Competition Creates Competition

Inside a takeover process

1 · The setting · 2–3 minutes

A stronger bidder usually discourages a rival from entering. This paper asks when it instead makes the target's stock price more informative and brings another buyer into the sale.

An illustrative sale process

A listed software company announces a strategic review. One buyer is ready to bid. A second sees a possible fit but must investigate the technology, assess contracts and arrange financing. It watches the target's shares before deciding whether to pay for that work.

Preparation means paying for the work needed to learn acquisition value and submit an executable offer. Interest or a confidentiality agreement alone does not establish this commitment. A buyer that declines to prepare stays out; it cannot bid on its prior valuation.

The model and information sets

The prepared incumbent knows its acquisition value \(R\). The challenger initially does not know its own value \(\theta\). The seller commits to a cash second-price auction with reserve \(p\):

R\sim U[0,r],\qquad \Pr(\theta=\ell)=\Pr(\theta=h)=\tfrac12, \qquad 0\lt p\lt\ell\lt r\lt h.

A higher \(r\) shifts incumbent values upward; it is not a higher announced bid. There is one challenger, with preparation cost \(C\):

C=\begin{cases}c_L&\text{with probability }\rho,\\c_H&\text{with probability }1-\rho,\end{cases} \qquad 0\le c_L\lt c_H,\quad 0\lt\rho\lt1.

An investor observes \(\theta\) and trades target shares, without control rights. Market makers observe total order flow; the challenger observes only the resulting price and its cost. Paying \(C\) reveals \(\theta\) and permits truthful bidding. Values, cost and noise demand are independent; agents are risk neutral and standalone target value is zero.

The buyer can know its own business while investors know more about the target's customers or technology. The benchmark isolates this information channel. An extension gives the buyer and investor distinct, imperfect signals.

The sale opportunity is publicly understood. The seller commits to a cash auction and a minimum acceptable price. One bidder is prepared; the challenger is still deciding.
An informed investor trades target shares alongside noise demand. Investor information concerns what the target would be worth to the challenger.
Competitive market makers observe total order flow. They price target proceeds, anticipating how the resulting price affects preparation.
The challenger observes the stock price and its own preparation cost. It pays to learn its value when expected acquisition profit covers that cost.
Prepared buyers bid their values in the second-price auction. Only bids meeting the reserve are admissible.
The highest admissible bidder wins and pays the greater of the reserve and the competing bid. Without an admissible bidder, no sale occurs.

Select a stage to see the information available at that point.

The opportunity must be public while the challenger can still act. A strategic review or open contest can provide this window; a confidential sale revealed only after participation is settled cannot. The auction isolates payment incentives, rather than describing every takeover literally.

Imprivata's proxy distinguishes initial interest from further diligence. It motivates the preparation margin, but does not establish that prices caused entry. The software-company example above is illustrative.

Institutional mapping and related literature

In the model	In the record	What the record must show
The sale rule is announced and the opportunity is publicly understood	A public strategic review, a disclosed approach, or an open contest	Visibility dated separately from the challenger's decision. A target that traded only during confidential negotiations does not qualify.
A prepared incumbent of strength \(r\)	A lead buyer already able to bid: its disclosed initial proposal, financing capacity, measurable complementarity with the target	Measured before the challenger decides. A final offer already reflects entry.
The investor trades and the price \(P\) forms	Target shares trading while the opportunity is visible	Returns and trading activity timestamped before the participation decision.
The challenger pays \(C\) to learn \(\theta\)	Commitment to substantive diligence, or a proposal that requires costly preparation	A confidentiality agreement alone does not establish preparation. A firm named by an adviser is not an entrant.
Entry \(\mathsf E\)	Commitment to investigation or substantive bidding	Not the count of public offers, a narrower outcome.
High-value ownership \(\mathsf O_H\)	A challenger that wins	Winning alone is not evidence of a high acquisition value.

From the model to the record. Eligibility and measurement rules of the pilot in Online Appendix D.

Positioning in the literature

Feedback effects. In Dow, Goldstein and Guembel (2017) a firm's investment decision shapes the incentive to produce information about its stock. Edmans, Goldstein and Jiang (2015) show that corrective real decisions can discourage trading on bad news under rational pricing. I ask how rival strength changes the information sensitivity of the claim and, through it, participation. Evidence that prices affect takeover activity (Edmans, Goldstein and Jiang 2012) runs from prices to control and does not identify a prospective challenger learning its own value; learning from announcement returns (Luo 2005) happens after the participation margin studied here.

Auction entry and information. The auction-entry literature makes the bidder pool endogenous (Levin and Smith 1994; Gentry and Stroup 2019) and shows how selective entry affects the choice of sale procedure (Roberts and Sweeting 2013). Bidders' own information acquisition responds to the format (Persico 2000); here the informed party is an investor outside the auction, trading a payoff that differs from the entrant's profit. Bidder learning about own values and competitors (Pernoud and Gleyze 2026), post-auction feedback in security-payment design (Liu and Bernhardt 2022), and bidder-pool choice with correlated values (Carlin et al. 2026) are the closest recent work.

Takeover models with trading. Other models connect trading to later stages of a deal: negotiations with stock-market feedback (Betton et al. 2014) and payment choice within an initiated deal (Lin, Ma, Yang and Zhu 2025). The investor here plays no role in tendering (Cornelli and Li 2002), and the model excludes toeholds and free riding (Bulow, Huang and Klemperer 1999; Grossman and Hart 1980). Endogenous investor research would add manipulation incentives through the real decision that responds to the price (Goldstein and Guembel 2008).

Two returns to information

2 · The mechanism · 4 minutes

The challenger cares about the acquisition value it keeps after paying for the target. Target shareholders receive that payment. The auction links these different payoffs to preparation and informed trading.

A stronger incumbent

The direct effect

Lower acquisition profitThe challenger expects to keep less after paying for the target.

Less reason to prepare at a fixed belief.

The information effect

Target payments depend more on challenger valueInformation becomes more valuable to the investor.

Informed trading becomes worthwhileThe stock price reveals information the challenger can use.

Favorable prices can justify costly preparation.

The participation reversal occurs when the second effect outweighs the first. Proposition 2 gives conditions on preparation costs and trading costs that establish this comparison.

One payment rule, two incentives

Let \(g_H(r),g_L(r)\) be the challenger's expected acquisition profits before preparation costs, conditional on its value. At posterior belief \(\mu(P)=\Pr(\theta=h\mid P)\), its decision is

\begin{gathered} B_r(\mu)=g_L(r)+\mu[g_H(r)-g_L(r)],\\ \text{prepare iff }B_r(\mu(P))\ge C. \end{gathered}

Let \(t_H,t_L\) be expected target proceeds when the challenger prepares. If \(R\le\ell\), either challenger type wins and pays the same amount. If \(R\gt\ell\), a high-value challenger wins and pays \(R\); a low-value challenger loses, and the incumbent pays \(\ell\). Thus

\begin{gathered} \Delta_T(r)\equiv t_H-t_L=\mathbb E[(R-\ell)_+] =\frac{(r-\ell)^2}{2r},\\ \frac{d\Delta_T}{dr}\gt0,\qquad \left.\frac{\partial B_r(\mu)}{\partial r}\right|_\mu\lt0. \end{gathered}

Here \((x)_+=\max\{x,0\}\). Stronger competition lowers acquisition profit at every fixed belief, but makes target proceeds more sensitive to challenger value. The second effect raises the value of the investor's information.

Why the price can change preparation

The investor chooses an order \(q\in[-1,1]\) to maximize expected profit conditional on \(\theta\), paying \(k|q|\), with \(k\gt0\). Independent noise demand is \(Z\sim\operatorname{Laplace}(0,b)\), \(b\gt1\). For terminal target proceeds \(V_T\), competitive pricing requires

\begin{gathered} X=q+Z,\qquad P(X)=\mathbb E[V_T\mid X],\\ \pi_I=q[V_T-P(X)]-k|q|. \end{gathered}

An equilibrium jointly determines investor orders, prices, Bayesian beliefs and optimal preparation under truthful bidding. Investor deviations may use any order or mixture, evaluated against the candidate price and preparation schedules. Comparing incumbent strengths instead solves those schedules again.

Noise bounds the market maker's posterior \(\mu_X=\Pr(\theta=h\mid X)\), under every order strategy:

m\le\mu_X\le M,\qquad m=\frac{1}{1+e^{2/b}},\quad M=1-m.

Price posteriors inherit these bounds. If \(c_L\lt B_r(m)\), the low-cost challenger always prepares. With preparation probability \(e(P)\ge\rho\) and no-preparation proceeds \(t_0\), rational pricing implies

\begin{gathered} P=t_0+e(P)[t_L-t_0+\Delta_T\mu_X],\\ \mu_X=\frac{P-t_0-e(P)(t_L-t_0)}{e(P)\Delta_T}. \end{gathered}

The positive denominator lets the challenger recover the market's posterior from the price alone. This preparation floor sustains state-sensitive proceeds. Rational pricing incorporates expected preparation but leaves an investor who knows \(\theta\) an informational advantage; the question is whether it covers trading costs.

Scroll horizontally to see the full figure.

Figure 1. At fixed beliefs, challenger profit falls as the incumbent strengthens, while the target-payoff spread rises. The horizontal axis is incumbent strength \(r\); \(\Delta_T\) is the expected difference in target proceeds across challenger types, and \(B_r\) is expected challenger profit before preparation costs. The three profit curves hold beliefs fixed at the lower bound, the prior and the upper bound. Values are per target share. Hover for values.

Payoff derivations, equilibrium details and the explorer

Values, preparation, and the sale

A listed target with standalone value zero is sold in a cash second-price auction with reserve \(p\), the minimum acceptable bid. All values and costs are per share. The incumbent is already prepared and knows its value \(R\), drawn uniformly from \([0,r]\). A higher \(r\) shifts the distribution toward higher values.

The challenger's value \(\theta\) is either \(\ell\) or \(h\), each with prior probability one half. After observing the stock price, it learns its preparation cost: \(c_L\) with probability \(\rho\), and \(c_H\) otherwise. Paying that cost reveals \(\theta\) and permits bidding. The benchmark assumes

0\lt p\lt\ell\lt r\lt h. \tag{1}

Bids are truthful and weakly dominant; the winner pays the larger of the reserve and the highest competing bid. An indifferent challenger enters, a tie rule that matters at one posterior plateau in the results.

Trading and pricing

An investor observes \(\theta\) and submits an order \(q\in[-1,1]\) in a small unit with no control rights, paying a linear trading cost \(k|q|\). Its profit and aggregate order flow are

\begin{gathered} q\{V_T-P(X)\}-k|q|,\qquad X=q+Z,\\ f(z)=\frac{1}{2b}e^{-|z|/b},\qquad b\gt 1, \end{gathered} \tag{2}

where \(V_T\) is the terminal payoff of a target share and \(Z\) is independent Laplace noise. Competitive market makers observe aggregate flow and set

P(X)=\mathbb E[V_T\mid X], \tag{3}

anticipating preparation and auction outcomes. The challenger observes \(P\), not \(X\).

Notation

The notation separates what the target receives from what the challenger keeps. Expected target proceeds are \(t_0\) without preparation and \(t_H,t_L\) when the challenger prepares and has a high or low value. The corresponding challenger profits, before preparation costs, are \(g_H,g_L\).

The target-payoff spread is \(\Delta_T=t_H-t_L\). At a belief \(\mu=\Pr(\theta=h)\), expected challenger profit is \(B_r(\mu)=g_L+\mu(g_H-g_L)\). A posterior is this probability after observing new information.

Equilibrium notion

An equilibrium specifies conditional order distributions \(\sigma_H,\sigma_L\), a measurable price function, beliefs given the price, optimal preparation, and truthful bidding. The investor can mix and can deviate to any order in \([-1,1]\); uniqueness claims below refer to trading and on-path entry under truthful bidding and allow arbitrary mixed orders and every continuous deviation.

Benchmark parameters

\(h\): 10
\(\ell\): 1
\(p\): 0.5
\(\rho\): 0.25
\(c_L\): 1
\(c_H\): 6
\(b\): 2
\(k\): 0.02
\(r_0\): 1.2
\(r_1\): 3
\(r_2\): 3.6

Acquisition profits and target proceeds

A high-value challenger wins and pays \(\max\{p,R\}\); with a low-value challenger the target receives \(\max\{p,\min(R,\ell)\}\). Integrating over the uniform incumbent gives

\begin{aligned} t_0&=p\left(1-\frac p r\right),& t_H&=\frac r2+\frac{p^2}{2r},\\ t_L&=\ell-\frac{\ell^2-p^2}{2r},& g_H&=h-\frac r2-\frac{p^2}{2r},\\ g_L&=\frac{\ell^2-p^2}{2r},& \Delta_T(r)&=\frac{(r-\ell)^2}{2r}. \end{aligned} \tag{4}

The spread rises in \(r\) and both gross profits fall in \(r\). A stronger incumbent lowers the challenger's profit at every belief, which is deterrence, and makes target proceeds more sensitive to challenger quality, which is what an informed investor cares about.

Derivatives (5)

\begin{aligned} B_r(\mu)&=g_L+\mu(g_H-g_L),\\ \Delta_T'(r)&=\frac12-\frac{\ell^2}{2r^2}\gt 0,\\ g_H'(r)&=-\frac12+\frac{p^2}{2r^2}\lt 0, \qquad g_L'(r)=-\frac{\ell^2-p^2}{2r^2}\lt 0. \end{aligned} \tag{5}

Proposition 1 (competition and the two returns to information).

For any continuous incumbent distribution, a first-order stochastic strengthening weakly increases the spread of target proceeds between a high- and a low-value challenger and weakly decreases the challenger's gross acquisition profit at every fixed posterior.

Full statement

Let the incumbent's value have a continuous distribution \(F\) on \([0,\bar r]\) with \(0\lt p\lt\ell\lt\bar r\lt h\), and extend \(F\) by unity above its support. A first-order stochastic strengthening of \(F\) weakly increases the spread of target proceeds between a high- and a low-value challenger and weakly decreases the challenger's gross acquisition profit at every fixed posterior, where

\begin{aligned} \Delta_T(F)&=\mathbb E_F[(R-\ell)_+] =\int_\ell^{\bar r}[1-F(u)]\,du,\\ G_\theta(F)&=\mathbb E_F[(\theta-\max\{p,R\})_+] =\int_p^\theta F(u)\,du. \end{aligned} \tag{6}

Both comparisons are strict when the change in \(F\) has positive integral over the corresponding range.

The payment rule explains the result. Below \(\ell\), the incumbent loses to either challenger type, and both types pay the same amount. Above \(\ell\), a high-value challenger wins and pays \(R\), while a low-value challenger loses and the incumbent pays \(\ell\). The difference in target proceeds is therefore \((R-\ell)_+\). A stronger incumbent increases this difference while reducing the challenger's profit. The bargaining comparison shows how a different payment rule can change the first effect.

Proof sketch

The realized sale rule fixes the payoff formulas. Without entry the target receives \(p\,\mathbf 1\{R\ge p\}\); with a high-quality challenger it receives \(\max\{p,R\}\); with a low-quality challenger it receives \(\max\{p,\min(R,\ell)\}\) and the challenger obtains \((\ell-\max\{p,R\})_+\). Truthful bidding is weakly dominant conditional on every competing bid, so nothing depends on a conjectured shading strategy. Integrating against the uniform density gives (4); differentiating gives (5).

For general \(F\), the high and low target payments coincide when \(R\le\ell\) and differ by \(R-\ell\) otherwise, so their difference is \((R-\ell)_+\). Tonelli's theorem on the nonnegative indicators gives both integrals in (6). A stronger distribution has a smaller CDF, which raises the survival integral and lowers the profit integral; a positive integral difference makes each comparison strict, and a weighted average at a fixed posterior preserves the profit ordering. Appendix A.2 and Online Appendix A.2 give the measure formulation.

What the challenger learns from the price

Noise demand limits what any order can reveal. Under Laplace noise, the likelihood ratio of any two feasible orders is at most \(e^{2/b}\). The market maker's posterior therefore stays within \([m,M]\), whatever the investor's strategy. At \(b=2\), these bounds are \(m=0.268941421\) and \(M=0.731058579\).

The upper bound limits preparation. Once \(B_r(M)\lt c_H\), even the most favorable attainable belief cannot justify the high cost. The lower bound helps explain price informativeness. When \(c_L\lt B_r(m)\), the low-cost challenger always prepares. Preparation probability is then at least \(\rho\), and the price reveals the market maker's posterior. Propositions A.1 and A.2 establish these bounds and the inference result.

How the price reveals the posterior (7) to (10)

For arbitrary mixed orders, define

\begin{aligned} a_H(x)&=\int f(x-q)\,d\sigma_H(q),\qquad &a_L(x)&=\int f(x-q)\,d\sigma_L(q),\\ \mu_X(x)&=\frac{a_H(x)}{a_H(x)+a_L(x)},\qquad &m&=\frac1{1+e^{2/b}},\quad M=1-m. \end{aligned} \tag{7}

For any two orders \(q,q'\in[-1,1]\) the Laplace density obeys

e^{-2/b}\le\frac{f(x-q)}{f(x-q')}\le e^{2/b}, \tag{8}

and integrating over the conditional order distributions preserves the inequalities, so \(m\le\mu_X\le M\) and the price-based posterior inherits the bounds. Averaging over preparation costs gives the entry rule and the candidate competitive price

\begin{aligned} e_r(\mu)&=\rho+(1-\rho)\mathbf 1\{B_r(\mu)\ge c_H\},\\ P_r(\mu)&=t_0+e_r(\mu)[t_L-t_0+\Delta_T\mu]. \end{aligned} \tag{9}

Entry is bounded below by \(\rho\gt 0\) and the increment in proceeds is strictly increasing in \(\mu\), so the price is strictly increasing in the posterior and can jump where high-cost preparation becomes worthwhile. Sufficiency does not depend on this construction: writing \(e(P)\) for entry conditional on the observed price, competitive pricing in any candidate equilibrium implies

\begin{aligned} P&=t_0+e(P)[t_L-t_0+\Delta_T\mu_X],\\ \mu_X&=\frac{P-t_0-e(P)(t_L-t_0)}{e(P)\Delta_T}. \end{aligned} \tag{10}

and the positive denominator makes \(\mu_X\) a measurable function of \(P\).

Informed trading incentives

The price already reflects the preparation and proceeds that market makers expect. The investor's remaining advantage comes from knowing challenger value. For a purchase on good news or a sale on bad news, this advantage equals preparation probability times the target-payoff spread times residual uncertainty. It lies between \(\rho m\Delta_T\) and \(\Delta_T\) for every candidate order distribution. Comparing these bounds with trading costs establishes the unique trading outcomes used in the main result.

Residual advantages (11)

\begin{aligned} A_H(x)&=\mathbb E[V_T\mid H,x]-P(x)=e_r(\mu_X(x))\Delta_T[1-\mu_X(x)],\\ A_L(x)&=P(x)-\mathbb E[V_T\mid L,x]=e_r(\mu_X(x))\Delta_T\mu_X(x),\\ \rho m\Delta_T&\le A_H(x),A_L(x)\le\Delta_T. \end{aligned} \tag{11}

Both bounds hold for every candidate order distribution, because \(e_r\ge\rho\) and \(\mu_X\in[m,M]\).

Explore the benchmark formulas

Move the slider to change incumbent strength \(r\), holding the other parameters fixed. The readouts update acquisition payoffs, information bounds and the main result's conditions. The plotted preparation probability assumes full investor orders; away from validated benchmarks, it need not describe an equilibrium.

What the paper establishes

3 · The result · 4 minutes

Proposition 2 · Sufficient conditions for the reversal

Compare \(\ell\lt r_0\lt r_1\lt h\), holding all other primitives fixed. Under the model above, suppose:

0\le c_L\lt B_{r_1}(m). \tag{A1}

A preparation floor. Even unfavorable prices leave low-cost preparation worthwhile.

B_{r_0}(1/2)\lt c_H\lt B_{r_1}(M). \tag{A2}

Information can recruit the high-cost challenger. The weak-incumbent prior does not justify preparation, but favorable strong-incumbent prices can.

\Delta_T(r_0)\lt k\lt\left(1-\frac1b\right)\rho m\Delta_T(r_1). \tag{A3}

Trading incentives switch. Trading costs exceed every informational return in the weak economy but lie below a global bound on marginal trading profits in the strong one.

Then the unique trading and on-path preparation outcomes satisfy

\begin{aligned} r_0:&\quad(q_H,q_L)=(0,0),\quad\mathsf E(r_0)=\rho,\\ r_1:&\quad(q_H,q_L)=(1,-1),\quad\mathsf E(r_1)\gt\rho,\\ &\quad\mathsf O_H(r_1)\gt\mathsf O_H(r_0). \end{aligned}

Here \(q_H,q_L\) are orders conditional on high and low challenger value, \(\mathsf E\) is preparation probability and \(\mathsf O_H\) is the probability that a high-value challenger acquires the target. The strong price experiment strictly Blackwell dominates the uninformative weak experiment. These comparisons hold on a nonempty open set of primitives, allowing arbitrary mixed orders and every continuous deviation. Off-path pricing rules need not be unique.

Why uniqueness holds. Against the weak incumbent, the investor's gross advantage per unit is at most \(\Delta_T(r_0)\lt k\), so no trade is optimal. Against the strong incumbent, the preparation floor and noise bound make every increase in a correctly signed order profitable. Full orders are therefore necessary in every equilibrium; the proof constructs the associated price and preparation schedules. Favorable prices recruit high-cost preparation more often when challenger value is high.

25.0%prepare against the weak incumbent
\(r=1.2\), uninformative price

52.3%prepare against the strong incumbent
\(r=3\), informative price

32.4%high-value challenger ownership
strong-incumbent benchmark

Hold the information fixed

Hold the investor's informative orders fixed across economies: stronger competition reduces preparation. Let orders adjust in equilibrium: information changes, and the comparison reverses.

Information when the buyer decides	Weak incumbent	Strong incumbent
Equilibrium information	25.0%	52.3%
Same informative orders	56.2%	52.3%
Price withheld from the buyer	25.0%	25.0%

Preparation probabilities in the benchmark. Holding orders fixed isolates the information effect; these orders would not be chosen in equilibrium against the weak incumbent. When the price is withheld, the equilibrium is solved again. Percentages are rounded for reading; exact CSV values remain available on hover.

Preparation, bidding and ownership are distinct. In the benchmark, preparation rises from 25.0% to 52.3%, while high-value ownership rises from 12.5% to 32.4%. These are model probabilities, not empirical estimates.

A still stronger incumbent can make even the best attainable price insufficient to justify high-cost preparation. Participation then returns to 25.0%. The theorem compares specified economies; it does not establish a smooth hump. At intermediate strengths, equilibria can coexist. The certificates and numerical searches below retain that distinction.

Full theorem, preparation probabilities and proof

Proposition 2

(i) With the weak incumbent \(r_0\), the unique equilibrium trading outcome is \(q_H=q_L=0\), the price carries no information, and entry equals \(\rho\). (ii) With the strong incumbent \(r_1\), the unique equilibrium trading outcome is \((q_H,q_L)=(1,-1)\), the price is informative, entry strictly exceeds \(\rho\), and the probability that the high-value challenger acquires the target is strictly higher than at \(r_0\). The strong-incumbent price experiment strictly Blackwell dominates the weak-incumbent experiment at the same noise law. (iii) For any \(r_2\in(r_1,h)\) with \(c_L\lt B_{r_2}(m)\), \(B_{r_2}(M)\lt c_H\), and \(k\lt(1-1/b)\rho m\Delta_T(r_2)\), the unique trading outcome is again \((1,-1)\), but entry returns to \(\rho\). These comparisons hold on a nonempty open set of primitives. Uniqueness refers to trading and on-path entry under truthful bidding and allows arbitrary mixed orders and every continuous deviation.

The full-order candidate (12)

\begin{aligned} \tau&=\frac{c_H-g_L}{g_H-g_L},& x^*&=\frac b2\log\frac\tau{1-\tau},\\ \alpha_H&=1-\frac12e^{(x^*-1)/b},& \alpha_L&=\frac12e^{-(x^*+1)/b},\\ \mathsf E&=\rho+\frac{1-\rho}{2}(\alpha_H+\alpha_L),& \mathsf O_H&=\frac12[\rho+(1-\rho)\alpha_H]. \end{aligned} \tag{12}

\(\tau\) is the posterior at which the expensive challenger is indifferent, \(x^*\) the order flow at which the full-order posterior reaches it, and \(\alpha_H,\alpha_L\) the probabilities that flow exceeds \(x^*\) given a high- or a low-value challenger.

Proof sketch

The proof has six steps.

Bound beliefs. The likelihood-ratio bound in (8) places every market-maker posterior in \([m,M]\). The price posterior inherits these bounds.
Recover information from prices. Low-cost preparation keeps the denominator in (10) positive, so the challenger can recover the market maker's posterior from the price.
Rule out trading against the weak incumbent. Any correctly signed order of size \(s\) earns at most \(s[\Delta_T(r_0)-k]\lt0\). A wrong-signed order also loses money. Zero trade is the only best response.
Establish full orders against the strong incumbent. The Laplace derivative bound and (A3) make profit strictly increasing in order magnitude across the feasible interval.
Construct the informative equilibrium. Full orders imply a posterior and a strictly increasing price function. The preceding bounds verify pricing and investor optimality.
Compare preparation and ownership. Condition (A2) puts the high-cost threshold between the prior and the upper posterior bound. Favorable prices induce additional preparation, more often for a high-value challenger. Strict inequalities make the result robust to small parameter changes.

For the still stronger incumbent in part (iii), trading remains informative, but \(B_{r_2}(M)\lt c_H\) rules out high-cost preparation at every price. Appendix A.3 contains the full proof.

Certified equilibria and numerical searches

Proposition 3 · computer-assisted. At three benchmark strengths, an informative equilibrium with full purchases and partial sales coexists with no trade. Outward interval arithmetic encloses each equilibrium and bounds every unilateral deviation. These certificates prove existence at the stated nodes; they do not establish an exhaustive correspondence.

Strength	Short-order magnitude v	Preparation probability E	Certificate verified
r = 1.55	[0.46031618, 0.46031620]	[0.545052889808945713520237620328749119481683845800618298332444, 0.545052892132246689741896675903819707329977318624990770537445]	yes
r = 1.60	[0.70747537, 0.70747539]	[0.548756306279402553094473342650650704987833563703042353794586, 0.548756308461269927283787391337035286049034141128488454660113]	yes
r = 1.65	[0.90333198, 0.90333201]	[0.551360798856776788533769270145071166432320955743300255818651, 0.551360801970062274597481435420114471003452102733480241341781]	yes

Certified enclosures for the short-order magnitude and preparation probability. Intervals retain their outward endpoints.

Scroll horizontally to see the full figure.

Figure 2. Preparation and investor orders across incumbent strengths. Shading denotes analytical uniqueness regions. Small axis ticks mark searched nodes with multiple distinct accepted continuations. Other curves are numerical diagnostics, with gaps at missing nodes and unmatched roots. Diamonds mark the computer-assisted certificates. The open diamond identifies a mixed diagnostic, not a new interval proof. Hover for evidence status; use the legend to show or hide a series.

The revised search retains a mixed candidate as a numerical diagnostic. Finite-support and tested-deviation checks do not establish exact mixed-equilibrium existence or close its between-grid deviation bound. Open search coverage remains visible in the data; a missing branch does not prove nonexistence. At the high-cost preparation ceiling, the Laplace plateau still induces preparation under the tie rule. Participation drops only strictly above that ceiling.

Boundary	Role	Value	Definition
r(k)	no trade uniquely optimal below	1.220997512	r(k): below this strength Delta_T < k and no trade is the unique outcome (sufficient bound)
r_N	no trade exists up to	1.747877538	r_N = r(2k/rho): exact boundary of pooling existence, rho Delta_T/2 = k
r_U	full orders uniquely optimal above	2.837416964	r_U: (1-1/b) rho m Delta_T = k; above it full orders are the unique outcome (sufficient bound)
r_C	high-cost entry infeasible above	3.592658519	r_C: B_r(M) = c_H; above it expensive entry is infeasible in every equilibrium (tie rule retained at equality)

Existence boundaries, sufficient uniqueness bounds and the preparation ceiling answer different questions.

Benchmark tables and comparisons across strengths

	Weak (r = 1.2)	Strong (r = 3)	Very strong (r = 3.6)
t₀, proceeds without a challenger	0.291667	0.416667	0.430556
t_H, proceeds with a high-value challenger	0.704167	1.541667	1.834722
t_L, proceeds with a low-value challenger	0.687500	0.875000	0.895833
g_H, gross profit of a high-value challenger	9.295833	8.458333	8.165278
g_L, gross profit of a low-value challenger	0.312500	0.125000	0.104167
Δ_T = t_H − t_L, target-payoff spread	0.016667	0.666667	0.938889
B_r(1/2), expected gross profit at the prior	4.804167	4.291667	4.134722

Auction-stage payoffs.

	High-value order q_H	Low-value order q_L	Preparation E	High-value ownership O_H	Target proceeds R_T	Evidence
Panel A. Equilibrium
Weak incumbent (r = 1.2)	0	0	0.250000	0.125000	0.392708	analytical
Strong incumbent (r = 3)	1	-1	0.522757	0.324192	0.872392	analytical
Very strong incumbent (r = 3.6)	1	-1	0.250000	0.125000	0.664236	analytical
Panel B. Information controls
Frozen informative orders, r = 1.2	1	-1	0.562178	0.350606	0.520039	control
Frozen informative orders, r = 3	1	-1	0.522757	0.324192	0.872392	control
Price hidden from the challenger, r = 1.2	0	0	0.250000	0.125000	0.392708	analytical
Price hidden from the challenger, r = 3	1	-1	0.250000	0.125000	0.614583	analytical

Equilibrium outcomes and information controls. Evidence class and experiment role are separate.

	r = 1.2	r = 3	r = 3.6	Across r
Δ_T, target-payoff spread	0.016667	0.666667	0.938889	↑
B_r(1/2), gross profit at the prior	4.804167	4.291667	4.134722	↓
B_r(M), gross profit at the posterior ceiling	6.879843	6.217155	5.997311	↓
τ, threshold belief for high-cost preparation	0.633117	0.705000	0.731392	↑
x*, threshold order flow	∞	0.871222	∞
E, total entry	0.250000	0.522757	0.250000	↑ then ↓
O_H, high-value challenger ownership	0.125000	0.324192	0.125000	↑ then ↓
R_T, expected target proceeds	0.392708	0.872392	0.664236	↑ then ↓

Comparisons across the three declared benchmark strengths.

Implications and next steps

4 · Implications · 2–3 minutes

The information channel does not require investors to be better informed overall than the buyer. In the private-information extension, the buyer's signal is more accurate than the investor's, yet the price adds useful information. A favorable private signal may leave preparation unattractive on its own; combined with a favorable price, it can make the investment worthwhile. This is why the target-specialist interpretation matters: different knowledge can be useful even to a knowledgeable acquirer.

The reversal also survives the paper's logistic-noise example, continuously distributed preparation costs and a narrower gap between acquisition values. These comparisons preserve the mechanism, but change its size. How often a sufficiently favorable price occurs matters as well as how much the price could reveal. The extensions and their conditions are available below.

Why sale terms matter

Holding incumbent strength at the strong benchmark, letting the challenger observe the price raises target proceeds and acquisition surplus net of preparation costs relative to withholding the price. This is a comparison of information access under a given sale rule. It does not establish that strengthening the incumbent always increases welfare, or that the benchmark auction is the seller's best mechanism.

A reserve changes what a buyer must pay and what target shares reveal. If it excludes a low-value challenger, it changes how target proceeds differ across challenger types, and therefore changes the investor's incentive to trade. Bargaining can divide the surplus differently again. The paper's bargaining result characterizes acquisition-stage payments; a participation result for that institution would require solving its trading and preparation game.

Choosing sale terms requires more than selecting the largest revenue in a table. When preparation can fall to zero, a single price may pool several order-flow observations. Different pooling rules can then give the buyer different information even with the same investor orders. The paper establishes a family of such outcomes. A seller must account for the resulting orders, prices, beliefs and preparation decisions together.

What I would pursue next

The next theoretical step is to solve the seller's choice of terms while accounting for the possible trading, pricing and preparation outcomes. A related question is whether committing before trading improves on revising terms after prices are observed. Empirical work would begin by establishing the public decision window, separating interest from substantive preparation, and dating the information buyers could observe. The paper reports a research design, not an estimated price-to-entry effect.

The contribution is a link between the payment rule and the information that recruits bidders. A stronger competitor can make target shares more informative precisely when the prospective buyer's direct return to preparation falls.

Robustness: noise, costs and private information

The paper checks the reversal with a different noise distribution, continuously distributed preparation costs, a smaller acquisition-value gap, and a buyer whose private signal is more accurate than the investor's. The argument retains a bound on what orders can reveal and positive preparation at every price. The size of the effect depends on how often a price is favorable enough to justify high-cost preparation.

Noise and preparation costs

For the trading argument, the key restriction on noise is a bound on the derivative of its log density. Logistic noise satisfies this restriction. Proposition A.5 therefore preserves the unique trading outcomes and the preparation comparison.

The magnitude changes. Preparation against the strong incumbent is 0.301509 under logistic noise, compared with 0.522757 under Laplace noise. The logistic threshold lies 1.495369 noise standard deviations from zero, so favorable observations are rare. Proposition A.6 also preserves parts (i) and (ii) of Proposition 2 when preparation costs are continuously distributed over bands of half-width 0.1, under either noise law.

Scroll horizontally to see the full figure.

Figure 3. Posterior tails and high-cost entry. Panel (a) plots \(\Pr(\mu_X\ge\tau)\) against \(M-\tau\) under full orders at the strong benchmark strength; panel (b) plots implied entry. At zero threshold distance the Laplace plateau induces entry under the tie rule, whereas the logistic bound is unattained. The plotted tail probability is nevertheless defined there: zero mass and baseline preparation are closed points.

Notes

Scale \(b=2\), rather than variance, is held fixed. These are fixed-profile comparisons; Online Appendix C.5 records equilibrium validation for each implied cost. The figure does not establish a Blackwell ranking.

Acquisition values

With \(h=2\) and \(\ell=1\), every strict inequality of Proposition 2 holds and entry rises from 0.250000 to 0.526805 between \(r_0=1.05\) and \(r_1=1.5\). What the reversal needs is the placement of incumbent strength relative to the low acquisition value, not a large value gap.

Complementary private information

A more accurate private signal need not contain all the information in the price. A buyer may know its integration plans, while a specialist investor knows more about the target's customers or technology. The extension represents this distinction with conditionally independent signals of accuracy \(d\) for the buyer and \(a\) for the investor.

Positive preparation keeps the price informative about the public posterior. Proposition A.7 gives sufficient conditions for the same unique trading and preparation outcomes without requiring \(a\gt d\). With buyer accuracy \(d=75\%\) and investor accuracy \(a=70\%\), preparation rises from 0.850000 to 0.879438. A favorable private signal alone does not justify the high preparation cost against the weak incumbent; combined with a favorable price, it does against the strong one.

	Entry weak	Entry strong	O_H weak	O_H strong	Minimum margin	Basis
Panel A. Noise and preparation costs (r = 1.2 and 3)
Laplace, cost atoms	0.250000	0.522757	0.125000	0.324192	2.41×10^-3	analytical
Laplace, cost mixture	0.250000	0.522715	0.125000	0.324148	2.41×10^-3	analytical
logistic, cost atoms	0.250000	0.301509	0.125000	0.161994	2.41×10^-3	analytical
logistic, cost mixture	0.250000	0.301374	0.125000	0.161854	2.41×10^-3	analytical
Panel B. Moderate values (h = 2, ℓ = 1; r = 1.05 and 1.5)
Moderate values	0.250000	0.526805	0.125000	0.327032	8.01×10^-4	analytical
Panel C. Complementary private information (r = 1.1 and 2.3)
Investor accuracy a = 0.70, buyer accuracy d = 0.75	0.850000	0.879438	0.425000	0.448942	1.45×10^-3	analytical

Table 3. Robustness and complementary private information.

Welfare, bargaining and reserve comparisons

Sale terms reach participation through the buyer's expected payment and through the information the market supplies. This section varies the second channel at fixed competition, then asks which payment rules produce the opposition behind Proposition 2.

Access to prices

Hold incumbent strength at \(r_1\). Compare the feedback equilibrium with an economy in which the challenger cannot observe the price. This represents a buyer able to see the market's response to a public opportunity versus one that must commit without that information.

Full investor orders arise in both economies, so trading costs are the same. Price access changes which high-cost challengers prepare, and each additional entrant expects a profit that covers its preparation cost. Proposition A.9 establishes higher target proceeds, 0.872392 rather than 0.614583, and a net acquisition-surplus gain of 0.080218. This ranks access to information under a fixed sale rule, not incumbent strengths or alternative mechanisms.

Separating information from the price level

A deterministic dividend of 0.257809 added to the traded claim in the price-hidden economy matches mean prices across the two economies while leaving \(V_T-P\), trading incentives, and entry unchanged. Entry still differs, so the gap is information, not the price level. The dividend is a diagnostic payment, outside acquisition surplus and outside the seller's feasible terms.

	Price observed	Price hidden	Gain
Panel C. Access to prices at r = 3
Target proceeds R_T	0.872392	0.614583	0.257809
Net acquisition surplus W	2.382301	2.302083	0.080218

Access to prices at the strong benchmark strength. Expected target proceeds and net acquisition surplus with the price observed and with it hidden.

The payment rule

The payment rule determines whether a stronger incumbent increases the target-payoff spread. In the benchmark auction, the competing bid sets the payment when both bidders clear the reserve. Now consider a negotiated sale with verifiable values and no reserve.

The highest-value buyer acquires the target. A sale to the next-best buyer at its value provides an enforceable fallback. The seller and winner split the surplus above that fallback through Nash bargaining, with seller share \(\eta\):

\begin{aligned} T_\eta(R,\theta)&=(1-\eta)\min\{R,\theta\}+\eta\max\{R,\theta\},\\ G_{\theta,\eta}&=(1-\eta)\mathbb E[(\theta-R)_+],\\ \Delta_\eta&=\eta(h-\ell)+(1-2\eta)\mathbb E[(R-\ell)_+]. \end{aligned} \tag{14}

Proposition A.8 shows that a stronger incumbent weakly reduces challenger profit for every seller share \(\eta\). It raises the target-payoff spread when \(\eta\lt1/2\), leaves it unchanged at \(\eta=1/2\), and lowers it when \(\eta\gt1/2\).

For \(R\gt\ell\), the high-value winner's payment rises with \(R\) at rate \(1-\eta\). After a low-value challenger loses, the incumbent's payment rises at rate \(\eta\). The spread widens precisely when the first rate is larger. The second-price rule with no reserve corresponds to \(\eta=0\). This is a result about acquisition payments; establishing a preparation reversal under bargaining would also require solving trading and preparation, which the paper does not do.

Scroll horizontally to see the full figure.

Figure 4. Bargaining and the division of acquisition surplus. Panel (a) plots \(\Delta_\eta\) against seller weight \(\eta\) for weak and strong incumbents; panel (b) plots conditional challenger profits on a log scale. At \(\eta=1\) profits are zero, so that endpoint is omitted only from the log panel; panel (a) retains it.

Notes

Zero-reserve verifiable-value institution, \(h=10\), \(\ell=1\), and uniform incumbents with \(r=1.2\) or \(r=3\). Values are per target share. The comparison concerns acquisition-stage payoffs; trading and entry under this institution are not solved.

Reserve comparisons

A reserve is the minimum payment the seller will accept. A reserve above the low challenger value excludes that type from bidding. Target proceeds in the low-value state then equal proceeds without preparation, increasing the difference between high- and low-value states. This can make informed trading worthwhile even against a weak incumbent.

The numerical comparison allows values to vary within bands of half-width \(\varepsilon_V=0.05\) around \(\ell\) and \(h\). Raising the reserve from 0.5 to 1.1 increases revenue against both incumbents and supports informative trading in both economies. Against the weak incumbent, revenue rises from 0.392665 to 0.432173 (Table 4).

This is a feasible improvement, not an optimal reserve. Several equilibrium continuations coexist at many reserves. Taking the largest revenue found at each reserve does not by itself solve the seller's problem.

	Preparation	Sale	Two admissible bidders	Target proceeds	Basis
Panel A. Binary values
r = 1.2, reserve p = 0.5	0.250000	0.687500	0.145833	0.392708	analytical
r = 1.2, reserve p = 1.01	0.543573	0.443232	0.053595	0.452756	analytical
r = 3, reserve p = 0.5	0.522757	0.920460	0.435631	0.872392	analytical
r = 3, reserve p = 1.01	0.513373	0.770226	0.210611	0.987487	analytical
Panel B. Atomless values with class information (ε_V = 0.05)
r = 1.2, reserve p = 0.5	0.250000	0.687500	0.145833	0.392665	analytical
r = 1.2, reserve p = 1.1	0.540309	0.391610	0.028025	0.432173	analytical
r = 3, reserve p = 0.5	0.522760	0.920460	0.435634	0.872367	analytical
r = 3, reserve p = 1.1	0.511638	0.749292	0.200293	1.014500	analytical

Table 4. Validated fixed-reserve comparisons. Preparation, sale and the presence of two admissible bidders are separate outcomes.

Proposition A.10 · analytical. At the declared high-reserve example, full investor orders support a family of rational price pools. A pooled price changes the buyer's posterior and its willingness to prepare, even though investor orders are fixed. The seller therefore cannot identify a continuation by orders alone. Appendix A.7 and Online Appendix C.6b give the construction and the validated family.