Inside a takeover process
1 · The setting · 2 minutes
An illustrative sale process
A listed software company announces a strategic review. One bidder is ready to bid. A second sees a possible fit but must investigate the technology, check the contracts and arrange financing first. Before it pays for that work, it watches the target's shares.
Preparation is the costly work that turns interest into an executable offer. A bidder that does not prepare stays out of the sale; it cannot bid on a guess. Preparation costs are sunk, and they differ across bidders: a bidder with capacity in place or prior familiarity with the target prepares cheaply, and others do not.
Who knows what
One bidder is already prepared; I call it the incumbent bidder, or the incumbent. It is the party that approached the target or a bidder that has completed diligence. The incumbent knows its acquisition value \(R\). Its strength is the public distribution \(F\) of \(R\), and a first-order stochastic shift of \(F\) strengthens the incumbent. A second bidder, the potential rival bidder, or the rival, has not decided whether to prepare. The rival belongs to a high or a low quality class, \(j\in\{H,L\}\), and its value \(\theta\) has a class distribution \(G_j\). It knows neither its class nor its value. After it sees the stock price, it learns its preparation cost:
Paying \(C\) reveals \(\theta\) and permits a bid. The seller has committed to a cash second-price auction with reserve \(p\), so each prepared bidder bids its value.
An outside informed trader with sector expertise knows the rival's class and remains uncertain about the rival's value. It holds no position and has no control rights. Trade happens in one public order slot before the rival decides. With probability \(\lambda\) the informed trader fills the slot with a sale, no order or a purchase of one unit, at cost \(k\) for a nonzero order. Otherwise a liquidity trader fills it:
No one sees who filled the slot. Competitive market makers see the order and set the price at the expected terminal value of a share. The rival sees the price and its own cost, and nothing else.
- The sale opportunity is public. The seller commits to a cash second-price auction with reserve \(p\). One bidder is prepared; the rival has not decided.
- The informed trader learns the rival's class \(j\). The rival, the market makers and the incumbent do not.
- With probability \(\lambda\) the informed trader places its order; otherwise a liquidity trader sells, abstains or buys. The order is public; the identity of its trader remains hidden.
- Market makers price the order. They anticipate how the price will change the rival's decision and therefore the sale proceeds.
- The rival reads a belief about its class from the price, learns its cost, and prepares when expected acquisition profit covers that cost.
- The incumbent bids its value. A prepared rival learns its value and bids it. The highest admissible bidder wins and pays the larger of the reserve and the other admissible bid.
Select a stage to see who knows what at that point.
The sale must become public before the set of bidders is fixed; only then can a potential rival watch the price while it decides. A disclosed approach, an announced review or an open contest can create that window. Imprivata's 2016 definitive proxy shows the stage the model describes: an unsolicited approach, outreach to other buyers, and indications of interest that still depend on diligence.
The model in full, the example and related literature
All agents are risk neutral, the target's standalone value is zero, and values and costs are per share. The pair consisting of class and the rival's value, the incumbent's value, the preparation cost, and the slot draws are mutually independent. Bids are truthful and weakly dominant. Without a rival, the incumbent buys at the reserve when its value meets it, so expected proceeds without entry are \(t_0=p[1-F(p)]\). They are positive when the incumbent can meet the reserve, as in the example. A zero-cost rival prepares after every price, because its gross profit is an auction rent and never negative. The classes satisfy the class ranking: \(G_H\) dominates \(G_L\), and both lie above the reserve. A path is a family \(F_r\) of continuous incumbent distributions indexed by a strength \(r\), in which a higher \(r\) is a first-order stochastic strengthening.
An equilibrium gives each class a distribution over the three orders, a posterior and a price after each order, the rival's belief and preparation at each price, and truthful bids. The informed trader may mix and may place a wrong-signed order. A deviation by the informed trader takes the candidate's prices and entry as given; a comparison across incumbent strengths solves for the equilibrium again in each economy. When ties arise, an indifferent informed trader may mix over its best orders, and an indifferent high-cost rival prepares; these are the default tie rules. The other two rules are that an indifferent informed trader abstains and that an indifferent high-cost rival may prepare with any probability.
The example
The example has two point-mass classes, the high class at \(h\) and the low class at \(\ell\), so class and value coincide. The incumbent is uniform on \([0,r]\), so a higher \(r\) is a stronger incumbent, and \(0\lt p\lt\ell\lt r\lt h\) at every strength of its path. Money is measured in units of the low value. The slot has \(1+u=\mathrm e\), where \(u=\lambda/b\) is the informed presence relative to the liquidity mass \(b=(1-\lambda)\beta\) in one direction.
- \(h\)
- 10
- \(\ell\)
- 1
- \(p\)
- 1/2
- \(\pi\)
- 1/2
- \(\rho\)
- 1/4
- \(c_H\)
- 6
- \(k\)
- 1/50
- \(\beta\)
- 0.294655
- \(\lambda\)
- 0.336122
- \(r_0\)
- 6/5
- \(r_1\)
- 3
- \(r_2\)
- 18/5
| In the model | In the record | What the record must show |
|---|---|---|
| The sale opportunity is public | A disclosed approach, an announced strategic review or an open contest | Visibility dated before the rival's decision |
| Incumbent strength | What the public can infer about the incumbent's value: a strategic acquirer with clear synergies, or a leveraged financial buyer | Measured before the rival decides |
| The order slot | The first prespecified public order opportunity after the sale becomes public, such as the opening auction of the next trading day | The sign of the net imbalance; an imbalance below a size fixed in advance is the zero order |
| Preparation | The start of substantive diligence, or a proposal that requires costly preparation | Dated after the price; public offers are a narrower outcome |
| The informed trader | Outside flow with a view on the rival's fit, such as traders with sector expertise | No stake and no role in the sale |
Related literature
The paper belongs to the literature on the real effects of financial markets (Bond, Edmans and Goldstein 2012). The closest antecedent is Dow, Goldstein and Guembel (2017), where a firm's investment decision feeds back into the incentive to produce information about its stock; here the sale rule divides acquisition surplus between a traded claim and a rival that decides whether to prepare. Edmans, Goldstein and Jiang (2015) show that corrective real decisions can discourage trade on bad news under rational pricing; here the real decision is the rival's entry itself. Firms learn from peers' prices (Foucault and Frésard 2014), and target prices affect takeover activity (Edmans, Goldstein and Jiang 2012); here a potential rival learns about its own acquisition value from the target's price before it commits.
The auction-entry literature makes the bidder pool endogenous (Samuelson 1985; McAfee and McMillan 1987; Levin and Smith 1994; Gentry and Stroup 2019). Under selective entry, a bidder's information before entry indexes the distribution of its value, and paying the entry cost reveals the value (Roberts and Sweeting 2013). Here that information is the target's stock price, which moves with the orders of an informed trader who knows the rival's class. The slot follows the alternative-arrival structure of Dow and Gorton (1997), and the unit order menu, the presence probability and the unit cost follow Edmans, Goldstein and Jiang (2015).
Two returns to information
2 · The mechanism · 3 minutes
The rival uses information to assess its acquisition profit. The informed trader uses the same class information to value the target's shares. The auction connects these two returns: a stronger incumbent reduces the rival's profit and makes target proceeds more sensitive to its class.
The direct effect
Lower acquisition profitA stronger incumbent reduces expected profit at each belief about the rival's class.
Preparation pays at fewer costs for a fixed belief.
The information effect
Proceeds depend more on the rival's classThe informed trader earns more from its knowledge of the class.
Informed trade can cover its costA purchase carries favorable class information to the rival through the price.
A favorable price can justify costly preparation.
One auction, two claims
The symbols \(g_H\) and \(g_L\) denote expected acquisition profit before the preparation cost, and \(t_H\) and \(t_L\) denote expected target proceeds with preparation. At belief \(\mu=\Pr(j=H)\), the rival prepares when expected profit \(B(\mu)=g_L+\mu(g_H-g_L)\) covers its cost. The target-payoff spread \(\Delta_T=t_H-t_L\) measures how much more shareholders receive with a high-class rival.
Under the second-price rule, a higher incumbent value moves part of the class value gap into target proceeds. At each value level between the reserve \(p\) and incumbent value \(R\), the incumbent outbids the rival and pays its bid. The remainder stays as the rival's profit. Proposition 1 applies to every continuous incumbent distribution. A stronger incumbent never raises the probability that a prepared rival wins. For every pair of classes, it therefore weakly reduces both \(g_H\) and \(g_L\), and gross profit falls weakly at every belief. Under the class ranking it weakly raises \(\Delta_T\), and the two gaps sum to a constant:
The right side does not depend on the incumbent; in the example it is \(h-\ell\) at every strength \(r\). A stronger incumbent leaves the value gap between the classes unchanged and moves it from the rival's profit to target proceeds. The class ranking requires first-order stochastic dominance of \(H\) over \(L\), with all values in both classes at or above the reserve.
What one order can reveal
A purchase can come from a liquidity trader in either class, so it leaves uncertainty about the class. The same applies to a sale and to no order. Each liquidity purchase or sale has probability \(b=(1-\lambda)\beta\), and the informed trader adds at most \(\lambda\) to an order's probability. With \(\beta\le1/3\), the posterior after every order obeys the same bounds under every pure or mixed strategy of the informed trader:
When the high class buys and the low class sells, a sale, no order and a purchase yield posteriors \(m(\pi)\), \(\pi\) and \(M(\pi)\), respectively. In the example's slot at the prior one half, the top posterior is \(M=0.731059\). When \(g_H\ge g_L\) and gross profit at the top posterior falls below \(c_H\), no order recruits a high-cost rival, in or out of equilibrium. This is the exclusion bound. In the example it holds against every incumbent stronger than the entry ceiling \(r_C\).
The zero-cost rival prepares after every order, so entry stays at least \(\rho\). With a positive spread, the competitive price \(P(x)=t_0+e(x)[t_L-t_0+\Delta_T\mu(x)]\) reveals the market makers' posterior. To see why, two orders with the same price induce the same entry. Since entry is positive, the price formula then requires the same posterior. When \(g_H\gt g_L\), gross profit covers the high cost exactly when the belief reaches the preparation threshold \(\tau=(c_H-g_L)/(g_H-g_L)\), so high-cost preparation follows exactly the orders whose posterior reaches \(\tau\); an indifferent high-cost rival prepares.
When informed trade pays
Market makers price the preparation response to each order. The informed trader earns from the class information that the price does not reveal. A purchase in class \(H\) and a sale in class \(L\) have gross returns
where \(\kappa(\pi)=\min\{m(\pi),1-M(\pi)\}\). Each return equals entry times the spread times residual uncertainty about the class. Both bounds hold under every candidate profile, and the lower one is the trading bound. If the trading cost is below \(\rho\kappa\Delta_T\), a purchase in \(H\) and a sale in \(L\) are strictly optimal. If the cost exceeds \(\Delta_T\), both classes abstain. A sale in \(H\) or a purchase in \(L\) always loses after costs. Each return depends only on the posterior and entry after its own order, so the informed trader's problem splits by class.
When the spread is positive and \(g_H\gt g_L\), the informed trader's orders, the posteriors and the rival's preparation depend on the auction only through two numbers: the target-payoff spread \(\Delta_T\), which sets what an informed order earns, and the preparation threshold \(\tau\), which sets which posteriors recruit a high-cost rival. Two auctions with the same spread and threshold, and the same other primitives, have the same equilibrium orders, posteriors and entry. A stronger incumbent never narrows the spread, and it never lowers the threshold while a rival known to be in the low class earns at most the high cost, \(g_L\le c_H\). The two numbers can therefore pull in opposite directions: a wider spread raises what a recruiting purchase earns, and a higher threshold makes recruitment harder.
The example's payoffs and the bounds in detail
A prepared rival with value \(\theta\ge p\) pays \(\max\{p,R\}\) if it wins. If it loses, the incumbent pays \(\theta\). Target proceeds are therefore \(\max\{p,\min(R,\theta)\}\), and expected proceeds plus the rival's profit satisfy \(t(\theta)+g(\theta)=\theta\). In the example, with point-mass classes and a uniform incumbent, the payoffs are
The derivatives are \(\Delta_T'(r)=\tfrac12-\ell^2/(2r^2)\gt0\), \(g_H'(r)=-\tfrac12+p^2/(2r^2)\lt0\) and \(g_L'(r)=-(\ell^2-p^2)/(2r^2)\lt0\), and the preparation threshold is \(\tau(r)=[2rc_H-(\ell^2-p^2)]/(2rh-r^2-\ell^2)\). For a general continuous \(F\), define \(D=G_L-G_H\). Then
The reserve term vanishes under the class ranking. Without value atoms at the reserve, two conditions characterize a weak increase in the spread under every first-order stochastic increase in incumbent value. The high class must dominate above the reserve, and both classes must have the same probability of a value below it.
Expected proceeds without entry, \(t_0\), cancel from the informed trader's returns because they enter both the price and the terminal payoff. These proceeds are positive in the example; the zero preparation cost is a separate assumption. The likelihood ratio of each order lies between \(1/(1+u)\) and \(1+u\), which gives the posterior bounds. Since \(1-M(\pi)=m(1-\pi)\), the lower bound uses \(\kappa(\pi)=m(\min\{\pi,1-\pi\})\). This quantity is largest at the prior one half, where it equals \(1/(2+u)\). Away from one half, the price already favors one class and reduces that class's advantage.
Competition creates competition
3 · The main result · 3 minutes
A stronger incumbent can induce rival entry through a more informative stock price. I compare three economies that differ only in incumbent strength: \(F_0\), a stronger \(F_1\), and a stronger still \(F_2\). At \(F_0\), informed trade cannot cover its cost, so the price carries no class information. At \(F_1\), informed trade pays and a purchase recruits a high-cost rival. At \(F_2\), informed trade still pays, while acquisition profit falls too far for a purchase to recruit.
Proposition 2 · Competition creates competition
Suppose the classes satisfy the class ranking, \(F_1\) is a continuous first-order stochastic strengthening of the continuous distribution \(F_0\), and \(F_2\) strengthens \(F_1\) in the same sense. Let \(0\lt\pi,\rho,\lambda\lt1\) and \(0\lt\beta\lt\tfrac12\), and assume the high-cost window (H1), the trading-cost window (H2) and the exclusion condition (H3):
Each economy has a unique equilibrium outcome under all four tie rules. At \(F_0\), the informed trader abstains in both classes and entry is \(\rho\). At \(F_1\), class \(H\) buys, class \(L\) sells, and only a purchase recruits the high-cost rival. Entry is \(\rho+(1-\rho)(b+\pi\lambda)\gt\rho\), and high-class ownership rises exactly when \(e_HW_H(F_1)\gt\rho W_H(F_0)\), where \(e_H=\rho+(1-\rho)(b+\lambda)\) is entry in class \(H\). At \(F_2\), both classes trade, the three prices differ, and entry returns to \(\rho\).
What each condition does, and the full statement
The high-cost window (H1) sets entry through acquisition profit. Gross profit at the prior falls short of the high cost against \(F_0\), so an uninformative price recruits no high-cost rival; gross profit at the top posterior exceeds the high cost against \(F_1\), so a purchase can recruit one. The trading-cost window (H2) fixes trading before any order profile is known: by the trading bound, every correct order earns at most \(\Delta_T(F_0)-k\lt0\) at \(F_0\) and at least \(\rho\kappa\Delta_T(F_1)-k\gt0\) at \(F_1\). The exclusion condition (H3) stops recruitment at \(F_2\), where gross profit at the top posterior falls short of the high cost.
The trading-cost window is nonempty exactly when \(\Delta_T(F_0)/\Delta_T(F_1)\lt\rho\kappa(\pi)\): competition must widen the spread enough between the two incumbent distributions. The condition involves no unit of money, and a prior away from one half narrows the window, because the price already leans toward the favored class.
Extra preparation favors the high class: a purchase occurs with probability \(b+\lambda\) in \(H\) and \(b\) in \(L\). High-class ownership multiplies entry in class \(H\) by the probability \(W_H(F)\) that a prepared high-class rival wins, and a stronger incumbent can lower that probability. The ownership condition holds whenever the stronger incumbent leaves the high class a good chance to win, \((1+\rho)W_H(F_1)\ge2\rho W_H(F_0)\gt0\), and the high class places a purchase more often than not, \(b+\lambda\gt\tfrac12\). When class \(H\) outbids \(F_1\), it outbids \(F_0\) too, ownership rises from \(\pi\rho\) to \(\pi e_H\), and class \(H\)'s share of the takeovers that go to the rival does not fall; it rises strictly when a low-class rival beats the weak incumbent with positive probability.
Class \(H\) outbids every incumbent against which a price recruits when it has a decisive advantage, a property of values and costs alone. With point-mass classes and uniform incumbents, the advantage holds exactly when \(c_H\ge(h^2-p^2)/(2h)\); in the example this floor is 399/80, below the high cost.
Every comparison is strict, so the outcome in each economy does not depend on how ties are broken. With the example's classes and reserve, the inequalities are strict between continuous functions of \((r_0,r_1,r_2,c_H,k,\rho,\pi,\lambda,\beta)\), so they hold on an open set of these parameters that contains the example. Small changes within this set preserve the unique order regimes and the direction of the entry comparison.
The example
The example satisfies all three conditions at \(r_0=6/5\), \(r_1=3\) and \(r_2=18/5\). From \(r_0\) to \(r_1\), expected acquisition profit at the prior falls from 4.804167 to 4.291667, while the target-payoff spread rises from 0.016667 to 0.666667. The price becomes informative, and entry more than doubles:
Between the first two economies, high-class ownership rises from 0.125000 to 0.324401, and the high class's share of the takeovers that go to the rival rises from 0.545455 to 0.830691.
Corollary 1 gives the slots at which this pattern holds. At the prior one half and the example's auction and costs, (H1) to (H3) hold exactly when the top posterior lies between the preparation thresholds of the strong and the very strong incumbent, \(\tau(r_1)=141/200\lt M\lt\tau(r_2)=4245/5804\); equivalently, \(82/59\lt u\lt2686/1559\). On this set class \(H\) outbids every incumbent on the path, entry rises from \(\rho\) at \(r_0\) to \(\rho+(1-\rho)(b+\pi\lambda)\) at \(r_1\) and returns to \(\rho\) at \(r_2\), and high-class ownership rises from \(\pi\rho\) to \(\pi[\rho+(1-\rho)(b+\lambda)]\).
The preparation threshold \(\tau(r)\) exceeds the prior along the whole path, so an uninformative price recruits no high-cost rival. It rises with strength until it meets the top posterior at the entry ceiling \(r_C\approx3.592659\), just below \(r_2\). The exclusion condition holds at every strength between \(r_C\) and \(h\), and parts (i) and (ii) of Proposition 2 hold without it. The band for \(M\) is narrow because class \(H\) outbids every incumbent on the path: its expected payment \(r/2+p^2/(2r)\) rises by less than one half per unit of strength, a small change next to the profit gap \(g_H-g_L\), so the threshold rises slowly. The example's top posterior lies 0.000334 below the band's upper edge.
Hold information fixed, and entry cannot rise
To isolate the information effect, I fix the joint distribution of the class and a signal that the rival observes, the class value distributions and the cost distribution. Given the class, the signal tells the rival nothing more about either value; the incumbent's value is independent of the class and the signal, and the two values are independent given both. The cost is independent of the class, both values and the signal. A stronger incumbent then lowers gross profit at every belief, so it can only remove a preparation incentive. Entry falls by exactly the probability that the cost lies between gross profit against the two incumbent distributions. In the order slot, this experiment fixes \(\lambda\), \(\beta\), \(\pi\), the cost distribution and both order distributions, and the price then reveals the same posterior after each order at every strength.
| The example | Entry at \(r_0\) | Entry at \(r_1\) | Ownership at \(r_0\) | Ownership at \(r_1\) |
|---|---|---|---|---|
| Equilibrium (Proposition 2) | 0.250000 | 0.522757 | 0.125000 | 0.324401 |
| Full orders held fixed | 0.522757 | 0.522757 | 0.324401 | 0.324401 |
| The rival does not see the price | 0.250000 | 0.250000 | 0.125000 | 0.125000 |
The three equilibria of the example
| \(r_0=6/5\) | \(r_1=3\) | \(r_2=18/5\) | |
|---|---|---|---|
| Orders of classes \(H\) and \(L\) | \((0,0)\) | \((1,-1)\) | \((1,-1)\) |
| Posterior after a sale | 0.500000 | 0.268941 | 0.268941 |
| Posterior after a purchase | 0.500000 | 0.731059 | 0.731059 |
| Price after a sale | 0.392708 | 0.576074 | 0.610002 |
| Price after no order | 0.392708 | 0.614583 | 0.664236 |
| Price after a purchase | 0.392708 | 1.362372 | 0.718471 |
| Threshold \(\tau(r)\) | 0.633117 | 0.705000 | 0.731392 |
| Entry | 0.250000 | 0.522757 | 0.250000 |
| High-class ownership | 0.125000 | 0.324401 | 0.125000 |
| Expected target proceeds | 0.392708 | 0.872531 | 0.664236 |
Between the three economies
4 · Coexistence and research · 3 minutes
Between the weak and the strong incumbent of Proposition 2, two equilibria can hold against the same incumbent. In an uninformative market the informed trader does not trade, the price carries no information, and entry stays at its floor. In the informative market that coexists with it, a purchase recruits a high-cost rival. Propositions 3 and 4 use the default tie rules: an indifferent informed trader may mix over its best orders, and an indifferent high-cost rival prepares.
Uninformative and informative markets
Proposition 3 · Uninformative and informative markets
Under the default tie rules, suppose the classes satisfy the class ranking, and take an incumbent distribution with a positive spread \(\Delta_T\), \(g_H\gt g_L\) and preparation threshold \(\tau\). An uninformative market and an informative market coexist if and only if
The informative market is then unique: class \(L\) abstains, only a purchase recruits, and entry is
which rises strictly with the spread while class \(H\) mixes between a purchase and abstention, and is constant once class \(H\) buys for sure. Along a path, the strengths at which an uninformative market coexists with a sure purchase form an interval, and every other strength with coexistence lies below it. When the product of the spread and \(g_H-c_H\) does not fall with strength, all strengths with coexistence form one interval, at every prior.
The threshold must lie above the prior and within reach of a purchase. The spread must be wide enough that a purchase that just reaches the threshold covers the trading cost, and narrow enough that no order against an uninformative price earns more than it. In an uninformative market entry stays at its floor after every order, so an informed order earns at most \(\rho\max\{\pi,1-\pi\}\Delta_T\); the upper bound on the spread keeps that below \(k\). An informative market changes the entry response to a purchase, and with it the return to trade.
While class \(H\) mixes, a recruiting purchase earns exactly the trading cost, so a wider spread needs a more informative purchase. The purchase rate rises, and entry with it, until class \(H\) buys for sure. A stronger incumbent never narrows the spread and never lowers the threshold while \(g_L\le c_H\), so the two numbers pull in opposite directions, and the economy can enter the region, leave it and enter it again. When class \(H\)'s gross profit exceeds the high cost against the weakest incumbent, the whole set stays in one piece if the spread grows, in proportion, at least as fast as that excess shrinks.
The example meets this condition, so at every prior its strengths with coexistence form one interval (Corollary 2). At the prior one half, with the slot in the band of Corollary 1, an equilibrium exists at every strength from \(r_0\) to \(r_2\), and coexistence runs from \(r_A\approx1.393958\), where a purchase that just reaches the threshold earns the trading cost, to \(r_N\approx1.747878\), where a unit order against an uninformative price earns it. Entry in the informative market is
Between \(r_A\) and \(r_F\), class \(H\) buys at the rate \(b[\Delta_T(r)/k-2]\), which rises with the spread, and entry rises from 0.454185 at \(r_A\) to 0.506112 at 1.45. From \(r_F\approx1.467127\), class \(H\) buys for sure when present, and entry stays at 0.522757 up to the entry ceiling \(r_C\). Above the ceiling, entry with a full purchase equals \(\rho\).
The marked strengths, the equilibrium set and other priors
The informed purchase rate is the probability of an informed purchase conditional on class \(H\); dividing it by \(\lambda\) gives the purchase probability conditional on the presence of the informed trader. In the informative market, entry is 0.459588 at 1.40. Above \(r_N\) the full purchase remains, and on \((r_N,r_J)\) mixed orders at the entry floor join it. The low class abstains up to \(r_N\), mixes on \((r_N,r_S)\) and sells for sure from \(r_S\approx2.124148\), where a full sale earns the trading cost.
Each marked strength solves one condition below. Here \(J(r)=\Delta_T(r)[1-\tau(r)]\) is the gross return of a purchase that just reaches the preparation threshold:
| Strength | Condition | Example | Equilibria on the path |
|---|---|---|---|
| \(r_A\) | \(J(r)=k\) | 1.393958 | N below; at \(r_A\), N and T |
| \(r_F\) | \(\Delta_T(r)(1-M)=k\) | 1.467127 | N and R on \((r_A,r_F)\) |
| \(r_N\) | \(\rho\Delta_T(r)/2=k\) | 1.747878 | N and F on \([r_F,r_N]\) |
| \(r_J\) | \(\rho J(r)=k\) | 1.966097 | F and Q on \((r_N,r_J)\) |
| \(r_S\) | \(\rho\Delta_T(r)m=k\) | 2.124148 | F alone from \(r_J\); full sales from \(r_S\) |
| \(r_C\) | \(\tau(r)=M\) | 3.592659 | F without recruitment above \(r_C\) |
Coexistence also holds away from the even prior. In the example's slot it starts at the same strength \(r_A\) for every prior between \(\approx0.396175\) and \(\approx0.640739\), and it starts later, where the prior reaches the threshold, for priors up to \(\approx0.650059\). Below \(\approx0.396175\), even the top posterior cannot reach the threshold once a purchase can pay for itself; from \(\approx0.650059\) on, the uninformative market ends before the prior falls below the threshold.
The other tie rules change the intermediate path. When an indifferent informed trader abstains, the low class has no qualifying sale rate on \((r_N,r_S]\), so no equilibrium exists there, and coexistence shrinks to \((r_F,r_N]\). When an indifferent high-cost rival may prepare with any probability, the threshold pair is an equilibrium on \([r_A,r_J]\), and entry after a purchase at \(r_C\) can take any value between \(\rho\) and one. Proposition 2 rests on strict comparisons and gives the same three outcomes under all four tie rules.
Competition creates research
The informed trader can also pay to learn the class. Before it learns the class, it chooses a research intensity \(\lambda\in[0,\bar\lambda]\) at cost \((a/2)\lambda^2\), and research succeeds with probability \(\lambda\). Success reveals the class and gives the informed trader the slot; after failure, a liquidity trader fills it. Market makers and the rival do not observe the research choice, so they price and decide at a conjectured intensity. More research adds informed orders and removes liquidity orders, so it reduces the camouflage of the informed trader's own orders. The prior is one half.
Consider the equilibrium in which class \(H\) buys after a success and a purchase recruits a high-cost rival. Class \(L\) abstains while a sale at the entry floor cannot pay its cost, \(\rho\Delta_T\le2k\); beyond that point it sells just often enough that its sale breaks even, and once a sure sale pays, it sells for sure. A sale never recruits, and a sale that breaks even adds nothing to the value of research, so at an interior research choice
where \(m_\lambda\) is the bottom posterior at intensity \(\lambda\). The left side rises with \(\lambda\) and the right side falls, so when \(\Delta_T\gt2k\) the two sides meet at one intensity \(\lambda_R(\Delta_T)\), and research is the smaller of \(\lambda_R\) and the cap.
Proposition 4 holds for every incumbent distribution with a positive spread, \(g_H\gt g_L\) and \(\tau\gt\tfrac12\). This recruiting equilibrium exists exactly when \(\Delta_T\gt2k\) and the posterior after a purchase, at the research the informed trader chooses, reaches the threshold. Research rises strictly with the spread until it reaches the cap and then stays there. Entry, \(\rho+(1-\rho)[\beta+(1/2-\beta)\lambda]\), and entry in class \(H\) rise with research, while entry in class \(L\) falls, so the rivals who prepare are more often of class \(H\). When class \(H\) outbids the incumbent, high-class ownership is half of entry in class \(H\) and rises with research. Take a path whose spread starts below and reaches the value at which research hits its cap, with the threshold below one up to that strength. When the recruitment margin, the value of a success less the marginal cost of research at the research that just lets a purchase recruit, rises strictly along it, from negative against the weakest incumbent to positive where research reaches its cap, the equilibrium holds exactly on one interval of strengths. On it research rises to the cap and stays there up to the last strength at which the threshold does not exceed the top posterior at the cap.
In the example (Corollary 3), set \(a=1/40\) and the cap \(\bar\lambda\approx0.336122\), the example's presence probability. Zero research, with entry \(\rho\), is an equilibrium exactly on \([r_0,r_N]\). Recruiting research is an equilibrium exactly from \(r_R\approx1.501023\) to the entry ceiling \(r_C\approx3.592659\): research is below the cap up to \(r_{\bar\lambda}\approx1.677633\) and at the cap from there, and entry rises from 0.500846 at \(r_R\) to 0.522757 at \(r_{\bar\lambda}\) and keeps that value up to \(r_C\). Floor-entry research, in which both classes trade after a success, no order recruits and entry is \(\rho\), is an equilibrium exactly on \((r_N,r_Q)\) below the cap, with \(r_Q\approx2.166995\), and on \((r_C,r_2]\) at the cap. There is no other equilibrium. The unique equilibrium thus has zero research at \(r_0\), recruiting research at the cap at \(r_1\) and floor-entry research at the cap at \(r_2\); at \(r_1\) and \(r_2\) it is the equilibrium of Proposition 2.
At each of 1.55, 1.60 and 1.65, zero research and recruiting research below the cap coexist. Along the recruiting equilibrium, research rises from 0.237901 through 0.279037 to 0.316683, and entry rises from 0.507630 through 0.513965 to 0.519763. A stronger incumbent widens the spread, so a purchase earns more, the informed trader chooses more research, and the posterior after a purchase rises further above the threshold.
Research in detail
At fixed prices and entry, the best research choice is unique: it equals expected net trading profit after a success divided by \(a\), capped at \(\bar\lambda\). In the recruiting equilibrium a purchase has probability \(\beta+(1/2-\beta)\lambda\), so entry has slope \((1-\rho)(1/2-\beta)\) in research for every \(\beta\). Research adds informed purchases in \(H\) and removes liquidity purchases in both classes; the first effect is larger.
At \(r_R\), a full purchase that just reaches the threshold also pays for its research, \(\Delta_T(1-\tau)=k+2a\lambda_E\), where the recruitment intensity \(\lambda_E\) is the research at which a full purchase just reaches the threshold; there research is 0.193853. The low class starts to sell at \(r_N\) and sells for sure from \(r_S\); research is at its cap before either strength, so in the example the sale's profit leaves research unchanged. Recruiting research coexists with zero research on \([r_R,r_N]\) and with floor-entry research on \((r_N,r_Q)\), where floor-entry research rises towards 0.234257. It is the only equilibrium on \([r_Q,r_C]\), and floor-entry research at the cap is the equilibrium above \(r_C\).
Recruiting research starts later than the informative market of Corollary 2, which starts at \(r_A\), because a purchase must pay for research as well as for trade. With a fixed presence probability the purchase rate adjusts through a mixture of the high class; with research the intensity adjusts, and the high class buys for sure after a success.
At \(\beta=0.294655\), the pattern of Corollary 3 at \(r_0\), the three intermediate strengths, \(r_1\) and \(r_2\) holds with strict margins exactly when the cap lies between the recruitment intensities at \(r_1\) and \(r_2\), about 0.290539 and 0.336721, and the cost lies in a window that depends on the cap. At the example's cap the window is \((0.021469,0.036763)\). With \(1+u=\mathrm e\), the cap \(\bar\lambda=u\beta/(1+u\beta)\) and \(a=1/40\), the pattern holds for every \(\beta\) above about 0.236157 and up to 1/3.
Robustness, welfare and sale terms
5 · Extensions · 2 minutes
The reversal survives changes to the model
The rise in entry survives other distributions of preparation costs, with or without atoms. Let the cost mix a low component with weight \(\rho\) and a high component. Retain the trading-cost window (H2), place every low cost strictly below \(B_{F_1}(m(\pi))\), and place every high cost strictly between \(B_{F_0}(\pi)\) and \(B_{F_1}(M(\pi))\). Parts (i) and (ii) of Proposition 2 then hold. The low component prepares at every feasible belief, so entry stays at least \(\rho\) and the trading bound applies unchanged. With an atomless high component, the drop at the entry ceiling becomes a ramp: full-order entry is continuous and nonincreasing in strength.
For uniform incumbents, every prior and every pair of classes that satisfy the class ranking and in which the high class has the higher mean admit an open set of strengths and costs with the reversal. This set satisfies (H1) to (H3) and the ownership condition, whatever the reserve, the zero-cost probability and the slot. The construction puts the weak and the strong incumbent just above the lowest value at which the classes start to differ above the reserve, where the spread vanishes while the profit gap stays near the full class gap. With more than two classes, prices reveal the posterior mean of class proceeds. Under a separation condition, every high-group class buys, every low-group class sells, and the reversal holds.
A price can add information even when the rival has a more accurate private signal than the informed trader. The rival knows its integration plans, and the informed trader knows the target's customers or product market. Two binary signals, independent given the class, represent the two sources; at the prior one half, under strict conditions on the costs, the weak incumbent has an uninformative market and entry \(\rho\), and the strong incumbent has full orders and entry above \(\rho\). In the signal example, with its own costs and strengths, the informed trader's accuracy is \(\alpha_I=0.70\) and the rival's accuracy is \(\alpha_C=0.75\). Entry rises from 0.850000 to 0.879797. Without the rival's signal, entry stays at 0.850000 at both strengths, so the rise needs both sources of information.
Price access raises proceeds and surplus
Hold the incumbent at \(F_1\) and compare price access with a hidden price, under the same sale mechanism. Under the hypotheses of Proposition 2, the trading bound forces unique full orders in both economies, even when the rival cannot use the price and entry stays at \(\rho\), so trading costs coincide. Price access adds high-cost preparation after a purchase, where expected acquisition profit exceeds the preparation cost, and it raises expected target proceeds and acquisition surplus net of preparation costs. In the example at \(r_1\), expected proceeds rise from 0.614583 to 0.872531, a gain of 0.257948, and net surplus rises by 0.081960. With research at the cap, the gains are the same. Acquisition surplus excludes transfers between bidders, shareholders and traders.
A shareholder who holds through the sale receives the higher expected proceeds. Inside the recruitment region, where high-cost preparation does not pay at the prior, price access adds preparation. When gross profit at the prior exceeds the high cost and gross profit after a sale falls below it, an informative price deters high-cost preparation after a sale.
Sale terms
The payment rule determines how incumbent strength changes the spread. Consider verifiable values and a zero reserve, with a sale to the losing bidder at its value as an enforceable fallback. The seller and the winner divide the surplus above this fallback by Nash bargaining, with seller weight \(\omega\in[0,1)\). For point-mass classes with \(0\lt\ell\lt\bar r\lt h\) and incumbent support \([0,\bar r]\), a stronger incumbent never raises the rival's profit and moves the spread only in the direction of the sign of \(1-2\omega\). The move is strict when \(\mathbb E[(R-\ell)_+]\) changes strictly and \(\omega\ne1/2\); at \(\omega=1/2\) the spread is \((h-\ell)/2\) against every incumbent. In the example, the spread under this rule changes by \(13/20\,(1-2\omega)\) from \(r_0\) to \(r_1\). The second-price auction is the case \(\omega=0\): the winner pays the losing bidder's value.
A reserve that excludes the low class can widen the spread, because low-class preparation then adds nothing to target proceeds. Informed trade can then pay against both incumbents. If informed trade is full and only a purchase recruits at both strengths, entry is flat in incumbent strength, and strict deterrence needs the stronger incumbent to push gross profit at the top posterior below the high cost. In the example, the reserve 101/100 gives entry 0.522757 at both strengths. The reserve 31/20 also excludes the weak incumbent, and entry falls as the incumbent strengthens; strict deterrence holds for reserves between about 1.325270 and 1.792723.
The extensions in numbers
In the example, low costs range from zero to 1/5, and high costs are uniform on \([c_H-\varepsilon,c_H+\varepsilon]\). Every half-width below 0.002689 preserves all three parts of Proposition 2. At \(\varepsilon=1/10\), entry is 0.522757 at \(r_1\), the value under the two-point distribution, and 0.382711 at \(r_2\), where gross profit at the top posterior lies 0.002689 below \(c_H\), inside the band of high costs.
For general classes, the construction places the first two incumbent strengths just above the lowest value at which the class distributions differ. There the target-payoff spread approaches zero, while the rival's profit gap approaches the full difference in class means. With wide uniform classes, entry rises from 0.250000 to 0.573175, and high-class ownership rises although the high class can lose to the strong incumbent. Point masses at the same means leave entry at 0.250000. A prepared rival's profit is convex in its value, so an increase in risk within a class never lowers gross profit at any belief. A four-class economy near the example satisfies the separation condition with distance 0.178871.
The signal example keeps the slot, values, reserve and prior of the example. It sets \(\rho=0.85\), \(c_H=7.14\), \(k=0.015\), \(r_0=1.1\) and \(r_1=2.3\). Against the weak incumbent, a favorable private signal leaves high-cost preparation unprofitable. Against the strong incumbent, a favorable private signal and a favorable price together justify preparation.
In the example itself, the conditions hold with \((\alpha_I,\alpha_C)=(0.94,0.505)\). Entry rises from 0.250000 to 0.386933, and without the rival's signal it is 0.250000 at both strengths. At the example's costs and strengths, the strong trading bound needs an accurate signal for the informed trader, and the weak bound then fails for a rival's signal at least as accurate; the signal example meets the conditions with a more accurate signal for the rival at other costs and strengths.
The informed trader's expected gross profit at \(r_1\), 0.037665, is a transfer from liquidity traders. If one order unit is a given number of shares per share of float, initial shareholders gain in total relative to an economy without the informed trader whenever that number is below 6.848394, even if they bear the whole transfer. A deterministic external dividend of 0.257948 in the hidden-price economy equalizes the mean prices and leaves the orders and entry unchanged, so the price's information content distinguishes the two economies.
Outside the recruitment region, set \(c_H=12/5\) against the strong incumbent of the example. Price access then deters preparation after a sale and changes proceeds by -0.173917 and net surplus by -0.013505. The reserve 31/20 lowers entry from 0.522757 to 0.250000, and proceeds exceed their levels at the reserve 1/2. With a reserve between \(\ell\) and \(r_0\), the top posterior recruits at both strengths and entry is flat. Narrow uniform classes give the flat pattern at the reserve 11/10.
| Reserve | Strength | Entry | Sale | Two admissible bidders | Expected proceeds |
|---|---|---|---|---|---|
| 1/2 | \(r_0\) | 0.250000 | 0.687500 | 0.145833 | 0.392708 |
| \(r_1\) | 0.522757 | 0.920459 | 0.435631 | 0.872531 | |
| 101/100 | \(r_0\) | 0.522757 | 0.431371 | 0.051364 | 0.440564 |
| \(r_1\) | 0.522757 | 0.772548 | 0.215186 | 0.994384 | |
| 31/20 | \(r_0\) | 0.522757 | 0.324401 | 0 | 0.502822 |
| \(r_1\) | 0.250000 | 0.547917 | 0.060417 | 0.893073 |
What the data should show
6 · Empirical implications · 2 minutes
The predictions concern publicly visible sale processes in which one bidder is prepared and a rival has not yet committed to diligence. Each applies to one target in one public state and states its conditions. Under full two-sided trade, class \(H\) buys and class \(L\) sells, each with probability one, and the informed share of nonzero orders is \(u/(2+u)\) at every prior. In the example at the prior one half, full two-sided trade is the unique outcome at every strength between \(r_S\), where a full sale earns the trading cost, and \(h\), a range that contains \(r_1\), the entry ceiling and \(r_2\).
Entry and incumbent strength
When the price reaches the rival before it decides, a stronger incumbent can raise the probability that the rival prepares: by 0.272757 from \(r_0\) to \(r_1\) in the example. When the price does not reach the rival, a stronger incumbent can only lower it. Against a strong enough incumbent, entry can return to its floor while the price stays informative. The first and third comparisons are Proposition 2; the second is the fixed-information control.
The purchase impact carries the entry response
Under full two-sided trade at a prior below the preparation threshold, the purchase impact \(P(1)-P(0)\), weighted by the frequency of a purchase, exceeds the sale impact \(P(0)-P(-1)\), weighted by the frequency of a sale, exactly when a purchase recruits the rival. The weighted difference equals the expected price before the order opportunity minus the price after no order, so the test uses only prices and order frequencies. At the prior one half the two frequencies are equal, and the test compares the two impacts directly. The price gap \(P(1)-P(-1)\) has a class component, \(\rho\Delta_T[M(\pi)-m(\pi)]\), and an entry component, which is positive exactly when a purchase recruits; the whole entry component sits in the purchase impact. In the example at \(r_1\), the price gap is 0.786299 and its entry component is 90.2% of it; with a hidden price it would be 0.077020. At \(r_2\) the price gap is 0.108469, the class component alone, and at the entry ceiling it falls by 0.861835.
Prices separate what order counts cannot
Under full two-sided trade the mean order is \(\lambda(2\pi-1)\), so order counts identify the informed trader's presence when the prior is known and differs from one half. The sale impact identifies presence at any known prior, one half included: when entry after a sale and after no order sits at its floor and the payoff scale \(\rho\Delta_T\) is known, the sale impact weighted by the frequency of a sale, divided by \(\rho\Delta_T\), equals \(\pi(1-\pi)\lambda\). At the prior one half, the counts depend on the slot only through the volume of nonzero orders, so prices separate slots that counts cannot.
Orders predict the winner
Under full two-sided trade, a prepared rival wins more often after a purchase than after a sale exactly when there is a positive-probability region in which a high-class value beats the incumbent with a higher probability than a low-class value. Above the entry ceiling, orders no longer predict entry, and they still predict which bidder wins. In the example class \(H\) outbids every incumbent, and the gap is 0.308078 at \(r_1\) and 0.333751 at \(r_2\). The covariance of the order with entry is 0.272757 at \(r_1\) and zero at \(r_2\), while its covariance with a win by the rival stays positive. Among prepared rivals, the share of the high class at each price is the posterior that the price implies.
The informed share, announcements, two clusters and measurement
The gap in win frequencies is \(M(\pi)-m(\pi)\) times the difference between the classes' chances of beating the incumbent. With the prior and the slot fixed and full two-sided trade at each strength compared, when class \(H\) outbids every incumbent the gap does not narrow as the incumbent strengthens, and it widens wherever the stronger incumbent takes wins from the low class.
The sale impact and the mean order \(\lambda(2\pi-1)\) together identify both the presence probability and the prior. In the example at \(r_1\), the informed shares consistent with the order counts and the informed trader's incentives range from about 0.250299 to 0.760000. A slot with informed share 0.444444 has the example's volume and entry, but its price gap at \(r_1\) is 0.778935 against 0.786299.
With a uniform incumbent and the prior and the slot fixed, the entry component of the price gap rises with strength while full two-sided trade and recruitment persist, when \(3p^2\lt\min\{\underline\theta,\underline r\}^2\), where \(\underline\theta\) is the lowest value of the rival and \(\underline r\) the lowest strength compared; the example meets this condition. As the incumbent strengthens, entry adds more to the price through competition and less by rescuing a sale that the incumbent alone would not complete.
For a positive spread and \(g_H\gt g_L\), if a public event after the price reveals only whether the rival prepared, the mean price change over preparation reports is positive exactly when preparation was still uncertain after some order, at every prior and in every equilibrium: 0.160592 at \(r_1\) in the example. The mean change over both reports, weighted by their probabilities, is zero. Under full two-sided trade at a prior below the threshold, an order opportunity that passes with no order is no news about the class and, when a purchase recruits, bad news about entry: at \(r_1\) the price after no order minus the expected price before the opportunity is -0.257948, and the size of this drop is the weighted difference of the price-impact test.
At one public strength strictly between \(r_F\) and \(r_N\), sale processes can form two clusters. In one the market is uninformative: the mean order is zero and the three prices coincide. In the other the high class buys and the low class abstains; the mean order is \(\lambda/2\), or 0.168061 in the example, the rival prepares after every purchase, and a sale executes above the price after no order, by 0.005363 at 1.60.
Preparation corresponds to the start of substantive diligence or a proposal that requires costly preparation; public offers are a narrower outcome. Incumbent strength is measured from information that is public before that decision, and the price must precede it. The order is the first prespecified public order opportunity after the sale becomes public, for example the opening auction on the first trading day after an approach is disclosed: its time is fixed in advance, and it takes place whether or not an order arrives. The sign of its net imbalance is the order, and an imbalance below a size fixed in advance records the zero order. A sample of first executed trades omits the zero order, so it cannot identify the volume, the effect of an inactive opportunity or the informed share. Target returns before a bid partly reflect anticipation of that bid (Schwert 1996), so each test dates the price before the rival's decision.
A stronger incumbent moves the class value gap from the rival's profit into target proceeds. Because a liquidity order can match the informed trader's direction, the informed trader keeps an advantage after every order. A wider spread can make that advantage worth the trading cost, so the price recruits a rival that a weaker incumbent could not attract.