Unit VII J

Received: 30 March 2021 | Accepted: 13 February 2024

DOI: 10.1111/jfir.12393

OR I G I NA L A R T I C L E

Toeholds and information quality in common‐value takeover auctions

Anna Dodonova

Telfer School of Management, University of

Ottawa, Ottawa, Ontario, Canada

Correspondence

Anna Dodonova, Telfer School of

Management, University of Ottawa, 55 E.

Laurier, Ottawa, ON, K1N 6N5, Canada.

Email: Dodonova@telfer.uottawa.ca

Funding information

Social Sciences and Humanities Research

Council of Canada, Grant/Award Number:

435‐2019‐0034

Abstract

In this article I analyze the effect of the sensitivity of firm

value on the information available to potential acquirers

in common‐value takeover auctions with toeholds. I show

that the quality of information does not affect equilibrium

when bidders have equal toeholds but has a significant

effect when toeholds are different. My article demon-

strates that increasing the relative information quality of

the bidder with a smaller toehold makes both bidders bid

more aggressively and leads to a higher price. I also analyze

the combined effect of toeholds and information quality

on equilibrium bidding strategies and discuss ways target

shareholders can increase the expected final price.

J E L C L A S S I F I C A T I ON

C72, D44, G34

1 | INTRODUCTION

When the exchange medium in a takeover is limited to cash, the takeover contest is often modeled as an English

outcry auction. There are, however, several features that distinguish takeover auctions, one being the diverse

ownership of the target and the ability of the potential acquirer to accumulate a certain percentage of the target's

shares, a toehold, before making a tender offer. Burkart (1995) considers a private‐value takeover auction with two

strategic bidders in which one or both bidders own a fraction of the target firm. He shows that a bidder with a

toehold bids above her valuation of the target firm, but this overbidding does not affect the strategy of the bidder

without a toehold, who still bids up to her valuation. In general, such overbidding benefits the target firm, although

J Financ Res. 2024;47:1229–1244. wileyonlinelibrary.com/journal/JFIR | 1229

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and

reproduction in any medium, provided the original work is properly cited.

© 2024 The Authors. Journal of Financial Research published by Wiley Periodicals LLC on behalf of The Southern Finance

Association and the Southwestern Finance Association.

the increase in the final price is moderate. Singh (1998) also investigates private‐value auctions with toeholds but

allows the winning bidder to renege on her bid. Given this possibility, the bidder with the larger toehold never quits

the auction whereas the bidder without the toehold shades her bids, which in turn leads to a lower expected price.

Dodonova (2012) and Hounwanou (2018) analyze the combined effect of both toeholds and preemptive jump

bidding on the takeover auction process.

Bulow et al. (1999) consider common‐value takeover auctions with toeholds, which are more appropriate for

takeover contests with financial bidders or when strategic bidders have similar plans for the target firm. Similar to

private‐value auctions, in common‐value auctions an increase in a toehold makes the bidder bid more aggressively.

However, contrary to private‐value auctions, even small toeholds may have a large effect on bidding strategies and

the auction's outcome. In particular, a bidder with a larger toehold tends to bid more aggressively, whereas a bidder

with a smaller toehold usually shades her bids. Given the symmetric value function that satisfies the requirement

that the bidder with the higher signal has higher marginal revenue, Bulow et al. (1999) show that unequal toeholds

reduce the expected final price. As one of the remedies to increase the target's revenue, they suggest the target

firm offers some of its equity to the bidder with the smaller toehold at a substantial discount or even for free.

Empirical findings (see, e.g., Asquis & Kieschnick, 1999; Betton & Eckbo, 2000) support Bulow et al.'s (1999)

hypothesis that unequal toeholds lead to a lower takeover premium. The hypothesis that the size of the toehold

positively affects the probability of winning the auction (for both private‐ and common‐value auctions) is also

supported by Choi (1991). Finally, Le and Schultz (2007) show that the bidding firm's share price reaction to the

takeover offer positively depends on its toehold.

In this article, I allow bidders’ information about the target firm to have different relative quality; that is,

I assume that the firm's value is more sensitive to higher quality information so that bidders with higher quality

information can better predict the firm's value. Such relative quality may take two forms: A bidder with higher

quality information may have information about an aspect of the firm's operations that greatly affects the firm's

value or she may have a large quantity of unique information not available to the other bidder. To incorporate

relative information quality, I extend Bulow et al.'s (1999) model to allow an asymmetric value function and, in

particular, to allow one of the bidders to have more valuable information about the target's value. Although the

effect of information quality on takeover contests has been studied previously, such studies usually focus on

information about the private‐value component of the bidder's valuation (Dionne et al., 2009; Dionne et al., 2015;

Povel & Singh, 2006) or, when combined with the effect of toeholds, on the effect that larger toeholds may have on

information precision (Povel & Sertsios, 2014).

To model relative information quality, I consider a “wallet”‐type value function and assume that the bidders’

signals may be weighted differently in determining the common value of the target firm, so that one of the bidders

has more valuable (or higher quality) information about the target's value than the other bidder has. Bulow et al.

(1999) discuss a possible extension of their model to an asymmetric value function and argue that in such a case, a

bidder with a larger toehold is still more likely to win the auction but equal toeholds do not necessarily lead to

higher expected price. The main contribution of this article is in explicitly modeling the takeover auction with an

asymmetric wallet function and analyzing the combined effect of both the toehold and information sensitivity.

I analyze whether there is a trade‐off between toehold and information, how a change in relative information

quality affects the bidding behavior, and how the target firm can use its ability to disclose information to maximize

its expected revenue. I also discuss how the conditions that make the target willing to offer deeply discounted

shares to one of the bidders depend on relative information quality.

Consistent with Bulow et al. (1999), I show that when bidders have identical toeholds, information quality does

not affect the bidding strategies and expected price. Indeed, when both bidders have the same signal, information

quality does not affect the object's value. As a result, if in an auction without toeholds a bidder knows that her

opponent bids up to the value of her signal, she will use the same strategy. An equal increase in both bidders’

toeholds has an identical effect on their bidding functions, keeping their bidding functions independent of

information quality.

1230 | JOURNAL OF FINANCIAL RESEARCH

When bidders’ toeholds are different, the bidder with the larger toehold has more incentives to overbid and one can

no longer assume that the auction ends with a tie when both bidders have identical signals. As a result, the target's

expected value at the time one of the bidders drops out of the auction depends on whether the exiting bidder has higher

or lower quality information. I show that in unequal toeholds, relative information quality has a significant effect on

equilibrium bidding strategies and the target's revenue. In particular, I show that strategically releasing information to the

bidder with the smaller toehold, and consequently increasing her relative information quality, increases the expected

selling price. Indeed, in equilibrium, the bidder with the larger toehold overbids whereas the bidder with the smaller

toehold usually shades her bids. An increase in the relative information quality of the smaller toehold bidder makes the

value more sensitive to her signal and less sensitive to the signal of her opponent. This reduces her exposure to

the winner's curse and leads to an increase in her bidding function. Such behavior, in turn, increases the incentive of the

bidder with a larger toehold to bid up the price above the expected value of the target and, as a result, leads to an upper

shift in her bidding function. An increase in both bidders’ bidding functions leads to a higher expected final price.

I also show that when the target firm has full control over information revelation, it can always achieve a higher

expected selling price when toeholds are different than when toeholds are the same or there are no toeholds at all.

In addition, when the target has no control over the relative information quality but can offer free shares to one of

the bidders, I show that when the bidder with a larger toehold has lower quality information, offering her free

shares (and hence increasing the difference between toeholds even further) increases the expected price.1

Although my model considers two financial bidders involved in a common‐value takeover auction in which only

cash bids are allowed and the target's only objective is to maximize the expected final price, the reality of the

takeover process is much more complex. One example that can be closely related to the model described in this

article is an auction in which a financial bidder with a large toehold competes against the leveraged buyout (LBO)

proposed by the firm's management, such as the 2011 Dell Inc. LBO. Without intent to incorporate the target firm

into the other businesses of either bidder, the contest can be considered to be a common‐value auction and one

can argue that the firm's management has greater expertise in the firm's valuation. However, even with strategic

bidders, as in the 2018 cash‐only regulated auction between Comcast and 21st Century Fox for British broadcaster

Sky, a common component of the target's valuation can exist and bidders may have expertise in different areas of

the target's operations based on their own business or, in case of 21st Century Fox, through past involvement with

the target firm, of which it owned 39% of shares.

2 | MODEL

Consider a common‐value takeover auction in which two potential acquirers (“bidders”) bid for a single target firm

(“object”). The value of the object in case of a successful takeover is given by V S S αf S α f S( , ) = ( ) + (1 − ) ( )1 2 1 2 , where

∈S [0, 1]i is the private information (“signal”) observed by bidder i, f S( )i is an increasing continuously differentiable

function such that f (0) = 0 and f (1) = 1, and α is a parameter that measures the relative importance (or quality) of

the first bidder's information and satisfies α0 < < 1. To simplify notations, denote μ E f S E V S S≡ ( ( )) = ( ( , ))i 1 1 to be

the expected value of the target firm in case of a successful takeover. Because any distribution of Si can be

converted into a uniform distribution by the appropriate change in f S( )i , assume, without loss of generality, that Si is

uniformly distributed on the [0, 1] interval. The stand‐alone value of the target firm (i.e., its value in case no takeover

takes place) is normalized to zero.2 In addition, assume that bidder i owns a fraction θi (“toehold”) of the target firm

with θ θ≥ > 01 2 . The auction design is a clock‐style English auction, and once one of the bidders drops out of the

1With a symmetric value function, that is, when both bidders have equal information precision, Bulow et al. (1999) show that the seller should offer free

shares to the bidder with a smaller toehold and decrease the difference between toeholds. 2For example, one can assume that the successful acquirer will be able to implement value‐improving changes into the target firm's operations that the

current management is either unable or unwilling to do.

TOEHOLDS AND INFORMATION QUALITY | 1231

auction, the target firm's shareholders (“seller”) accept the offer by the remaining bidder at the price at which the

losing bidder has dropped out.3 I am looking for an equilibrium in which the strategy of bidder i, that is, the price at

which she drops out of the auction, is given by a strictly increasing differentiable bidding function B S( )i i .

This auction setup is very closely related to Bulow et al.'s (1999) model except that I do not require the value

function to be symmetric. In contrast, my goal is to study how the asymmetry of the value function, and in particular

the relative information quality α, affects the equilibrium behavior and profit allocation, and to investigate any

possible measures (such as strategically disclosing information or offering discounted shares to one of the bidders)

the target firm can implement to increase the expected final price.

To find the equilibrium bidding function B S( )i i , denote S B B SΦ ( ) = ( ( ))j i j i i −1 to be a function such that bidder i with

signal Si wins the auction if and only if S S< Φ ( )j j i . Therefore, when bidder i with signal Si chooses at which price level

to drop out, she effectively chooses the signal level of the other bidder Sj such that she wins against a bidder with

signal below Sj. Therefore, the expected profit of the first bidder with signal S1 can be written as:

{ }∫π S αf S α f x θ B x dx θ S B S( ) = max ( ( ) + (1 − ) ( ) − (1 − ) ( )) + (1 − ) ( ) . S

S

1 1 0

1 1 2 1 2 2 2 2

2 (1)

Similarly, the expected profit of the second bidder with signal S2 can be written as:

{ }∫π S αf x α f S θ B x dx θ S B S( ) = max ( ( ) + (1 − ) ( ) − (1 − ) ( )) + (1 − ) ( ) . S

S

2 2 0

2 2 1 2 1 1 1 1

1 (2)

The solution to optimization problems (1) and (2) yields

dB S

dS θ S B S αf S α f S

( ) =

1

(1 − ) ( ( ) − ( ) − (1 − ) ( ))

2 2

2 1 2 2 2 1 2 (3)

dB S

dS θ S B S αf S α f S

( ) =

1

(1 − ) ( ( ) − ( ) − (1 − ) ( )).

1 1

1 2 1 1 1 1 2 (4)

Because in equilibrium one must have B B(1) = (1) = 11 2 , Equations (3) and (4) have a unique solution and the

equilibrium bidding strategies are given by the following theorem:

Theorem 1. In a unique equilibrium with strictly increasing differentiable bidding functions, the equilibrium

bidding strategies are given by

∫B S θ S

αf x α f x x dx( ) = 1

(1 − ) ( ( ) + (1 − ) (Φ ( )))(1 − )

θ S

θ 1 1

2 1 1/

1

2 1/ −1

2 1

2 (5)

∫B S θ S

αf x α f x x dx( ) = 1

(1 − ) ( (Φ ( )) + (1 − ) ( ))(1 − ) ,

θ S

θ 2 2

1 2 1/

1

1 1/ −1

1 2

1 (6)

where

S B B S SΦ ( ) ≡ ( ( )) = 1 − (1 − ) .i j i j j j θ θ−1 /j i (7)

Proof. See the Appendix. □

3In case both bidders drop out simultaneously, assume the item is allocated at random, although this assumption is not important because with continuous

bidding functions, the probability that both bidders drop out simultaneously is zero.

1232 | JOURNAL OF FINANCIAL RESEARCH

3 | ANALYSIS

In a case of a symmetric value function (α = 0.5), Bulow et al. (1999) show that the bidder with the larger toehold is

more likely to win and bids up to a higher price than her competitor with the same signal but smaller toehold, and

that a bidder's bidding function increases with the size of her toehold for any value of her signal S < 1i . As one can

see from Equation (5)–(7), the introduction of asymmetry does not affect these results, so the following proposition

can be stated:

Proposition 1. The bidder with the larger toehold bids more than the bidder with the same signal but a smaller

toehold and is expected to win the auction more often, that is, if θ θ>1 2, then B x B x( ) > ( )1 2 for any ∈x (0, 1),

and st bidder winsPr(1 ) > 1/2. Also, an increase in the bidder's toehold increases her bidding function, that is,

B S θ∂ ( )/∂ > 0i i i for any ∈S [0, 1)i .

Proof. See the Appendix. □

Intuitively, the bidder with the larger toehold has more incentives to bid up the price because she has more to

gain from the price increase if she loses and less to pay for the remaining shares if she wins. As a result, she not only

bids more given the same signal, but she is also more likely to win the auction. An increase in a bidder's toehold

always makes that bidder bid more aggressively because she benefits both from selling a larger toehold at a higher

price when she loses and from having to buy fewer shares if she wins. The effect of bidder j's toehold θj on the

other bidder's bidding function B S( )i i is ambiguous and may depend on the relative information quality α, the value

function ⋅f ( ), and the bidder's own signal Si. In particular, although the desire to sell a nonzero toehold incentivizes

the bidder to bid up the price, the fact that the other bidder may bid above her signal results in a winner's curse and

the desire to bid lower. The larger the toehold is, the more important the first effect is. As a result, the bidder with a

smaller toehold faces a stronger winner's curse and may end up bidding below her signal, especially when her signal

is high. Such underbidding increases the other bidder's conditional expected value of the target for any current bid

level, making her bid even more, which leads to a spiral effect.

Although many people's first impulse may be to believe that the bidder with higher quality information has an

advantage and is more likely to win the auction whereas her opponent is more likely to bid more cautiously,

Equation (5)–(7) show that this is not the case. The effect of relative information quality on one's bidding behavior

(not just the magnitude but also the direction of the change in one's bidding function) depends on which bidder has

a larger toehold. Furthermore, as Equation (7) shows, the winner of the auction is determined only by the bidders’

signals S1 and S2 and the ratio of their toeholds θ θ/2 1, and does not depend on the relative information quality. In

addition, when bidders have equal toeholds θ θ θ≡ =1 2, the bidders’ bidding functions become independent of α

and are given by ∫B S θ S f x x dx( ) = (1/ (1 − ) ) ( )(1 − )i i i θ

S θ1/ 1 1/ −1

i . The latter also implies that the final price is not

affected by the relative information quality when θ θ=1 2. The following proposition summarizes the preceding

discussion:

Proposition 2. The relative information quality α does not affect the identity of the winning bidder given both

bidders’ signals. Furthermore, if both bidders have equal toeholds, θ θ=1 2, the relative information quality does

not affect equilibrium bidding strategies B S( )i i and the expected final price.

To understand the intuition behind Proposition 2, consider an auction with no toeholds. In such an auction, the

symmetric equilibrium strategy of any bidder is to bid up to the value of her own signal f S( )i . The relative quality of

the signal does not affect the bidding decision because it does not affect the value V at the critical point where

S S=1 2. Because an equal increase in toeholds has the same effect on both bidders’ strategies, in equilibrium, bidder

TOEHOLDS AND INFORMATION QUALITY | 1233

i wins if and only if S S>i j; therefore, S SΦ ( ) =i j j for θ θ=1 2. The latter implies that α has no effect on V at the critical

point S S=1 2, making the equilibrium bidding strategy and the expected final price independent of α.

When bidders have different toeholds, the effect of information quality on bidding functions depends on

whose toehold is the largest. Substituting (7) into (5) and (6), while noting that x x xΦ ( ) < < Φ ( )1 2 for θ θ>1 2, leads to

the following result:

Proposition 3. An increase in a bidder's relative information quality increases her bidding function when she

has a smaller toehold than the other bidder and decreases her bidding function when her toehold is larger, that is,

if θ θ>1 2, then B S α∂ ( )/∂ < 0i i for ∈i {1, 2} and S ≠ 1i .

Proposition 3 states that better information quality does not necessarily lead to more aggressive bidding,

whereas a bidder with less precise information does not always shade her bids. Intuitively, two things affect bidding

functions in opposite directions: the desire to bid up the price to be able to sell the existing toehold and the winner's

curse that the winner may experience when her opponent bids above her signal. The larger the toehold, the more

important the first effect is and, consequently, the more the bidder is willing to bid above her expected value. Such

overbidding makes the winner's curse more severe for the bidder with the smaller toehold, often leading her to

shade her bids. A decrease in the relative information quality of the bidder with a smaller toehold (who suffers from

winner's curse because of the aggressive bidding of her opponent) makes the winner's curse problem more severe,

leading to additional shading of her bids. Such lower bidding reduces the possible gain that the bidder with the

larger toehold can receive from bidding up the price, making her reduce her bids too.

Figure 1 shows bidding functions B S( )i i for the linear value function V S S αS α S( , ) = + (1 − )1 2 1 2 when θ = 10%1 ,

θ = 2%2 and three relative information quality values: α = 0.2, α = 0.5, and α = 0.8. As can be seen from this figure,

an increase in the relative information quality of the bidder with a larger toehold (and thus a decrease in the relative

information quality of the bidder with a smaller toehold) makes both bidders decrease their bids, and this effect is

economically significant for the chosen toehold values. Note that when bidders have equal toeholds θ θ θ≡ =1 2 and

the value function is linear, both bidding functions are given by B S θ θ S θ( ) = /(1 + ) + /(1 + )i i i , that is, by a straight

line connecting B θ θ(0) = /(1 + )i and B (1) = 1i , and they are independent of α.

If shareholders of the target firm were able to affect the relative information quality by strategically releasing

information to one of the bidders, they would be able to increase their expected revenue by revealing the

information to the bidder with a smaller toehold, as the following proposition, derived directly from Proposition 3,

shows:

Proposition 4. If θ θ>1 2, the final price negatively depends on the first bidder's relative information quality

α for any realized signals S1 and S2, and consequently, the expected final price decreases with α.

Bulow et al. (1999) show that when the value function is symmetric and the size of the toeholds is sufficiently

small, the expected price becomes smaller as the difference between toeholds becomes larger. As a possible way to

increase the expected price, Bulow et al. (1999) suggest the target firm offer its shares to the bidder with the

smaller toehold at a discount or even for free. As Proposition 4 shows, a cheaper way to increase the expected price

would be to provide additional information to that bidder.

Using Theorem 1 and Equations (1) and (2), the expected bidders’ profits π S( )i i and the expected price P can be

written as (see the Appendix for the proof):

  

  ∫π S θ B α x f x dx( ) = (0) + 1 − (1 − ) ′ ( )

S

1 1 1 1 0

θ

θ 1 1

2 (8)

  

  ∫π S θ B α x f x dx( ) = (0) + (1 − ) 1 − (1 − ) ′ ( )

S

2 2 2 2 0

θ

θ 2 2

1 (9)

1234 | JOURNAL OF FINANCIAL RESEARCH

∫ ∫ ∫ ∫

P μ E π S π S

θ θ

μ θ B θ B x f x dxdy α x x f x dxdy

θ θ

= − ( ( ) + ( ))

1 − −

= − (0) − (0) − (1 − (1 − ) ) ′ ( ) − ((1 − ) − (1 − ) ) ′ ( )

1 − −

y θ θ y θ θ θ θ

1 1 2 2

1 2

1 1 2 2 0

1

0 /

0

1

0 / /

1 2

2 1 2 1 2 1

(10)

where

  

  

  

  ∫B B

θ αf x α f x x dx(0) = (0) =

1 1 − (1 − ) + (1 − ) ( ) (1 − ) .θ1 2

1 0

1 1 −1

θ

θ 2

1 1 (11)

When θ θ>1 2, Propositions 3 and 4 show that both the bidding functions and the expected price negatively

depend on the first bidder's relative information quality α. As can be seen from Equation (9), the expected profit of

the second bidder also negatively depends on her rival's information quality. Indeed, an increase in α reduces both

the price the second bidder receives for her toehold when she loses and the gain she receives when she wins. The

effect of α on the profit of the first bidder is ambiguous. On one hand, because an increase in α leads to lower

bidding functions, she receives less for her toehold when she loses. On the other hand, her profit from winning

positively depends on her information quality. When her signal S1 is small, the probability of her losing the auction is

high, and the first effect dominates the second. When S1 is high, the second effect dominates the first, and her

expected profit increases with α. Ex ante (before the first bidder observes her signal), however, she is more likely to

win the auction, and her ex ante expected profit positively depends on her relative information quality. In other

F IGURE 1 Effect of information quality α on bidding functions when θ = 10%1 , θ = 2%2 , and the value function is linear V S S αS α S( , ) = + (1 − )1 2 1 2.

TOEHOLDS AND INFORMATION QUALITY | 1235

words, although the first bidder, on average, benefits when her information quality improves, she does not do so all

the time, and higher quality information hurts her when her signal is low and she is bidding up the price in hopes of

losing and selling her toehold. The following proposition summarizes the preceding discussion:

Proposition 5. If θ θ>1 2, the expected profit of the second bidder positively depends on her own information

quality α(1 − ) for any signal S2, that is, dπ S dα( )/ < 02 2 . An increase in the first bidder's information quality α

positively affects her own profit when her signal S1 is high and negatively when it is low; that is, there is

∈S̄ (0, 1)1 such that π S dα( )/ < 01 1 for any S S< ¯ 1 1 and π S dα( )/ > 01 1 for any S S> ¯

1 1. Furthermore, an increase in

the first bidder's information quality α increases her ex ante expected profit, that is, dE π S dα( ( ))/ > 01 1 .

Proof. See the Appendix. □

Propositions 2 and 4 state that the ability of the seller to affect relative information quality α by strategically

releasing information to one of the bidders has no effect on the final price when both bidders have equal toeholds

but increases the price when the information is released to the bidder with the smaller toehold. Furthermore, the

benefit of this information release is higher when the toeholds are substantially different from each rather than

when the toeholds are similar. In particular, the following result is true:

Proposition 6. Let λ θ θ= / < 12 1 . In this case, for sufficiently small toeholds, a positive effect that an increase

in the second bidder's information quality α(1 − ) has on the expected price becomes smaller when λ increases

either through an increase in the first bidder's toehold or through a decrease in the second bidder's toehold; that

is, there is θ̄ > 0 such that | < 0 P

α λ θ ∂

∂ (1 − )∂ =constant

2

2 and | < 0 P

α λ θ ∂

∂ (1 − )∂ =constant

2

1 for any θ θ< ¯ 1 .

Proof. See the Appendix. □

How effective can the strategic information release be and can it overcome any possible negative effect on the

selling price that unequal toeholds can bring? For θ θ>1 2, Proposition 4 and Equation (10) imply that when the

target firm has full control over the bidders’ relative information quality, it must make the second bidder's

information as valuable as possible, that is, make α as small as possible. Furthermore, this strategy is more effective

when the toeholds are substantially different from each other, that is, when the toehold ratio λ θ θ= /2 1 is small.

When both toeholds are small, the incentive to bid up the price to sell the toeholds is weak and an increase in the

expected selling price when both bidders have low signals is outweighed by the decrease in price when the bidders

have high signals. Hence, when both toeholds are small and the target firm is able to control the information, it

prefers unequal toeholds to equal or no toeholds. The following proposition summarizes this result:

Proposition 7. If the target firm has full control over the relative information quality α, then for any λ < 1 there

is θ such that for any θ θ<1 the expected price is higher when θ λθ θ= <2 1 1 than when θ θ=2 1.

Proof. See the Appendix. □

Figure 2 shows the effect of the first bidder's relative information quality α on the expected final price when

the bidders’ value function is linear V S S αS α S( , ) = + (1 − )1 2 1 2 for equal large toeholds θ θ= = 10%1 2 , equal small

toeholds θ θ= = 2%1 2 , unequal toeholds with a high toehold ratio θ = 10%1 , θ = 2%2 , and unequal toeholds with a

small toehold ratio θ = 5%1 , θ = 2%2 . It illustrates the results of Propositions 2 and 4 that α does not affect the

expected price for equal toeholds but has a negative effect when the first bidder has a larger toehold. It also

illustrates the results of Propositions 6 and 7 that the effect of α is larger when the toeholds are less equal, and

1236 | JOURNAL OF FINANCIAL RESEARCH

when the seller has full control over α, she prefers unequal toeholds. Furthermore, for a given set of parameters,

Figure 2 shows that information quality has a greater effect on the expected price than the toehold ratio does.

Can the target firm's shareholders increase the expected price if they have no control over relative information

quality α? Bulow et al. (1999) show that when the value function is symmetric, that is, α = 0.5, and both toeholds

are small, they can do so by offering discounted or free shares to the bidder with the smaller toehold, hence

reducing the difference in toeholds. When one of the bidders has an information advantage, this strategy is no

longer optimal. According to Propositions 6 and 7, shareholders of the target firm prefer a situation in which the

bidder with the larger toehold has an information disadvantage, and this disadvantage has the highest positive

effect on the target's price when the toeholds are significantly different from each other. Hence, one can expect

that given a sufficient difference in information quality, shareholders of the target firm may benefit from offering

free shares to the bidder with the larger toehold but information disadvantage and, by doing so, increase the

difference between toeholds even further. The following proposition provides the exact condition for this strategy:

Proposition 8. Let θ λθ θ= <2 1 1. Then there is θ such that for any θ θ<1 there is α λ( ) and ε λ θ( , ¯) that satisfy

α λ0 < ( ) < 1 and ε λ θlim ( ( , ¯)) = 0 θ̄→0

such that dπ dλ/ < 0s for any α α λ ε λ θ< ( ) − ( , ¯) and > 0 dπ

dλ s for any

α α λ ε λ θ> ( ) + ( , ¯), where π P θ θ= × (1 − − )s 1 2 is the expected profit of the nonbidding (i.e., excluding both

bidders) target firm shareholders.

Proof. See the Appendix. □

F IGURE 2 Effect of information quality α on the expected price for different values of toeholds and linear value function V S S αS α S( , ) = + (1 − )1 2 1 2.

TOEHOLDS AND INFORMATION QUALITY | 1237

In particular, Proposition 8 states that when the first bidder has a larger toehold, that is, θ θ>1 2, although both

toeholds are sufficiently small, there is a critical value of information quality α λ¯ ( ) that may depend on the current

toehold ratio λ θ θ= /2 1 such that shareholders of the target firm can increase their profit by offering free shares to

the first bidder (hence increasing the difference between toeholds even further) if the first bidder's information

quality is less than its critical value α λ¯ ( ), but they should offer free shares to the second bidder if the first bidder's

information precision is above α λ¯ ( ).

For a linear value function V S S αS α S( , ) = + (1 − )1 2 1 2, one can use Equation (10) for λ θ θ= /2 1 and infinitesimal

θi to find the explicit formula for α λ λ λ̄ ( ) = (2 + 1)/(3 + 3). For this linear example, Figure 3 illustrates the seller's

decision to provide free shares to either the first (with a larger toehold) or second (with a smaller toehold) bidder as

a function of the toehold ratio λ θ θ= /2 1.

As can be seen from Figure 3, even a small information disadvantage for the bidder with the larger toehold

makes the nonbidding target firm's shareholders willing to offer free or discounted shares to that bidder and, as a

result, to make the toeholds even less equal. Furthermore, when α ≤ 1/3, the target firm's shareholders want to

make the toeholds as distinct as possible so that the ratio λ θ θ= /2 1 becomes zero. Note, however, that Figure 3 and

Proposition 8 assume that both toeholds are sufficiently small, so the target cannot decrease λ to an arbitrarily small

number by offering discounted shares to the first bidder because it makes θ1 too large to ignore the additional

effects in (10). An alternative approach to decreasing λ θ θ= /2 1 while keeping toeholds sufficiently small is buying

back shares from the second bidder at a premium and thus reducing θ2. This approach, however, may be hard to

implement because the second bidder can always replenish her toehold by buying shares on the open market.

F IGURE 3 Target's decision to offer free or discounted shares to one of the bidders as a function of the toehold ratio ∈λ θ θ= / [0, 1]2 1 and the first bidder's relative information precision α in the case of a linear value function V S S αS α S( , ) = + (1 − )1 2 1 2 and small toeholds.

1238 | JOURNAL OF FINANCIAL RESEARCH

The model provides several testable empirical implications related to the relative size of toeholds and

information asymmetry. Although toeholds are easily observable, the asymmetry in the information amount and

quality is not. Furthermore, although the target firm can use voluntary disclosure, which would reduce the

asymmetry in information amount and quality, it may not be easy to observe a targeted information disclosure

that either reveals information to one of the bidders or includes only the information already known to only

one bidder.

There are at least two proxies that can be used to measure the information asymmetry among bidders. First, a

vast body of research has studied the general accounting and disclosure practice of the target firm and developed

several measures of transparency. A low degree of transparency is likely to result in a situation where one of the

acquirers can learn more important information about the target than the other. In terms of the model's parameters,

low transparency implies that α is different from 0.5, but it is not possible to say if it is close to 0 or 1 based on the

degree of transparency alone. Assuming deviation from 0.5 in both directions is possible, Proposition 4 (as

illustrated by Figure 2) implies:

Empirical Implication 1: The volatility of the takeover bid premium is lower for a higher degree of the target's

firm transparency.

Furthermore, it is likely that the toehold and information quality are positively related because either a potential

acquirer with better information is more likely to acquire a toehold or a firm with a larger toehold is more willing to

spend resources on acquiring more information. As a result, for low‐transparent targets a high value of α is likely to

be observed, which implies:

Empirical Implication 2: Transparency of the target firm positively affects the average takeover premium.

In addition, because a target firm may decrease the information advantage of the better informed bidder by

voluntary information disclosure during the takeover attempt, one can potentially test:

Empirical Implication 3: The voluntary information disclosure during the takeover attempt should more often

be done by low‐transparent target firms and it should lead to a higher takeover premium.

Second, one of the acquirers can have better access to information about the target firm by either being

involved in business transactions with the target in the past or having its representative on the board of

directors. This provides a better proxy for information asymmetry than the target's transparency because it

allows mapping the information advantage with toeholds. Hence, one can test the hypothesis derived from

Proposition 4:

Empirical Implication 4: For a given toehold ratio, the takeover premium is lower when a bidder with a larger

toehold also has an information advantage.

When an identified insider, that is, a bidder with an information advantage, has a larger toehold, the target firm

wants to decrease that bidder's relative information quality by a voluntary information discloser that reduces the

information asymmetry between bidders. Thus, Propositions 4 and 5 lead to the following:

Empirical Implication 5: When a bidder with an information advantage has a larger toehold, the target firm is

more likely to make voluntary information disclosure during the takeover attempt. Such disclosure leads to an

increase in the expected takeover premium (hence, the current target firm's price), a decrease in the expected

profit (thus, a share price) of the bidder with an information advantage, and an increase in expected profit

TOEHOLDS AND INFORMATION QUALITY | 1239

(thus, a share price) of the other bidder. When a disclosure is made when the bidder with an information

advantage has a smaller toehold, the effect on the share prices of all parties is the opposite.

The first three implications are not unique to my model. For example, a low degree of the target's firm

transparency that can be resolved by potential acquirers through thorough examination may result in higher

volatility of the takeover bid premium (Empirical Implication 1). Such a negative effect of target transparency on the

takeover premium is documented by Feijóo et al. (2015). Similarly, more transparent accounting data allow potential

acquirers to better estimate the target's value, leading to higher premia (Raman et al., 2013), which is consistent

with Empirical Implication 2. By the same logic, low‐transparent firms could increase the takeover premium by

voluntarily disclosing relevant information (Empirical Implication 3). Empirical Implications 4 and 5, however, are

more specific to my model and are yet to be tested.

4 | CONCLUSION

This article extends Bulow et al.'s (1999) model for common‐value takeover auctions with toeholds by explicitly

incorporating the asymmetry in the sensitivity of firm value to the bidder's information. It shows that information

quality does not affect the equilibrium bidding functions and expected target price when both bidders have identical

toeholds; however, it has a significant effect on both bidders when their toeholds are not the same. Also, an increase in

the relative information quality of the bidder with a smaller toehold makes both bidders bid more aggressively and

leads to a higher expected target price. As a result, the target firm may increase its expected revenue by strategically

releasing information to the bidder with a smaller toehold. I further show that the greater the difference between

toeholds is, the more beneficial such strategic information release will be. Furthermore, if such information release can

lead to a large effect on the difference in information quality, the target firm's shareholders always prefer bidders to

have different toeholds rather than having identical toeholds or no toeholds at all. Finally, I analyze how the change in

toehold ratio affects the expected target's revenue and show that when the bidder with the larger toehold has an

information disadvantage, the target firm's shareholders may benefit by offering that bidder discounted shares and

increasing her toehold, thus making the difference between the toeholds even greater.

ACKNOWLEDGMENTS

This research was supported by Social Sciences and Humanities Research Council of Canada (SSHRC) Grant

435‐2019‐0034.

ORCID

Anna Dodonova http://orcid.org/0000-0002-9855-9384

REFERENCES

Asquis, D., & Kieschnick, R. (1999). An examination of initial shareholdings in tender offer bids. Review of Quantitative

Finance and Accounting, 12, 171–189. Betton, S., & Eckbo, B. (2000). Toeholds, bid jumps, and expected payoffs in takeovers. Review of Financial Studies, 13,

841–882. Bulow, J., Huang, M., & Klemperer, P. (1999). Toeholds and takeovers. Journal of Political Economy, 107, 427–454. Burkart, M. (1995). Initial shareholdings and overbidding in takeover contests. Journal of Finance, 50, 1491–1515. Choi, D. (1991). Toehold acquisitions, shareholder wealth, and the market for corporate control. Journal of Financial and

Quantitative Analysis, 26, 391–407. Dionne, G., La Haye, K., & Bergeres, A. (2015). Does asymmetric information affect the premium in mergers and

acquisitions? Canadian Journal of Economics, 48, 819–852. Dionne, G., St‐Amour, P., & Vencatachellum, D. (2009). Asymmetric information and adverse selection in Mauritian slave

auctions. Review of Economic Studies, 76, 1269–1295.

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Dodonova, A. (2012). Toeholds and signaling in takeover auctions. Economics Letters, 117, 386–388. Feijóo, L., Kaprielyan, M., Madura, J., & Viale, A. (2015). Target valuation complexity and takeover premiums. International

Journal of Banking, Accounting and Finance, 6, 151–176. Hounwanou, D. (2018). Participation costs and inefficiency in takeover contests. Decision Analysis, 15(1), 1–10. Le, H. T., & Schultz, E. (2007). Toeholds and the bidder shareholder wealth effects of takeover announcements. Australian

Journal of Management, 32, 315–344. Povel, P., & Sertsios, G. (2014). Getting to know each other: the role of toeholds in acquisitions. Journal of Corporate

Finance, 26, 201–224. Povel, P., & Singh, R. (2006). Takeover contests with asymmetric bidders. Review of Financial Studies, 19, 1399–1431. Raman, K., Shivakumar, L., & Tamayo, A. (2013). Target's earnings quality and bidders’ takeover decisions. Review of

Accounting Studies, 18, 1050–1087. Singh, R. (1998). Takeover bidding with toeholds: the case of the owner's curse. Review of Financial Studies, 11, 679–704.

How to cite this article: Dodonova, A. (2024). Toeholds and information quality in common‐value takeover

auctions. Journal of Financial Research, 47, 1229–1244. https://doi.org/10.1111/jfir.12393

APPENDIX

Proof of Theorem 1. To solve (3) and (4), note that dividing (4) by (3) while taking S S= Φ ( )2 2 1 , and dividing

(4) by (3) while taking S S= Φ ( )1 1 2 yields

B S

B S

θ S

θ S

′ ( )

′ (Φ ( )) =

(1 − Φ ( ))

(1 − ) ,

i i

j j i

i j i

j i

(A1)

where B x′ ( )i is the first derivative of B x( )i with respect to its argument. At the same time, taking a full

differential of B S B S( ) = (Φ ( ))i i j j i yields

B S

B S

d S

dS

′ ( )

′ (Φ ( )) =

Φ ( ) .

i i

j j i

j i

i

(A2)

Substituting (A2) into (A1) results in a differential equation

d S

dS

θ S

θ S

Φ ( ) =

(1 − Φ ( ))

(1 − ) .

j i

i

i j i

j i

(A3)

Because in equilibrium a bidder with signal S = 1i never wants to lose the auction, this leads to boundary

condition Φ (1) = 1j and a unique solution to (A3):

S SΦ ( ) = 1 − (1 − ) .j i i θ θ/i j (A4)

Substituting (A4) into (3) and (4) and solving the corresponding ordinary differential equations with a

boundary condition B (1) = 1i results in (5) and (6). □

Proof of Proposition 1. Note that if θ θ>1 2 then x x xΦ ( ) < < Φ ( )1 2 . Therefore,

≫

≫

∫

∫ ∫

∫

B x θ x

αf y α f y y dy

θ x f y y dy

θ x f y y dy

θ x αf y α f y y dy B x

( ) = 1

(1 − ) ( ( ) + (1 − ) (Φ ( )))(1 − )

1

(1 − ) ( )(1 − ) >

1

(1 − ) ( )(1 − )

1

(1 − ) ( (Φ ( )) + (1 − ) ( ))(1 − ) = ( ).

θ x θ

θ x θ

θ x θ

θ x θ

1 2

1/

1

2 1/ −1

2 1/

1 1/ −1

1 1/

1 1/ −1

1 1/

1

1 1/ −1

2

2 2

2 2

1 1

1 1

(A5)

TOEHOLDS AND INFORMATION QUALITY | 1241

Substituting (7) into (5) and (6) immediately demonstrates that B S( )i i increases with θi for any Si. Finally,

x xΦ ( ) < Φ ( )1 2 implies

st bidder wins S x x x nd bidder wins S xPr(1 | = ) = Φ ( ) > Φ ( ) = Pr(2 | = )1 2 1 2 (A6)

And, therefore, st bidder wins nd bidder winsPr(1 ) > 1/2 > Pr(2 ). □

(8)–(11)>Proof of Equations (8)–(11). Applying the envelope theorem to Equation (1) results in

∫ dπ S

dS α df S

dS dx αf S S

( ) =

( ) = ′ ( )Φ ( ).

S1 1

1 0

Φ ( ) 1

1 1 2 1

2 1 (A7)

Therefore,

∫π S π α f x x dx( ) = (0) + ′ ( )Φ ( ) . S

1 1 1 0

2

1 (A8)

Because π θ B(0) = (0)1 1 1 , Equation (A8) is the same as Equation (8). Similarly, Equation (9) follows from

applying the envelope theorem to (2). Equation (10) follows from (8) and (9) by integrating (8) and (9) over all

S1 and S2, respectively, and noticing that the total profit of all bidders and the remaining θ θ(1 − − )1 2

shareholders is equal to the target's expected value μ. Finally, to show that B B(0) = (0)1 2 , rewrite the integral

in (6) by changing the variable of integration y x= 1 − (1 − )θ θ/2 1 . □

Proof of Proposition 5. From Proposition 2, B S α∂ ( )/∂ < 0i i for ∈i {1, 2} and S ≠ 1i . Using this result for B (0)i ,

Equation (9) immediately leads to dπ S dα( )/ < 02 2 . Because dπ S dα( )/ < 02 2 for any S2, it follows that

dE π S dα( ( ))/ < 02 2 . Furthermore, because E π S E π S P θ θ μ( ( )) + ( ( )) + × (1 − − ) =1 1 2 2 1 2 does not depend on α

and dP dα/ < 0, it follows that dE π S dα( ( ))/ > 01 1 . From (8) and (11), it follows that

  

  

  

  

     

  

  ∫ ∫

dπ S

dα f x f x dx x f x dx

( ) = 1 − (1 − ) − 1 − (1 − ) + 1 − (1 − ) ′ ( ) .

S1 1

0

1

0

θ

θ

θ

θ

θ

θ 2

1

1

2 1 1

2 (A9)

Note that dπ 0 /dα < 0( )1 and, from (A9), dπ S /dα( )1 1 is a continuous increasing function of S1. Furthermore,

because dE π 1 /dα > 0( ( ))1 , there is ∈S̄ (0, 1)1 such that dπ S /dα < 0( )1 1 for any S S< ¯ 1 1 and dπ S /dα > 0( )1 1 for

any S S> ¯ 1 1. □

Proof of Proposition 6. I prove this proposition for the case when θ const=1 . The proof for θ const=2 can

be done in a similar way. To simplify notations, I use o (1) to denote an infinitesimal function when θ̄

converges to zero, and I use the term const to represent any positive constant, not necessarily the same one

even in the same equation. From (10), it follows that

P

α λ λ A A A A

A

λ A

A

λ A

A

λ A

A

λ A

∂

∂(1 − )∂ =

∂

∂ ( + ) =

∂

∂ + ∂

∂ + ∂

∂ + ∂

∂ ,

2

1 2 3 4 1

2 2

1 3

4 4

3 (A10)

where

A λ θ

λ θ o=

(1 + )

1 − (1 + ) = (1)1

1

1

(A11)

∫A f x f x x dx= ( (1 − (1 − ) ) − ( ))(1 − )λ θ2 0

1 1 −1

1 (A12)

1242 | JOURNAL OF FINANCIAL RESEARCH

A λ θ

o= 1

1 − (1 + ) = 1 + (1)3

1

(A13)

( )∫ ∫A x x f x dxdy= (1 − ) − (1 − ) ′ ( ) . y

λ 4

0

1

0 λ 1 (A14)

Because f x( ) and f x′( ) are continuous on [0,1], they must be bounded. Therefore, A4 is finite and

⋅∫ ∫A const x dx const θ o| | < (1 − ) = = (1) y

θ2 0

1

0

1 −1

11 (A15)





 















( )( )

∫ ∫

∫

A

λ f x x x dx const x d

x

const λ

x dx const λ λ

const

∂

∂ = (1 − (1 − ) )ln(1 − )(1 − ) ≤ ≤ (1 − )

1

1 −

≤ 1 + 1

− 1 + (1 − ) ≤ ≤ 1 +

1

− 1 + − 2 +

≤ .

λ θ λ

θ λ

θ

θ λ

θ θ

2

0

1 1 −1+

0

1 1 −1+

1 0

1 1 −3+

1 1

1 1

1

1

1 1

(A16)

From (A1), (A13), and (A14) it follows that

A

λ

θ λ θ θ λ θ

λ θ o

∂

∂ =

(1 − (1 + ) ) + (1 + )

(1 − (1 + ) ) = (1)

1 1 1 1 1

1 2

(A17)

A

λ

θ

λ θ o

∂

∂ = (1 − (1 + ) )

= (1) 3 1

1 2

(A18)

( )∫ ∫ A

λ x x λ x f x dxdy

∂

∂ = (1 − ) − (1 − ) ln(1 − ) ′ ( ) < 0.

y λ4

0

1

0

−2 λ 1 (A19)

Using (A11)–(A19), Equation (A10) leads to

  

   

 

  P

α λ A

A

λ

A

λ lim

∂

∂(1 − )∂ = lim

∂

∂ = ∂

∂ < 0.

θ θ¯→0

2

¯→0 3

4 4 (A20)

□

Proof of Proposition 7. From Proposition 6 it follows that   

  

   

 ( ) ( )P Plim sup | = lim lim |

θ α θ λθ

θ α θ λθ

→0 =

→0 →1 =2 1 2 1 . Thus,

using (1), it follows that

≫   

  ∫ ∫ ∫ ∫P θ λθ x f x dxdy xf x dxdy P θ θlim (sup ( | = )) = 1 − (1 − ) ′ ( ) ′ ( ) = lim ( | = ).

θ α

y y

θ→0 2 1

0

1

0 0

1

0 →0 2 1

θ

θ 2

1

(A21)

□

Proof of Proposition 8. Consider a specific value of λ < 1 and denote

   

 ( )

Q α

d π

dλ ( ) ≡

lim |

. θ

S θ θ ¯→0

/ =constant2 1 (A22)

TOEHOLDS AND INFORMATION QUALITY | 1243

From Equation (10) it follows that

( ) ( )∫ ∫ ∫ ∫ Q α

d x f x dxdy

dλ α d x x f x dxdy

dλ ( ) =

(1 − ) ′ ( ) −

((1 − ) − (1 − ) ) ′ ( ) .

y λ y λ λ 0

1

0 0

1

0 1/

(A23)

Note that Q (0) < 0, Q (1) > 0, and dQ α /dα > 0( ) . Hence, there is a threshold ᾱ such that an increase in λ

increases πS for any α α ε θ> ¯ + ( ¯) and decreases πS for any α α ε θ< ¯ − ( ¯) where ε θ( ¯) is an infinitesimal function

of θ̄, that is, ε θlim ( ( ¯)) = 0θ̄→0 . □

1244 | JOURNAL OF FINANCIAL RESEARCH

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