of the presence of an AA; as in the three-firm case, without an AA, there would be merger to monopoly for all parameter combinations. This is a case of firms not merging in order to get less concentration: when the AA will only allow a single merger anyway, the firms prefer no merger at all.

  28. 28.

    That range is \(\phantom {\dot {i}\!}\left [ 9+\frac {3}{5}\sqrt {105},\frac {117}{7}+\frac {18}{7} \sqrt {37}\right ] \approx \left [ 15.1,32.4\right ] \).

  29. 29.

    In another range of values of a, \(\phantom {\dot {i}\!}\left [ 7+\sqrt {21},9+\frac {3}{5}\sqrt {105 }\right ) \approx \left [ 11.6,15.1\right ) \), there will be complete monopoly anyway for \(k=\frac {1}{3}\), but firm 1 benefits from getting in late and so chooses not to merge at the first opportunity. This is the bargaining-power motive for not merging. Symmetry is not a restrictive assumption to make when discussing this motive. By introducing firm asymmetry, we are still able to show, though, that this motive is present, when an antitrust authority is around, both when firm 1 is relatively large (\(k>\frac {1}{3}\)) and when it is relatively small (\(k<\frac {1}{3}\)).

  30. 30.

    The exact formula for the curve splitting the two CM1 and CM2 regions in Fig. 7 is given in (A10) in the Appendix

  31. 31.

    At c = 0, we have essentially the cost function used by Fumagalli and Vasconcelos (2009).

  32. 32.

    There are alternative approaches to the modeling of myopia in the antitrust authority in the existing literature. First, Motta and Vasconcelos (2005) limit the discussion of the myopic antitrust authority to the case where a single merger is carried out, whereas we allow for two mergers to happen even when the antitrust authority thinks there will be only one. Secondly, Nocke and Whinston’s (2010) model has product-market competition following each merger decision, which is different from both Motta and Vasconcelos (2005) and us. Thus, for them, myopia means for the antitrust authority to disregard any developments following the immediately subsequent product-market competition.

  33. 33.

    A little caveat is in order here, since the prevalence of pill sweetening is slightly reduced, as just discussed.

  34. 34.

    Barros finds, in a three-firm oligopoly where merger to monopoly is not allowed, that a big asymmetry leads to a merger between the two most efficient firms while a medium-sized asymmetry leads to a merger between the most and the least efficient firm.

  35. 35.

    One could, of course, try to take the analysis a step further by letting all three firms have different marginal costs. Along the lines of Barros (1998), again, one could for example think of letting the three firms have constant marginal costs equal to c1 = c, c2 = c + δ, and c3 = cδ, where now δ ∈ (0,c). This would, however, complicate the analysis a lot, and there would not be any obvious choice of a move sequence.

  36. 36.

    The incidence of pill sweetening is not affected by this. The only change is that the incidence of the bargaining-power motive for not merging is slightly reduced.

  37. 37.

    See Farrell and Katz (2006) for an overview of the literature on welfare standards to use in antitrust.

  38. 38.

    See the Online Appendix for an analysis of the four-firm case.

  39. 39.

    See the analysis of the four-firm case, in the Online Appendix, for a case where In may consist of more than two nodes.

  40. 40.

    There are two solutions to the equation \(\phantom {\dot {i}\!}a=a_{CO}^{A}\left ( k\right ) \). We report here only the one that is at least partly within Z. The other one is always outside Z and therefore irrelevant for our analysis and not reported. The same consideration holds for later cases.


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We are grateful for helpful comments from an anonymous referee. In addition, we are grateful for comments from, and discussions with, Jonas Björnerstedt, Sven-Olof Fridolfsson, Kai-Uwe Kühn, Volker Nocke, Lars Persson, Åsa Rosén, Christian Schultz, Lars Sørgard, Frode Steen, Helder Vas-concelos, and Jon Vislie, as well as participants at seminars at BI Norwegian Business School, the Universities of Bergen and Oslo, the Research Institute of Industrial Economics (Stockholm), and the Norwegian School of Economics, and at EARIE in Valencia, the Norwegian Economists' Meeting in Oslo, IIOC in Arlington, the BECCLE Conference in Bergen, NORIO in Reykjavik, and CRESSE at Rhodes. We have received able research assistance from Esther Ann Bøler, Ella Getz Wold, Eirik Brandsås, Dana Øye, and Anne Killi. Part of Fumagalli's research was carried out while she was affiliated with IGIER and IEFE, both at Bocconi University. Nilssen thanks IGIER and IEFE for their hospitality. During this research, Nilssen was associated with the ESOP Centre at the University of Oslo; ESOP received support from the Research Council of Norway through its Centres of Excellence funding scheme, project number 179552. Views expressed in this article are not necessarily in line with the views of the Norwegian Competition Authority.

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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.Footnote 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 nN 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 nN 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 nN. 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, Γ4A = {PO, CM}. Denote by \(\phantom {\dot {i}\!}V_{\xi \iota }^{n}\subset Z\) the relevant region of the parameter space at node nN 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, hIn, and lh, such that l ∈Φξ and h ∈Φι.Footnote 39

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:

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

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, Γ4A = {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 ifFootnote 40

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


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

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 = Γ4A = {PO, CM}. The firms prefer to merge if

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


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

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 = Ω4A, 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 Γ3A = {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

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

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

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

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 = Ω3A.

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., Γ2A = {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 = Ω2A.

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: Γ1A = {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

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

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,

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

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

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

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

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

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.

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

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.

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Fumagalli, E., Nilssen, T. Sweetening the Pill: a Theory of Waiting to Merge. J Ind Compet Trade 19, 351–388 (2019).

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  • Mergers
  • Merger policy
  • Sequential mergers
  • Antitrust policy

JEL Classification

  • L11
  • L13
  • L41
  • G34