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

• A disclosed approach, an announced strategic review or an open contest can create one if the opportunity becomes public before the buyer set is fixed
• Only then can a prospective challenger watch the price while it decides
• Strength = the public distribution of the incumbent's value, not an announced bid
• Imprivata's proxy separates an unsolicited approach, outreach to potential buyers, and indications of interest conditional on diligence, which documents a costly preparation stage; the three predictions say how to test whether prices draw buyers into it Imprivata 2016; Boone and Mulherin 2007; Gentry and Stroup 2019

Whether a deal fits is a question about its chronology; Online Appendix D shows how to date it (Section 2.1).

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• Pilot (Online Appendix D): reconstruct from disclosure records the timing of approaches, public visibility, buyer contacts, diligence, proposals and final selection
• Separate when an event occurred from when it first became public
• First requirement: a publicly understood sale opportunity that stays contestable while a challenger decides
• Outcome: the decision to prepare; prices must precede it
• Caution: returns before a bid also reflect its anticipation Schwert 1996

Online Appendix D sets out the pilot design for the predictions of Section 7.

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The paper does not solve toeholds, so it makes no statement on the direction of a toehold effect.

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• The challenger knows its own integration plans; a specialist investor may know the target's customers, technology and product demand
• Different facts about the same acquisition match
• Preparation (diligence) is what reveals the exact value to the challenger
• The benchmark's perfectly informed investor and uninformed challenger make the mechanism visible; Section 5.3 removes both extremes
• The reversal holds with challenger accuracy $d=0.75$ above investor accuracy $a=0.70$ (Prop. A.7, analytical)

The signal economy Back

• Investor signal $T$ (accuracy $a$), challenger signal $Y$ (accuracy $d$); the challenger uses $\Pr(H\mid P,\,Y=y)$, $P$ = stock price
• Entry is state dependent: $e_H$, $e_L$ = entry probability when the challenger is high / low value; the price still reveals the market posterior
• Declared example $a=0.70$, $d=0.75$: entry 0.850 → 0.879 (analytical, Prop. A.7); $\rho=0.85$, $c_H=7.14$, $k=0.015$, strengths 1.1 vs 2.3

The price matters when the challenger's own signal is informative but not decisive.

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• Definition: order distributions $\sigma_H,\sigma_L$ (mixing allowed; any deviation $q\in[-1,1]$); a rational price that anticipates entry; Bayesian beliefs from the price; optimal entry; truthful bids
• Order flow $X=q+Z$, Laplace noise $Z$ with scale $b$; stock price $P(X)$ = expected target payoff given $X$
• Two comparisons kept apart: a unilateral deviation is evaluated against fixed price and entry schedules; a cross-economy comparison re-solves them
• Uniqueness refers to trading and on-path entry under truthful bidding

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Proposition 1 (analytical) holds for any $F$ on $[0,\bar r]$, $\ell<\bar r<h$; strict when the change in $F$ has positive integral over the relevant range.

The closed forms are the uniform case; the two signs are not a uniform-distribution artifact.

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Payment stage only: an entry reversal under bargaining is not solved, and first-price equilibria are not solved.

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• Weak: gross advantage per unit $\le\Delta_T(r_0)<k$, so every nonzero order loses and the posterior stays at $1/2$
• Strong: with $\Pi_\theta(s)$ the expected residual per unit at order size $s$,

• Holds against every candidate price and entry schedule, including mixed orders: global, not first-order
• Investor manipulation (in-model): both residual advantages in eq. (11) are positive under every candidate order distribution, so a wrong-signed order, such as a low-value investor buying to fake good news and trigger entry, has negative gross payoff and still pays $k$
• Incumbent or target manipulation and toeholds: outside the model (A4)

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• $q\in[-1,1]$ is a normalized small trading unit, not ownership of the target, measured in the same units as the noise (scale $b$); the investor has no initial position
• $k$ is an execution or position-carrying friction, distinct from adverse-selection price impact, which competitive pricing already generates
• Why a corner, unlike Kyle: the per-unit advantage is at least $\rho m\Delta_T$ because beliefs stay in $[m,M]$, and one more unit erodes it by at most a fraction $1/b$ (the haircut)
• So marginal profit stays above $k$ over the whole interval, and full orders are the unique best response (A10)
• Against the weak incumbent the same bounds make every order lose

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• Under any mixed orders the posterior from order flow is bounded (Laplace likelihood ratio within $e^{\pm2/b}$):

• $t_0$ = expected target proceeds without a challenger
• The price is strictly increasing in $\mu$, because entry $\ge\rho>0$ and $\Delta_T>0$
• Atoms and entry jumps allowed; no differentiability needed (Props. A.1–A.2, analytical)

The price is a sufficient statistic for the market's belief, so the challenger loses nothing by seeing the price rather than the flow.

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• Both noise densities $f$ satisfy $|f'|\le f/b$, so the global trading bounds hold for both
• Laplace: the posterior reaches $[m,M]$ at finite flow; logistic: only as flow $\to\infty$ ($x^*\approx5.42$, about 1.5 noise s.d. from the center)
• Entry 0.302 vs 0.523; atomless costs ($\varepsilon_C=0.1$): 0.523 and 0.301. Common scale $b$, not common variance; not a Blackwell ranking (analytical, Props. A.5–A.6)

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• The low-cost floor keeps entry $\ge\rho$ after every price: the price reveals the posterior, the investor's edge is at least $\rho m\Delta_T$, and information can only add entry
• If cheap preparation fails only at bad prices, bad prices deter it and flows can pool at one price
• If cheap preparation fails at the prior, no preparation and no trade form an equilibrium
• Theorem margin at the benchmark: 1.37
• The floor alone gives entry 0.250 at both strengths when the price is hidden
• Atomless cost supports keep it (Prop. A.6, analytical)

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1. Weak economy: no order covers $k$; no trade; entry $\rho$
2. Strong economy: the global bound (A10) forces full orders
3. Under full orders the high-cost window puts the belief threshold inside $(1/2,M)$, so good news crosses it with positive probability; $\alpha_H>\alpha_L$ (good news is likelier when the challenger is high value, $\alpha_\theta=\Pr(\text{order flow crosses the entry threshold}\mid\theta)$) tilts extra entry to high values
4. Blackwell: a constant kernel maps the strong experiment to the weak one; no state-independent kernel maps the constant experiment to the strong one

Part (iii): with the floor and profitable trading still in place at $r_2$, $B_{r_2}(M)<c_H$ removes expensive entry.

What does the work: step 2, a bound that holds against every candidate schedule.

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• Nonemptiness for every $h>\ell$ comes from a construction: take $r_0<r_1$ close to $\ell$, then $k$, $c_H$, $c_L$ strictly inside their windows
• A mathematical construction, not slack at the benchmark and not an effect-size claim; moderate values $h=2$: minimum margin $8.01\times10^{-4}$
• Part (iii) is not claimed for every $h>\ell$

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stronger incumbent → wider $\Delta_T$ → informed trading pays → informative price → good news clears $c_H$ → the expensive challenger enters

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Units are arbitrary; magnitudes are not calibrated; the signs are the theorem; nonemptiness holds for every $h>\ell$ (A17).

A moderate economy with $h=2$ gives a rise of similar size (0.250 → 0.527), so the benchmark scale is a declaration, not the mechanism.

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• $\tau$ = belief threshold for costly preparation; $\alpha_\theta$ = probability that order flow crosses the entry threshold; entry $\mathsf E$ = Pr(challenger prepares); high-value ownership $\mathsf O_H$ = Pr(high-value challenger acquires the target)
• The high-cost window puts $\tau\in(1/2,M)$ and $x^*\in(0,1)$; $\alpha_H>\alpha_L$
• Above the ceiling strength $r_C\approx3.59$, $\tau>M$; the left-limit entry at $r_C$ is 0.506
• Expected target proceeds 0.393 / 0.872 / 0.664 at $r_0$ / $r_1$ / $r_2$

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The reversal (frames 9–10) compares economies in which trading and on-path entry are unique, so no selection is needed; multiplicity arises only between them (frame 12).

• Low type: global strict concavity; a sign change of its marginal profit at its own order brackets the root (0.000000000114 and $-0.000000000096$ at $r=1.55$)
• High type: cover margin strictly positive over the whole bracket (last column)
• Not proved: uniqueness of the informative profile, absence of mixed equilibria, a branch between nodes

computer-assisted (Proposition 3): interval arithmetic on exact decimal inputs

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• Prop. A.3 (analytical): at any fixed information experiment and cost law, entry weakly falls with strength; within fixed orders, not across order profiles that change with $r$
• The frozen weak profile is a control, not an equilibrium: full orders lose money there because $\Delta_T(r_0)<k$; at $r_1$ it coincides with the equilibrium

Price hidden: the challenger enters only at low cost (high-cost window); at $r_1$ the investor still trades full orders.

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• Invariance diagnostic: add a dividend $D_0=0.257809$ (the revenue gain from price access) to the traded claim in the price-hidden economy. Mean prices then match, but entry does not, so the information matters, not the price level. (numerical diagnostic)
• The revenue gain itself, 0.258, is analytical
• Prop. A.9 at $r_1$: each extra entry is chosen only when expected gross profit covers its cost; the allocation gain includes that profit and sales that would otherwise fail the reserve; transfers excluded
• The dividend is not a sale mechanism and is excluded from surplus; no ranking of strengths or mechanisms

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No trade unique for $r<r_P\approx1.22$, an equilibrium up to $r_N\approx1.75$; full orders unique above $r_U\approx2.84$; expensive entry impossible above $r_C\approx3.59$ (analytical, Prop. A.4; shaded: unique); full correspondence open.

Order sizes Back

No trade is $(0,0)$. Certified nodes: $|q_L|\approx$0.460, 0.707, 0.903 (computer-assisted); the curve through them and the mixed candidate are numerical diagnostics; no branch between nodes is claimed.

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• Minimum theorem margins: base $2.41\times10^{-3}$, moderate $8.01\times10^{-4}$, signals $1.45\times10^{-3}$
• Distinct parameter vectors, not one joint calibration; all rows analytical (Props. 2, A.5–A.7); entry $\mathsf E$ = Pr(challenger prepares), $\mathsf O_H$ = high-value ownership; pp from unrounded probabilities

The rise from $r_0$ to $r_1$ holds in every row; the $r_2$ fall is shown for the benchmark only.

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Value classes: $p=0.5$ vs $p=1.1$ ($\varepsilon_V=0.05$)

Binary values: $p=0.5$ vs $p=1.01$. Analytical at the listed nodes.

• Preparation, sale and two admissible bidders come apart

A feasible improvement, not an optimal reserve: optimal terms are open.

Same orders, different prices Back

Prop. A.10 (analytical existence): at reserve $p=7$ and $r_0$, the same full orders $(1,-1)$ support price pools below a cutoff $\kappa\in[-\log2,\,0]$.

• Same orders, different prices, beliefs, and entry
• Does not arise on the benchmark support

A continuation must include the price rule, so the seller's problem cannot be solved over orders alone.

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