Competition Creates Competition:
Stock Prices and the Discovery of Takeover Bidders

Austin Li

Department of Economics, University College London

Does a stronger incumbent always keep the rival out?

Why the answer matters

SellerA public sale tries to find a rival bidder. A stronger incumbent can help find one, or end the search.

RivalPreparation costs money before a bid is possible. The stock price can show whether preparation is worth its cost.

PriceStock trade comes before preparation. A price move can cause entry, not only forecast it.

One extra bidder can have a first-order effect on sale proceeds (Bulow and Klemperer, 1996). Here the sale rule stays fixed. The object is entry: rival preparation.

The received answer: strength cannot raise entry

When information before preparation stays fixed

The rival pays to learn its value before it can bid.
A stronger incumbent weakly cuts the rival’s expected gross profit at a fixed belief.
Preparation then no longer covers some costs. Entry cannot rise.
Fishman (1988): a chosen preemptive bid. Here: a public shift in the incumbent’s value distribution. The realised value stays private.

This holds when information before preparation stays fixed.

Fishman (1988); Hirshleifer and Png (1989). Auction entry: Samuelson (1985); McAfee and McMillan (1987); Levin and Smith (1994); Gentry and Stroup (2019).

Where the paper sits: price feedback meets costly entry

Strand	Core idea	Difference here
Price feedback — Dow, Goldstein and Guembel (2017, closest antecedent); Edmans, Goldstein and Jiang (2015)	Price guides a real decision	The decision is rival preparation
Costly entry — Fishman (1988); Roberts and Sweeting (2013)	Preparation cost deters entry	Synergy news in the price can rise with strength
Takeover trade — Edmans, Goldstein and Jiang (2012); Kyle and Vila (1991)	Trade moves target prices	An outside informed trader with no initial position or target control

An outside informed trader can put synergy news in the price. Incumbent strength can raise the return to that trade.

Order slot (frame 6): alternative arrival from Dow and Gorton (1997); unit orders, presence and order cost from Edmans, Goldstein and Jiang (2015).

This paper: a stronger incumbent can recruit the rival

A stronger incumbent weakly cuts the rival’s gross profit at each belief. It can also widen the shareholder spread. Informed trade can then pay, and the price can recruit the rival.

Non-monotone entry: entry can rise from the weak to the strong incumbent, then return to its floor. Each case has a unique equilibrium outcome (Proposition 2).
Value split: strength weakly cuts rival gross profit and weakly widens the shareholder spread (Proposition 1).
Information control: with information before preparation fixed, a stronger incumbent cannot raise entry.

Extensions: costly research by the informed trader; gains in target proceeds from a visible price; a purchase-impact prediction.

Roadmap: Model $\rightarrow$ Mechanism and main result $\rightarrow$ Fixed-information control $\rightarrow$ Extensions and prediction.

Players and the sale rule

Players
Timing
Information
Actions
Payoffs
Equilibrium

A listed target has standalone value $0$ . Values and costs are per share. All agents are risk neutral.

Incumbent: incumbent bidder, already prepared: it knows its value $R$ and faces no further preparation cost. $R\sim F$ ; $F$ is public, continuous on $[0,\bar r]$ , and others do not observe the draw before bids. Stronger means a replacement $\widetilde F$ with $\widetilde F(y)\le F(y)$ for each $y$ .
Rival: potential rival bidder: sees the price, then decides whether to prepare. Synergy class: $j=H$ (good synergy) or $j=L$ (poor synergy), with prior $\pi=\Pr(j=H)\in(0,1)$ . Its value $\theta$ has distribution $G_j$ in class $j$ ; each $G_j$ has a finite mean.
Class ranking: $G_H(y)\le G_L(y)$ for each $y$ — class $H$ draws weakly higher values. Also $\theta\ge p$ almost surely under each $G_j$ , where $p>0$ is the auction reserve below. The sale and the rival’s identity are public; its synergy class is unknown to the rival itself.
Preparation cost $C$: $C=0$ with probability $\rho\in(0,1)$ ; otherwise $C=c_H>0$ , the high cost ( $H$ in $c_H$ means high, not class $H$ ). Pay $C$ to learn $\theta$ and to bid; a rival who does not pay cannot bid. (A zero-cost rival always prepares, so $\rho$ is the entry floor.)
Sale: the seller commits before trade to a cash second-price auction with reserve $p>0$ . Sale proceeds go to target shareholders.

The pair $(j,\theta)$ , the value $R$ , the cost $C$ , and the order-slot draws are mutually independent. Neither bidder trades target shares. The next frame adds the order slot: informed trader, liquidity trader, competitive market makers.

Timing: one order slot, then preparation

Players
Timing
Information
Actions
Payoffs
Equilibrium

Seller commits: auction, reserve $p$

Order slot: outcome $X$

Market makers set $P(X)$

Rival sees $P$ , learns $C$ , prepares or stays out

Rival, if prepared, learns and bids $\theta$ ; payoffs

Incumbent’s distribution $F$ is public

Incumbent bids $R$

One order slot comes before preparation. Its outcome is $X\in\{-1,0,1\}$ : $-1$ a sale order, $0$ no order, $1$ a purchase.
With probability $\lambda\in(0,1)$ — the informed presence — the slot goes to an outside informed trader. It observes only the synergy class $j$ , not $\theta$ , and chooses an order $q\in\{-1,0,1\}$ ; each nonzero order costs $k>0$ .
With probability $1-\lambda$ , a liquidity trader fills the slot and submits $-1$ , $0$ , $1$ with probabilities $\beta$ , $1-2\beta$ , $\beta$ , where $0<\beta\le 1/3$ .
Competitive market makers observe the outcome $X$ only. They set $P(X)=\mathbb{E}[V_T\mid X]$ , where $V_T$ is realised target proceeds per share. In equilibrium that expectation already reflects how rival entry responds to the price.

Information and choices

Players
Timing
Information
Actions
Payoffs
Equilibrium

Player	Observes	Chooses
Seller	public primitives	auction and reserve $p$ , before trade
Incumbent	its value $R$	bid $R$ if $R\ge p$
Informed trader (slot with probability $\lambda$ )	class $j$ only	order $q$ ; cost $k$ if $q\ne0$
Liquidity trader (slot with probability $1-\lambda$ )	no private information	order drawn with $(\beta,\,1-2\beta,\,\beta)$
Market makers	outcome $X$ , not who traded	price $P(X)=\mathbb{E}[V_T\mid X]$
Rival	price $P$ (not $X$ on its own), then cost $C$ ; knows neither $j$ nor $\theta$ before preparing	prepare or stay out; if it prepares, learn $\theta$ , then bid $\theta$

Public primitives (frames 5 and 6) are common knowledge; the table lists each player’s own information.

Order belief: $\mu(x)=\Pr(j=H\mid X=x)$ — the belief of an observer who sees the realised outcome $X=x$ .
Price belief: $\bar\mu(\bar P)=\Pr(j=H\mid P(X)=\bar P)$ — the rival’s belief. One price pools all outcomes with that price.

Payoffs

Players
Timing
Information
Actions
Payoffs
Equilibrium

Fix a prepared rival with value $\theta\ge p$ , and average over $R$ . Shareholders receive the auction payment $\max\{p,\min(R,\theta)\}$ ; its mean is $t(\theta)$ . With $(z)_+=\max\{z,0\}$ , the rival’s gross auction profit is $g(\theta)=\mathbb E_R[(\theta-\max\{p,R\})_+]$ , before cost $C$ . After averaging over $R$ , $t(\theta)+g(\theta)=\theta$ exactly.
$\mathbb E_j$ denotes expectation under $G_j$ . Class means are $t_j=\mathbb E_j[t(\theta)]$ and $g_j=\mathbb E_j[g(\theta)]$ . With no rival entry, the incumbent alone pays $p$ if $R\ge p$ ; expected proceeds are $t_0=p[1-F(p)]$ .
Shareholder spread: $\Delta_T=t_H-t_L$ — extra expected target proceeds in class $H$ versus class $L$ if the rival prepares. Rival gross profit at belief $\mu$ is $B(\mu)=g_L+\mu(g_H-g_L)$ , the $\mu$ -weighted mean of class gross auction profits, before $C$ . Net gain from preparation: $B(\bar\mu)-C$ .
The informed trader with order $q$ earns $q[V_T-P(q)]-k|q|$ , where $P(q)$ is the price after outcome $X=q$ .

Equilibrium with competitive prices

Players
Timing
Information
Actions
Payoffs
Equilibrium

Solution concept: perfect Bayesian equilibrium with break-even prices from competitive market makers.

Orders: let $e(x)$ be the probability that the rival prepares after realised outcome $X=x$ — the rival responds to the price $P(x)$ , not to $x$ itself. An informed trader who sees class $j$ gets expected net payoff $U_j(q)=q[t_0+e(q)(t_j-t_0)-P(q)]-k|q|$ from chosen order $q$ ; the term $t_0+e(q)(t_j-t_0)$ is expected target proceeds in class $j$ . For each $j$ , the informed trader puts positive probability only on orders $q$ that maximise $U_j(q)$ , with the schedules $P(\cdot)$ and $e(\cdot)$ at their equilibrium values.
Competitive prices: $P(x)=\mathbb{E}[V_T\mid X=x]=t_0+e(x)[t_L-t_0+\Delta_T\,\mu(x)]$ . With entry, expected proceeds are $t_L+\mu(x)\Delta_T$ ; without entry they are $t_0$ . Zero-cost preparation below gives the entry floor $e(x)\ge\rho$ .
Beliefs: the order belief $\mu(x)=\Pr(j=H\mid X=x)$ follows Bayes’ rule. The rival’s price belief $\bar\mu(\bar P)=\Pr(j=H\mid P(X)=\bar P)$ pools all outcomes with that price. It can differ from $\mu(x)$ for any one pooled outcome. Liquidity orders give each outcome positive probability in each class, so Bayes’ rule pins down order beliefs.
Preparation and bids: a rival with $C=0$ prepares at each price. With $C=c_H$ , it prepares if and only if $B(\bar\mu)\ge c_H$ . A prepared rival bids $\theta$ ; the incumbent bids $R$ subject to the reserve.

Strength weakly cuts rival profit, weakly widens the spread

Proposition 1

Let $F$ and $\widetilde F$ be continuous incumbent value distributions, with $\widetilde F(y)\le F(y)$ for every $y$ .

The new incumbent draws high values more often.

Hold the reserve and the finite-mean class distributions fixed.

(a) For $j\in\{H,L\}$ and every $\mu\in[0,1]$ ,

g_j(\widetilde F)\le g_j(F),\qquad B_{\widetilde F}(\mu)\le B_F(\mu).

Here $B_F(\mu)=g_L(F)+\mu[g_H(F)-g_L(F)]$ .

(b) Under the class ranking,

G_H(y)\le G_L(y)\quad\text{for every }y,\qquad G_H(p^-)=G_L(p^-)=0,

the shareholder spread satisfies $\Delta_T(\widetilde F)\ge\Delta_T(F)$ .

Class $H$ draws weakly higher values, and all meet the reserve.

Since values in both classes meet the reserve almost surely, $\Delta_T(F)+[g_H(F)-g_L(F)]=\mathbb E_H[\theta]-\mathbb E_L[\theta]$ . The right side does not depend on $F$ .

Strength can shift part of the fixed class-value gap from rival gross profit to the spread.

Beliefs stay in a band, and the price reveals the belief

Belief bounds

Define $\quad b=(1-\lambda)\beta,\qquad u=\dfrac{\lambda}{b},\qquad m(\pi)=\dfrac{\pi}{\pi+(1-\pi)(1+u)},\qquad M(\pi)=\dfrac{\pi(1+u)}{\pi(1+u)+1-\pi}.$

Call $u$ the mass ratio, $m(\pi)$ the band floor and $M(\pi)$ the top belief.

For every pure or mixed order profile and every $x\in\{-1,0,1\}$ ,

0<\beta\le\frac13\quad\Longrightarrow\quad m(\pi)\le\mu(x)\le M(\pi).

With $\beta\le1/3$ , liquidity orders cover every outcome, including no order.

The price belief $\bar\mu(\bar P)$ obeys the same bounds.

Suppose $\Delta_T>0$ , $g_H>g_L$ and $\rho\in(0,1)$ . Under Bayes’ rule, optimal preparation, competitive prices and truthful bids,

\begin{aligned}&P(x)=P(x')\ \Longrightarrow\ \mu(x)=\mu(x'),\\&\Pr(H\mid P(X)=P(x))=\mu(x).\end{aligned}

Preparation rule. When $g_H>g_L$ , define the preparation threshold $\tau=(c_H-g_L)/(g_H-g_L)$ .

$B(\mu)\ge c_H\iff\mu\ge\tau$ .

With the default tie rule — a rival at the threshold prepares —

\begin{aligned}&C=c_H:\quad\text{prepare exactly when }\mu(x)\ge\tau,\\&e(x)=\begin{cases}\rho,&\mu(x)<\tau,\\1,&\mu(x)\ge\tau.\end{cases}\end{aligned}

A liquidity trader can place any order, so no order reveals the synergy class for certain.

Gross gains from correct orders lie between two spread bounds

Trading bounds

Let $\Delta_T\ge0$ and $k>0$ . Take any order profile with Bayes beliefs, competitive prices, truthful bids, and $e(x)\in[\rho,1]$ .

The bounds hold for every order profile, optimal or not.

The gross gains from the correct orders — a purchase in class $H$ and a sale in class $L$ — are

K_H=e(1)\Delta_T[1-\mu(1)],\qquad K_L=e(-1)\Delta_T\mu(-1).

Define $\kappa(\pi)=\min\{m(\pi),1-M(\pi)\}$ .

$\kappa(\pi)$ lower-bounds the doubt each correct order leaves: $1-\mu(1)$ after a purchase, $\mu(-1)$ after a sale.

For either class $j\in\{H,L\}$ ,

\rho\kappa(\pi)\Delta_T\le K_j\le\Delta_T.

Net of the order cost, a wrong-signed order earns at most $-k<0$ ; abstention earns zero. The informed trader who sees class $H$ buys if $K_H>k$ and abstains if $K_H<k$ . The informed trader who sees class $L$ sells if $K_L>k$ and abstains if $K_L<k$ . At $K_j=k$ , the correct order and abstention are both optimal, and any mixture of the two is optimal.

Each gross gain is entry times the shareholder spread times the doubt the order leaves. Net gain subtracts $k$ , so doubt alone does not pay.

Strength can raise entry, then return it to its floor

Proposition 2

Classes have finite means and satisfy $G_H\le G_L$ , $G_H(p^-)=G_L(p^-)=0$ . Take $F_0$ weak, $F_1$ strong, and $F_2$ very strong, so $F_2\le F_1\le F_0$ . Fix the class distributions and $p>0$ ; $\pi,\rho,\lambda\in(0,1)$ ; $\beta\in(0,1/2)$ (the $\beta\le 1/3$ bound for no order is not needed); and $c_H,k>0$ . Write $\Delta_i=\Delta_T(F_i)$ and $B_i=B_{F_i}$ . Suppose

$\begin{array}{ll}\mathrm{(H1)}&B_0(\pi)<c_H<B_1(M(\pi)),\\\mathrm{(H2)}&\Delta_0<k<\rho\kappa(\pi)\Delta_1,\\\mathrm{(H3)}&B_2(M(\pi))<c_H.\end{array}$

(H1) Rival gross profit at the prior $\pi$ is below $c_H$ at $F_0$ ; at the top belief $M(\pi)$ it is above $c_H$ at $F_1$ .

(H2) Order cost stops informed trade at $F_0$ , not at $F_1$ ; $\Delta_2\ge\Delta_1$ keeps trade on at $F_2$ .

(H3) At $F_2$ , rival gross profit at the top belief $M(\pi)$ remains below $c_H$ .

Orders by class $(H,L)$	Beliefs by outcome $(-1,0,1)$	Entry by outcome $(-1,0,1)$
$F_0$ weak: $(0,0)$	$(\pi,\pi,\pi)$	$(\rho,\rho,\rho)$
$F_1$ strong: $(1,-1)$	$(m(\pi),\pi,M(\pi))$	$(\rho,\rho,1)$
$F_2$ very strong: $(1,-1)$	$(m(\pi),\pi,M(\pi))$	$(\rho,\rho,\rho)$

Let $\mathsf E_i$ denote total entry at $F_i$ . Then

\mathsf E_0=\mathsf E_2=\rho<\mathsf E_1=\rho+(1-\rho)(b+\pi\lambda).

The outcome is unique in each of the three economies, under all four pairs of tie rules.

A wider shareholder spread pays the informed trader to trade, and at $F_1$ a purchase recruits the high-cost rival. At $F_2$ , rival gross profit at the top belief stays below $c_H$ .

Proof idea: nonzero-order bounds and (H2) pin orders in every candidate equilibrium; Bayes then gives beliefs, including $\mu(0)=\pi$ ; Proposition 1 with (H1) and (H3) pins entry. Steps: B2. Open set of parameters: frame 14. Exact form: B1.

Example: the top belief clears only the strong threshold

Corollary 1

The class values are point masses: $\theta=h$ in class $H$ and $\theta=\ell$ in class $L$ , with $(h,\ell,p)=(10,1,1/2)$ . The incumbent has $R\sim\mathrm{Unif}[0,r]$ ; a larger $r$ means a stronger incumbent.

Write $\tau(r)$ for the preparation threshold at strength $r$ . The three strengths are $(r_0,r_1,r_2)=(6/5,3,18/5)$ : weak, strong and very strong. At $\pi=1/2$ , in this family, conditions (H1)–(H3) hold exactly when the top belief after a purchase lies between the two thresholds:

\mathrm{(H1)}\text{--}\mathrm{(H3)}\quad\Longleftrightarrow\quad 141/200=\tau(r_1)<M(1/2)<\tau(r_2)=4245/5804.

The left inequality says a purchase at $r_1$ recruits the high-cost rival; the right inequality says no order at $r_2$ does.

The example sets $u=\mathrm e-1$ , where $\mathrm e$ is Euler’s number, only for a closed form: $M(1/2)=\mathrm e/(1+\mathrm e)\simeq0.731059$ , inside the interval.

Any slot whose top belief lies in $(\tau(r_1),\tau(r_2))$ gives the same pattern.

Against the strong incumbent, a purchase lifts the belief above the threshold; against the very strong one, the threshold rises just above that belief.

Other inputs: $(\pi,\rho,c_H,k)=(1/2,1/4,6,1/50)$ ; liquidity chance $\beta=0.294655$ (exact decimal); presence from $u=\lambda/[(1-\lambda)\beta]$ . Inputs and regions in full: B10.

With fixed information, strength cannot raise entry

Fixed information

Fix the joint distribution of the class $j$ and the signal $S$ that the rival observes. Fix the class distributions $G_j$ , the reserve, and the preparation-cost distribution. In economy $i$ , conditional on $(j,S)$ , the values $\theta$ and $R$ are independent, with distributions $G_j$ and $F_i$ . The cost $C$ is independent of the class $j$ , the values $\theta$ and $R$ , and the signal $S$ .

What the rival can learn does not change. Given the class, the signal adds no news about the values. The preparation cost carries no news about the class, the values or the signal.

Let a rival whose gross profit exactly equals its preparation cost prepare. If $F_1\le F_0$ , entry cannot rise:

\mathsf E_1\le\mathsf E_0,

where $\mathsf E_i$ is the probability of preparation in economy $i$ .

Entry in the example	Weak	Strong
Full model, orders solved again	At floor $\rho$	Above floor $\rho$
Full informed orders imposed (control)	Above floor $\rho$	Above floor $\rho$
Stock price hidden from the rival	At floor $\rho$	At floor $\rho$

In the middle row, the rival still reads the stock price, and full informed orders (buy in class $H$ , sell in class $L$ ) are imposed at both strengths; at the weak strength they are not optimal.

With fixed information, a stronger incumbent weakly lowers rival gross profit at each belief, so it can only remove reasons to prepare.

Costly research rises with the spread, up to the cap

One change from frame 6. The informed trader now privately chooses its research level, its presence $\lambda\in[0,\bar\lambda]$ , at cost $(a/2)\lambda^2$ , with $a>0$ and $0<\bar\lambda<1$ . Market makers and the rival anticipate its equilibrium level. With chance $\lambda$ it learns the synergy class and fills the slot. Otherwise the liquidity trader fills the slot.

Proposition 4

Assume the class ranking, $\pi=1/2$ , $\Delta_T>0$ , $g_H>g_L$ and $\tau>1/2$ .

At the prior, rival gross profit falls short of $c_H$ .

An indifferent informed trader may mix; an indifferent high-cost rival prepares. At presence $\lambda$ and prior $\pi=1/2$ , write $M_\lambda$ for the top belief after a purchase and $m_\lambda=1-M_\lambda$ for the band floor. For $\Delta_T>2k$ , interior research $\lambda_R(\Delta_T)\in(0,1)$ is the unique root of the first-order condition in B12. Equilibrium research is $\widehat\lambda=\min\{\lambda_R(\Delta_T),\bar\lambda\}$ .

The recruiting equilibrium — the informed trader buys when it learns class $H$ , and only a purchase recruits — exists exactly when

\Delta_T>2k\qquad\text{and}\qquad\tau\le M_{\widehat\lambda}.

The spread pays for trade, and the top belief reaches $\tau$ .

Research $\widehat\lambda$ rises strictly with $\Delta_T$ below the cap $\bar\lambda$ , and equals $\bar\lambda$ thereafter. Total entry is

\mathsf E=\rho+(1-\rho)[\beta+(1/2-\beta)\widehat\lambda].

As research rises, total entry $\mathsf E$ and entry in class $H$ rise strictly, while entry in class $L$ falls strictly.

A wider shareholder spread pays for research until the cap. More research sorts entry toward good synergy.

First-order condition, branches, class entry formulas, and the research figure: B12. Coexistence with fixed presence $\lambda$ : B3.

At $F_1$ , a visible price raises expected proceeds and net surplus

Price access

Hold the incumbent at $F_1$ (strong) under Proposition 2’s hypotheses. Compare price access with an economy where the rival sees neither the price nor the order. Hold all other primitives fixed; prices stay competitive in both economies.

Let $\mathcal R$ denote expected target proceeds and $\mathcal W$ expected acquisition surplus net of preparation costs. Superscripts $1$ and $0$ denote price access and access to neither price nor order, respectively. In both economies the informed orders are uniquely full: buy in class $H$ , sell in class $L$ . Then

\mathcal R^1>\mathcal R^0\qquad\text{and}\qquad\mathcal W^1>\mathcal W^0.

Price access raises entry only after a purchase, from $\rho$ to $1$ , because $B_1(M(\pi))>c_H$ .

The price reveals the top belief $M(\pi)$ after a purchase. The high-cost rival then prepares only where rival gross profit covers $c_H$ .

Exact differences, example values, and the paper’s separate payment-rule and reserve comparisons: B4.

Weighted purchase impact reveals recruitment

Weighted price impact

Assume the class ranking, $\Delta_T>0$ , $g_H>g_L$ , full two-sided trade (the informed trader who sees class $H$ buys; the informed trader who sees class $L$ sells), and $\pi<\tau$ .

At the prior, rival gross profit falls short of $c_H$ .

Define order frequencies $\bar f(x)=\Pr(X=x)$ , the purchase impact $I_+=P(1)-P(0)$ and the sale-order impact $I_-=P(0)-P(-1)$ .

Then $\bar f(1)=b+\pi\lambda$ , $\bar f(-1)=b+(1-\pi)\lambda$ , and

\begin{aligned}&\bar f(1)I_+-\bar f(-1)I_->0\\&\Longleftrightarrow\ e(1)>\rho.\end{aligned}

At $\pi=1/2$ , the order frequencies are equal, so

I_+>I_-\quad\Longleftrightarrow\quad e(1)>\rho.

Recruitment adds an entry part to the purchase price. Weight by order frequency, and that part makes purchase impact exceed sale-order impact.

With no recruitment ( $e(x)=\rho$ after each outcome), orders still predict the winner when the win chance is higher in class $H$ ( $W_H>W_L$ ): B13. The test needs prices dated before preparation starts, and a record of the no-order outcome.

Competition can create competition

A stronger incumbent can induce rival entry through a more informative stock price.

Main result. Entry rises against the strong incumbent and returns to its floor against the very strong one (Proposition 2; its hypotheses hold on an open set of parameters).
Discipline. Strength weakly cuts rival gross profit and weakly widens the shareholder spread. Fix what the rival can learn, and strength cannot raise entry.
Implication. At the strong incumbent, a visible price raises expected target proceeds and expected surplus net of preparation costs. In data, recruitment is equivalent to frequency-weighted purchase impact exceeding frequency-weighted sale-order impact.

In the recruiting equilibrium with costly research, a wider shareholder spread also raises research until the cap, and sorts entry toward good synergy.

Proposition 2 in exact form

Setup: the synergy distributions satisfy the class ranking (good synergy first-order dominates poor synergy; each value meets the reserve $p$ ) and have finite means. The incumbent’s value distribution $F$ is continuous on $[0,\bar r]$ , with finite upper bound $\bar r$ . Distributions $F_0$ , $F_1$ , $F_2$ satisfy $F_2\le F_1\le F_0$ , so each step is a first-order stochastic strengthening. Fixed: prior $\pi$ , entry floor $\rho$ and presence $\lambda$ , each in $(0,1)$ ; liquidity chance $\beta\in(0,1/2)$ ; high cost $c_H>0$ ; order cost $k>0$ . Write $b=(1-\lambda)\beta$ for the liquidity mass in one direction, and $\kappa(\pi)=\min\{m(\pi),1-M(\pi)\}$ for the common lower bound on residual doubt, with band floor $m(\pi)$ and top belief $M(\pi)$ .
With $B_i$ rival gross profit and $\Delta_i$ shareholder spread $\Delta_T$ against $F_i$ : (H1) $B_0(\pi)<c_H<B_1(M(\pi))$ . (H2) $\Delta_0<k<\rho\,\kappa(\pi)\,\Delta_1$ . (H3) $B_2(M(\pi))<c_H$ .
Weak $F_0$ : informed orders $(0,0)$ by synergy class (good, poor); belief $\mu(x)=\pi$ after each order $x$ ; entry $e(x)=\rho$ after each order; total entry $\rho$ ; one stock price. Strong $F_1$ : orders $(1,-1)$ ; beliefs after a sale, no order, a purchase are $(m(\pi),\pi,M(\pi))$ ; entry is $(\rho,\rho,1)$ ; prices rise strictly; total entry is $\rho+(1-\rho)(b+\pi\lambda)>\rho$ ; entry under good synergy is $\rho+(1-\rho)(b+\lambda)$ , and under poor synergy $\rho+(1-\rho)b$ . Very strong $F_2$ : orders and beliefs as at $F_1$ ; entry $\rho$ after each order; total entry $\rho$ ; three distinct prices.
Ownership: $W_j(F)=\mathbb{E}_j[F(\theta)\mathbf{1}\{\theta\ge p\}]$ is the win chance of a prepared rival in case $j$ , where $\theta$ is its value and the indicator $\mathbf{1}\{\theta\ge p\}$ equals one when the value meets the reserve $p$ , and zero otherwise. Good-synergy ownership rises from $F_0$ to $F_1$ exactly when $e_H W_H(F_1)>\rho\,W_H(F_0)$ , with $e_H$ entry under good synergy at $F_1$ . If $W_H(F_1)=1$ , good-synergy ownership rises from $\pi\rho$ to $\pi e_H$ , and the good-synergy share of rival wins rises strictly exactly when $W_L(F_0)>0$ .
The outcome in each economy is unique under each of the four tie-rule pairs. With the example synergy values and reserve of frame 14, (H1) to (H3) hold together on an open set of inputs.

Why each outcome is unique

Posterior bounds, entry and prices

Lemma A.1 (order probabilities and beliefs), for any order plans: each order has positive probability in each synergy class. For a sale or a purchase, the chance of that order in class $j$ lies between $b$ and $b+\lambda$ , where $b=(1-\lambda)\beta$ is the liquidity mass in one direction and $\lambda$ is presence. With $u=\lambda/b$ , each nonzero order has likelihood ratio between $1/(1+u)$ and $1+u$ , so the belief $\mu$ after that order lies in $[m(\pi),M(\pi)]$ , where $M(\pi)=\pi(1+u)/[\pi(1+u)+1-\pi]$ and $m(\pi)=\pi/[\pi+(1-\pi)(1+u)]$ . The no-order case lies in the same band when $\beta\le1/3$ . Under full orders, good synergy buys and poor synergy sells, the beliefs after a sale, no order, a purchase are $(m(\pi),\pi,M(\pi))$ . A belief at a price lies between the smallest and largest beliefs of the orders with that price.
Lemma A.2 (entry and prices): entry $e(x)\ge\rho$ after each order $x$ , with entry floor $\rho\in(0,1)$ . The price after order $x$ is $P(x)=t_0+e(x)[t_L-t_0+\Delta_T\,\mu(x)]$ , with proceeds $t_0$ without entry and $t_L$ under poor synergy. When $\Delta_T>0$ and rival gross profit rises with the belief $\mu$ , equal prices give equal beliefs, so the price reveals $\mu(x)$ ; entry is $\rho$ when $\mu(x)<\tau$ and $1$ when $\mu(x)\ge\tau$ , with threshold $\tau$ (at $\mu(x)=\tau$ , any entry in $[\rho,1]$ can hold when an indifferent high-cost rival may prepare with any probability); prices rise strictly with $\mu$ .

Why each outcome is unique

Trading bounds and the four-step proof

Lemma A.3 (informed trading): the gain before order cost $k$ from a correct purchase under good synergy is $K_H=e(1)\,\Delta_T\,[1-\mu(1)]$ ; from a correct sale under poor synergy, $K_L=e(-1)\,\Delta_T\,\mu(-1)$ . Under every order plan, $\rho\,\Delta_T[1-M(\pi)]\le K_H\le\Delta_T$ and $\rho\,\Delta_T\,m(\pi)\le K_L\le\Delta_T$ . With $\kappa(\pi)=\min\{m(\pi),1-M(\pi)\}$ , each gain lies between $\rho\,\kappa(\pi)\,\Delta_T$ and $\Delta_T$ ; the purchase bound uses $1-M(\pi)$ and the sale bound uses $m(\pi)$ . A wrong-signed order earns at most $-k$ . Good synergy buys when $K_H>k$ and stays out when $K_H<k$ ; poor synergy compares $K_L$ with $k$ in the same way.
Four-step proof: 1. Weak: each gain is at most $\Delta_T$ , below $k$ , so no informed order; the belief stays at the prior; entry stays at $\rho$ . 2. Strong: each gain is at least $\rho\,\kappa(\pi)\,\Delta_T$ , above $k$ , so full orders hold under every candidate; beliefs are $(m(\pi),\pi,M(\pi))$ . 3. A purchase reaches $M(\pi)$ , rival gross profit there strictly exceeds the high cost, so entry after a purchase is $1$ ; after other orders entry stays at $\rho$ . 4. Very strong: full orders stay, but rival gross profit at $M(\pi)$ falls short of the high cost, so entry returns to $\rho$ after each order.

1Weak: gain cap below

k

K_j\le\Delta_0<k

2Strong: gain floor above

k

K_j\ge\rho\kappa(\pi)\Delta_1>k

3Purchase reaches

M(\pi)

B_1(M(\pi))>c_H

4Very strong:

M(\pi)

falls short

B_2(M(\pi))<c_H

Each comparison is strict, so no tie rule changes the outcome.

Informative and uninformative markets at one strength

Proposition 3 · Paper §4.3 · Default tie rules

\pi<\tau\le M(\pi),\qquad \frac{k}{1-\tau}\ \le\ \Delta_T\ \le\ \frac{k}{\rho\max\{\pi,1-\pi\}}

Inside this region, two markets coexist. Uninformative market: no informed trade, one price, entry $\rho$ . Informative market: only a purchase recruits, and entry is
$\rho+(1-\rho)\min\Bigl\{\frac{(1-\pi)\,b\,\Delta_T}{k},\ b+\pi\lambda\Bigr\}$
Entry in the informative market rises strictly with $\Delta_T$ while the informed trader mixes on good synergy. Entry stays flat once the informed trader buys for sure on good synergy. The informative market is unique inside the region.
Self-confirming logic, market by market. Expect no recruitment: one order earns at most the floor return, so silence pays. Expect recruitment: a purchase lifts entry, so trade pays.

Scope: synergy ranking, $\Delta_T>0$ , $g_H>g_L$ , default tie rules. In general, strengths with coexistence need not form one interval. Interval results: Backup B3, page 3.

Between the Proposition 2 economies, an uninformative and an informative equilibrium can coexist at one strength.

Informative and uninformative markets at one strength

Corollary 2 · Paper §4.3

Entry against strength $r$ . Shaded band: coexistence interval $[r_A,r_N]$ .

Informative entry starts at $0.454185$ when $r_A\approx1.394$ opens the band. Uninformative entry stays at $0.250000$ .
Informative entry rises to $0.522757$ at $r_F\approx1.467$ , where the informed trader buys for sure on good synergy. Informative entry then stays flat.
The uninformative market holds through $r_N\approx1.748$ . Informative entry holds through $r_C$ , then drops to its floor.

Along one path, entry can climb, sit flat, then fall — with a jump when the informative market opens.

Full equilibrium list and marked strengths $r_J$ , $r_S$ : Backup B3, page 3.

Informative and uninformative markets at one strength

Coexistence in full: interval parts and marked strengths

Strength	Value in the example	Role
$r_A$	$1.393958$	first strength with an informative market
$r_F$	$1.467127$	first strength with a certain purchase in it
$r_N$	$1.747878$	last strength with an uninformative market
$r_J$	$1.966097$	the full purchase is alone from here
$r_S$	$2.124148$	the informed trader sells for certain on poor synergy from here
$r_C$	$3.592659$	entry ceiling: the top belief stops short of $c_H$ above $r_C$

Before $r_A$ , a purchase cannot pay at any mix rate, so no informative market exists
On $[r_A,r_F)$ , the mix rate adjusts with strength and entry rises with it
From $r_F$ to $r_N$ , a full purchase coexists with an uninformative market and entry in it is constant
Past $r_N$ , an uninformative market fails: a unit order against the uninformative price pays. Mixed orders at the entry floor join the full purchase on $(r_N,r_J)$ . From $r_J$ , the full purchase is alone

In general, strengths with coexistence need not form one interval. Sure-purchase strengths form one interval. Under the nonfalling product $\Delta_T(g_H-c_H)$ , the whole set forms one interval at each prior.

Price access and sale terms

Price access and sale terms · Paper §7

At the strong incumbent $F_1$ , under (H1)–(H3), price access raises expected target proceeds and surplus net of preparation costs.

Why: informed trader orders stay full when the price hides, so trading cost matches in both worlds. Access adds only preparation where profit covers the cost.

	Expected target proceeds	Surplus net of preparation costs
Price seen	0.872531	2.634044
Price hidden	0.614583	2.552083

Scope: inside the recruitment region, where the high cost does not pay at the prior. Outside that region, an informative price can deter preparation after a sale. Detail: Backup B4, page 2.

Payment rule (separate bargaining set-up: zero reserve, point-mass classes, $R<h$ ). Competition moves the spread with the sign of $1-2\omega$ , where $\omega$ is seller bargaining weight. Second price is $\omega=0$ : full weight on the runner-up bid.

Reserve. A reserve that excludes poor synergy can make the price informative at both strengths. Entry then sits flat in strength. Strict deterrence needs still more.

Sale terms set the payment to win and the information in the stock price before entry.

Price access and sale terms

Sale terms in full: access, payment rule, reserves

Price access (A.7): at $r_1$ , proceeds gain equals $0.257948$ and net surplus gain equals $0.081960$
Orders and trade costs are the same in the two access cases at $r_1$ . Access adds preparation only after a purchase
Reserve (A.9): at the reserve $101/100$ , which excludes poor synergy, full orders hold at both strengths and entry is flat
Strict deterrence needs reserve $31/20$ , a level that excludes poor synergy and the weak incumbent
Payment rule (A.8): zero reserve, point-mass classes, $R<h$ . With seller share $\omega$ , the spread is $\Delta_\omega=\omega(h-\ell)+(1-2\omega)\mathbb{E}[(R-\ell)_+]$ , so its response weakly has the sign of $1-2\omega$ ; strictly when $\omega\ne1/2$ and $\mathbb{E}[(R-\ell)_+]$ changes strictly

Exact differences at $F_1$ , with $\mathcal R$ expected target proceeds and $\mathcal W$ expected surplus net of preparation costs; superscripts $1$ price access, $0$ neither price nor order:

\begin{aligned}\mathcal R^1-\mathcal R^0&=(1-\rho)\{\pi(b+\lambda)(t_H-t_0)+(1-\pi)b(t_L-t_0)\}>0,\\\mathcal W^1-\mathcal W^0&=(1-\rho)(b+\pi\lambda)\,[B_1(M(\pi))-c_H+pF_1(p)]>0.\end{aligned}

Preparation, sale, and two admissible bidders are three distinct events once a reserve can exclude. The seller’s best reserve: B17.

Who the informed trader is

The informed trader is an outsider with sector expertise. It holds no net position in the target before trade. It has no control rights, and it cannot acquire the target.
Its information concerns the rival’s synergy class. It forms that view from its own research on the target’s customers, technology, and demand.
It observes the synergy class, good synergy ( $H$ ) or poor synergy ( $L$ ). It gets no further news about the rival’s value. In the point-value example, the class fixes the value.
Neither bidder trades target shares. The informed trader stands outside the sale and earns only its trading payoff: sale proceeds per share minus the stock price on a purchase, the reverse on a sale, less order cost $k$ on a nonzero order.

Scope

Scope checks · Paper §6 · Each row keeps the slot and varies one part

Change one part	Result
Any preparation-cost distribution	Under (H2), low costs below $B_1(m(\pi))$ and high costs between $B_0(\pi)$ and $B_1(M(\pi))$ , with atoms or without atoms, keep the rise. An atomless high-cost component turns the ceiling drop into a ramp.
Any ranked classes with class $H$ higher in mean, any prior	For uniform incumbent values, an open set of strengths and costs delivers the reversal. Point-mass synergy values are not essential.
Many synergy classes	The reversal holds under a separation condition. Without separation, one price can carry two synergy mixes.
Rival has its own signal	At prior $1/2$ , the rise can hold when the rival’s accuracy $\alpha_C$ exceeds the informed trader’s accuracy $\alpha_I$ . In the stated signal example, entry needs both signals.

The slot carries the mechanism. The shapes around the slot can change.

Formal statements and the six-economy table: Backup B6, pages 2 and 3. Continuous order sizes: Backup B15.

Scope

Scope in full: costs, synergy classes, separation, signals

Cost distributions (A.3)

The reversal holds with atoms or without atoms; an atomless high component turns the drop at the entry ceiling into a ramp

Many synergy classes (A.5)

The result holds under a separation condition on synergy proceeds; without it one price can pool two beliefs

Without separation, two orders can carry two synergy mixes at one price.

General values, priors (A.4)

Proposition 2 holds for every ranked pair with class $H$ higher in mean, and every prior, at suitable uniform incumbent values and costs, under (H1)–(H3) with $m(\pi)$ , $M(\pi)$

Private signals (A.6)

In the example, accuracies $(\alpha_I,\alpha_C)=(0.94,0.505)$ give entry $0.250000\to0.386933$ ; without the rival’s signal entry is $0.250000$ at both strengths

Scope

Panel A. Primitives
Economy	Poor synergy	Good synergy	$\pi$	$c_H$	$k$	$r_0$	$r_1$	$r_2$
Moderate gap	$1$	$2$	1/2	89/100	1/500	21/20	3/2
Narrow bands	$U[19/20,21/20]$	$U[199/20,201/20]$	1/2	6	1/50	6/5	3	18/5
Low prior	$1$	$10$	3/10	9/2	1/100	11/10	3	18/5
High prior	$1$	$10$	7/10	36/5	1/100	11/10	3	18/5
Wide classes	$U[3/5,7/5]$	$U[1,19]$	7/10	67/10	1/50	1	9/2	11/2
Same means, points	$1$	$10$	7/10	67/10	1/50	1	9/2	11/2
Panel B. Outcomes
Economy	Entry $\mathsf{E}_0$	Entry $\mathsf{E}_1$	Good-synergy ownership $\mathsf{O}_{H,1}$
Moderate gap	0.250000	0.522757	0.324401
Narrow bands	0.250000	0.522757	0.324401
Low prior	0.250000	0.472339	0.194641
High prior	0.250000	0.573175	0.454162
Wide classes	0.250000	0.573175	0.419819
Same means, points	0.250000	0.250000	0.175000

The order slot in full

The order slot · Paper §2.3

Informed trader — fills the slot with probability

\lambda

; sees synergy

j

; picks

q\in\{-1,0,1\}

; a nonzero order costs

k>0

Liquidity trader — fills the slot with probability

1-\lambda

; sells, does nothing, or buys with chances

\beta,\ 1-2\beta,\ \beta

, where

0<\beta\le 1/3

Order slot — one public order

X\in\{-1,0,1\}

; no observer sees who filled it

Market makers — see

X

only; set

P=\mathbb{E}[V_T\mid X]

, where

V_T

is the sale proceeds per share

Rival — sees the price

P

and cost

C

; sees neither

X

nor

j

A liquidity order in one direction arrives with probability $b=(1-\lambda)\beta$ . Set $u=\lambda/b$ . The number $1+u$ caps the likelihood ratio of one order.

Alternative arrival: Dow and Gorton (1997). Unit orders and unit cost: Edmans, Goldstein and Jiang (2015). Zero-profit pricing: Glosten and Milgrom (1985); Kyle (1985).

Payoffs and Proposition 1 in full

Payoff objects · Paper §2.4 and §3.1

Take a prepared rival with value $\theta\ge p$ . It pays $\max\{p,R\}$ when it wins. The incumbent pays $\theta$ when it wins.

t(\theta)=\mathbb{E}_R\bigl[\max\{p,\min(R,\theta)\}\bigr],\qquad g(\theta)=\mathbb{E}_R\bigl[(\theta-\max\{p,R\})_+\bigr],\qquad t(\theta)+g(\theta)=\theta

$\mathbb{E}_R$ averages over the incumbent’s value $R\sim F$ .

Synergy means: $t_j=\mathbb{E}_j[t(\theta)]$ and $g_j=\mathbb{E}_j[g(\theta)]$ for $j\in\{H,L\}$ . Without entry, $t_0=p[1-F(p)]$ .
Shareholder spread: $\Delta_T=t_H-t_L$ . Read it as: how much more shareholders get with good synergy than with poor synergy.
At belief $\mu=\Pr(H)$ , expected rival gross profit is $B(\mu)=g_L+\mu(g_H-g_L)$ .
Win chance: $W_j(F)=\mathbb{E}_j[F(\theta)\mathbf 1\{\theta\ge p\}]$ . A good-synergy rival outbids $F$ when $W_H(F)=1$ .

The traded claim and the rival’s claim sum to the same value. Strength moves the split.

Payoffs and Proposition 1 in full

Proposition 1 · Paper §3.1

Proposition 1

Let $\tilde F\le F$ be a first-order stochastic strengthening of the distribution $F$ .

(a) For any class distributions with finite means, gross auction profit $g_H$ and $g_L$ weakly fall. Rival profit $B(\mu)$ weakly falls at each fixed belief $\mu$ .

(b) Under the class ranking $G_H\le G_L$ , the shareholder spread $\Delta_T$ weakly rises.

Under the class ranking, $\Delta_T(F)+g_H(F)-g_L(F)=\mathbb{E}_H[\max\{\theta,p\}]-\mathbb{E}_L[\max\{\theta,p\}]$ , with the rival’s value $\theta$ and reserve $p$ .

Intuition: one class value gap is fixed, and strength moves its split between target proceeds and rival gross profit.

Payoffs and Proposition 1 in full

Proposition 1 in full: forms, strict signs, equal admissibility

With class gap $D(y)=G_L(y)-G_H(y)$ : $\Delta_T(F)=pF(p)D(p^-)+\int_p^\infty[1-F(y)]D(y)\,dy$ and $g_j(F)=\int_p^\infty[1-G_j(y)]F(y)\,dy$ .
Profit falls strictly exactly when $\int_p^\infty[1-G_j(y)][F(y)-\tilde F(y)]\,dy>0$ . Spread rises strictly exactly when $\int_p^\infty[F(y)-\tilde F(y)]D(y)\,dy>0$ .
Without the class ranking, the reserve term $pF(p)D(p^-)$ can reverse the spread sign. With no class atom at the reserve, the spread weakly rises under every strengthening exactly when class $H$ dominates above the reserve and the classes have equal admissibility, $G_H(p^-)=G_L(p^-)$ .

Equilibrium in full

Tuple: $\mathcal{E}=(\sigma_H,\sigma_L,\mu,P,\bar\mu,\zeta)$ , with truthful bids. Order distribution in class $j$ : $\sigma_j$ on $\{-1,0,1\}$ . Order belief: $\mu(x)=\Pr(j=H\mid X=x)$ . Stock price after order $x$ : $P(x)$ . Belief at a price: $\bar\mu(\bar P)=\Pr(j=H\mid P(X)=\bar P)$ . High-cost preparation probability at price $\bar P$ : $\zeta(\bar P)$ . Cost $C=0$ prepares at each price.
Liquidity probabilities: $\eta(-1)=\eta(1)=\beta$ and $\eta(0)=1-2\beta$ . Order probability in class $j$ : $f_j(x)=(1-\lambda)\eta(x)+\lambda\sigma_j(x)$ , with presence $\lambda$ . Entry after order $x$ : $e(x)=\rho+(1-\rho)\zeta(P(x))$ , with entry floor $\rho$ .
(E1) Order beliefs by Bayes’ rule: $\mu(x)=\pi f_H(x)/[\pi f_H(x)+(1-\pi)f_L(x)]$ after each order $x$ , with prior $\pi$ .
(E2) Beliefs at a price by Bayes’ rule: $\bar\mu(\bar P)=\pi\sum_{x:\,P(x)=\bar P}f_H(x)\big/\sum_{x:\,P(x)=\bar P}[\pi f_H(x)+(1-\pi)f_L(x)]$ at each price $\bar P$ that occurs.
(E3) Preparation: $\zeta(\bar P)=1$ when rival gross profit $B(\bar\mu(\bar P))>c_H$ , and $\zeta(\bar P)=0$ when $B(\bar\mu(\bar P))<c_H$ , with high cost $c_H$ . At equality, the tie rule applies.
(E4) Competitive prices: $P(x)=\mathbb{E}[V_T\mid X=x]$ after each order $x$ , with sale proceeds per share $V_T$ . In payoff objects, $P(x)=t_0+e(x)[t_L-t_0+\Delta_T\,\mu(x)]$ , with shareholder spread $\Delta_T$ and class proceeds $t_L$ , $t_0$ from frame 8.
(E5) Orders: $\sigma_j(q)>0$ only when the order $q$ maximizes the informed trader’s class- $j$ payoff $U_j(q)=q[t_0+e(q)(t_j-t_0)-P(q)]-k|q|$ , with order cost $k$ . Prices $P$ and entry $e$ stay at the stated functions.
(E6) Truthful bids: each prepared bidder with value at or above reserve $p$ bids its value.
Tie rules, default pair: an indifferent informed trader may mix over its best orders, and an indifferent rival at cost $c_H$ prepares. Other rules: an indifferent informed trader abstains, or an indifferent rival at cost $c_H$ prepares with any probability.

The example in full

Input	Value	Meaning
$h$ , $\ell$	$10$ , $1$	good-synergy and poor-synergy values
$p$	$1/2$	reserve price
$\pi$ , $\rho$	$1/2$ , $1/4$	prior on good synergy; probability of zero cost
Costs $0$ , $c_H$ , $k$	$0$ , $6$ , $1/50$	preparation costs; cost of a nonzero order
$\beta$ , $\lambda$	$0.294655$ , $0.336122$	liquidity order chance; informed presence chance
$u$ , $b$	$1.718282$ , $0.195615$	mass ratio $u=\lambda/b$ ; here $1+u=\mathrm{e}$
$r_0,r_1,r_2$	$6/5$ , $3$ , $18/5$	weak, strong, stronger still

Object	Region	Example
Mass ratio $u$ (Corollary 1)	$(1.389831,1.722899)$	$1.718282$
Top belief $M$ at prior $1/2$	$(0.705000,0.731392)$	$0.731059$
Liquidity chance $\beta$	$0<\beta\le 1/3$	$0.294655$
Presence chance $\lambda$ at that $\beta$	from $u=\lambda/((1-\lambda)\beta)$	$0.336122$
Prior $\pi$ for Proposition 2	$(0.467850,0.500424)$	$1/2$
Research cost $a$ (Proposition A.1)	$(0.021469,0.036763)$	$1/40$
Research cap $\bar\lambda$	$(0.290539,0.336721)$	$0.336122$

Each region is cut out by strict inequalities, so the results hold on an open set of inputs.

Outcomes and controls in full

Corollary 1 · Paper §4.2

	Weak $r_0=6/5$	Strong $r_1=3$	Very strong $r_2=18/5$
Rival profit at the prior, $B_r(1/2)$	4.804167	4.291667	4.134722
Shareholder spread, $\Delta_T(r)$	0.016667	0.666667	0.938889
Informed trader orders (good synergy, poor synergy)	$(0,\ 0)$	$(1,\ {-1})$	$(1,\ {-1})$
Price	Uninformative	Informative	Informative
Entry	0.250000	0.522757	0.250000
Good-synergy ownership	0.125000	0.324401	0.125000

At each step, rival gross profit falls and the shareholder spread rises. Entry rises against the strong incumbent and returns to its floor against the very strong one.

In the example at prior $1/2$ , (H1)–(H3) hold exactly when the top belief sits between the two thresholds: $\tau(r_1)<M(1/2)<\tau(r_2)$ . Region and margins: Backup B10. Fixed-information control: Backup B11, pages 2 and 3. Full payoff and outcome tables: Backup B11, pages 4 to 6.

Outcomes and controls in full

Fixed-information control · Paper §4.2 and Proposition A.2

Fix the distribution of synergy and signal, the synergy value distributions, and the cost distribution. A stronger incumbent cannot raise entry.

Entry falls by exactly the chance that the preparation cost $C$ sits between rival gross profit at the two strengths.

Entry	Weak $r_0$	Strong $r_1$
Full model, orders re-solved	0.250000	0.522757
Full orders held fixed at each strength	0.522757	0.522757
Rival cannot see the price	0.250000	0.250000

Row 2 at $r_0$ is a control, not an equilibrium. In the slot, fixed order distributions give the same belief after each order at each strength.

The rise needs the signal itself to change with strength. Frozen news gives flat entry, then a fall past the ceiling.

Outcomes and controls in full

The exact fall of entry under fixed information · Proposition A.2

Proposition A.2 (Fixed information)

Fix the joint distribution of class $j$ and signal $S$ , the class value distributions $G_j$ , and an independent cost distribution for preparation cost $C$ . Conditional on $(j,S)$ , the values $\theta$ and $R$ are independent with distributions $G_j$ and $F_i$ in economy $i$ , with signal belief $\mu(S)=\Pr(j=H\mid S)$ . For a strengthening from $F_0$ to $F_1$ , total entry $\mathsf{E}$ falls by exactly $\mathsf{E}_0-\mathsf{E}_1=\Pr\{B_1(\mu(S))<C\le B_0(\mu(S))\}\ge0$ , with rival gross profit $B_i$ at $F_i$ .

The rival still reads the stock price $P$ here. The fixed object is the joint distribution of the signal and the class.

Outcomes and controls in full

Proposition 2 in full: tuples, payoff tables, control panels

Bounds for every profile: belief band $m\le\mu\le M$ ; gain per unit at most $\Delta_T$ ; entry at least $\rho$
Weak $r_0$ : $\Delta_T(r_0)<k$ makes each nonzero order lose in expectation. An uninformative market is the sole outcome; entry is $\rho$
Strong $r_1$ : $k<\rho\kappa\,\Delta_T(r_1)$ makes each correct order pay under every profile; a wrong-signed order always loses. Orders are $(1,-1)$
Entry at $r_1$ : a buy order gives belief $M$ with $B_{r_1}(M)>c_H$ , so cost $c_H$ prepares after a buy order. Entry exceeds $\rho$
Return at $r_2$ : trade still pays, but $B_{r_2}(M)<c_H$ keeps cost $c_H$ out at every price. Entry is $\rho$

Ownership: with $W_H(F)=\mathbb{E}_H[F(\theta)\mathbf{1}\{\theta\ge p\}]$ , good-synergy ownership rises exactly if $[\rho+(1-\rho)(b+\lambda)]W_H(F_1)>\rho W_H(F_0)$ .

Share rises exactly when $e_HW_H(F_1)W_L(F_0)>e_LW_H(F_0)W_L(F_1)$ ; in the example, share rises from $0.545455$ to $0.830691$ .

Outcomes and controls in full

Under full orders, a good-synergy rival faces a buy order with chance $\lambda+b$ , and a poor-synergy rival with chance $b$ .

e_H=\rho+(1-\rho)(\lambda+b),\qquad e_L=\rho+(1-\rho)b,\qquad \text{entry}=\pi e_H+(1-\pi)e_L

At $r_1$ , cost $c_H$ prepares after a buy order and only after a buy order. Synergy entry rates are $e_H$ and $e_L$
At the example’s prior, entry is $0.522757$ and good-synergy ownership is $0.324401$
At $r_0$ , an uninformative market gives an uninformative price, so entry is $0.250000$ and ownership is $0.125000$

At $r_2$ , no price covers $c_H$ , so entry is $0.250000$ and ownership is $0.125000$ , with full orders
$B_r(\pi)$ equals $4.804167$ , $4.291667$ , $4.134722$ at $r_0$ , $r_1$ , $r_2$

	$r_0=6/5$	$r_1=3$	$r_2=18/5$
Proceeds without entry, $t_0$	0.291667	0.416667	0.430556
Proceeds with a good-synergy rival, $t_H$	0.704167	1.541667	1.834722
Proceeds with a poor-synergy rival, $t_L$	0.687500	0.875000	0.895833
Shareholder spread, $\Delta_T$	0.016667	0.666667	0.938889
Rival profit, good synergy, $g_H$	9.295833	8.458333	8.165278
Rival profit, poor synergy, $g_L$	0.312500	0.125000	0.104167
Rival profit at the prior, $B_r(1/2)$	4.804167	4.291667	4.134722
Preparation threshold, $\tau(r)$	0.633117	0.705000	0.731392

Outcomes and controls in full

	$r_0=6/5$	$r_1=3$	$r_2=18/5$
Panel A. Equilibrium (Proposition 2)
Informed orders, good synergy and poor synergy	$(0,0)$	$(1,-1)$	$(1,-1)$
Belief after a sale	0.500000	0.268941	0.268941
Belief after no order	0.500000	0.500000	0.500000
Belief after a purchase	0.500000	0.731059	0.731059
High-cost preparation	none	after a purchase	none
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
Entry $\mathsf{E}$	0.250000	0.522757	0.250000
Good-synergy ownership $\mathsf{O}_H$	0.125000	0.324401	0.125000
Expected target proceeds	0.392708	0.872531	0.664236
Panel B. Full orders held fixed (Proposition A.2)
Entry $\mathsf{E}$	0.522757	0.522757
Good-synergy ownership $\mathsf{O}_H$	0.324401	0.324401
Expected target proceeds	0.503998	0.872531
Panel C. The rival does not see the price
Informed orders, good synergy and poor synergy	$(0,0)$	$(1,-1)$
Entry $\mathsf{E}$	0.250000	0.250000
Good-synergy ownership $\mathsf{O}_H$	0.125000	0.125000
Expected target proceeds	0.392708	0.614583

Research in full

Proposition 4 · Paper §5 · Prior $1/2$

One change only. Before it learns the synergy, the informed trader picks research $\lambda\in[0,\bar\lambda]$ at cost $(a/2)\lambda^2$ . Success has chance $\lambda$ , reveals the synergy, and fills the slot. Failure hands the slot to liquidity.

More research adds informed trader orders and removes liquidity cover. The prior is $1/2$ throughout this block.

At research $\lambda$ , write $M_\lambda$ for the top belief after a purchase. Recruitment intensity $\lambda_E(\tau)$ solves $M_\lambda=\tau$ : a full purchase recruits exactly at or past this research level.

In the recruiting equilibrium, research $\min\{\lambda_R(\Delta_T),\bar\lambda\}$ rises strictly with $\Delta_T$ until the cap, then stays at the cap.
Entry $\rho+(1-\rho)[\beta+(1/2-\beta)\lambda]$ rises with research. Good-synergy entry rises with research. Poor-synergy entry falls.
The recruiting equilibrium exists exactly when $\Delta_T>2k$ and $\tau\le M_\lambda$ at $\lambda=\min\{\lambda_R(\Delta_T),\bar\lambda\}$ .

Interior research $\lambda_R(\Delta_T)$ solves $2a\lambda=\Delta_T m_\lambda-k+(\rho\Delta_T m_\lambda-k)_+$ , with band floor $m_\lambda=1-M_\lambda$ . Branches: Backup B12, page 3. Scope: default tie rules, $\tau>1/2$ .

A wider spread pays for more research. More research brings in a good-synergy rival more often.

Research in full

Research corollary · Paper §5

Research $\lambda$ and entry against strength $r$ . Research cost $a=1/40$ . Cap $\bar\lambda=0.336122$ .

Recruiting research opens at $r_R\approx1.501$ . Entry there is $0.500846$ .
Research reaches the cap at $r_{\bar\lambda}\approx1.678$ . Entry reaches $0.522757$ and holds to $r_C$ .
Zero research exists on $[r_0,r_N]$ , beside recruiting research on $[r_R,r_N]$ . At $r_1$ , $r_2$ only the cap remains.

At the cap, the research economy is the Proposition 2 economy.

Floor-entry research branch, cap region, and three-strength table: Backup B12, pages 3 and 4.

Research in full

Research in full: first-order condition, branches, cap region

2a\lambda=\Delta_T m_\lambda-k+(\rho\Delta_T m_\lambda-k)_+

Research picks $\lambda$ in $[0,\bar\lambda]$ before the slot, at cost $\tfrac{a}{2}\lambda^2$ (Proposition 4)
In the example, $a=1/40$ and the cap is $\bar\lambda=0.336122$ . The marked strengths are $r_R\approx1.501023$ , $r_{\bar\lambda}\approx1.677633$ , $r_N\approx1.747878$ , $r_Q\approx2.166995$ , $r_C\approx3.592659$
Zero research, with entry $\rho$ , is an equilibrium exactly on $[r_0,r_N]$
Recruiting research is an equilibrium exactly on $[r_R,r_C]$ : research rises from $0.193853$ to the cap on $[r_R,r_{\bar\lambda})$ and stays at the cap on $[r_{\bar\lambda},r_C]$ ; entry rises from $0.500846$ to $0.522757$ and keeps that value up to $r_C$

Zero research on $[r_0,r_N]$ ; recruiting research on $[r_R,r_C]$ ; floor-entry research on $(r_N,r_Q)$ and $(r_C,r_2]$ , with $r_Q\approx2.167$ .

Research in full

Floor-entry research, in which the informed trader trades in both synergy classes after a success, no order recruits and entry is $\rho$ , is an equilibrium exactly on $(r_N,r_Q)$ and on $(r_C,r_2]$ ; there is no other equilibrium
At $1.55$ , $1.60$ and $1.65$ there are exactly two equilibria: zero research, and recruiting research below the cap (the informed trader buys on good synergy after a success and abstains on poor synergy). Research is $0.237901$ , $0.279037$ , $0.316683$ . Entry is $0.507630$ , $0.513965$ , $0.519763$
The unique equilibrium has zero research at $r_0$ , recruiting research at the cap at $r_1$ and floor-entry research at the cap at $r_2$ , where entry is $\rho$

Costly research with an alternative arrival follows Dow and Gorton 1997.

Predictions in full

Predictions · Paper §8 · Full two-sided trade, prior below $\tau$

Weight each price impact by how often its order occurs. Weighted purchase impact beats weighted sale impact exactly when a purchase recruits the rival.

At prior $1/2$ , order frequencies match. Compare raw impacts: $P(1)-P(0)$ against $P(0)-P(-1)$ .
The price gap $P(1)-P(-1)$ splits into a synergy part, $\rho\Delta_T[M(\pi)-m(\pi)]$ , and an entry part. The entry part is positive exactly under recruitment, and it sits wholly in the purchase impact.

Entry can rise with incumbent strength only when the price lands before preparation starts. Hide the price, and entry can only fall.
Past the entry ceiling, orders stop predicting entry. Orders keep predicting the winner when $W_H>W_L$ .
At prior $1/2$ , order counts cannot separate slot types. Prices can.

Date the price before preparation starts. Record the zero order.

In the example at $r_1$ , the entry part is $90.2\%$ of the price gap. Informed-trader share and announcement detail: Backup B13, page 2.

Test recruitment in prices first. The sharpest test needs no auction object.

Predictions in full

Mean order under full two-sided trade is $\lambda(2\pi-1)$ . Counts name informed presence only when the prior is known and differs from $1/2$ .
At prior $1/2$ , sale impact plus volume name presence when entry sits at its floor after a sale and after no order, with payoff scale $\rho\Delta_T$ known.
For a prepared rival, the winner gap after purchase against sale equals $[M(\pi)-m(\pi)](W_H-W_L)$ , with win chance $W_j$ in class $j$ . The prediction is strict only when $W_H>W_L$ .
A report that reveals preparation alone carries a positive mean price change exactly when preparation stayed uncertain after some order. Both reports together average to zero.
At one middle strength, sale processes can form two clusters: uninformative prices with mean order zero, or informative prices with mean order $\lambda/2$ .
Count preparation starts, not public offers alone. Date incumbent strength from public records before the preparation decision.

Predictions in full

Weighted price impact and the winner · Proposition A.10

Let order frequency be $\bar f(x)=\Pr(X=x)$ , purchase impact $I_+=P(1)-P(0)$ , and sale impact $I_-=P(0)-P(-1)$ . Proposition A.10: under full two-sided trade, class $H$ buys and class $L$ sells with probability one, and prior $\pi<{}$ threshold $\tau$ , the weighted difference $\bar f(1)I_+-\bar f(-1)I_-$ is positive exactly when entry after a purchase exceeds the floor, $e(1)>\rho$ . At prior $\pi=1/2$ the frequencies agree, and the raw impacts $I_+$ and $I_-$ compare directly. The entry part of the price gap $P(1)-P(-1)$ sits in purchase impact $I_+$ .
Under the same full two-sided trade, where entry $e(x)=\rho$ after each order, the order $X$ does not predict entry. It still predicts the winner exactly when the win chance $W_H$ exceeds $W_L$ , with $W_j=\mathbb{E}_j[F(\theta)\mathbf{1}\{\theta\ge p\}]$ for a prepared rival of class $j$ ; the indicator $\mathbf{1}\{\theta\ge p\}$ is one when the value meets the reserve. For a prepared rival, the win-probability gap after a purchase against a sale is $[M(\pi)-m(\pi)](W_H-W_L)$ .

References and the full literature table

What each strand gives the paper, and what the paper adds

Strand	Closest work	Builds on	Adds
Prices guide real decisions	Dow, Goldstein and Guembel (2017); Edmans, Goldstein and Jiang (2015); Bond, Edmans and Goldstein (2012); Foucault and Frésard (2014)	The price as an information source; real decisions feed back into the incentive to produce information	The decision is a rival’s entry, and the sale splits one value between shareholders and that rival
Costly entry into auctions	Fishman (1988); Hirshleifer and Png (1989); Samuelson (1985); McAfee and McMillan (1987); Levin and Smith (1994); Roberts and Sweeting (2013); Gentry and Stroup (2019)	Costly preparation, deterrence and selective entry	Pre-entry information that moves with competition
Takeover prices and trading	Edmans, Goldstein and Jiang (2012); Kyle and Vila (1991); Luo (2005); Betton, Eckbo, Thompson and Thorburn (2014)	Target prices shape takeovers; informed orders hide in liquidity trade; bidders learn from prices after a bid	An outside informed trader with no stake informs a rival bidder before it prepares

References and the full literature table

Admati and Pfleiderer (2009). Rev. Financ. Stud. 22.

Betton, Eckbo, Thompson and Thorburn (2014). J. Finance 69.

Bond, Edmans and Goldstein (2012). Annu. Rev. Financ. Econ. 4.

Boone and Mulherin (2007). J. Finance 62.

Bulow and Klemperer (1996). Amer. Econ. Rev. 86.

Dow and Gorton (1997). J. Finance 52.

Dow, Goldstein and Guembel (2017). J. Eur. Econ. Assoc. 15.

Edmans, Goldstein and Jiang (2012). J. Finance 67.

Edmans, Goldstein and Jiang (2015). Amer. Econ. Rev. 105.

Fishman (1988). RAND J. Econ. 19.

Foucault and Frésard (2014). J. Financ. Econ. 111.

Gentry and Stroup (2019). J. Financ. Econ. 132.

Gorbenko and Malenko (2014). J. Finance 69.

Hirshleifer and Png (1989). Rev. Financ. Stud. 2.

Kyle and Vila (1991). RAND J. Econ. 22.

Levin and Smith (1994). Amer. Econ. Rev. 84.

McAfee and McMillan (1987). Econ. Letters 23.

Luo (2005). J. Finance 60.

Roberts and Sweeting (2013). Amer. Econ. Rev. 103.

Samuelson (1985). Econ. Letters 17.

Schwert (1996). J. Finance 51.

Continuous order sizes

The slot set-up uses unit orders $\{-1,0,1\}$ , so every order is on path and no off-path belief enters
A menu of continuous order sizes needs a convention for the sizes that no synergy class uses, as in Edmans, Goldstein and Jiang 2015
Additive liquidity flow also keeps every size on path
With Laplace noise of scale $\xi$ and full unit orders, the top belief equals the slot value $M(\pi)$ when $1+u=\mathrm{e}^{2/\xi}$
The two set-ups then share the belief bounds. They differ in the probability that each synergy class reaches the top belief

Unit orders keep prices on path at every order.

Notation

Symbol	Gloss	First defined
$R$ , $F$	incumbent’s value $R$ ; its distribution $F$ , with $R\sim F$ in the general model; in the paper’s example (frames 14–16 and their backups), $R$ is uniform on $[0,r]$	frame 5
$j$ , $\pi$ , $\theta$ , $G_j$	synergy class $j\in\{H,L\}$ ; prior $\pi=\Pr(j=H)$ ; rival’s value $\theta$ ; class value distribution $G_j$ for $\theta$ given $j$	frame 5
$p$	reserve $p$ of the cash second-price auction	frame 5
$C$ , $c_H$ , $\rho$	baseline preparation cost $C\in\{0,c_H\}$ (B11 also allows any fixed independent cost distribution); high cost $c_H$ ; entry floor $\rho=\Pr(C=0)\in(0,1)$	frame 5
$X$ , $q$ , $\lambda$ , $\beta$ , $k$	slot outcome $X$ ; the informed trader’s chosen order $q$ ; presence $\lambda$ ; liquidity probability $\beta$ ; order cost $k$	frame 6
$P$ , $V_T$ , $\mathbb{E}[\cdot]$	stock price $P(X)=\mathbb{E}[V_T\mid X]$ ; target proceeds per share $V_T$ ; expectation $\mathbb{E}[\cdot]$	frame 6
$\mu(x)$ , $\bar\mu(\bar P)$	order belief $\mu(x)=\Pr(j=H\mid X=x)$ ; belief at a price $\bar\mu(\bar P)=\Pr(j=H\mid P(X)=\bar P)$ . The two agree under Lemma A.2, with $\Delta_T>0$ , $g_H>g_L$ , and $e(x)\ge\rho$	frames 7, 9, 11
$t(\theta)$ , $g(\theta)$	expected target proceeds $t(\theta)$ ; gross auction profit $g(\theta)$ , at value $\theta\ge p$	frame 8
$t_j$ , $g_j$ , $t_0$	class means of $t(\theta)$ , $g(\theta)$ in class $j$ ; proceeds with no entry $t_0=p[1-F(p)]$	frame 8
$\Delta_T$ , $B(\mu)$	shareholder spread $\Delta_T=t_H-t_L$ ; rival gross profit $B(\mu)=g_L+\mu(g_H-g_L)$ at belief $\mu$ (before $C$ )	frame 8
$e(x)$ , $\mathsf{E}$	entry after order $x$ , $e(x)\ge\rho$ ; total entry $\mathsf{E}$ , the probability that the rival prepares	frame 9, B11

Notation

Symbol	Gloss	First defined
$U_j(q)$	payoff from order $q$ ; informed trader sees class $j$	frame 9
$\tilde F$ , $F_i$	strengthening $\tilde F\le F$ in B8; distributions $F_0$ , $F_1$ , $F_2$ in B1, with $B_i$ , $\Delta_i$ at $F_i$	B8, B1
$b$ , $u$ , $m(\pi)$ , $M(\pi)$	liquidity mass $b=(1-\lambda)\beta$ ; relative informed mass $u=\lambda/b$ ; band floor $m(\pi)$ ; top belief $M(\pi)$	frame 11, B2
$\tau$ , $\kappa(\pi)$	threshold $\tau=(c_H-g_L)/(g_H-g_L)$ when $g_H>g_L$ ; residual doubt $\kappa(\pi)=\min\{m(\pi),1-M(\pi)\}$	frame 12, B1
$K_H$ , $K_L$	gross returns of a purchase in class $H$ and a sale in class $L$ , before order cost $k$	frame 12, B2
$h$ , $\ell$ , $r$	point-mass values $h$ (class $H$ ) and $\ell$ (class $L$ ); strength $r$ with $R$ uniform on $[0,r]$	frame 14
$S$ , $\mu(S)$	fixed signal $S$ ; signal belief $\mu(S)=\Pr(j=H\mid S)$	frame 15, B11
$a$ , $\bar\lambda$ , $\lambda_R$ , $M_\lambda$ , $m_\lambda$	research cost coefficient $a>0$ in $(a/2)\lambda^2$ ; research cap $\bar\lambda\in(0,1)$ ; interior research $\lambda_R(\Delta_T)$ ; top belief $M_\lambda$ and band floor $m_\lambda$ at presence $\lambda$	frame 16, B12
$\bar f(x)$ , $I_+$ , $I_-$ , $W_j$	order frequency $\bar f(x)=\Pr(X=x)$ ; purchase impact $I_+=P(1)-P(0)$ ; sale-order impact $I_-=P(0)-P(-1)$ ; win chance $W_j$ of a prepared class- $j$ rival	frame 18, B13
$\sigma_j$ , $\eta$ , $f_j(x)$ , $\zeta$	order distribution $\sigma_j$ ; liquidity probabilities $\eta$ ; order probability $f_j(x)$ in class $j$ ; high-cost preparation probability $\zeta$ at a price	B9
$e_H$ , $e_L$ , $W_j(F)$	entry in class $H$ , class $L$ ; win chance $W_j(F)=\mathbb{E}_j[F(\theta)\mathbf{1}\{\theta\ge p\}]$ against distribution $F$	frame 16, B1, B13

The seller's choice of reserve in the example

Each strength at its own best reserve

A reserve near $h$ sells only to a ready rival with good synergy, so it earns $\rho\pi h$ . That gain depends on the ready share $\rho$ .

		Weak			Strong			Very strong
$\rho$	$k$	$p$	Proceeds	Entry	$p$	Proceeds	Entry	$p$	Proceeds	Entry
1/4	1/50	near $h$	1.25	0.25	near $h$	1.25	0.25	near $h$	1.25	0.25
1/5	1/50	near $h$	1.00	0.20	1.325	1.069	0.491	1.906	1.038	0.20
1/10	0.017	1.793	0.519	0.427	1.325	1.044	0.427	1.849	0.968	0.10

For $\rho$ between 0.104 and 0.2145, with $k$ inside (H2), entry at the best reserves is at its floor, then above its floor, then back at its floor.
At or below $\ell$ , the reserve does not move the spread. As $p\to0$ , the same pattern holds with a very strong incumbent above $r=3.685$ .

The seller’s choice limits the channel without removing it.

Point values. A reserve near $h$ is a limit. Narrow bands give the same pattern at $\rho=3/20$ ; the exact range is open. Calculation from the reserve formulas of the appendix; not a result of the paper.

Competition Creates Competition:Stock Prices and the Discovery of Takeover Bidders

Competition Creates Competition:
Stock Prices and the Discovery of Takeover Bidders