Optimal equity auctions with heterogeneous bidders

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ScienceDirect Journal of Economic Theory 166 (2016) 94–123 www.elsevier.com/locate/jet

Optimal equity auctions with heterogeneous bidders Tingjun Liu Faculty of Business and Economics, The University of Hong Kong, Pokfulam Road, Hong Kong Received 24 January 2014; final version received 15 August 2016; accepted 17 August 2016 Available online 24 August 2016

Abstract I analyze the effects of heterogeneity in terms of bidders’ valuation distributions and standalone values in equity auctions—auctions in which bidders offer equities rather than cash. Heterogeneity misaligns equity bids’ face values, monetary values, and bidder types, making the seller’s revenues sensitive to the auction design. Given heterogeneity, I identify the mechanism that maximizes the expected revenue among all incentive-compatible mechanisms of equity auctions. I show how different sources of heterogeneity alter the optimal design, revenues, and allocational efficiency. Unlike optimal cash auctions (Myerson, 1981), where bidders’ standalone values are irrelevant, the allocation of optimal equity auctions favors bidders with lower standalone values, because the seller can extract a larger proportion of their rents. By contrast, when bidders differ only in valuation distributions, optimal equity auctions feature more efficient allocations than optimal cash auctions. © 2016 Elsevier Inc. All rights reserved. JEL classification: D44; D82 Keywords: Auctions; Bidder heterogeneity; Equity bids; Mechanism design

1. Introduction Equity auctions have the core feature that bidders pay with equity shares (rather than cash) that entitle the seller to a fraction of the value of the combined entity of the bidder and the auctioned asset. Such auctions are widespread: Andrade et al. (2001) report 58% of mergers and E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jet.2016.08.005 0022-0531/© 2016 Elsevier Inc. All rights reserved.

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acquisitions are paid entirely in equity, and 70% involve at least some equity1; Skrzypacz (2013) reports oil and gas lease auctions typically feature equity payments in the form of royalties; and venture capital financing, procurement auctions, and lead-plaintiff auctions all use securities with predominantly equity components. The equity-auction literature has focused on ex-ante identical bidders with the same standalone values, investment costs, and distributions of valuations. In their seminal papers, Hansen (1985) and Riley (1988) obtain an important finding that equity bids can generate higher expected revenues than cash bids; and DeMarzo et al. (2005) (henceforth DKS) study general security-bid auctions and derive an elegant irrelevance result that standard (e.g., first- and second-price) formats of equity auctions yield the same expected revenue when bidders are ex-ante identical. While the assumption of ex-ante identical bidders simplifies the analysis and provides many insights into the workings of equity auctions, an understanding of the impact of bidder heterogeneity is of practical importance. Bidders in auctions often differ ex ante in their characteristics, such as standalone values or distributions of valuations.2 For example, in takeover auctions, bidders usually have different market values. In project-rights auctions, bidders could face different opportunity or financing costs. In addition, the findings of Gorbenko and Malenko (2014) suggest that bidders (e.g., strategic versus financial) in takeover auctions draw valuations from different distributions. I show bidder heterogeneity is important for the design and performance of equity auctions—more so than for those of cash auctions. The key reason is that unlike cash bids, the monetary values of equity bids are not transparent: the values depend not only on the bids’ face values (i.e., equity fractions), but also on bidders’ observable characteristics (e.g., standalone values) and private types (e.g., valuations). Concretely, in a takeover auction, an acquirer’s offer to pay a fraction θ of the merged firm’s equity has a monetary value of θ (VA + VT + s), where VA and VT are the standalone market values of the acquirer and target, respectively, and s is the synergy gain the acquirer can realize in the target, which is typically the acquirer’s private information. By contrast, the value of a cash offer does not vary with a bidder’s standalone value or synergy value. Bidder heterogeneity exacerbates this lack of transparency, affecting the performance of equity auctions.3 If bidders are ex-ante identical, expected revenue is insensitive to the auction design, and standard equity formats always generate higher expected revenues than cash auctions (Hansen, 1985; Riley, 1988; DKS). By contrast, if bidders are heterogeneous, expected revenues for different auction formats can vary widely and are sensitive to the nature of the heterogeneity. Indeed, standard equity-auction formats can even generate lower revenues than cash auctions—highlighting the need for equity-auction design. I contribute to the literature by investigating the optimal design of equity auctions with heterogeneous bidders. Despite allowing for bidder heterogeneity, I obtain a tractable formulation for the expected revenue in incentive-compatible mechanisms of equity auctions. Among all such mechanisms, I identify the one that maximizes expected revenue. I show how bid1 Cash constraints and institutional rigidities (e.g., existing leverage, bankruptcy risk concerns, and tax considerations) can lead to the use of equities; see Faccio and Masulis (2005) and Eckbo et al. (2015), among others. 2 Although heterogeneity in standalone values is not relevant for cash auctions, it is for equity auctions. 3 Heterogeneity affects the evaluation of equity offers. When bidders are ex-ante identical and employ symmetric strategies, bids’ face values, monetary values, and bidder types align: higher-type bidders submit bids with higher face and monetary values. This alignment facilitates the winning bid selection. By contrast, when bidders differ ex ante, this alignment breaks down and the bid ranking becomes ambiguous.

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der heterogeneity alters the optimal design, and derive the implications of different sources of heterogeneity. In my model, risk-neutral bidders compete to acquire a target in a takeover auction in which payments take the form of equities. Bidders privately observe their synergies with the target, which are distributed independently. The bidders’ and target’s standalone values and the synergy distributions are common knowledge. I solve for the optimal selling mechanism, incorporating two sources of heterogeneity: bidders may differ in their standalone values and distributions of synergies. Although I place the model in the context of takeover auctions, the results apply generally to the sale of indivisible assets through equity payments.4 I adopt Myerson’s (1981) classic mechanism-design approach for cash auctions, making important adjustments to account for the dependence of equity bids’ values on bidders’ private types. The relevant object for the seller is a bidder’s virtual valuation, which measures the rent a seller can extract from a bidder. The bidder with the highest virtual valuation wins in the optimal equity auction whenever that virtual valuation exceeds the target’s standalone value. I derive a bidder’s virtual valuation in equity auctions, showing how it depends on the bidder’s synergy value, standalone value, and distribution of synergy. I establish five key features of optimal equity auctions. First, their allocations depend on bidders’ standalone values. Given the same synergy distribution, the probability of winning is higher for bidders with lower standalone values—heterogeneity in standalone values results in inefficiencies in allocations (the highest-synergy bidder sometimes loses) in optimal equity auctions. This result contrasts sharply with that in optimal cash auctions (Myerson, 1981), in which bidders’ standalone values are irrelevant. To understand why, note that bidders derive informational advantages from having private information about their synergies, where the synergies affect their winning payoffs. Cash and equity auctions differ in the impact of synergies on bidders’ winning payoffs. At the margin, a bidder in a cash auction is the residual claimant: for a given bid, a $1 increase in the synergy translates into a $1 increase in the winning payoff. By contrast, for a given equity bid, an increase in the synergy is shared in proportion to the fraction of equity that was surrendered. Consequently, a bidder’s informational advantage in an equity auction is reduced and is scaled by the equity stake the bidder would retain upon winning. When bidders of different standalone values participate in an equity auction, lower-standalone-value bidders will have less informational advantage because they bid away greater and retain smaller equity stakes. Thus, a seller can extract a larger proportion of rents from lower-standalone-value bidders, making it optimal to let them win more often.5 Optimal equity and optimal cash auctions can lead to very different allocations. Consider a two-bidder example in which synergies are i.i.d. uniform on [1, 2], and the standalone value of the target is 3. In optimal cash auctions, both bidders are equally likely to win, regardless of their standalone values. By contrast, in optimal equity auctions, a “small” bidder with a standalone value of 3 is 23% more likely to win when facing a “large” bidder with a standalone value of 6, and 44% more likely to win when facing a “huge” bidder with a standalone value of 12. The contrasts highlight how allocations in optimal equity auctions favor bidders with lower standalone values. 4 For example, results carry over to project-rights auctions upon replacing standalone values with investment costs, and synergies with net present values. 5 In short, the intuition is to reward weaker (in terms of informational advantages) bidders that the seller can better exploit.

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The second key feature of optimal equity auctions is that heterogeneity in bidders’ synergy distributions induces fewer inefficient allocations (i.e., less social-welfare loss) than it does in optimal cash auctions. In other words, the two forms of bidder heterogeneity have different impacts on allocational efficiency: heterogeneity in standalone values makes optimal equity auctions less efficient than optimal cash auctions, but synergy-distribution heterogeneity does the opposite. Intuitively, heterogeneity introduces differences in bidders’ informational advantages, and inefficient allocations exploit such differences. In contrast to heterogeneity in standalone values, synergy-distribution heterogeneity introduces differences in bidders’ informational advantages in both types of auctions; and equity bids reduce bidders’ informational advantages, reducing the degree of their differences. As a result, synergy-distribution heterogeneity leads to fewer inefficiencies in optimal equity than in optimal cash auctions. The third key feature of optimal equity auctions is that losing bidders never pay. If bidders pay upon losing, their payments upon winning would decrease, raising the winner’s retained equity share. Because bidders’ informational advantages scale with their retained shares upon winning, the seller’s revenues fall. Thus, having only the winner pay is optimal. This result contrasts with optimal cash auctions in which losing bidders may also pay (e.g., all-pay formats). A bidder in a cash auction retains all of its equity upon winning, thereby retaining the full extent of its informational advantage whether or not it would pay upon losing. Thus, payments by losing bidders do not affect the rents the seller is able to extract in cash auctions. The fourth key feature of optimal equity auctions is that the probability of a sale is typically higher than in optimal cash auctions. In both optimal cash and optimal equity auctions, the seller retains the asset when bidders’ synergies are not sufficiently positive, because the rent a seller can extract from a bidder is less than the bidder’s synergy, and the seller optimally extracts only positive rents. Because the seller is able to extract more from equity than from cash bids, the threshold synergy value (i.e., reserve price) is generally lower in optimal equity than in optimal cash auctions, leading to a higher trading probability in optimal equity auctions. Finally, optimal equity auctions exhibit clean limiting properties. When bidders’ standalone values greatly exceed the possible value of the asset, both the allocation and seller revenue in optimal equity auctions approach those in optimal cash auctions. Intuitively, when bidders are large, the fraction of equity they pay upon winning is tiny: bidders converge to be the residual claimants of their synergies again, as in cash auctions. Differences in bidders’ informational advantages between equity and cash auctions vanish and the optimal equity-auction structure converges to that of an optimal cash auction. At the other limit when bidders’ standalone values are far less than the asset’s, optimal equity auctions achieve efficient allocations and the seller extracts all rents. Intuitively, bidders offer almost all of their equities if their standalone values are low. Bidders’ informational advantages—and hence rents—approach zero, allowing the seller to keep all surplus. Given these properties, optimal equity auctions respond to both sources of bidder heterogeneity and maximally exploit the features of equity bids. Consequently, they always generate higher expected revenues than optimal cash auctions, regardless of how much bidders differ ex ante. This result formalizes the strong intuition about the advantages of equity bids that the prior literature has derived (Hansen, 1985; Riley, 1988; DKS). Importantly, the key to the revenue superiority of optimal equity auctions lies in the fact that they simultaneously adjust for both forms of bidder heterogeneity; by contrast, any equity-auction format that only adjusts for one form of

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bidder heterogeneity would generate lower revenues than optimal cash auctions when the other form of bidder heterogeneity is substantial.6 My analysis of optimal equity auctions also sheds light on suboptimal equity-auction formats. Because optimal equity auctions extract the maximum possible revenue of any equity-auction format, they provide a benchmark for evaluating the merits of alternative formats. My work provides guidance for research on how standard equity-auction formats can be modified to account for bidder heterogeneity in simple, albeit slightly suboptimal ways. En route to deriving the optimal design, I obtain a tractable formulation for expected revenues, which provides guidance for how a seller should set optimal reserve prices for any (possibly suboptimal) equity-auction format. On a technical level, my work contributes to mechanism-design methodology. It generalizes the approaches for cash auctions to settings in which payments are equities whose values depend on bidders’ private information. My findings show the associated optimal mechanism is a broader mechanism whose outcomes reduce to those of optimal cash auctions only in limiting situations. The underlying connection between optimal equity and optimal cash auctions lies in the virtual valuation, which drives the optimal design in the two types of auctions. The virtual valuation in equity auctions subsumes its cash-auction counterpart, degenerating to the latter as bidders’ standalone values increase unboundedly. The rest of the paper proceeds as follows. Section 2 reviews the literature. Section 3 describes the model. Section 4 solves for optimal equity auctions. Section 5 derives their properties and implications. Section 6 concludes. Appendix A contains proofs. Appendix B provides further details on the analysis. 2. Related literature Myerson (1981) pioneered the mechanism-design approach and derives optimal cash auctions by formulating the concept of virtual valuations. A vast literature has since analyzed optimal mechanisms in various cash-auction settings.7 By contrast, I examine equity auctions. What complicates the analysis is that the approaches for deriving optimal cash mechanisms do not directly apply, because equity bids’ values depend on bidders’ private types.8 Nonetheless, through a set of transformations of bidders’ incentive conditions and of the seller’s objective function, I show the concept of virtual valuation extends, and via this concept, the optimal design can again be formulated. The virtual valuation in equity auctions has a finer structure than its cash-auction counterpart, and I derive the rich set of implications of this structure. Hansen (1985) was the first to examine equity auctions, demonstrating how equity bids can generate higher revenues than cash bids in second-price auctions. Cremer (1987) shows full rent extraction with negative transfers. Riley (1988) shows adding royalties to cash payments in6 Existing analyses of equity auctions do not adjust for heterogeneity in valuation distributions. They suboptimally adjust for heterogeneity in standalone values. 7 Among others, the literature has examined optimal cash mechanisms with risk-averse bidders (Maskin and Riley, 1984), resale (Calzolari and Pavan, 2006), asymmetrically informed bidders (Povel and Singh, 2006), and in dynamic settings with information arriving over time (Pavan et al., 2014). Studies have also examined cash-auction revenues in many other situations, including preemptive bidding (Fishman, 1988), sequential auctions (Bernhardt and Scoones, 1994), bidder cross-shareholdings (Dasgupta and Tsui, 2004), and club bidding (Marquez and Singh, 2013). 8 In cash auctions, Myerson (1981) shows allocation and payment can be separately determined, which simplifies the optimization. In equity auctions, however, a constraint arises in the form of an integral equation that links the allocation and payment, complicating the analysis.

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creases revenues. Other papers that study equity and security bidding include Samuelson (1987), Rhodes-Kropf and Viswanathan (2000), Zheng (2001), Board (2007), Povel and Singh (2010), Che and Kim (2010), Kogan and Morgan (2010), Gorbenko and Malenko (2011), Abhishek et al. (2015), and Cong (2015). Skrzypacz (2013) surveys the security-bid-auction literature. These papers focus on standard auction formats with symmetric bidders.9 By contrast, my paper incorporates bidder asymmetries and shows how they can significantly alter the auction design and affect the revenues. DKS provide a general analysis of security-bid auctions. They show that when the seller restricts bids to an ordered set and uses a standard auction format, steeper securities yield higher revenues, and the first-price auction with call options yields the highest revenue over a general set of auction mechanisms. They also establish revenue equivalence among standard formats when bidders are ex-ante identical and securities are linear (e.g., equities). Their paper and mine solve for optimal selling mechanisms under orthogonal constraints: they consider ex-ante identical bidders and incorporate general classes of securities, whereas I examine equity auctions and incorporate heterogeneous bidders. One feature of equity auctions is that the winner’s type is partially verifiable ex post, as the seller receives a fraction of post-auction cash flows that are correlated with the winner’s type. Relatedly, Mylovanov and Zapechelnyuk (2014) determine the optimal allocation in a labor market setting in which bidders compete for a job, the winner’s type is verifiable after hiring, and a penalty can be imposed on the winner based on the verification. Their model considers ex-ante identical bidders with an exogenously fixed payment (wage). By contrast, my paper considers heterogeneous bidders and solves the joint optimization of allocation and payment. Cremer and McLean (1985, 1988) study mechanism design with correlated valuations, showing all surplus can be extracted. Mezzetti (2004) shows an efficient mechanism exists in an environment in which bidders with interdependent valuations submit bids in two stages. In both papers, verifiable information about bidders’ types is revealed before all bids are submitted: with correlated valuations, a bidder’s own valuation conveys information about others’ types; and Mezzetti assumes bidders observe their realized decision-outcome payoffs—which reveal information about other bidders’ types—before submitting their second-stage bids, and bids from both stages determine payments. By contrast, in my paper, all bids are submitted before any information revelation; information about the winner’s type is revealed only once post-auction cash flows are generated, which is typically long after the bidding concludes. Liu (2012) examines an equity-auction format in which bidders submit monetary bids that are paid with equity after the auction, at a price determined ex post in a competitive market that observes the bidding. Bids reveal information about bidders’ private types, thereby influencing post-auction equity prices and hence bidder payoffs. The paper studies how such endogenous information revelation feeds back to affect bidding and equilibrium outcomes. By contrast, my present paper studies how a seller can make the best use of equity payments in light of their correlation with bidder types, and how bidder heterogeneity affects the seller’s solution.

9 When bidders have different standalone values, Hansen (1985) constructs a revised second-price equity auction that adjusts for the standalone values and achieves efficient allocations. However, as we have shown, when either source of bidder heterogeneity is present, allocational efficiency is not optimal for the seller: optimal equity auctions generate strictly higher expected revenues.

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3. The model A group of n ≥ 1 risk-neutral bidders competes to acquire an indivisible asset—a target firm. Bidder i (i = 1, ..., n) has standalone value Vi > 0 and values the target at xi , where xi is the target’s standalone value VT plus the synergy gains bidder i can realize upon the acquisition. obIf bidder i acquires the target, the value of the joint firm is Vi + xi . Bidder i privately  serves xi , independently drawn from cumulative distribution Fi with support x i , x¯i , where  0 ≤ x i < x¯i < ∞. Fi admits a continuous density fi ≡ Fi and fi (xi ) > 0 for all xi ∈ x i , x¯i . The standalone values VT , Vi , and distributions Fi (i = 1, ..., n) are common knowledge. The contribution of my paper is to introduce heterogeneity in Vi and Fi , allowing them to differ across bidders.10 Definition 1 (Heterogeneity). Bidders are ex-ante identical if and only if Vi = Vj and Fi = Fj for all i, j . Bidders are heterogeneous if they are not ex-ante identical. The target is sold via an equity auction. An equity auction maps the vector of bids into (1) a vector of winning probabilities and (2) a vector of equity shares bidders pay, where the winner pays with shares of the joint firm and losers pay with shares of their standalone firms. Thus, an equity-auction mechanism (, W, Q) consists of: (1) a set of possible bids (or more broadly, strategy options) i for each i ∈ B, where B ≡ {1, 2, ..., n} denotes the set of bidders; (2) a winning rule W :  → Rn+1 ≥0 , where Wj (θ) is the probability that bidder j ∈ A ≡ {0, 1, 2, ..., n} wins when bids are θ (with j = 0 representing the situation in which the target is not sold) and j ∈A Wj (θ ) = 1 for all bids θ ∈ ; and (3) an equity-retention rule Q :  × A → Rn≥0 , where Qi (θ, j ) is the fraction of equity bidder i ∈ B retains (i.e., 1 − Qi is the fraction bidder i pays) when bids are θ and bidder j ∈ A wins, and Qi (θ , j ) ∈ [0, 1]

(1)

for all θ, i, j . Further, denote by μ :  × A → Rn the associated monetary transfer, where μi (θ , j ) is the value of the payment of bidder i ∈ B when bids are θ and j ∈ A wins. Then  (1 − Qi (θ, i)) (Vi + xi ) if i = j . (2) μi (θ , j ) = if i = j (1 − Qi (θ, j )) Vi Examples of equity auctions. Equity auctions as defined above encompass infinitely many possible forms. To illustrate, I describe two standard forms. In a standard first-price equity auction, each bidder submits a bid θi ∈ [0, 1], the highest bid wins, and ties are broken randomly. The winner pays an equity fraction that equals its own bid, and losing bidders do not pay. Thus, if bidder i wins the auction with bid θi , its payment is θi (Vi + xi ), which is also the seller’s revenue. In a standard second-price equity auction, each bidder submits a bid θi ∈ [0, 1], and the highest bid wins with ties broken randomly. The winner pays an equity fraction that equals the second-highest bid, and losing bidders do not pay. Thus, if bidder i wins and the second-highest bid is θj , the winner’s payment—and hence the seller’s revenue—is θj (Vi + xi ). 10 Although my paper focuses on heterogeneous bidders, the optimal auction design also applies to ex-ante identical bidders, including the case of a single bidder (n = 1).

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3.1. Discussions and numerical examples Equity auctions exploit the fact that the cash flows generated by the auctioned asset or project are ex-post verifiable. Therefore, their realization can be used as a basis for payment. Because the realized cash flows are correlated with the winning bidder’s ex-ante private type, equity payments’ values are tied to the winner’s type, placing my model outside the standard Myerson (1981) framework. I assume for simplicity that bidders know their ex-post valuations (i.e., the future cash flows) with certainty. More generally, my model applies to settings in which a bidder’s valuation, conditional on its private type, is stochastic. Because equity payments’ values are proportional to the realized valuations and agents are risk neutral, all results hold if we define a bidder’s type to be the expected value of the realized valuations. That is, my analysis of equity auctions extends when the winner’s type is only partially, and not fully, revealed ex post. My assumption that xi ≥ 0 is consistent with bidders’ rationality considerations: suppose bidder i with xi < 0 participates in an equity auction and wins; then its profit is Qi (θ , i) xi − (1 − Qi (θ , i)) Vi , which, by (1) and Vi > 0, is strictly negative. If i loses, its profit is nonpositive. Thus, bidder i’s expected profit would be strictly negative if it has any chance of winning. Hence, individual rationality rules out xi < 0. The value of winner i’s equity payment depends on both xi and Vi , as (2) shows. Consequently, both sources of heterogeneity (Definition 1) are relevant. Heterogeneity breaks the revenue equivalence among standard (e.g., first- and second-price) equity auctions. This is reminiscent of how revenue equivalence in cash auctions breaks down with asymmetric bidders. However, the impact of asymmetries for equity auctions is larger. In cash auctions, only heterogeneity in bidders’ valuation distributions affects revenues. By contrast, in equity auctions, both sources of heterogeneity do. Moreover, they have different implications for the auction design as I will show. Bidder heterogeneity also affects the relative performance of equity versus cash auctions. Standard equity formats always generate higher expected revenues than cash auctions absent bidder heterogeneity, but not when bidders differ ex ante. The two examples below illustrate such a contrast. Example 1 (Ex-ante identical bidders). Two bidders and the asset have the same standalone value V1 = V2 = VT = 3. The bidders’ valuations for the asset are uniformly distributed over [4, 5]. First consider a standard second-price cash auction: bidders make cash offers, and the highest bidder wins, paying the second-highest offer. Bidding their actual valuations is a dominant strategy for bidders: for instance, if x1 = 5 and x2 = 4, bidder 1 bids 5 and wins, paying bidder 2’s bid of 4. Integrating over x1 and x2 yields expected revenues of E [min {x1 , x2 }] = 4.33. Now consider the standard second-price equity auction. As in a second-price cash auction, submitting a bid whose monetary value (conditional on the bidder winning) equals the bidder’s i valuation is a dominant strategy. Thus, i will bid Vix+x . For instance, if x1 = 5 and x2 = 4, i

bidder 1 bids 85 and bidder 2 bids 74 . Thus, bidder 1 wins and pays bidder 2’s bid, corresponding to a value of 47 × (5 + 3) = 4.57. Integrating over x1 and x2 yields expected revenues of   min{x1 ,x2 } E (3 + max {x1 , x2 }) 3+min{x = 4.53. } ,x 1 2 This example is based on Hansen (1985) and demonstrates how the second-price equity auction generates higher revenues than the second-price cash auctions with ex-ante identical

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bidders.11 As Hansen (1985) and DKS show, the gains come from the fact that with the same equity bid, a higher-type bidder pays more in monetary terms. Thus, equity bids effectively lower the difference between the winner’s valuation and that of the second-highest bidder, and because this difference corresponds to bidders’ rent, reducing it benefits the seller. Example 2 (Heterogeneous bidders). The asset has a standalone value of 3, and two bidders have the same standalone value of 18. One bidder’s valuation forthe asset is uniformly distributed over  1 [3, 6] and the other’s is uniformly distributed over 4 4 , 4 34 . The key feature of Example 2 is that bidders differ in their valuation distributions. In cash auctions, Myerson (1981) identifies the optimal mechanism that adjusts for such differences. In this example, these optimal adjustments generate an expected revenue of 4.51. In equity auctions, by contrast, existing theories do not prescribe such adjustments. The standard second-price equity auction that does not adjust for valuation distributions generates an expected revenue of only 4.26. However, the seller can increase expected revenue in the second-price equity auction by imposing appropriate reserve prices (although such a format is still not the optimal mechanism that I will formulate with heterogeneous bidders). Specifically, let {0 ≤ ri ≤ 1}ni=1 be the discriminatory reserve prices, and let θi be the bid submitted. If θi ≥ ri , bidder i wins and pays  highest  an equity fraction max ri , maxj =i θj ; if θi < ri , no bidder wins. Corollary 5 in Appendix B derives the reserve prices that maximize the expected revenue. With these optimal reserve prices, the expected revenue from the second-price equity auction rises to 4.48 but is still less than the expected revenue from the optimal cash auction. If bidders also differ in their standalone values, such differences do not affect cash auctions but further complicate equity auctions. Equity formats that do not properly adjust for standalone values (in addition to not adjusting for valuation distributions) would generate even lower revenues. These examples motivate the following questions: When bidders differ ex ante, what forces drive equity auctions’ revenues? What is the optimal mechanism that generates the highest expected revenue? How does the optimal mechanism respond to different forms of bidder heterogeneity? Does it always outperform optimally designed cash auctions, even if heterogeneity is substantial? My formal analysis provides answers to these questions. 4. Mechanism design with equities In section 4.1, I derive a tractable formulation for the expected revenue in any incentivecompatible equity mechanism. In section 4.2, I identify the equity mechanism that maximizes the expected revenue. 4.1. Expected revenues in any incentive-compatible equity mechanism Let f (x) ≡ ni=1 fi (xi ) denote the joint density of x ≡ (x1 , x2 , ..., xn ), and for all i ∈ B, let f−i (x−i ) ≡ k =i fk (xk ) denote the joint density of x−i ≡ (x1 , ..., xi−1 , xi+1 , ..., xn ), and let 11 In this example, the second-price cash auction (with no reserve price) is also the optimal cash auction; thus, the second-price equity auction generates higher revenues than cash auctions of any format.

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  χi ≡ x i , x¯i denote the set of bidder i’s valuations. Let χ ≡ ×nk=1 χk denote the product of bidders’ valuation sets, and let χ −i ≡ ×k =i χk denote the product of these sets excluding i. In cash auctions, the revelation principle (Myerson, 1981) shows that for any equilibrium of a mechanism, allocations and monetary transfers can be replicated by a truthful equilibrium of some direct mechanism in which the set of bids coincides with the set of values (i.e., i = χi for all i). In such direct mechanisms, the outcomes depend only on bidders’ reported types. In equity auctions, monetary transfers depend on both reported and actual types, in contrast to cash auctions. Nonetheless, because the equity-retention rule Q does not depend on bidders’ actual types, the revelation principle extends to equity auctions: Lemma 1. Given an equity-auction mechanism and an equilibrium for that mechanism, a direct mechanism (χ, W, Q) exists in which (1) W and Q depend only on bidders’ reported values, (2) it is an equilibrium for each bidder to report its value truthfully, and (3) the outcomes (W, Q, μ), where μ is given by (2), are the same as in the given equilibrium of the original mechanism. Thus, we consider a direct-revelation mechanism without loss of generality. Denote by vi (xi , zi ) bidder i’s expected profit when it has xi but reports zi , and all other bidders report truthfully; then   vi (xi , zi ) = (Vi + xi ) Qi (zi , x−i , i) − Vi Wi (zi , x−i ) f−i (x−i ) dx−i − ωi (zi ) , (3) χ −i

where ωi (zi ) ≡

j =iχ

Vi (1 − Qi (zi , x−i , j )) Wj (zi , x−i ) f−i (x−i ) dx−i .

(4)

−i

The first term on the right-hand side of (3) is bidder i’s expected profit absent payments upon losing, and ωi (xi ) is the reduction from payments upon losing. To express (3) more concisely, denote by Gi (zi ) bidder i’s winning probability when it reports zi and all others report truthfully: Gi (zi ) = Wi (zi , x−i ) f−i (x−i ) dx−i . (5) χ −i

Analogously, denote by qi (zi ) the expected fraction of the merged entity that bidder i retains contingent upon winning if it reports a value zi and all others report truthfully: qi (zi ) Gi (zi ) = Qi (zi , x−i , i) Wi (zi , x−i ) f−i (x−i ) dx−i . (6) χ −i

Then (3) becomes

  vi (xi , zi ) = (Vi + xi ) qi (zi ) − Vi Gi (zi ) − ωi (zi ) .

(7)

Denoting bidder i’s equilibrium expected profit by ui (xi ) ≡ vi (xi , xi ) ,

(8)

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the incentive-compatibility condition gives ui (xi ) = max vi (xi , zi ) ,

(9)

zi

which, by (7) and the general envelope theorem (see Milgrom and Segal, 2002), yields  ui (xi ) = ui x i +

xi (10)

qi (t) Gi (t) dt. xi

Equation (10) is instructive. The term qi is less than 1 in equity auctions, whereas it is 1 in cash auctions. Thus, (10) shows the differential rents a high-valuation bidder earns over a low-valuation bidder in equity auctions are less than those in cash auctions. It follows that under appropriate boundary conditions, bidders’ overall rents are lower, and the seller’s revenues are higher. This result reflects the insights of Hansen (1985) and DKS that because equity (or general security) bids tie payments to the winner’s actual type, a higher-type bidder will pay more even with the same bid, which benefits the seller as implied by the linkage principle (Milgrom and Weber, 1982). Equation (10) also reveals that a bidder’s (differential) rent is proportional to qi , the extent to which the bidder retains its equity upon winning. Intuitively, a high-valuation bidder can always deviate by mimicking a low-valuation bidder and, upon winning, enjoy higher rents (due to its higher valuation) than the low-valuation bidder. Such a rent difference is scaled by the fraction of equity the bidder would retain upon winning.12 To prevent such a deviation, in equilibrium, the differential rents a high-valuation bidder earns over a low-valuation bidder scale accordingly. This scaling property underlies many of the results in the paper. Bidder i’s contribution to the seller’s expected profit is x¯i πs,i =

x¯i Gi (xi ) (xi − VT ) fi (xi ) dxi −

xi

ui (xi ) fi (xi ) dxi ,

(11)

xi

where the first term is the bidder’s contribution to the expected increase in social welfare, and the second is the bidder’s expected profit. Summing over these contributions yields the seller’s total expected profit. Adding VT , the seller’s expected revenue is πs =

n

πs,i + VT .

(12)

i=1

Plugging in (10), the second term in (11) becomes ⎞ ⎛ x¯i x¯i xi ⎟ ⎜  ui (xi ) fi (xi ) dxi = − ⎝ui x i + qi (t) Gi (t) dt ⎠ d (1 − Fi (xi )) xi

xi

 = ui x i +

xi

x¯i (1 − Fi (xi )) qi (xi ) Gi (xi ) dxi , xi

12 The rent difference is the valuation difference multiplied by the equity fraction the low-valuation bidder retains upon winning.

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where integration by parts is used. Thus, x¯i πs,i =

x¯i Gi (xi ) (xi − VT ) fi (xi ) dxi −

xi

 (1 − Fi (xi )) qi (xi ) Gi (xi ) dxi − ui x i . (13)

xi

The right-hand side of (13) depends on qi (xi ). Recall that in cash auctions, qi (xi ) is always 1, regardless of xi . By contrast, in equity auctions, qi (xi ) is not only less than 1 but also not constant, and its functional form depends on both of the primitive functions of the mechanism, Wi and Qi , via (6). This feature complicates the analysis of equity auctions and gives rise to implications that differ from those in cash auctions. Plugging (7) into (8) and equating with the right-hand side of (10) yields: 



 (Vi + xi ) qi (xi ) − Vi Gi (xi ) − ωi (xi ) = ui x i +

xi qi (t) Gi (t) dt,

(14)

xi

which imposes a constraint on the function qi in the form of an integral equation. Through a set of transformations of the constraint (14) and the seller’s objective function, the following theorem eliminates the function qi from (13) and expresses the optimization program in a form that does not contain the equity-retention function Qi . Theorem 1 (Revenue decomposition and existence of virtual valuation in equity auctions). In any incentive-compatible mechanism of equity auction, the seller’s expected revenue (12) decomposes: πs = πs,a + πs,b + πs,c , where πs,a ≡ −

n

i=1

(15)

⎧ ⎫ ⎪ ⎪ x¯i ⎨ ⎬  1  (1 − Fi (xi )) dxi + 1 , ui x i ⎪ ⎪ ⎩ Vi + x i ⎭ xi

⎫ ⎧ ⎪ ⎪ n x¯i ⎬ ⎨ ω (x ) xi ω (t)

i i i πs,b ≡ − + dt dxi , (1 − Fi (xi )) ⎪ (Vi + t)2 ⎪ ⎭ ⎩ Vi + xi i=1 x

πs,c ≡

i



n χ

(16)

(17)

xi



Wi (x) φi (xi ) + W0 (x) VT f (x) dx,

(18)

i=1

and φi (xi ) is the virtual valuation defined in (19). Definition 2. The virtual valuation in equity auctions is  x¯ Vi xii (1 − Fi (t)) dt Vi (1 − Fi (xi )) φi (xi ) ≡ xi − . − (Vi + xi ) fi (xi ) (Vi + xi )2 fi (xi )

(19)

 The first term πs,a is bounded by bidders’ rationality constraints: because ui x i ≥ 0  x¯ and x i (1 − Fi (xi )) dxi > 0, the maximum possible value for πs,a is zero, which obtains if i

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 ui x i = 0 for all i. The second term πs,b represents contribution to the expected revenue from losing bidders’ payments. all terms on the right-hand side of (4) are nonnega xBecause (xi ) i ωi (t) tive, ωi (xi ) ≥ 0 and ωVii +x + dt ≥ 0 for all xi . Thus, the maximum possible value 2 x i i (Vi +t) for πs,b is zero, which obtains if losing bidders do not pay. The third term πs,c represents n  contribution from the auction’s allocations. Note the term Wi (x) φi (xi ) + W0 (x) VT equals VT +

n 

i=1

Wi (x) (φi (xi ) − VT ). Hence, allocating only to the bidder with maximal φi (xi ) is op-

i=1

timal, provided that φi (xi ) exceeds the asset’s standalone value VT . This would maximize the expression at every point x and thus would maximize πs,c . To better understand Theorem 1, consider the corresponding cash-auction result for comparison. In cash auctions, Myerson (1981) develops the concept of virtual valuation, showing it represents the rents the seller can extract from a bidder who offers cash. Myerson (1981) obtains the form of the virtual valuation as a function of the bidder’s actual valuation and its distribution:

ψi (xi ) ≡ xi −

1 − Fi (xi ) . fi (xi )

(20)

When bidders offer securities, on the other hand, because of the complication that security bids’ values depend on bidders’ private types, the classic cash-auction approaches do not directly apply, and a priori, whether virtual valuations can even be defined in such settings is unclear. Theorem 1 demonstrates an important result that the concept of virtual valuation holds in equity auctions; namely, a function exists for an equity-offering bidder in terms of the bidder’s private type and any common knowledge (e.g., bidders’ valuation distributions and standalone values), which corresponds to the rents the seller can extract. Section 4.2 shows optimal equity auctions can again be formulated via this solution concept. The virtual valuation for an equity auction has a clean representation (equation (19)) and entails a finer structure than its cash-auction counterpart, with an additional dependence on the bidder’s standalone value. In section 5, I derive the implications of this structure, showing it is crucial for understanding how heterogeneity alters the optimal auction design.13 Theorem 1 relates to the elegant result of DKS that all symmetric and increasing mechanisms generate the same expected revenues when securities are linear (e.g., equities). The existence of virtual valuations in equity auctions, as Theorem 1 establishes, implies the expected revenue depends only on the allocations (given appropriate boundary conditions), which is consistent with the insights of DKS. Technically, due to the linearity of equities, the expected revenue and bidder i’s payoff depend on the payment function (1 − Qi ) through its reduced form (1 − qi ) in an affine fashion. Making appropriate transformations and exploiting the envelope theorem, this affine feature allows the seller’s objective to be expressed in terms of virtual valuations. In a separate work, I extend the analysis to show that virtual valuations can be defined for all linear securities.

13 Theorem 1 also provides guidance for how to set optimal reserve prices for suboptimal formats to maximize their performance (e.g., Corollary 5 in Appendix B).

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4.2. Optimal equity mechanisms An optimal equity mechanism maximizes the expected revenue (equation (12)) subject to the incentive compatibility (equation (9)) and individual rationality (ui (xi ) ≥ 0 for all i and xi ). I derive all optimal equity mechanisms under the following condition, which I impose through the rest of the paper.   Assumption 1. The design problem is regular: φi (·) strictly increases over x i , x¯i for all i. The regularity condition holds as long as the distribution fi (·) does not decrease too quickly. Satisfying this condition is typically easier than satisfying its analogue in cash auctions. In both equity and cash auctions, a bidder’s virtual valuation is less than its actual valuation xi , and the difference represents the bidder’s rents. Because xi increases in itself at a rate of 1, the regularity condition holds unless this difference (between the actual and virtual valuations) increases in xi with a rate of more than 1. Because the bidder captures less rent in equity auctions, this difference—and its rate of change—is generally smaller, and hence the regularity condition is more likely to hold. In particular, when the standalone value of the bidder is much lower than that of the seller, the seller extracts almost full rents (Corollary 4) and the regularity condition holds for any distribution fi (·). Assumption 2 of section 5 provides a simple sufficient condition for regularity. Under the regularity condition, an incentive-compatible and individually rational mechanism exists that maximizes πs,a , πs,b , and πs,c simultaneously; it follows that the mechanism maximizes the sum of πs,a , πs,b , and πs,c and is therefore optimal. Below, I first present a necessary condition for optimality. Lemma 2. Losing bidders never pay in the optimal equity mechanism: Qi (x, j ) = 1 for all i = j . Lemma 2 reflects the intuition that payments by losing bidders tend to reduce payments upon winning, increasing the amount of equity the winning bidder retains. Because informational rents scale with the equity retention, bidders’ rents rise, reducing the seller’s revenues. This result contrasts with optimal cash auctions, in which losing bidders may also pay. Intuitively, because bidders in cash auctions retain all of their equities when they win, whether they would pay upon losing does not affect their informational advantages—and hence does not affect the seller’s revenues. Proposition 1 (The set of all optimal equity mechanisms). A direct-revelation mechanism is optimal if and only if (i) the winning rule is     1 if φi (xi ) > maxj =i φj xj and φi (xi ) ≥ VT    Wi (x) = , (21) 0 if φi (xi ) < maxj =i φj xj or φi (xi ) < VT where φi (xi ) is the virtual valuation in (19), for all i and x, (ii) losing bidders do not pay, and (iii) the equity-retention rule upon winning satisfies xi Vi Gi (t) Vi Gi (xi ) Qi (xi , x−i , i) Wi (xi , x−i ) f−i (x−i ) dx−i = + dt Vi + xi (Vi + t)2 χ −i

xi

for all i and xi , where Gi (·) is given by (5), and (1) holds for all x, i, j .

(22)

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Corollary 1 (An optimal mechanism). The following constitutes an optimal equity mechanism: the winning and payment rules in (i) and (ii) of Proposition 1, and the equity-retention rule Qi (x, i) =

Vi , Vi + yi (x−i )

(23)

where

  !    yi (x−i ) ≡ φi−1 max max φj xj , VT

(24)

j =i

is the minimum value of xi that corresponds to a virtual valuation exceeding VT and allows i to win against x−i , and φi−1 (·) denotes a bounded inverse of φi (·): ⎧  ⎪ if x < φi x i ⎨x i φi−1 (x) ≡ x¯i . (25) if x > φi (x¯i ) ⎪      ⎩ y ∈ x i , x¯i s.t. φi (y) = x if x ∈ φi x i , φi (x¯i ) Part (i) of Proposition 1 shows optimal equity mechanisms select the bidder with the highest virtual valuation as the winner, provided it exceeds VT . This result reflects the intuition that the virtual valuation represents the rents the seller can extract from a bidder, and it is optimal to extract only rents exceeding the standalone value of the auctioned asset (equation (18)). Part (ii) of Proposition 1 reflects the suboptimality of losing bidders’ payments (Lemma 2). Part (iii) of Proposition 1 establishes a constraint on the equity-retention rule, which ensures the incentive compatibility and individual  rationality of the mechanism and the boundary condition required by optimality (i.e., ui x i = 0). Note (22) does not restrict the equity-retention rule to a specific form. Thus, the optimal mechanism can be constructed in multiple ways with different equity-retention forms, as long as (22) holds. Equation (23) is one such construction; the associated mechanism is analogous to the second-price auction in that the winner’s payment does not depend on its own bid, and truth-telling is weakly dominant. Implementations. Optimal equity mechanisms in Proposition 1 are derived using the revelation principle, and hence they take a form in which bidders report their values. It would be interesting to explore how to implement these mechanisms in auctions in which bidders bid equity fractions they would pay, and the auction rules retain some of the key features of standard formats. In the special case in which bidders are ex-ante identical, implementations of optimal equity auctions are simple: all standard equity auctions (in which losers do not pay) with an optimal reserve price are optimal. Corollary 2. (i) When bidders are ex-ante identical, standard first- and second-price equity auctions with a reserve price of

φi−1 (VT )

Vi +φi−1 (VT )

are optimal equity auctions.

(ii) With a single bidder (n = 1), making a take-it-or-leave-it offer of

φi−1 (VT )

Vi +φi−1 (VT )

is optimal.

When bidders are heterogeneous, standard equity auctions (even with optimal discriminatory reserve prices) are no longer optimal. Implementations of optimal equity auctions are more complicated, which I investigate in a separate study.

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Extensions. From a mechanism-design perspective, equity auctions impose a constraint that the monetary transfers are linear functions of bidder types. Extensions could involve allowing for cash in addition to equity payments, or payment rules that depend nonlinearly on cash flows. If the mechanism designer could choose payment functions that depend on the realized cash flows in arbitrary forms without restrictions, the optimal mechanism would be trivial: the seller could extract full rents by using negative cash payments, as Cremer (1987) has shown. Concretely, the seller can “buy” bidders by reimbursing the winner’s standalone value or investment cost, ask for bidders’ types, sell to the highest type and keep all revenues. Negative cash transfers effectively lower bidders’ standalone values; and we previously established that the informational advantages of bidders decline as standalone values decrease, vanishing as standalone values go to zero (see also Corollary 4). I do not allow for negative cash payments. In practice, such payments are not observed. In addition, to extract full rents, the cash amount would need to be no less than the standalone value of the largest bidder—and bidders in takeovers tend to be significantly larger than the seller (target). Moreover, DKS show how moral hazard considerations can preclude negative cash payments. My working-paper version considers a model in which bidders pay with combinations of equity and nonnegative cash, showing that accepting only equity payments with no cash components is optimal for the seller. DKS study bidding with general securities whose values may derive nonlinearly from the underlying cash flows. 5. Properties and implications The virtual valuation φi (xi ) is a central component of this paper’s analysis. I first examine its limiting behaviors. Corollary 3. limVi →∞ φi (xi ) = ψi (xi ), where ψi (xi ) is the virtual valuation for cash auctions. Corollary 3 reflects the intuition that the fraction of equity a large bidder pays upon winning is close to zero; thus, the bidder would retain almost all of its equity, as in cash auctions. Because a bidder’s informational advantage is proportional to the equity retention, the seller is able to extract the same amount of rents as in cash auctions. The corollary adds flexibility to the results. In optimal equity auctions, all bidders offer equity. However, Corollary 3 suggests the limit Vi → ∞ corresponds to bidder i offering cash. When all bidders’ standalone values increase unboundedly, optimal equity auctions reduce to optimal cash auctions: both the allocation and the seller’s revenue approach those in optimal cash auctions.14 Furthermore, upon taking the limit Vi → ∞ for only a subset of bidders, the formulation of optimal equity auctions applies to a “hybrid” setting in which some bidders must offer cash while others offer equity (the working-paper version provides a formal treatment). Corollary 3 corresponds to the lower bound on the expected revenues optimal equity auctions can generate. In another limiting scenario, the upper bound obtains: Corollary 4. limVi →0 φi (xi ) = xi . 14 In addition, when all bidders’ standalone values are large, the revenue decomposition for equity auctions (equation (15)) reduces to its cash-auction counterpart; see Corollary 6 in Appendix B.

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Corollary 4 corresponds to full rent extraction. Intuitively, a bidder with a low standalone value would offer nearly 100% of its equity. Bidders’ informational advantages—and hence rents—approach zero, allowing the seller to keep all surplus. In the general case of finite standalone values, φi (xi ) is given by (19). Note that in both cash and equity auctions, the virtual valuation is less than xi and the difference represents the bidder’s i (xi ) rents. In cash auctions, this difference is 1−F fi (xi ) . In equity auctions, the difference has two components: the second and third terms in (19). The second term is the corresponding second i term in the virtual valuation for cash auctions scaled by a factor, ViV+x , which is the fraction of i equity the bidder would keep to break even upon winning. This factor represents a “royalty-rate” effect: consider a setting of constant royalty rates in which winner i pays with a combination of cash and a fixed fraction αi of equity; similar algebra as in section 4.1 shows the virtual i (xi ) valuation exists in this setting and is xi − (1 − αi ) 1−F fi (xi ) for bidder i, where (1 − αi ) is the bidder’s retained equity fraction upon winning.15 i The factor ViV+x endows the virtual valuation with a scaling property, making it decrease in i the extent to which the bidder would retain its equity upon winning. This feature is the driving force for the limiting behaviors of the virtual valuation examined earlier, and it has additional implications I will describe. The third term in the virtual valuation is a correction to the fact that the actual fraction of i equity the bidder retains upon winning is less than ViV+x and is not a constant. This term is i particularly sensitive to the upper tails in the valuation distribution: from (19), its ratio to the  x¯i 1 1 second term in the virtual valuation is Vi +x multiplied by a factor, xi (1 − Fi (t)) dt, 1−F (x ) i i i where the factor measures the extent of information asymmetry, or the “fatness” of the tails, in the upper portion of the valuation distribution. This third term is a unique feature of equity auctions that differs from cash auctions. Its sensitivity to the high tails reflects the underlying difference between equity and cash auctions in that equity bids tie payments to the winner’s actual value, whereas cash bids do not. Thus, the seller’s rents, and hence the virtual valuations, are more sensitive to the distribution tails in equity than in cash auctions. In section 5.1, I show a sufficiently fat upper tail can cause this term to dominate the second term and significantly affect the optimal auction design. For typical valuation distributions, however, the extent of fat tails tends to be small, and the third term represents only a small correction. Below is a simple condition on the valuation distribution that restricts the size of the fat tails.   Assumption 2. The distribution fi (·) is nondecreasing over x i , x¯i for all i. This condition also guarantees the regularity condition in Assumption 1: Lemma 3. The design problem is regular under Assumption 2. Below, I derive a number of results under Assumption 2 (in the working-paper version, I show these results also hold under more general conditions). 15 The form of the virtual valuation with constant royalty rates can also be understood by noting (19) approaches  1−Fi xi Vi  in the limit that VT and Vi are much larger than the synergy xi − VT . Intuitively, under such a xi − V +V T i fi xi Vi limit, the fraction of equity the bidder retains is a constant V +V , independent of the bidder’s valuation. T i

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Lemma 4. Under Assumption 2, for any valuation xi such that ψi (xi ) ≥ VT , bidder i’s virtual valuation for equity auctions exceeds its cash-auction counterpart: φi (xi ) ≥ ψi (xi ), where strict inequality holds for xi < x¯i . Lemma 4 reflects the intuition that equity bids allow the seller to extract more rents than cash bids, which increases the virtual valuation. This result has implications for trading probabilities in optimal auctions. In both optimal equity and optimal cash auctions, if the highest virtual valuation among all bidders is below the asset’s standalone value VT , the seller retains the asset even when some bidders have actual valuations exceeding VT and a trade would result in social gains. Because virtual valuations are higher in equity than in cash auctions as Lemma 4 shows, socially beneficial trade occurs more frequently in optimal equity auctions: Proposition 2. Under Assumption 2, the probability that the asset is sold in optimal equity auctions is at least as high as in optimal cash auctions. The virtual valuation for equity auctions depends on the bidder’s standalone value: from (19), φi (xi ) depends on Vi in both the second and third terms. The second term (with its minus sign) decreases in Vi for all values of xi ; the third term, however, may either decrease or increase in Vi , depending on xi . When the third term is small, φi (xi ) decreases in Vi for all values of xi . Lemma 5. Under Assumption 2, for any bidder i and xi such that φi (xi ) ≥ VT , where strict inequality holds for xi < x¯i .

∂φi (xi ) ∂Vi

≤ 0,

Proposition 3. Suppose there are n ≥ 2 bidders, and two of them have the same valuation distribution but different standalone values. Under Assumption 2, the lower-standalone-value bidder has a higher probability of winning in optimal equity auctions than the other bidder. The intuition for Proposition 3 (and Lemma 5) reflects that upon winning, a bidder with a lower standalone value pays a larger and hence retains a smaller equity stake. Because bidders’ informational advantages scale with the equity retention, a seller can extract a larger proportion of rents from lower-standalone-value bidders, making it optimal to let these bidders win more often. More broadly, the intuition reflects the optimality of rewarding weaker (in terms of informational advantages) bidders that the seller can better exploit. Proposition 3 sharply contrasts with the results in optimal cash auctions, in which the allocations do not depend on bidders’ standalone values, because bidders in cash auctions retain 100% equity stakes, so that their informational advantages are independent of standalone values. Optimal equity and optimal cash auctions can lead to significantly different allocations. For instance, consider a two-bidder setting in which valuations are i.i.d. uniform on [4, 5], and the standalone values of the smaller bidder and the auctioned asset are both 3. In optimal cash auctions, each bidder is equally likely to win. By contrast, in optimal equity auctions, a bidder whose standalone value is half that of the larger bidder is 23% more likely to win, and a bidder whose standalone value is one quarter that of the larger bidder is 44% more likely to win. Such contrasts highlight the extent to which allocations of optimal equity auctions favor lower-standalone-value bidders relative to optimal cash auctions. In addition to responding to standalone values, optimal equity auctions also respond to bidders’ valuation distributions. Suppose bidders have the same standalone values and the means of their valuation distributions are the same, but the variances of the valuation distributions differ.

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Then, in the optimal auction, bidders whose valuation distributions have lower variances tend to have a higher probability of winning. The intuition is that the seller’s knowledge of such bidders’ valuations is more precise, which effectively gives the seller an outside option, allowing the seller to demand higher bids from other bidders. This intuition is similar to that for why setting a reserve price above a seller’s reservation value can allow the seller to extract more rents.16 The response of optimal equity auctions to bidders’ differing valuation distributions is in the same broad direction as that of optimal cash auctions, because virtual valuations in both auctions Vi i (xi ) share the same term, 1−F fi (xi ) . However, the magnitudes differ: due to the scaling factor Vi +xi , optimal equity auctions typically feature fewer inefficiencies when bidders differ only in valuation distributions. Intuitively, in equity auctions, a seller can extract more rents from bidders; hence, virtual valuations tend to be closer to bidders’ actual valuations, and differences in virtual valuations due to bidders’ differing valuation distributions fall. Because the seller optimally selects the winner based on virtual valuations, optimal equity auctions lead to more efficient allocations than optimal cash auctions when bidders differ only in valuation distributions. Formally, I define a strong notion of efficiency ordering: one mechanism is ex-post more efficient than another if it leads to higher social welfare at all valuation realizations. Definition 3. Mechanism A is ex-post more efficient than mechanism B if the following holds at all valuation realizations (x1 , x2 , ..., xn ): (i) if the asset is sold in B, either it is also sold in A and the winner’s valuation is no less than that in B, or it is not sold in A and the winner in B has a valuation below the asset’s standalone value VT ; (ii) if the asset is not sold in B, either the asset is not sold in A or the winner in A has a valuation exceeding VT . I show optimal equity auctions are ex-post more efficient than optimal cash auctions under the following conditions: Proposition 4. Suppose bidders have the same standalone values. Optimal equity auctions are ex-post more efficient than optimal cash auctions if (i) bidders’ valuations are uniformly distributed (their support may differ), or (ii) bidders’ and the asset’s standalone values are much larger than the synergies; that is, given any synergy distributions and constants VT∗ and V ∗ , let VT = kVT∗ and Vi = kV ∗ (i = 1, ..., n), where k is arbitrarily large, or (iii) bidders’ standalone values are much lower than the asset’s. Note the two forms of bidder heterogeneity have opposite impacts on the efficiency of the optimal auction: heterogeneity in standalone values makes optimal equity auctions less efficient than optimal cash auctions, whereas heterogeneity in distributions of bidder valuations leads to more efficient allocations. To understand this contrast, observe that bidder heterogeneity makes bidders’ informational advantages different from each other—and inefficient allocations exploit such differences. Because heterogeneity in valuation distributions affect bidders’ informational advantages in both cash and equity auctions, and equity auctions reduce bidders’ informational advantages—and hence the degree of their differences—optimal equity auctions lead to more efficient allocations than optimal cash auctions when bidders differ only in valuation distributions. In particular, when bidders’ standalone values are sufficiently low, optimal equity auctions 16 Alternatively, note bidders whose valuations are more precisely known have less informational advantage; hence, letting these bidders win with higher probabilities benefits the seller.

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are always efficient regardless of how bidders’ valuation distributions differ (because virtual valuations approach actual valuations), whereas optimal cash auctions are not, unless bidders’ valuation distributions are identical. By contrast, a bidder’s standalone value affects its informational advantage in equity but not in cash auctions, making it attractive to favor lower-standalone-value bidders in equity auctions. Thus, when bidders have identical valuation distributions but different standalone values, optimal cash auctions result in efficient allocations but optimal equity auctions do not. Optimal equity auctions take into account both sources of bidder heterogeneity and maximally exploit the features of equity bids. Consequently, they generate higher expected revenues than cash auctions of any format. Proposition 5. Optimal equity auctions generate higher expected revenues than optimal cash auctions, regardless of any heterogeneity in bidders. Note that if Assumption 2 holds, Proposition 5 would follow immediately from Lemma 4, which shows a bidder’s virtual valuation for an equity auction exceeds its cash-auction counterpart (provided ψi (xi ) ≥ VT ). Concretely, higher virtual valuations in equity auctions contribute to the greater revenues in two ways. First, at the same reserve prices (i.e., threshold bidder types), equity auctions yield higher revenues than cash auctions. Second, equity auctions optimally feature lower reserves, allowing the seller to extract additional revenues from bidder types that would contribute only negatively to the revenues of cash auctions. If Assumption 2 does not hold, however, a simple ordering of virtual valuations no longer exists: a bidder’s virtual valuation for an equity auction may be less than its cash-auction counterpart at certain valuations, as section 5.1 will show. Nonetheless, Proposition 5 still holds as a general result. Proposition 5 formalizes the strong intuition about the advantages of equity bids that the prior literature has derived. Importantly, with heterogeneous bidders, the key to the revenue superiority of optimal equity auctions lies in the fact that they simultaneously adjust for both sources of bidder heterogeneity. However, any equity-auction format that adjusts for only one source of bidder heterogeneity can generate lower revenues than optimal cash auctions when the other source of bidder heterogeneity is substantial. 5.1. Abnormalities Lemmas 4 and 5 and Propositions 2 and 3 are established under Assumption 2, which is a sufficient condition that restricts the upper tails of valuations. One might conjecture these results follow from the casual intuition that the seller can always extract more rents (i) from equity than cash bids and (ii) from bidders with lower than larger standalone values in equity auctions, and hence that no conditions need be imposed to ensure these results. I show such conjectures are false. Concretely, I show the following can obtain even when φi (·) and ψi (·) both strictly increase   on x i , x¯i : 1. A bidder’s virtual valuation in the equity auction can increase in its standalone value: ∂φi (x ∗ ) ∗ ∗ ∂Vi > 0 and φi (x ) > VT for some x . 2. The probability of trade can be lower in the optimal equity than in the optimal cash auction.

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3. A virtual valuation in the cash auction can exceed that in the equity auction: ψi (x ∗ ) > φi (x ∗ ) > VT for some x ∗ . I provide details on these seemingly counter-intuitive results in the working-paper version, showing they can arise when the valuation distribution has sufficiently fat upper tails that cause the third term in the virtual valuation to dominate the second. These results and the tail effects that underlie them highlight the richness in the implications the formulation of optimal equity auctions can generate. 6. Conclusions I analyze the impact of a pervasive, but little-studied, real-world feature of equity auctions: bidders usually differ ex ante in their characteristics, such as standalone values or valuation distributions. The analysis applies to the sale of non-divisible assets through equity payments, which encompasses a wide variety of economic situations. Bidder heterogeneity is more important for equity than for cash auctions. Central to this is the opacity of equity bids, the values of which depend on bidders’ private information. Heterogeneity exacerbates this opacity, making the seller’s revenue sensitive to the auction format. I determine the revenue consequences of bidder heterogeneity for different equity-auction designs, and derive the revenue-maximizing mechanism. I show how heterogeneity alters the optimal design, and obtain the distinct implications of different sources of heterogeneity. My analysis reveals a number of unique features of the optimal design that have no analogue in cash auctions. The most important of these features are (1) the preferential treatment of bidders with lower standalone values, (2) the opposing effects of the two forms of bidder heterogeneity on efficiency, (3) the suboptimality of payments by losing bidders, (4) the lower reserve prices than optimal cash auctions, and (5) the clean limiting properties when bidders are sufficiently large or small. These properties allow optimal equity auctions to respond to both sources of bidder heterogeneity and to maximally exploit the features of equities, generating revenues that always exceed optimal cash auctions, regardless of the nature of the bidder heterogeneity. Extending beyond equity auctions, my work highlights the importance of accounting for bidder heterogeneity in securities auctions. Drawing from the insights of the analysis of equity auctions, heterogeneity affects the design and performance of securities auctions more than those of cash auctions, because securities’ values depend on bidders’ private information. My paper’s findings regarding how to optimally adjust the auction structure for equity bids, and how such adjustments restore their revenue advantages over cash bids, provide a stepping stone for future research of such adjustments for broader security classes. Acknowledgments I would like to thank Alessandro Pavan (editor), an associate editor, an anonymous referee, Dan Bernhardt, and Robert Marquez for their valuable comments and suggestions. I also thank Hank Bessembinder, Jacques Cremer, Sudipto Dasgupta, Bobbie Goettler, Jennifer Huang, Andy Skrzypacz, Zhongzhi Song, Brian Viard, Tak-Yuen Wong, Fei Xie, and seminar participants at CKGSB, City University of Hong Kong, Peking University, Renmin University of China, SAIF, Singapore Management University, Shanghai University of Finance and Economics, and Tsinghua PBCSF for very helpful inputs.

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115

Appendix A. Proofs Proof of Lemma 1. Let (, Wo , Qo ) denote the original mechanism, and θi (xi ), i = 1, ...n, denote bidder i’s equilibrium bid. Let Wj (x1 , ..., xn ) = Wjo (θ1 (x1 ) , ..., θn (xn )) for all j ∈ A and Qi (x1 , ..., xn ; j ) = Qoi (θ1 (x1 ) , ..., θn (xn ) ; j ) for all i ∈ B and j ∈ A. The lemma follows from (2) and similar arguments as in cash auctions. 2 Proof of Theorem 1. I first show qi (·) satisfies  ui x i + ωi (x)

xi qi (xi ) Gi (xi ) =

(Vi + x)2

xi

 xi Vi Gi (x) Vi Gi (xi ) ui x i + ωi (xi ) dx + + + dx. Vi + xi Vi + xi (Vi + x)2 xi

(26) Divide both sides of (14) by (Vi + xi )2 to have  xi  ui x i + ωi (xi ) qi (xi ) Gi (xi ) Vi Gi (xi ) x i qi (t) Gi (t) dt − = + . Vi + xi (Vi + xi )2 (Vi + xi )2 (Vi + xi )2

(27)

Integrating (27) yields xi xi

qi (x) Gi (x) dx − Vi + x xi =

xi

xi = xi

xi =

 ui x i + ωi (x) (Vi + x)

 ui x i + ωi (x) (Vi + x)2  ui x i + ωi (x) (Vi + x)

2

xi

(Vi + x)2

xi

2

xi

Vi Gi (x)

xi dx + xi

xi dx −

⎛ ⎜ ⎝ ⎛ ⎜ ⎝

xi

 xi

dx −

dx

xi

x

⎞ ⎟ qi (t) Gi (t) dt ⎠

xi

x

1 (Vi + x)2

⎞ ⎟ qi (t) Gi (t) dt ⎠ d

xi

qi (t) Gi (t) dt Vi + xi

xi + xi



(28)

dx

1 Vi + x

! (29)

qi (x) Gi (x) dx, Vi + x

(30)

where integration by parts is used.17 Note, the last term on the right-hand side of (30) cancels the first term on the left-hand side. This cancellation is the consequence of and rationale for dividing (14) by (Vi + xi )2 . Thus, (30) simplifies to xi xi

 ui x i + ωi (x) (Vi + x)

2

 xi dx −

xi

qi (t) Gi (t) dt Vi + xi

xi + xi

Vi Gi (x) (Vi + x)2

dx = 0.

(31)





17 Note a technical subtlety that q (·), G (·), or ω (·) may not be differentiable over the full range of x , x¯ . Accordi i i i i

ingly, the proof utilizes integration that does not assume the differentiability of these quantities.

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Further, (14) yields  xi x qi (t) Gi (t) dt i

Vi + xi

 Vi Gi (xi ) ui x i + ωi (xi ) = qi (xi ) Gi (xi ) − − . Vi + xi Vi + xi

(32)

Note the left-hand side of (32) is the second term in (31), yielding (26) upon substitution. Rewrite (26) as  xi ui x i Vi Gi (t) Vi Gi (xi ) + δi (xi ) , qi (xi ) Gi (xi ) = + dt +  Vi + xi Vi + x i (Vi + t)2 where δi (xi ) ≡ x¯i πs,i =

ωi (xi ) Vi +xi

+

 xi

ωi (t) x i (Vi +t)2 dt,

x¯i − xi

= xi

and substitute into (13) to have

 Gi (xi ) (xi − VT ) fi (xi ) dxi − ui x i

xi

x¯i

(33)

xi

⎡ ⎢ Vi Gi (xi ) + (1 − Fi (xi )) ⎣ Vi + xi

xi

Vi Gi (t) (Vi + t)

xi

2

dt + 

 ui x i Vi + x i

⎤ ⎥ + δi (xi )⎦ dxi

(

)  (1 − Fi (xi )) Vi Gi (xi ) fi (xi ) xi − VT − dxi − ui x i τi (Vi + xi ) fi (xi ) x¯i

−

x¯i (1 − Fi (xi )) Li (xi ) dxi −

xi

(1 − Fi (xi )) δi (xi ) dxi ,

(34)

xi

 x¯i

 xi Vi Gi (t) 1 where τi ≡ V +x x i (1 − Fi (xi )) dxi + 1 and Li (xi ) ≡ x i (Vi +t)2 dt. Define Hi (xi ) ≡ i i   x¯i xi (1 − Fi (t)) dt. Integrating by parts and utilizing Hi (x¯ i ) = Li x i = 0 yields x¯i

x¯i (1 − Fi (xi )) Li (xi ) dxi = −

xi

Li (xi ) d (Hi (xi )) xi

x¯i =

Hi (xi ) d (Li (xi )) xi

x¯i = xi

Vi Hi (xi ) (Vi + xi )2

Gi (xi ) dxi .

Thus, (34) becomes x¯i πs,i = xi

 (φi (xi ) − VT ) Gi (xi ) fi (xi ) dxi − ui x i τi −

x¯i (1 − Fi (xi )) δi (xi ) dxi , (35) xi

T. Liu / Journal of Economic Theory 166 (2016) 94–123

117

where φi (xi ) is given by (19). Substituting (5) into (35) yields πs,i =

x¯i

 Wi (x) (φi (xi ) − VT ) f (x) dx − ui x i τi −

χ

(1 − Fi (xi )) δi (xi ) dxi .

(36)

xi

Substituting (36) into (12) and rearranging terms establishes the theorem. 2 Proofs of Lemma 2, Proposition 1, and Corollary 1. Step 1: I prove Corollary 1 by first showing truth-telling is an equilibrium. Compare the profits of bidder i with valuation xi between truthful reporting and over-reporting. (1) If truthful and over-reporting both result in winning, then by (23), profits are the same. (2) If truthful and over-reporting both result in losing, profits are but over-reporting results in winning,  If  truthful  reporting results in losing  zero. (3) i by (23). Hence, over-reporting is not max maxj =i φj xj , VT > φi (xi ), and Qi < ViV+x i profitable. Summarizing, truthful reporting weakly dominates over-reporting. A similar argument shows truthfulreporting also weakly dominates under-reporting.   Next, I show ui x i = 0 for all i in two cases. (1) Gi x i = 0. Then ui x i = 0 by (7).   i (2) Gi x i > 0. Because yi (x−i ) ≥ x i by (24), Qi ≤ ViV+x by (23), yielding ui x i ≤ 0 by (3). i Further, because a bidder can always ensure a zero profit low value,  by reporting a sufficiently  its equilibrium profit must be nonnegative. Thus, ui x i ≥ 0, yielding ui x i = 0. Therefore, πs,a (equation (16)) is zero. Because losers do not pay, πs,b (equation (17)) is also zero. Because the bidder with maximum φi (xi ) wins provided φi (xi ) > VT , by the arguments following Theorem 1, this mechanism maximizes πs,a , πs,b and πs,c simultaneously and hence is optimal. Step 2: I show the “if” part of Proposition 1 by assuming conditions (i) through (iii). Equations (6) and (22) yield Vi Gi (xi ) qi (xi ) Gi (xi ) = + Vi + xi

xi xi

Vi Gi (t) (Vi + t)2

(37)

dt.

Equation (7) and ωi = 0 yield vi (xi , xi ) − vi (xi , zi ) = (Vi + xi ) (qi (xi ) Gi (xi ) − qi (zi ) Gi (zi )) − Vi (Gi (xi ) − Gi (zi )) , which, by (37) yields zi − xi vi (xi , xi ) − vi (xi , zi ) = Vi Gi (zi ) + (Vi + xi ) Vi + zi

xi zi

Vi Gi (t) (Vi + t)2

dt.

(38)

Further, by condition (i), and Gi (·) is nondecreasing by (5). There x Wi (·, x−i ) isnondecreasing x fore, zi i Vi Gi (t)2 dt ≥ zi i Vi Gi (zi2) dt for both zi > xi and zi < xi . Thus, (38) yields (Vi +t)

(Vi +t)

zi − xi vi (xi , xi ) − vi (xi , zi ) ≥ Vi Gi (zi ) + (Vi + xi ) Vi + zi =0 for all zi , establishing truth-telling as an equilibrium.

xi zi

Vi Gi (zi ) (Vi + t)2

dt

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   i Next, note that if Gi x i > 0, then qi x i = ViV+x by (37), and ui x i = 0 by (7). By argui  ments similar to those in Step 1, ui x i = 0 for all i, and the mechanism is optimal. Step 3: I prove Lemma 2 and the “only if” part of Proposition 1. As Corollary 1 shows, a mechanism exists that simultaneously maximizes πs,a , πs,b , and πs,c ; an optimal mechanism cannot do worse, and therefore it must also simultaneously maximize πs,a , πs,b , and πs,c . By the arguments following Theorem 1, the and conditions (i) and (ii) of the proposition hold.  lemma The same arguments also yield ui x i = 0 for all i, establishing condition (iii) upon equating the right-hand sides of (33) and (6). 2 Proof of Lemma 3. Refer to (19). The first term xi increases in xi . The second term also increases in xi , because the numerator (1 − Fi (xi )) decreases in xi , the denominator (Vi + xi ) fi (xi ) increases in xi by Assumption 2, and the sign for the term is negative. Simi x¯ larly, the third term also increases in xi , because xii (1 − Fi (t)) dt decreases, (Vi + xi )2 fi (xi ) increases, and the sign is negative. The lemma follows. 2 Proof of Lemma 4. Note

 x¯ Vi xii (1 − Fi (t)) dt xi (1 − Fi (xi )) φi (xi ) − ψi (xi ) = − (Vi + xi ) fi (xi ) (Vi + xi )2 fi (xi )  x¯i xi (1 − Fi (xi )) x (1 − Fi (t)) dt ≥ − i (Vi + xi ) fi (xi ) (Vi + xi ) fi (xi ) (1 − Fi (xi )) ≥ (2xi − x¯i ) , (Vi + xi ) fi (xi )

(39)

where x¯i (1 − Fi (t)) dt ≤ (x¯i − xi ) (1 − Fi (xi ))

(40)

xi

is used, and strict inequality holds for xi < x¯i . Further, fi (·) is nondecreasing yields 1 − Fi (xi ) ≥ fi (xi ) (x¯i − xi ) .

(41)

Thus, the virtual valuation for cash auctions ψi (xi ) ≤ 2xi − x¯i , and ψi (xi ) ≥ VT yields 2xi ≥ x¯i + VT , establishing the lemma by (39). 2 Proof of Proposition 2. Follows from Lemma 4.

2

Proof of Lemma 5. By (19), ∂φi (xi ) 1 − Fi (xi ) xi 1 =− − 2 ∂Vi f (x ) (Vi + xi ) (Vi + xi )2 i i  x¯i 2Vi xi (1 − Fi (t)) dt + fi (xi ) (Vi + xi )3 1 − Fi (xi ) Vi + ≤− 2 fi (xi ) (Vi + xi ) (Vi + xi )3 xi

 x¯i xi

(1 − Fi (t)) dt fi (xi )

 x¯i xi

(1 − Fi (t)) dt fi (xi )

.

(42)

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119

Consider two cases. (1) xi ≥ 12 x¯i . Then (42) and (40) yield  x¯i ∂φi (xi ) 1 − Fi (xi ) xi 1 xi (1 − Fi (t)) dt ≤− + 2 2 ∂Vi fi (xi ) fi (xi ) (Vi + xi ) (Vi + xi ) 1 − Fi (xi ) 1 ≤ (x¯i − 2xi ) , (Vi + xi )2 fi (xi ) where strict inequality holds in the second line for xi < x¯i , establishing the lemma. (2) xi < 12 x¯i . Note φi (xi ) ≥ VT yields  x¯ Vi xii (1 − Fi (t)) dt (1 − Fi (xi )) Vi − VT , ≤ xi − 2 (Vi + xi ) fi (xi ) (Vi + xi ) fi (xi ) yielding, by (42), ∂φi (xi ) 1 − Fi (xi ) xi − VT xi (1 − Fi (xi )) Vi ≤− − + 2 ∂Vi fi (xi ) Vi + xi (Vi + xi ) (Vi + xi )2 fi (xi ) 1 1 − Fi (xi ) xi − VT + =− Vi + xi fi (xi ) Vi + xi  ! 1 1 − Fi (xi ) < xi − , Vi + xi fi (xi ) which, by (41), yields ∂φi (xi ) 1 < (xi − (x¯i − xi )) < 0, ∂Vi Vi + xi establishing the lemma.

2

Proof of Proposition 3. Assume Vi > Vj without loss. Denote by φ (V , x) the virtual valuation of a bidder who has standalone value V , valuation x, and the same valuation distribution as i and j . Then Vi φ (V , x) = φ (Vi , x) −

∂φ(t, x) dt. ∂t

(43)

V

  Claim 1: If φ (Vi , x) ≥ VT , then φ (V , x) ≥ VT for all V ∈ Vj , Vi . Suppose the claim is not true; then φ (V ∗ , x) < VT for some V ∗ ∈ [Vj , Vi ), and V ∗∗ ∈ (V ∗ , Vi ] exists such that φ (V ∗∗ , x) = VT and φ (V , x) < VT for all V ∈ [V ∗ , V ∗∗ ). But this contradicts Lemma 5 and (43), establishing Claim 1.  By Claim 1, (43), and Lemma 5, for all x ∗ ∈ [x i , x¯i ), if φ (Vi , x ∗ ) ≥ VT , then φ Vj , x ∗ > φ (Vi , x ∗ ). It is straightforward to establish the proposition. 2 Proof of Proposition 4. Denote the index of the winner by E if the asset is allocated in optimal equity auctions, and by C if the asset is allocated in optimal cash auctions. I now prove part (i). Under uniform distributions, (19) becomes φi (xi ) = xi −

V V 1 (x¯i − xi )2 , (x¯i − xi ) − 2 (V + xi )2 (V + xi )

(44)

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T. Liu / Journal of Economic Theory 166 (2016) 94–123

where V denotes a bidder’ standalone value, and the virtual valuation for cash auctions is ψi (xi ) = 2xi − x¯i . If the asset is not sold in optimal cash auctions, it is straightforward to prove part (i). I now consider the case in which the asset is sold in optimal cash auctions. By ψC (xC ) ≥ VT and Lemma 4, the asset is also sold in optimal equity auctions. Next, I show xC ≤ xE by contradiction. Instead, suppose xC > xE . The winning criteria ψC (xC ) ≥ ψC (xE ) and φE (xE ) ≥ φE (xC ) give x¯C − xC ≤ x¯E − xE + ,

(45)

where  ≡ xC − xE , and xE −

V (x¯E − xE ) 1 V (x¯E − xE )2 V (x¯C − xC ) 1 V (x¯C − xC )2 − x + ≥ 0. (46) − + C 2 (V + xE )2 2 (V + xC )2 (V + xE ) (V + xC )

By (45) and the assumption xC > xE , the left-hand side of (46)   V 1 V (x¯C − xC )2 − (x¯E − xE )2 ≤ − + 2 (V + xC ) (V + xC )2 V 1 V  (x¯C − xC + x¯E − xE ) ≤ − . + 2 (V + xC ) (V + xC )2

(47)

Further, assumption  > 0 and (46) yield V (x¯C − xC ) 1 V (x¯C − xC )2 V (x¯E − xE ) 1 V (x¯E − xE )2 ≥ , + + 2 (V + xC )2 2 (V + xC )2 (V + xC ) (V + xC ) yielding x¯C − xC > x¯E − xE . Thus, the second term in the right-hand side of (47) satisfies 1 V  (x¯C − xC + x¯E − xE ) V  (x¯C − xC ) < 2 (V + xC )2 (V + xC )2 xC < , (V + xC )

(48)

because ψC (xC ) ≥ VT yields x¯C < 2xC . By (48) and (47), the left hand-side of (46) is strictly negative. This contradicts (46), proving the proposition for part (i). i (xi ) I now prove part (ii) of the proposition. When k is sufficiently large, φi (xi ) = xi − δ 1−F fi (xi ) , ∗

where δ = V ∗V+V ∗ ∈ (0, 1). Similar to the proof of part (i), we only need to consider the case T in which the asset is sold in optimal cash auctions. Then ψC (xC ) ≥ VT . Because φC (xC ) ≥ ψC (xC ) ≥ VT , the asset is also sold in optimal equity auctions. The winning criteria yield xC −

1 − FC (xC ) 1 − FE (xE ) ≥ xE − fC (xC ) fE (xE )

and xE − δ

1 − FE (xE ) 1 − FC (xC ) ≥ xC − δ , fE (xE ) fC (xC )

yielding 1 − FE (xE ) 1 − FC (xC ) − ≥ xE − x C fE (xE ) fC (xC ) and

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121

1 1 − FE (xE ) 1 − FC (xC ) − . (xE − xC ) ≥ δ fE (xE ) fC (xC ) * + + * Thus, 1δ (xE − xC ) ≥ xE − xC , yielding 1δ − 1 (xE − xC ) ≥ 0. Because 1δ − 1 > 0, xE ≥ xC , establishing part (ii). Part (iii) follows from Corollary 4. 2 Proof of Proposition 5. Consider a generalized framework in which the seller can ask bidders to pay with any combination of cash and equity, where the cash payment is nonnegative. Represent the associated direct mechanism by (χ , W, Q, M), where M : χ × A → Rn≥0 is the cash-payment rule: Mi (x, j ) is the cash that bidder i pays when bidders report x and j wins. Then (7) holds upon replacing (4) with

ωi (zi ) ≡ Mi (zi , x−i , i) Wi (zi , x−i ) f−i (x−i ) dx−i + [Vi (1 − Qi (zi , x−i , j )) j =iχ

χ −i

+ Mi (zi , x−i , j )]Wj (zi , x−i ) f−i (x−i ) dx−i .

−i

(49)

Following similar procedures as in the main analysis, Theorem 1 holds with (49) replacing ωi in (17). Note optimal equity auctions are also optimal in this generalized framework because they simultaneously maximize πs,a , πs,b , and πs,c , and no other mechanism can do better. Because πs,b is zero in optimal equity auctions but is strictly negative with cash payments, by the same arguments as those following Theorem 1, the proposition holds. 2 Appendix B. Further details Optimal reserve prices in standard second-price auctions. Section 3 describes standard second-price equity auctions with discriminatory reserve prices. Utilizing Theorem 1, the following derives the optimal reserve prices that maximize the expected revenue. Corollary 5. The following reserve prices maximize the seller’s expected revenue in the secondprice equity auction: ri =

φi−1 (VT )

Vi + φi−1 (VT )

.

Proof of Corollary 5. Note reserve prices do  not affect the bidding strategies and the term πs,b in (15) is zero. At the reserve prices above, ui x i = 0 for all i, and πs,a obtains its maximum value of zero. Now I examine how reserve prices affect πs,c . Consider any realization of valuations x, and assume bidder k with valuation xk has the highest bid. Given the reserve prices, the asset is allocated if and only if xk ≥ φk−1 (VT ). Referring to (18), one has  φk (xk ) if xk ≥ φk−1 (VT ) n i=1 Wi (x) φi (xi ) + W0 (x) VT = . (50) if xk < φk−1 (VT ) VT The right-hand side of (50) is no less than would be obtained by any other reserve prices. Thus, the corollary holds. 2 Revenue decomposition when bidders are large. I show when bidders are large, the general revenue decomposition in equity auctions, (15), reduces to the corresponding revenue decomposition in cash auctions (Myerson, 1981; Krishna, 2010).

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Corollary 6. Suppose losing bidders’ payments are bounded: a constant c < ∞ exists such that Vi (1 − Qi (x, j )) ≤ c for x and i) = j . Then, when bidders are sufficiently large, πs,a ( all n   (equation (16)) approaches − ui x i , πs,b (equation (17)) approaches zero, and πs (equai=1

tion (15)) approaches the revenue decomposition in cash auctions:  

n n

 lim πs = − ui x i + Wi (x) ψi (xi ) + W0 (x) VT f (x) dx. Vi (i=1,...,n)→∞

i=1

χ

(51)

i=1

Proof of Corollary 6. Equation (4) yields

ωi (zi ) ≤ c Wj (zi , x−i ) f−i (x−i ) dx−i j =iχ



≤c

−i

f−i (x−i ) dx−i = c.

(52)

χ −i

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