The scope of auctions in the presence of downstream interactions and information externalities

We scrutinize the scope of auctions for firm acquisitions in the presence of downstream interactions and information externalities. We show that no mechanism exists that allows an investor to acquire a low-cost firm under incomplete information: a separating auction implies adverse selection and relies substantially on commitment to allocation and transfer rules. A pooling auction serves as a commitment device against ex-post opportunistic behavior and alleviates adverse selection. It can earn the investor a higher expected payoff than a separating auction, even when consistency is required as to qualify for a sequential equilibrium.

1. Procurement environments with incomplete information may generate different results. Focusing on procurement and modeling information asymmetries on the quality dimension, for example, Manelli and Vincent (1995) show that standard auction mechanisms are often inefficient, especially due to the adverse selection aspect of procurement environments. See, also, Klemperer (1999, 2004) and Krishna (2002) for reviews of the literature on auction theory.

2. For more details and some anecdotal evidence, see McAdams and Schwarz (2007), Vartiainen (2013), and Skreta (2015).

3. In reverse auctions (procurement auctions, subcontracting, etc.) a buyer asks potential sellers to quote prices for a particular contract.

4. Mergers and acquisitions have been the driving force of international integration and have increased substantially, especially in the post-deregulation era of the 1990s (e.g., see Andrade et al. 2001). According to UNCTAD (2014), close to 35% of all global foreign direct investment took place as cross-border mergers and acquisitions (valued at US$349bn in 2013). Also Boone and Mulherin (2007) show that auctions were employed in half of the takeovers of major U.S. firms in the 1990s. However, not all mergers and acquisitions are successful (see e.g., Gugler et al. 2003), indicating asymmetric information.

5. See Pagnozzi and Rosato (2016) for an alternative analysis of takeover auctions in which an entrant and some incumbent firms, first, compete in an auction to acquire a target firm, then compete by quantities on the product market. They also compare the takeover auction mechanism with bilateral negotiations. Their model, however, does not look at commitment to auction rules, nor does it consider asymmetric information.

6. Throughout the paper, we refer to an auction with a separating equilibrium as a separating auction, and we refer to an auction with a pooling equilibrium as a pooling auction.

7. See Krishna (2002, chapter 4) for a discussion on the impossibility of efficiency in a first-price auction with resale when all bids are made public.

8. We set up the model such that the investor is not allowed to acquire both firms. The reason is that local competition authorities would not permit the foreign firm to gain monopoly power.

9. We shall note that the assumption that the marginal costs are distributed independently is important for our results. This assumption implies that there is no correlation among cost signals. In the case of auctions with correlated private signals, the literature has shown that it would be possible to find efficient mechanisms in which the seller could extract all the surplus; see Crémer and McLean (1985, 1988).

10. For the role of partial ownership as a potential screening device, see Stähler (2014).

11. Colucci et al. (2015) develop a model where the buyer’s valuation of a seller’s input is private information of the buyer, and they discuss the incentives of the buyer whether to reveal this information. In their model, each seller knows all sellers’ costs, while costs are private information in our model.

12. Efficiency gains play an important role in the profitability of mergers and acquisitions: firms may benefit from a merger, provided sufficient efficiency gains are generated as in Perry and Porter (1985). These efficiency gains cannot be realized by a simple technology transfer, but only by an integration of the acquired firm into the acquirer’s firm boundaries. Since the combination of assets is key for cost savings, the investor will not able to sell or auction off any technology to one or both firms. Without efficiency gains, firms may not benefit from a merger if they compete in a market of strategic substitutes in the sense of Bulow et al. (1985) due to the merger paradox; see, for example, Salant et al. (1983), and Farrell and Shapiro (1990). Convex demand (Hennessy 2000), product differentiation (Lommerud and Sorgard 1997) and competition in a market of strategic complements (Deneckere and Davidson 1985) can overcome the merger paradox. The merger paradox does not apply here as the investor stays out of the market and earns zero profit if no acquisition takes place.

13. Scrutinizing the optimal entry mode when different options for foreign market entry are available is beyond the scope of this study; see Koska (2016) for a model that employs an ascending takeover auction in the case of complete information, and a second-price, sealed-bid, takeover auction in the case of incomplete information, and that looks at an investor’s choice between firm acquisition and greenfield investment.

14. Formally, we consider a perfect Bayesian equilibrium. We do not consider a potential repetition of an auction as in Skreta (2015). A well-known complication in these setups is the ratchet effect; see Freixas et al. (1985), and Laffont and Tirole (1988).

15. Our results would not change qualitatively if bids were (honestly) disclosed. However, we find this setup more compelling, also because the investor would always claim that the winning firm was just as productive as the losing firm unless she could credibly commit to reveal the bids truthfully.

16. See, for example, Krishna (2002) for a sketch of the general proof.

17. A pooling auction is strategically equivalent to a game in which an investor (i) can make simultaneous take-it-or-leave-it offers to both firms and (ii) has a certain belief structure on types depending on their unilateral acceptance. This should not be confused with take-it-or-leave-it acquisition offers to target firms that are desinged to separate types (see Koska and Stähler 2014).

18. This commitment problem is similar to the one mentioned in Spulber (1990), in which he points out that the bidding behavior and the efficiency of auctions is determined by the contractual commitment in procurement auctions.

19. In contrast to Jehiel and Moldovanu (2000) and Koska (2016), we do not assume revelation of firms’ private information after the auction and before product market interactions occur.

20. Note carefully that the equilibrium is not perfect as it does not imply Nash behavior at all nodes off the equilibrium path. A perfect Bayesian equilibrium is also a sequential equilibrium and vice versa if either player has at most two types or the sequence of the game is limited to two stages (Fudenberg and Tirole 1991), but our model does not meet any of these two conditions.

Authors and Affiliations

1. Department of Economics, Middle East Technical University (METU), 06800, Çankaya, Ankara, Turkey

   Onur A. Koska

2. School of Commerce, Division of Business, University of South Australia, Adelaide, SA, 5001, Australia

   Ilke Onur

3. Department of Economics, University of Tübingen, 72074, Tübingen, Germany

   Frank Stähler

4. CESifo, Munich, Germany

   Frank Stähler

5. University of Adelaide, Adelaide, SA, Australia

   Frank Stähler

Corresponding author

Correspondence to Onur A. Koska.

Appendix

1.1 A.1 The investor’s net payoff under complete information

Suppose that the cost types are public information such that \(c_i < c_j\), \(i\ne j\). The net value of firm acquisition to the investor is the difference between the aggregate payoff given by Eq. (3) and the acquisition price. Suppose that the investor has the full bargaining power and can acquire any firm by paying the firm its outside profits. Would the investor acquire the low-cost or the high-cost firm? Comparing the investor’s net gain when she acquires the low-cost firm to that when she acquires the high-cost firm shows that

for any \(c_i < c_j\), \(i\ne j\), so long as the parameter values are such that the inequality \((c_j-c_i)(1-\gamma )(2a-5(1+\gamma )(c_i+c_j))/9 \le 0\) holds. That is, the investor gains more by acquiring the low-cost firm than the high-cost firm so long as market size (a) is sufficiently small, or when there is no cost saving (i.e., when \(\gamma =1\), in which case the investor will be indifferent).

1.2 A.2 Proof of Proposition 1

\(c_{(1)}\) and \(c_{(2)}\) are the signals that are rearranged in ascending order: \(c_{(1)} < c_{(2)}\). If the design is given, such that the investor acquires the low-cost firm and competes against the high-cost firm, then the investor’s ex-ante expected operating profit from acquiring the low-cost firm, denoted \(\hat{v}^{(1)}(c_{(1)},c_{(2)})\), is equal to

where \(v^{(1)}(c_{(1)},c_{(2)})\) is derived from Eq. (3), and denotes the investor’s operating profit after having acquired the low-cost firm. \(F_2(c)\) is the lowest-order statistics, and given the uniform distribution \(F(c) = c\), we can show that \(F_2(c) = 2 F(c) - F(c)^2 = c(2 - c)\). If, however, the design is given, such that the investor acquires the high-cost firm and competes against the low-cost firm, then the investor’s ex-ante expected operating profit from acquiring the high-cost firm, denoted \(\hat{v}^{(2)}(c_{(1)},c_{(2)})\), is equal to

where \(v^{(2)}(c_{(1)},c_{(2)})\) is derived from Eq. (3), and denotes the investor’s operating profit after having acquired the high-cost firm. \(F_1(c) = F(c)^2 =c^2\) is the highest-order statistics. We are now ready to show that

which proves the first part of Proposition 1. In the second part of Proposition 1, it is noted that, as to realize \(\hat{v}^{(1)}(c_{(1)},c_{(2)})\), no implementable design that leads to truthful revelation exists. We do the proof by contradiction: we assume that there is a design in which the investor learns the types. In such a design, for all possible cost realizations, the low-cost firm is selected and the other firm learns - by not being selected - that the acquired firm has a lower cost. If such a design exists, using the Revelation Principle, we can confine the analysis to a design in which each target firm will send the investor a cost signal that should reveal, in equilibrium, the firm’s realized (true) cost. In an implementable design, each target firm truthfully reports its cost-type to the investor. Suppose that a target firm of type c sends cost signal \(\tilde{c}\) to the investor. The target firm’s expected profit is then given by

where the expression in brackets, derived from Eq. (2), is the firm’s profit when it competes against the lower-cost (acquired) firm, and \(T(\tilde{c})\) is the transfer from the investor to the target firm signaling type \(\tilde{c}\), given the other firm reports the true type. If the target firm sends cost signal \(\tilde{c}\), the probability of being a higher-cost firm (that is, the probability of the other firm being a lower-cost firm) is exactly equal to \(\tilde{c}\): with probability \(\tilde{c}\), the firm will not be selected and will realize the profit of a firm competing against a lower-cost firm. At the same time, the firm that is not selected learns that the selected firm has a lower cost than \(\tilde{c}\), so the expected ex-post marginal cost of the selected firm—which corresponds to \(E_{i}(c_{j})\) in Eq. (2)—is equal to \(\tilde{c} \gamma /2\). Consider now any two different types \(c^{\prime }, c^{\prime \prime } \in [0, 1]\). If a design exists, it must be incentive compatible, such that neither type has an incentive to mimic the other type:

Adding up these two inequalities should also be positive, although the result suggests

which is definitely negative for sufficiently low cost realizations (and/or for a sufficiently large a). Consequently, any implementable design in which the target firms will truthfully report their cost-types and the low-cost firm will be selected for acquisition will not exist, proving the second part of Proposition 1. Note that “Appendix A.3” shows that a mechanism (designed as an auction) will exist if the investor can commit to select the high-cost firm. In that case, the transfer, denoted by T in this section, is equivalent to the expected profit when the firm is selected for acquisition, that is, the probability of being selected times the quoted acquisition price.

1.3 A.3 Proof of Proposition 2

The proof consists of two parts: (i) incentive compatibility and (ii) individual rationality and the development of the optimal bids. As for incentive compatibility, consider two different types—as defined by private (marginal cost) information—\( c^{\prime },c^{\prime \prime }\in [0,1]\). A separating takeover auction must be incentive compatible, such that neither type has an incentive to mimic the other type:

Adding up the two inequalities, given by Eqs. (A.1) and (A.2), leads to

which is clearly positive for any \( c^{\prime },c^{\prime \prime }, \gamma \in [0,1]\), given \(a>2\). Hence, the sufficient condition for incentive compatibility is fulfilled.

We now turn to the optimal bids and individual rationality. If Condition 1 is fulfilled, firm i will win the auction by quoting \(\phi _{i} < \phi _{j}\), and will be paid \(\phi _{i}\). We denote by \(\Psi _{i}\) the probability that firm i wins the auction (that is, the probability that \(\phi _{i}<\phi _{j}\)). By the same token, the other firm wins the auction with complementary probability \( \left( 1-\Psi _{i}\right) \) (that is, the probability that \(\phi _{i}>\phi _{j} \)). If firm j wins the auction, firm i will have to compete against the investor in a Cournot duopoly, and its profit follows from Eq. (2), where \( E_{j}\left( c_{i}\right) =\Psi _{i}\left( \phi _{i}\right) \) and \( E_{i}\left( c_{j}\right) =\gamma \left( 1+\Psi _{i}(\phi _{i})\right) /2\). Firm i’s probability to win the auction \(\Psi _{i}\left( \phi _{i}\right) \) is determined by firm i’s productivity signal. Provided Condition 1 is fulfilled, if the investor acquires firm j, firm i updates its belief about firm j’s productivity such that \(E_{i}( c_{j}) =\gamma \int _{\Psi _{i}}^{1}(c_{j}/(1-\Psi _{i}))dc_{j}=\gamma \left( 1+\Psi _{i}\right) /2\), since the investor would have acquired firm j only if firm j had quoted a lower price, that is, only if firm j’s cost signal had been higher than that of firm i. By the same token, in equilibrium, \(E_{j}(c_{i}) =\Psi _{i}(\phi _{i})\), since the investor, acquiring firm j, observes the bid of the other firm, and so can invert the bidding function. If the firms bid according to their true productivity, then \(E_{j}(c_{i}) = c_{i}\). Firm i’s expected profit, denoted \(\hat{\pi }^a(c_i)\) where superscript a stands for the auction outcome, is thus equal to

where \(\Psi _{i}\left( \phi _{i}\right) \) coincides with the inverse of the price function. The price function \(\phi _{i}\left( \Psi _{i}\right) \) specifies firm i’s price demand, where \(\Psi _{i}\) represents firm i’s signal. Incentive compatibility requires \(\phi _{i}\left( \Psi _{i}\right) \equiv \phi _{i}\left( \Psi _{i}=c_{i}\right) \),^{Footnote 19}

Firm i has to quote a price that maximizes the expected profit given by Eq. (A.3). We simplify the notation by expressing equations without subscript i. The first-order condition, \(\partial \pi \left( \phi \right) /\partial \phi =0,\) is equal to

We assume that both firms follow the same strategy \(\phi \left( c\right) \), which is strictly decreasing in a firm’s marginal cost and has a well-defined inverse function. In equilibrium, the inverse of a firm’s price function is equal to the firm’s marginal cost.

Substituting \(c\equiv \Psi \left( \phi \left( c\right) \right) \) into the first-order condition gives

where \(\Psi \left( \overline{\phi }\right) \equiv 0\) such that \(\overline{\phi }\equiv (2a+\gamma )^{2}/36+(2a+\gamma )/18 \). Note that \(\overline{\phi }\) is the maximum price that the most efficient firm quotes in equilibrium. We can use Eq. (A.4) to characterize the firms’ quoted prices in equilibrium. Rewriting Eq. (A.4) as a differential equation,

and, by including the boundary condition \(\phi (0) =(2a+\gamma )^{2}/36+(2a+\gamma )/18 \), solving for \(\phi (c)\) gives the optimal price function:

Individual rationality is guaranteed: a firm’s expected profit when it participates in the auction, \(c \phi ^{*}\left( c,\gamma \right) + (1-c) (2a + (1+c) \gamma - 4c)^2/36\), is larger than its expected profit when it stays away from the auction, \((2a-(1/2) + \gamma - 3c)^2/36\). A firm can manipulate the post-auction market game by participating in the auction, and by pretending to be the lowest-cost firm (\(c^{\prime \prime }=0\) or \(c^{\prime }=0\) in Eqs. (A.1) and (A.2), respectively) which of course leads to a larger expected outside profit, \((2a + \gamma - 3c)^2/36\). Even in this case, \(c \phi ^{*}\left( c,\gamma \right) + (1-c)(2a + (1+c) \gamma - 4c)^2/36 > (2a + \gamma - 3c)^2/36\), provided \(a>2\).

1.4 A.4 Proof of Proposition 3

Recall that \(V^{i}(c_i, c_j)\), \(i \ne j \in \{1,2\}\), given by Eq. (7), denotes the investor’s expected profit after having acquired firm i and when competing against firm j. Firms quote prices \(\phi ^{*}(c_{i})\), \(i \in \{1,2\}\), in equilibrium, following the price function, given by Proposition 2. The investor’s commitment on acquiring the higher-cost firm is credible only if the investor acquires firm i when firm i’s quoted price is less than the price quoted by the other firm.

Let us suppose that firm i’s marginal cost is larger than that of firm j, such that \(c_{i}>c_{j}\), \(i \ne j \in \{1,2\}\), implying firm i and firm j will quote prices in equilibrium such that \(\phi ^{*}(c_{i})<\phi ^{*}(c_{j})\), \(i \ne j \in \{1,2\}\). We need to prove that \(V^{i}(c_i, c_j) > V^{j}(c_i, c_j)\), \(i \ne j \in \{1,2\}\), that is, the investor will acquire firm i in such a situation, even without binding commitment, which makes the auction self-enforcing. Equation (A.5) gives the difference between \(V^{i}(c_i, c_j)\) and \(V^{j}(c_i, c_j)\):

where \(C=c_{i}+c_{j}\), \(\alpha =16+8a+8\gamma -56a\gamma +6\gamma ^{2},\) and \(\beta =-48+\gamma \left( 32+31\gamma \right) \).

Equation (A.5) shows that \(V^{i}(c_i, c_j) > V^{j}(c_i, c_j)\) only if \(\left( \alpha +\beta C\right) >0\), given \(c_{i}>c_{j}\). We can see that \(\partial \left( \alpha +\beta C\right) /\partial C<0\) if \(\gamma <0.831\). Let us start from the case \(\gamma >0.831\), so \(\partial \left( \alpha +\beta C\right) /\partial C>0.\) It is obvious that \(V^{i}(c_i, c_j) - V^{j}(c_i, c_j) = 0\) if \(C=\widetilde{C}\), where \(\widetilde{C}=-\alpha /\beta \). Also, it is straightforward to show that \(\widetilde{C}>2\) for any given \(a>2\), and for \(\gamma \in \left[ 0.831,1\right] \). Therefore, for all \(c_{i}\in \left[ 0,1\right] \), \(i=\{1,2\}, C<\widetilde{C}\), implying that \(V^{i}(c_i, c_j) - V^{j}(c_i, c_j) < 0\). The investor’s profit will be larger if she acquires the lower-cost firm. So the investor’s commitment is not credible given that \(\gamma \) is sufficiently large such that \(\gamma >0.831\). If, however, \(\gamma <0.831\), then \(\partial \left( \alpha +\beta C\right) /\partial C<0\). In this case, we can see that \(\widetilde{C}<0<C\) for all \(c_{i}\in \left[ 0,1\right] , i=\{1,2\}\), for any given \(a>2\), and for \(\gamma \in \left[ 0.313,0.831\right] \). Consequently, the investor fails to commit credibly on acquiring the higher-cost firm when \(\gamma \in \left[ 0.313,0.831\right] \), or rather, when \(\gamma \in \left[ 0.313,1\right] \).

As for \(\gamma \in \left[ 0,0.313\right] \) at which \(\partial \left( \alpha +\beta C\right) /\partial C<0\), we find that \(\widetilde{C}<0<C\). The investor fails to commit credibly on acquiring the higher-cost firm, especially for some constellations of parameter values of a and \(\gamma \) (Region I in Fig. 1). Similarly, we find that \(\widetilde{C}>2>C\) in Region III, illustrated by Fig. 1, so the auction is self-enforcing in this region. As is illustrated by Fig. 1, \(\widetilde{C}\in \left[ 0,2\right] \) in Region II: the investor’s ex post behavior depends on the value of C. In Region II, the investor wants to acquire the lower-cost firm if the average industry marginal cost before the acquisition of a firm takes place, C / 2, is sufficiently large, such that \(C>\widetilde{C}\). Although a separating takeover auction can be self-enforcing if \(C<\widetilde{C}\), the firms do not know the average industry marginal cost at the time of the auction. Consequently, a separating takeover auction does not work in Region II.

1.5 A.5 Proof of Proposition 4

The difference between Eqs. (9) and (6) is equal to

for any \(\gamma \in [0,1]\) and \(a > 2\).

1.6 A.6 Proof of Proposition 5

Suppose that a target firm quotes a higher acquisition price \(\Phi ^{\prime }>\Phi ^*\). This higher acquisition price will make sure that this target firm will not be selected. The reason is simple. First, the acquisition price is higher; and second, the investor believes now that it will face a weak rival after having acquired the other firm. The expected profit of the target firm quoting \(\Phi ^{\prime }\), denoted \(\hat{\pi }^{p \prime }(c_i)\), and that of the target firm quoting \(\Phi ^*\), denoted \(\hat{\pi }^{p}(c_i)\) and given by Eq. (12), are compared in Eq. (A.6):

where the inequalities are the condition that this defection option is not profitable. This defection option should not be profitable for any cost-type. To determine the relevant condition, we define

where \(d\Lambda /dc_i = -(4 a - 6 c + 2 \gamma -3)/12\) and \(d^2\Lambda / dc_i^2 = 1/2\), showing that \(\Lambda (c_i)\) is convex in \(c_i\). Thus, the maximum of \(\Lambda \) is either \(\Lambda (0)\) or \(\Lambda (1)\). We can show that

because \(a > 2\). Thus, we find that the condition given by Eq. (A.6) holds for all cost-types if it is satisfied for the most productive firm, \(c_i = 0\):

Now suppose that the target firm quotes a lower acquisition price \(\Phi ^{\prime \prime } < \Phi ^*\). The investor has the option to accept this lower offer, but at the same time she updates her beliefs, such that she assumes that this is now a target firm with the highest marginal production cost, which is equal to 1. A remark on the rival firm is in order when this lower bid is accepted by the investor. Since bids are not revealed, the firm that is not selected by the investor will not be able to learn whether or not the winning bid was a deviation; therefore it will continue to assume that the firm that is selected by the investor has an expected cost realization of 1 / 2, which leads to the expected ex-post marginal cost of the acquired firm \(\gamma /2\). Therefore, the investor’s expected payoff from accepting the lower offer, denoted \(\hat{V}^{p\prime \prime }(\Phi ^{\prime \prime })\), is equal to

which we can compare with her expected payoff from rejecting \(\Phi ^{\prime \prime }\) (in which case she will update her beliefs accordingly) and accepting \(\Phi ^*\), denoted \(\hat{V}^{p\prime \prime }(\Phi ^*)\) and given by

A lower offer makes sense only if it will be accepted by the investor. Otherwise, the target firm loses, not only because the lower offer is declined, but also it will be considered as the highest-cost firm by the investor. The investor will accept the lower offer \(\Phi ^{\prime \prime }\) if \(\hat{V}^{p\prime \prime }(\Phi ^{\prime \prime }) > \hat{V}^{p\prime \prime }(\Phi ^*) \) which is the case when

At the same time, \(\Phi ^{\prime \prime }\) must be large enough to make the defecting firm better off such that \(\Phi ^{\prime \prime } > \hat{\pi }^{p}(c_i)\), where \(\hat{\pi }^{p}(c_i)\) is given by Eq. (12). Thus, we conclude that no target firm of any cost-type has an incentive to quote a lower acquisition price if

Note that \(\hat{\pi }^p(c_i)\), given by Eq. (12), increases with a decrease in \(c_i\). Therefore, if the acquisition price that would be accepted by the investor (\(\Phi ^{\prime \prime }\)) is too small even for the least productive target firm, such that \(\hat{\pi }^p(c_i = 1)>\Phi ^{\prime \prime }\), then it is not profitable for any firm to deviate. In summary, we can use the two conditions that are given by Eqs. (A.7) and (A.8), respectively, to show that a pooling equilibrium exists if these two conditions

hold at the same time, where \(\overline{\Phi } - \underline{\Phi } = (2\gamma (4 a-5 \gamma -2)+11)/24 > 0\) for any \(\gamma \in [0,1]\) and \(a>2\). This completes the proof that pooling equilibria exist, such that each target firm quotes the same acquisition price \(\Phi ^* \in [\underline{\Phi }, \overline{\Phi }]\).

1.7 A.7 Proof of Proposition 6

The difference between Eqs. (13) and (9) is equal to

for any \(\gamma \in [0,1]\) and \(a > 2\).

This shows that the best weak PBE earns the investor a higher expected payoff than the second-price separating auction, and since the second-price separating auction earns the investor a higher expected payoff than the first-price separating auction (see Proposition 4), our proof is complete.

1.8 A.8 Proof of Proposition 7

Any SE must allow for a range of bids in which the belief structure is consistent. The upper bound \(\overline{\Phi }\) results from the same line of reasoning as in the case of a weak PBE: it should not be more profitable for any firm to be regarded as a high-cost firm successfully underbidding the candidate equilibrium bid such that the investor would prefer this bid. This puts an upper bound on the bid as in the weak PBE. However, this is only a bound for the belief structure in the SE, and the equilibrium bid is unique, belongs also to the set of equilibrium bids of the weak PBE, but is larger than the lowest bid supported by a weak PBE. The reason is competition in the range in which the belief structure is given by the priors.

Suppose that both firms bid \(\tilde{\Phi }\) where \(\hat{\Phi } < \tilde{\Phi }\). The investor will immediately accept a marginally lower bid by any of the two firms because it does not change its belief structure and she can get a target firm for a cheaper price. The immediate implication is that bids work in a Bertrand manner in this range so that an equilibrium must imply \(\Phi _i = \hat{\Phi }\). At the same time, however, it should also not be profitable to quote a higher price in this range. A firm doing so will not change the belief structure of the investor, and while this firm will not be selected, it may be better off if the stand-alone profit is larger than the expected profit of competing against the rival for a partnership with the investor. Thus, the lower bound is now determined by the condition that \(\hat{\pi }^p(c_i)\) [see Eq. (12)] should not be smaller than the stand-alone profit under unrevised investor beliefs, that is \((2a - 1/2 + \gamma - 3 c_i)^2/36\). It is then straightforward to see that this constraint is fulfilled if it is met by the least-cost type \(c_i = 0\), leading to \(\hat{\Phi }\) as the equilibrium bid.

Along with consistency of beliefs discussed above, an SE requires that a sequence of purely mixed strategies exists such that the limit of this sequence is the SE.^{Footnote 20} Consider the following purely mixed strategies of each firm: choose \(\hat{\Phi }\) with probability \(\sigma _n = 1 - (1/n)\) and \(\Phi _i\) with probability \(1 - \sigma _n = (1/n)\) where \(\hat{\Phi } < \Phi _i\) and \(n \in \mathbb {N}\).

Thus, we define a sequence

The (constant) sequence of beliefs is consistent with the belief structure: as long as all bids are in the relevant range, the investor will stick to her priors.

A sufficient condition for \(\hat{\Phi }< \overline{\Phi }\) is \(\gamma \ge 1/3\) which guarantees existence of an SE because \(\lim _{n \rightarrow \infty }(\sigma _n, \mu _n) = (1, 1/2)\): a sequence exists that converges to the candidate equilibrium, and since the strategies are optimal given beliefs and the beliefs are derived from Bayes’ Rule for the optimal strategies, our proof for the SE is complete.

1.9 A.9 Proof of Proposition 8

Replacing \(\Phi ^{*}\) in Eq. (11) with \(\hat{\Phi }\), given by Proposition 7, yields the investor’s expected profit from firm acquisition via an SE pooling auction such that

The difference between Eqs. (A.9) and (6) is equal to

for any \(\gamma \in [0,1]\) and \(a > 2\), which proves the first part of Proposition 8.

The difference between Eqs. (A.9) and (9) is equal to

which is clearly positive for any \(\gamma \in [1/2,1]\) and \(a > 2\), which proves the second part of Proposition 8. As is clear from the equation above, \(\gamma \ge 1/2\) is only a sufficient condition for payoff dominance of the SE. In particular, we can solve for the threshold \(\gamma \) (as a function of \(a>2\)) at which the equation above is equal to zero, and hence above which the SE earns the investor a higher expected payoff than the second-price separating auction.

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