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\begin{document}
\title{Toeholds and Takeovers }
\author{Jeremy Bulow \\
%EndAName
Graduate School of Business, Stanford University, USA\\
Ming Huang\\
Graduate School of Business, University of Chicago, USA\\
and\\
Paul Klemperer\\
Nuffield College, Oxford, UK}
\date{February, 1998}
\maketitle
\vskip0.1in %\newpage
\noindent {\bf Abstract} \vskip20pt \noindent (forthcoming {\it Journal of
Political Economy}). Part ownership of a takeover target can help a bidder
win a takeover auction, often at a low price. A bidder with a ``toehold''
bids aggressively in a standard ascending auction because its offers are
both bids for the remaining shares and asks for its own holdings. While the
direct effect of a toehold on a bidder's strategy may be small, the indirect
effect is large in a common value auction. When a firm bids more
aggressively, its competitors face an increased winner's curse and must bid
more conservatively. This allows the toeholder to bid more aggressively
still, and so on. One implication is that a controlling minority shareholder
may be immune to outside offers. The board of a target may increase the
expected sale price by allowing a second bidder to buy a toehold on
favorable terms, or by running a sealed bid auction.
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{\sl Keywords}: Toeholds, Takeovers, Auctions, Mergers, Corporate
Acquisitions, Footholds, Winner's Curse, Common Value Auctions.
\noindent
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{\sl JEL numbers}: G34 (Mergers and Acquisitions), D44 (Auctions), G30
(Corporate Finance)%
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Acknowledgment: We are very grateful to the referee and editor, as well as
to our colleagues and seminar audiences, for helpful comments and
suggestions.
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\section{Introduction}
Buying a stake or ``toehold'' in a takeover target is a common and
profitable strategy.\footnote{%
Well-known empirical studies that discuss toeholds include Bradley, Desai
and Kim (1988), Franks and Harris (1989), Jarrell and Poulsen (1989), Stulz,
Walkling, and Song (1990), Jennings and Mazzeo (1993), Betton and Eckbo
(1997), Franks, Mayer and Renneboog (1997) and Jenkinson and Ljungqvist
(1997). These studies indicate that a large percentage of bidders own
toeholds, often of 10-20 percent or more, at the time they make offers.
(Betton and Eckbo's highly comprehensive data set of 1353 takeover attempts
shows that about half of the initial bidders have toeholds.) We know of no
data on options granted to friendly bidders such as Kohlberg, Kravis, and
Roberts in its offer for Borden or U.S. Steel in its offer for Marathon Oil,
or similar devices which can effectively serve as ``toehold substitutes''.
There is also little information on the differences in the types of bidders
who acquire toeholds and those that do not.} The potential acquirer can gain
either as a buyer who needs to pay a premium for fewer shares, or as a
losing bidder who sells out at a profit. Therefore a company that owns a
toehold has an incentive to bid aggressively, as every price it quotes
represents not just a bid for the remaining shares but also an ask for its
own holdings.
But this is the beginning of the story, not the end. Auctions of companies%
{\it , }at least when the bidders are ``financial'' buyers such as leveraged
buyout firms rather than ``strategic'' buyers such as customers, suppliers,
or competitors, are substantially common-value affairs. That is, differences
in perceptions about the value of a company will often stem primarily from
differences in expectations about the company's underlying business rather
than differences in the expectations of different bidders of their ability
to raise the value of the business.\footnote{%
For non-controlling shareholders, stocks are almost entirely common value
assets. For competing LBO groups, which are likely to apply similar
managerial and financing techniques to acquired companies, the common value
element probably dominates. When Wall Street analysts quote a company's
``break-up value'' they are essentially making common value estimates of the
value of a company's businesses.} The implication of common values is
dramatic.
When a toehold makes a bidder more aggressive, it increases the winner's
curse for a competitor. In a common value ascending auction, this will cause
the competitor to bid more conservatively.\footnote{%
In a private value ascending auction a non-toeholder will be unaffected by
an opponent's bidding; if a competitor has a toehold then it will become
more aggressive if it thinks there is less chance of its opponent dropping
out at any given price.}~\footnote{%
This observation has also been made by Bikhchandani (1988), in showing the
value of a reputation for aggressive bidding in common value auctions.} The
conservative competitor reduces the toeholder's winner's curse, allowing the
toeholder to bid more aggressively still, and so on. The change in bidding
strategies caused by a toehold will be much larger in a common value
ascending auction than in a private value auction.
Furthermore, it is not so much the change in the toeholder's own strategy
that raises its profitability as it is the induced change in competitors'
bidding that makes the toehold such an important strategic weapon. A bidder
makes tradeoffs in deciding to become more aggressive, but unambiguously
benefits from a competitor becoming more conservative.
While the intuition above is right when only one bidder has a toehold,
things get more complex when two bidders have shares. Now each bidder wears
both buyer and seller hats when quoting a price. A bidder who expects to
lose the auction, and is primarily in selling mode, may quote a higher price
against an opponent who has a large toehold and is therefore expected to be
very aggressive. So in our model an exogenous increase in a bidder's toehold
always increases its probability of winning and its expected profits, but
sometimes increases the average price it pays when it wins. Our results are
consistent with empirical findings that toeholds increase a bidder's chance
of winning a takeover battle (Walkling (1985), Betton and Eckbo (1997)) but
it is unclear whether they decrease (Jarrell and Poulsen (1989), Eckbo and
Langhor (1989)), increase (Franks and Harris (1989)), or have no effect on
(Stulz, Walkling, and Song (1990)) target returns.{\bf \ }By contrast, the
private-value models of Englebrecht-Wiggans (1994), Burkart (1995), and
Singh\ (1994) imply that toeholds should unambiguously raise bids and
prices, but that the effects should be relatively small. Only Hirshleifer
(1995) concludes as we do that even a small toehold can have a large effect
on the final price in a multiple-bidder takeover battle.\footnote{%
Hirshleifer (1995, Section 4.5) shows that in the special case of full
information, a small toehold can have a big effect on an ascending
private-value auction. The firm with the lower value will drop out at a
price just below the other bidder's valuation if it has a small toehold (and
if any bidding costs are small enough), but if it has no toehold it will bid
no further than its own valuation (and will withdraw from the bidding if
there are any bidding costs).}
Our model can explain why bidders sometimes seem to overpay for the
companies they take over, without appealing to stories of managerial hubris
or of management pursuing its own interests at the expense of shareholders.
Here, bidding ``too high'' maximizes a bidders' ex-ante expected profits
even though it sometimes loses money ex-post.\footnote{%
Burkart (1995) and Singh (1996) have made this point in the context of a
private-value auction, but in their models a small toehold has only a small
effect. Chowdhry and Nanda (1993) argue that an indebted firm may commit
itself to aggressive bidding (and so sometimes deter competition) by
committing to finance the acquisition through additional debt of equal or
senior priority, and that this might sometimes lead to overpayment.}
The model also implies that an ownership stake of significantly less than
50\% in a company may be sufficient to guarantee effective control; a
toehold may make it much less likely that an outside bidder will enter a
takeover battle. This result is consistent with Walkling and Long (1984) and
Jennings and Mazzeo (1993), who find that toeholds lower the probability of
management resistence; of Stulz, Walkling and Song (1990), who report much
larger toeholds in uncontested than in contested takeovers; and of Betton
and Eckbo (1997), who find that greater toeholds increase the probability of
a successful single-bid contest by lowering both the chance of entry by a
rival bidder and target management resistance.\footnote{%
Except that both Betton and Eckbo (1997) and Jennings and Mazzeo (1993) find
that very small toeholds lead to {\it more} target management resistance
than zero toeholds. This result would be explained if, as we argue next,
financial bidders are more likely to acquire toeholds and, because they have
no private-value advantage, are also more likely to be challenged.}
Our analysis also makes predictions that have not yet been tested, because
empirical work in the field has not distinguished between private-value and
common-value auctions. Since a toehold should have a lesser effect on a
private-value auction than a common-value auction, we believe that the
incentive for acquiring a toehold is much lower for a ``strategic'' bidder
than for a ``financial'' bidder. A financial bidder should generally not
compete with a strategic bidder unless it has a toehold or other financial
inducement.
Since a basic message of the analysis is that if just one bidder has a
substantial toehold then that bidder can expect large profits, we consider
two natural ways in which the management of the target company might seek to
even the contest.
One approach is to replace a conventional ascending-bid takeover auction
with a first-price auction in which bidders are permitted to make only a
single sealed ``best and final offer'' and the company is sold at the
highest bid.\footnote{%
While it may be legally difficult for a board to refuse to consider higher
subsequent offers, if it can award the highest sealed bidder a ``breakup
fee'', options to buy stock, or options to purchase some of the company's
divisions on favorable terms, then de facto it may create a first price
auction. (A ``break-up'' fee is a fee that would be payable to the highest
sealed bidder in the event that it did not ultimately win the company.) Thus
our analysis can justify the use of ``lock-up'' provisions to support the
credibility of a first-price auction. For previous analyses\ of the merits
of allowing ``lock-ups'' see Kahan and Klausner (1996) and the references
cited there.} Because a bidder's offer now affects the sale price only if
the bidder wins, there is no incentive to bid up the price purely in order
to ``sell high''. Therefore, with symmetric toeholds, bidders will be less
aggressive in a first-price auction and prices will be lower on average in
the first-price auction than in an ascending auction. However, with
asymmetric toeholds the large toeholder being more aggressive in an
ascending bid auction also means that the small toeholder becomes more
conservative on average. Since it is the lower of the two bids that
determines price, and the small toeholder is more likely to have the low
bid, with small asymmetric toeholds prices will be lower on average in an
ascending auction than in a first-price auction.
A second approach is to try to ``level the playing field'' by giving a
second bidder the opportunity to acquire stock at a low price, narrowing the
differences in toeholds. Doing so will make the auction for the company more
competitive. While it would not pay to sell stock cheaply to two symmetric
bidders, we show that the cost would be surprisingly small, because larger
toeholds lead to more aggressive bidding. Therefore, the increased
competition created by selling stock cheaply to only the smaller of two
asymmetric bidders can easily swamp the ``giveaway'' aspect of such a deal.
With small toeholds, it will always pay to subsidize the smaller toeholder
in this way.
While our primary focus is on auctions of companies, there are several
related problems to which our analysis can apply. Perhaps the most
interesting at the moment is the sale of ``stranded assets'' by public
utilities. In these sales of assets that are worth far less than book value,
state public utilities commissions promise to reimburse utilities'
shareholders for some percentage of the difference between the asset's sale
price and the book value. If the percentage reimbursement is 80 percent,
then the utility effectively has a toehold of 20 percent in the auctioned
asset.\footnote{%
That is, the utility is 20 cents better off if the asset is sold to someone
else for a dollar more, and is only 80 cents worse off if it must bid an
extra dollar to win the auction. This makes the utility's position
strategically identical to a toeholder with a 20 percent stake in our model.}
Other applications include the sharing of profits in bidding rings,\footnote{%
See McAfee and McMillan (1992) and Engelbrecht-Wiggans (1994).} creditors'
bidding in bankruptcy auctions,\footnote{%
See Burkart (1995).} and the negotiation of a partnership's dissolution.
\footnote{%
See Cramton, Gibbons, and Klemperer (1987).} More generally, the theory
lends insight into problems in which a losing bidder cares how much the
winner pays, as when a competitor in several auctions faces an aggregate
budget constraint.\footnote{%
The theory here is also closely related to other examples in which one
player has a small advantage (e.g. a small private-value advantage or a
reputational advantage) in an otherwise pure common-value auction; see
Bikhchandani (1988), Bulow and Klemperer (1997), and Klemperer (1997).}
There are two strands to the theoretical literature on toeholds. One strand,
originated by Shleifer and Vishny (1986) and including Hirshleifer and
Titman (1990) and Chowdhry and Jegadeesh (1994), focuses on the use of
toeholds by a single bidder to combat the free rider problem described by
Grossman and Hart (1980). Owning a toehold gives a bidder a profit from a
successful takeover, even if it has to pay the expected full value for any
shares bought in a tender offer. While a larger toehold increases the chance
that a tender offer will be successful, on average all of a bidder's profits
will be accounted for by gains on the toehold. A larger toehold reduces the
price a bidder will have to pay in the Shleifer and Vishny and Hirshleifer
and Titman models, but increases it in the Chowdhry and Jegadeesh model.
The second strand focuses on bidding contests and assumes away the free
rider problem. There are several justifications for this approach. The
ability of a bidder that acquires a supermajority of the stock to force out
non-tendering shareholders can eliminate the free-rider problem. Also, if
small minority stakes can be left outstanding, the loss of liquidity in
those shares can have the same effect in reducing their value as would
measures that directly oppress minority investors, giving bidders an extra
incentive to tender.
Engelbrecht-Wiggans (1994) has a private value model in which all bidders
are symmetric and have identical toeholds. Burkart (1995) and Singh (1996)
have private value models in which one bidder has a toehold and the other
does not. In all these models a small toehold has only a small effect,%
\footnote{%
But in Hirshleifer's (1995) model without asymmetric information a small
toehold has a large effect. See note 5.} and a bidder with a toehold bids
more aggressively so toeholds always raise prices. Of course, none of these
models can show how a toehold can make a competitor more conservative, and
so significantly raise a bidder's expected profits while lowering prices.%
{\bf \ }In contrast to the free-rider models, in these models and ours
bidders make profits beyond the direct gains on their toeholds.\footnote{%
The free-rider models provide a theoretical foundation for the conventional
wisdom that acquirers do not make profits on average, judged by their
subsequent stock market performance. However, Loughran and Vijh (1996) show
that acquirers who pay cash do make profits while those that issue stock
underperform the market, just as other non-acquiring equity issuers do. So
market prices may overstate the consideration paid in stock takeovers, and
market returns may understate the real profitability of these transactions.
(Similarly, Rau and Vermaelen (1996) show that ``value'' companies appear to
make profits on tender offers, while ``glamour'' companies, those whose
shares sell at a high multiple of book value, decline in the extended period
following the issuance of new equity in a takeover.) These papers are
therefore consistent with the ``bidding contest'' models of toeholds,
including ours, in which bidders make profits. Of course, there are many
non-public investors, such as private entrepreneurs and leveraged buyout
firms, who make a business of acquiring and reorganizing companies, and
appear to be very profitable on average.}
These private value models are probably most appropriate for auctions among
``strategic'' bidders whose differential valuations are not explained by
varying perceptions about what the target is worth on its own.\ However, we
would predict that because toeholds are of much greater importance to
``financial'' bidders competing in common value auctions, toeholds are much
more likely to be acquired by common-value bidders.
To focus clearly on the strategic effects we concentrate on the polar case
of pure common values. Of course, in reality takeover targets have both
private-value and common-value components, so our pure common-values model
yields some results that are quantitatively implausible,\footnote{%
For example, we find that bidders' probabilities of winning are in
proportion to their toeholds even when the toeholds are arbitrarily small.}
even though we believe they are qualitatively correct.\footnote{%
It can be checked that the equilibrium we find is continuous as small
private-value components are added. See also Bulow, Huang, and Klemperer
(1995) for the general partially common-value, partially private-value, case.%
} Our model also does not allow for the possibility of firms
``jump-bidding'', that is, discontinuously raising the bidding level to
intimidate opponents into quitting the auction, as is often observed in
practice. Jump-bidding is less likely when there are toeholds, since it is
harder to discourage an opponent with a toehold from bidding, but would
still arise if there were substantial bidding costs (including costs of
entering the auction), especially with smaller toeholds and private-value
components. Although we do not expect jump-bidding to affect our basic
results and intuitions, it would probably attenuate their quantitative
significance by making behavior closer to that in a first-price auction, so
this is a further reason for not taking our results too literally when
toeholds are small.\footnote{%
See Section 5 for our analysis of a first-price auction with toeholds, and
see Avery (1996) and Daniel and Hirshleifer (1996), for pure common-value
and pure private-value models, respectively, of jump-bidding in the absence
of toeholds.}
Section 2 sets out our basic ``common-values'' model of two bidders who have
toeholds in a target company, and also have private independent information
about the value of that company. Were the bidders to completely share
information, they would have the same valuation for the target.
Section 3 solves for the unique equilibrium of an ascending auction between
the bidders.\footnote{%
Note that with toeholds we obtain a unique equilibrium in the ascending
English auction even with pure common values. It is well-known that when
bidders have no initial stakes in the object they are competing for, there
is multiplicity of (perfect Bayesian) equilibria, but we show that (even
arbitrarily small) toeholds resolve this multiplicity.} Section 4 derives
its properties and shows that asymmetric toeholds tend to lower sale prices.
Sections 5 and 6 discuss how the management of the target company might
``change the game'' to reduce the advantage of the bidder with the larger
toehold (or only toehold) and so raise the expected sale price. Section 5
solves and analyzes the equilibrium in a common value first price auction,
while Section 6 considers the effect of offering stock cheaply or options to
the bidder with the smaller toehold to make the auction more competitive.
Section~7 extends our analysis to the case in which bidders' private signals
are of different informativenesses, and shows that most of our results are
unaffected.
Section 8 concludes.
\section{The Model}
Two risk-neutral bidders $i$ and $j$ compete to acquire a company. Bidder $k$
$(k=i,j)$ owns a share $\theta _{k}$ of the company, $0<\theta _{k}<{{{{{{{{{%
{\frac{1}{2}}}}}}}}}}},$ and observes a private signal $t_{k}$. Bidders'
shares are common knowledge\footnote{%
This assumption is consistent with takeover regulations that require bidders
to disclose their stakes.} %(See Burkart (1995) p. 5 for discussion.)
and exogenous.\footnote{%
Among the many factors that could affect the size of a bidder's toehold are
the liquidity of the company's shares, institutional constraints such as the
Williams Act and SEC rule 16(b) which may affect some bidders' ability to
retain profits if a toehold of 10\% or more is sold, the effect of
accumulating shares on the likelihood of arranging a friendly deal (as in
Freeman (1991)), the probability that management will find out that a
toehold is being accumulated and the range of management response, the risk
that information leakage about a potential offer will cause a pre-bid runup
in the stock price (Schwert (1996) shows that a pre-bid runup forces a
bidder to pay more to buy a company), and the amount of shares held by the
bidder prior to any decision to make an offer for the company (many
toeholders own large stakes accumulated years before a buyout offer).
%If accumulating a toehold increases the likelihood that
%information will leak about a potential offer, then accumulating a toehold
%may have some risks. %See also Ravid and Spiegel (1992).
} Bidders' signals are independent, so without loss of generality we can
normalise so that both the $t_{k}$ are uniformly distributed on [0,1]. That
is, a signal of $t_{k}=.23$ is more optimistic than 23\% of the signals $k$
might receive and less optimistic than 77\%. Conditional on both signals,
the expected value of the company to either bidder is $v(t_{i},t_{j}).$ We
assume $v(\cdot ,\cdot )$ has strictly positive derivatives $\partial
v/\partial t_{k}$ everywhere.
%We choose units so that $v(0,0)=0$ and $% v(1,1)=1.$
%We do not assume that $v(\cdot, \cdot )$ is symmetric in $t_i$ and
%$t_j.$ So, for example, one bidder's information may be more important than
%the other's.
The company is sold using a conventional ascending bid (i.e. English)
auction. That is, the price starts at zero and rises continuously. When one
bidder drops out, the other bidder buys the fraction of the company that he
does not already own at the current price per unit.\footnote{%
Thus all shareholders (including the two bidders) are assumed to be willing
to sell out to the highest bidder so we are ignoring any free-rider problems
of the kind discussed by Grossman and Hart (1980). Also, all offers are
assumed to be binding (which is supported by the legal environments of the
EC and US). Offers are for all the outstanding shares. (Partial offers are
legal under dominant US law but only if they are non-discriminating and we
would obtain similar results in this case.) See Burkart (1995) and McAfee,
Vincent, Williams, and Havens (1993), p. 461, for more legal details.} (If
bidders quit simultaneously we assume the company is allocated randomly at
the current price, though this assumption is unimportant.) Thus a (pure)
strategy for bidder $k$ is a price $b_{k}(t_{k})$ at which he will quit if
the other bidder has not yet done so.
We assume that $v(\cdot ,\cdot )$ is symmetric in $t_{i}$ and $t_{j}.$ We
define $i$'s ``marginal revenue'' as $MR_{i}(t_{i},t_{j})\equiv
v(t_{i},t_{j})-(1-t_{i}){{{{{{{{{{\frac{\partial v}{\partial t_{i}}}}}}}}}}}}%
(t_{i},t_{j}),$\footnote{%
In analysing our auction using marginal revenues, we are following Bulow and
Roberts (1989) who first showed how to interpret private-value auctions in
terms of marginal revenues, and Bulow and Klemperer (1996) who extended
their interpretation to common-values settings such as this one. Since the
marginal revenue of a bidder is exactly the marginal revenue of the customer
who is the same fraction of the way down the distribution of potential
buyers in the monopoly model, this interpretation allows the direct
translation of results from monopoly theory into auction theory and so
facilitates the analysis of auctions and the development of intuition about
them.} and assume that the bidder with the higher signal has the higher
marginal revenue, i.e., $t_{i}>t_{j}\Longrightarrow
MR_{i}(t_{i},t_{j})>MR_{j}(t_{i},t_{j})$. This is a standard assumption in
auction theory and monopoly theory; it corresponds to assuming that bidders'
marginal revenues are downward sloping in symmetric private-values auction
problems and the corresponding monopoly problems. The assumption is a much%
{\bf \ }stronger one for common-value auctions than for private-value
auctions, \footnote{%
See Bulow and Klemperer (1997) for discussion of when this assumption holds
in the common-value case. See also Myerson (1981), who calls this the
``regular'' case in his largely private-value analysis, Bulow and Roberts
(1989), who refer to this as downward-sloping marginal revenue in their
private-value analysis, and Bulow and Klemperer (1996), who also (more
loosely) refer to this as downward-sloping marginal revenue in the general
case.} but we note that the assumptions of this paragraph are only required
for Proposition~2 and 6.
We denote the price that the bidding has currently reached by $b$. We write
bidder $k$'s equilibrium profits, conditional on his signal, as $\pi
_k(t_k), $ and his unconditional profits (averaged across his possible
signals) as $\Pi _k$. We write the expected profits accruing to all the
shareholders except the two bidders as $\Pi _0.$
\section{Solving the Model}
In this section, we first establish the necessary and sufficient conditions
for the equilibrium strategies of our model (Lemmas~1 and 2), next solve for
the equilibrium (Proposition~1), and then calculate the expected revenue of
the bidders and the non-bidding shareholders.
By standard arguments, we obtain
\begin{lemma}
Bidders' equilibrium strategies must be pure strategies $b_{i}(t_{i})$ and $%
b_{j}(t_{j})$ that are continuous and strictly increasing functions of their
types with $b_{i}(0)=b_{j}(0)>v(0,0)$ and $b_{i}(1)=b_{j}(1)=v(1,1)$.%
\footnote{%
See Appendix for proof.}
\end{lemma}
\medskip
\noindent We can therefore define ``equilibrium correspondence'' functions $%
\phi_i(\cdot)$ and $\phi_j(\cdot)$ by $b_i\bigl(\phi_i(t_j)\bigr) = b_j(t_j)$
and $b_j\bigl(\phi_j(t_i)\bigr) = b_i(t_i)$. That is, in equilibrium, type~$%
t_i$ of $i$ and type~$\phi_j(t_i)$ of $j$ drop out at the same price, and
type~$t_j$ of $j$ and type~$\phi_i(t_j)$ of $i$ drop out at the same price.
So bidder~$i$ will defeat an opponent of type $t_j$ if and only if $t_j\leq
\phi_j(t_i)$, and $\phi_j(t_i)$ is type~$t_i$'s probability of winning the
company.
Given $i$'s bidding function $b_{i}(\cdot )$, for any type $t_{j}$ of $j$ we
can find $t_{j}$'s equilibrium choice of where to quit or, equivalently, $%
t_{j}$'s choice of which $t_{i}$ to drop out at the same time as, by
maximizing $t_{j}$'s expected revenues
\begin{equation}
\max_{t_{i}}\left\{ \int_{t=0}^{t_{i}}\Bigl[v(t,t_{j})-(1-\theta
_{j})b_{i}(t)\Bigr]\,dt+\theta _{j}(1-t_{i})b_{i}(t_{i})\right\} .
\label{a1}
\end{equation}
The term in the integral is $j$'s revenues from buying, and the second term
is $j$'s revenue from selling. Setting the derivative of (\ref{a1}) equal to
zero\footnote{%
Making this argument assumes $b_i(t_i)$ is differentiable. Strictly we
should proceed by noting that type $t_j=\phi_j(t_i)$ prefers quitting at $%
b_i(t_i)$ to $b_i(t_i+\Delta t_i)$. Therefore
\[
\theta_j \left[b_i(t_i+\Delta t_i)-b_i(t_i)\right] \left(1- {{{{{{{{{{\frac{%
\Delta t_i}{1-t_i}}}}}}}}}}}\right) \leq \left( {{{{{{{{{{\frac{\Delta t_i}{%
1-t_i}}}}}}}}}}}\right) \left[b_i(t_i)-v\bigl(t_i,\phi_j(t_i)\bigr)\right] +
o(\Delta b) + o(\Delta v),
\]
in which $o(\Delta b)$ and $o(\Delta v)$ are terms of smaller orders than,
respectively, $\Delta b \equiv b_i(t_i+\Delta t_i)-b_i(t_i)$ and $\Delta
v\equiv \Delta t_i \cdot {{{{{{{{{{\frac{\partial v}{\partial t_i}}}}}}}}}}}$%
. So
\[
\limsup_{\Delta t_i\to 0}{{{{{{{{{{\frac{ b_i(t_i+\Delta t_i)-b_i(t_i)}{%
\Delta t_i}}}}}}}}}}} \leq {{{{{{{{{{\frac{1}{\theta _j}}}}}}}}}}}\left( {{{{%
{{{{{{\frac{1}{1-t_i}}}}}}}}}}}\right) \left[b_i(t_i)-v\bigl(t_i,\phi_j(t_i)%
\bigr)\right].
\]
Using the fact that $j$'s type $\phi_j(t_i+\Delta t_i)$ prefers quitting at $%
b_i(t_i + \Delta t_i)$ to $b_i(t_i)$ yields the same equation except with
the inequality reversed and $\liminf$ instead of $\limsup$, so the right
derivative of $b_i(\cdot)$ exists and is given by (\ref{a2}). Examining the
incentives for $j$'s type $\phi_j(t_i)$ to quit at $b_i(t_i-\Delta t_i)$ and
for $j$'s type $\phi_j(t_i-\Delta t_i)$ to quit at $b_i(t_i)$ completes the
argument by showing the left derivative exists and is also given by (\ref{a2}%
).} and using the fact that $t_{j}=\phi _{j}(t_{i})$ in equilibrium yields
\begin{equation}
b_{i}^{\prime }(t_{i})={{{{{{{{{{\frac{1}{\theta _{j}}}}}}}}}}}}\left( {{{{{{%
{{{{\frac{1}{1-t_{i}}}}}}}}}}}}\right) \Bigl[b_{i}(t_{i})-v\bigl(t_{i},\phi
_{j}(t_{i})\bigr)\Bigr]. \label{a2}
\end{equation}
The logic is straightforward: given that the price has already reached $%
b_i(t_i)$, the benefit to $j$ of dropping out against type $(t_i+dt_i)$
instead of type $t_i$ is $\theta_j b_i^{\prime}(t_i)\, dt_i$ --- $j$'s
toehold times the increase in price per share earned by the later exit. The
cost is that with probability $dt_i/(1-t_i)$, $j$ will ``win'' an auction he
would otherwise have lost, suffering a loss equal to the amount bid less the
value of the asset conditional on both bidders being marginal.
It is easy to check that (\ref{a2}) and the corresponding condition for $%
b_{j}^{\prime }(t_{j})$ are sufficient for equilibrium, i.e., satisfy global
second-order conditions,\footnote{%
Assume, for contradiction, that at some bidding level type $t_i$'s optimal
strategy is to deviate to mimic type $t_i^{\prime }>t_i$. Observe that at
any point a higher type has a greater incentive than a lower type to remain
in the bidding (the potential gains from selling out at a higher price are
the same and the potential losses from being sold to are less). But the
derivation of the first-order condition demonstrates that a type slightly
below $t_i^{\prime }$ does not wish to stay in the bidding to mimic $%
t_i^{\prime }$ (see preceding note). So $t_i$ prefers to mimic this type
than to mimic $t_i^{\prime }$, which is a contradiction.} so we have:%
\footnote{%
Our working paper, Bulow, Huang, and Klemperer (1995), extends this lemma to
a more general setting in which the bidders' valuations of the target
company have both private- and common-value components, and proves existence
and uniqueness of equilibrium for the general case.}
\begin{lemma}
Necessary and sufficient conditions for the bidding strategies $b_{i}(t_{i})$
and $b_{j}(t_{j})$ to form a Nash equilibrium are that $b_{i}(\cdot )$ and $%
b_{j}(\cdot )$ are increasing functions that satisfy
\begin{equation}
b_{i}^{\prime }(t_{i})={{{{{{{{{\frac{1}{\theta _{j}}}}}}}}}}}{{{{{{{{{\frac{%
1}{1-t_{i}}}}}}}}}}}\left[ b_{i}(t_{i})-v\bigl(t_{i},\phi _{j}(t_{i})\bigr)%
\right] , \label{a}
\end{equation}
\begin{equation}
b_{j}^{\prime }(t_{j})={{{{{{{{{\frac{1}{\theta _{i}}}}}}}}}}}{{{{{{{{{\frac{%
1}{1-t_{j}}}}}}}}}}}\left[ b_{j}(t_{j})-v\bigl(\phi _{i}(t_{j}),t_{j}\bigr)%
\right] , \label{b}
\end{equation}
where
\[
\phi _{i}(\cdot )=b_{i}^{-1}\left( b_{j}(\cdot )\right) \quad \hbox{and}%
\quad \phi _{j}(\cdot )=b_{j}^{-1}\left( b_{i}(\cdot )\right) ,
\]
with boundary conditions given by
\begin{equation}
b_{i}(0)=b_{j}(0)>v(0,0), \label{bdry1}
\end{equation}
\begin{equation}
b_{i}(1)=b_{j}(1)=v(1,1). \label{bdry2}
\end{equation}
\end{lemma}
\medskip
\noindent Equation (\ref{a}) can be integrated to yield %\[
%\theta _jb_i^{\prime }(t_i)-\frac{b_i(t_i)}{1-t_i}=\frac{-v(t_i,\phi _j(t_i))%
%}{1-t_i}
%\]
\[
%\begin{array}{ll}
%\Rightarrow & \theta _j(1-t_i)^{\frac{-1}{\theta _j}}\frac d{dt_i}\left[
%(1-t_i)^{\frac 1{\theta _j}}b_i(t_i)\right] =-%\TeXButton{bg}
%{\displaystyle \frac{v(t_i,\phi _j(t_i))}{1-t_i}}\medskip\ \\
b_i(t_i)= {{{{{{{{{{\frac{1}{\theta_j}}}}}}}}}}}(1-t_i)^{- {{{{{{{{{{\frac{1%
}{\theta_j}}}}}}}}}}}} \left[ k-\int_0^{t_i}v(t,\phi _j(t))(1-t) ^{{{{{{{{{{%
\frac{1}{\theta _j}}}}}}}}}}-1}dt\right],
\]
where $k$ is a constant of integration. According to boundary condition (\ref
{bdry2}), it is given by
%But as $t_i\rightarrow 1,\frac 1{\theta _j}(1-t_i)^{\frac{-1}{\theta _j}
%}\rightarrow \infty $ so
$k=\int_0^1v(t,\phi _j(t))(1-t)^{{{{{{{{{{\frac{1}{\theta _j}}}}}}}}}}-1}dt$%
. So we have
\begin{equation}
b_i(t_i)={{{{{{{{{{\frac{\int_{t_i}^1v(t,\phi _j(t))(1-t) ^{{{{{{{{{{\frac{1%
}{\theta _j}}}}}}}}}}-1}dt}{\int_{t_i}^1(1-t) ^{{{{{{{{{{\frac{1}{\theta _j}}%
}}}}}}}}-1}dt}}}}}}}}}}}. \label{c}
\end{equation}
Define $H_k(t_k)$ to be bidder $k$'s hazard rate, that is, the instantaneous
rate at which bidder $k$ quits as the price rises divided by the probability
$k$ is still present. So $H_k(t_k)={{{{{{{{{{\frac{1/b_k^{\prime }(t_k)}{%
1-t_k}}}}}}}}}}} $ since types are distributed uniformly. Since $b_i(t_i)=b_j%
\bigl(\phi_j(t_i)\bigr)$, dividing equation (\ref{a}) by equation (\ref{b})
yields
\begin{equation}
{{{{{{{{{{\frac{H_i(t_i)}{H_j\bigl(\phi_j(t_i)\bigr)}}}}}}}}}}}= {{{{{{{{{{%
\frac{\theta _j}{\theta _i}}}}}}}}}}}. \label{d}
\end{equation}
Since boundary conditions (\ref{bdry1}) and (\ref{bdry2}) imply that $%
\phi_j(0)=0$ and $\phi_j(1)=1$, the unique solution to (\ref{d}) is
\begin{equation}
(1-t_j)^{\theta _j}=(1-t_i)^{\theta _i} \label{e}
\end{equation}
i.e.
\begin{equation}
\phi _j(t_i)=1-(1-t_i)^{\theta _i/\theta _j} \label{f}
\end{equation}
Substituting into (\ref{c}), we have:\newline
\noindent {\bf Proposition 1.} {\it \ There exists a unique Nash
equilibrium. In it bidder }$i${\it \ remains in the bidding until the price
reaches }
\begin{equation}
b_i(t_i)={{{{{{{{{{\frac{\int_{t_i}^1v\bigl(t,1-(1-t) ^{{{{{{{{{{\frac{%
\theta_i}{\theta _j}}}}}}}}}}}\bigr) (1-t) ^{{{{{{{{{{\frac{1-\theta _j}{%
\theta _j}}}}}}}}}}}dt}{\int_{t_i}^1(1-t)^{{{{{{{{{{\frac{1-\theta _j}{%
\theta _j}}}}}}}}}}} dt}}}}}}}}}}} \label{g}
\end{equation}
{\it and bidder }$j${\it 's strategy can be expressed symmetrically.}\newline
Note that our equilibrium is unique, in stark contrast to the case without
toeholds in which {\it every} different weakly increasing function $\phi
_{j}(t_{i})$ yields a distinct equilibrium, $b_{i}(t_{i})=v(t_{i},\phi
_{j}(t_{i}))=b_{j}(\phi _{j}(t_{i}))$ (see Milgrom (1981)). The reason is
that the toeholds determine a precise relationship for each bidder between
his opponent's hazard rate and the ``markup'' he will bid over what the
company would be worth conditional on his opponent being of the lowest
remaining type. Absent toeholds, these markups are zero and there is no
restriction on the ratio of the hazard rates at any price.\footnote{%
More precisely, without toeholds, the two bidders' optimization conditions
are degenerate and so cannot uniquely determine the two equilibrium
strategies. Introducing toeholds breaks this degeneracy, giving two distinct
optimization conditions which uniquely determine the equilibrium strategies.}
The easiest way to calculate bidder $i$'s profits is to note, by the
envelope theorem, that type $t_i+dt_i$'s profits can be computed to first
order as if he followed type $t_i$'s strategy, in which case he would earn $%
t_i$'s profits, except that the company is worth $%\TeXButton{bg}
{\displaystyle {{{{{{{{{{\frac{\partial v}{\partial t_i}}}}}}}}}}}(t_i,t_j)}%
\, dt_i $ more when he wins against a bidder with signal $t_j$, so
\[
{{{{{{{{{{\frac{d\pi_i(t_i)}{dt_i}}}}}}}}}}}=\int_{t_j=0}^{\phi _j(t_i)} {{{{%
{{{{{{\frac{\partial v}{\partial t_i}}}}}}}}}}}(t_i,t_j)\, dt_j
\]
which implies
\begin{equation} \label{h}
\begin{array}{lll}
\pi_i(t_i) & = & \displaystyle \pi _i(0)+\int_{t=0}^{t_i}\int_{t_j=0}^{\phi
_j(t)}{{{{{{{{{{\frac{\partial v}{\partial t}}}}}}}}}}}(t,t_j)\, dt_j\, dt
\\
& = & \displaystyle \theta_i b_i(0)+\int_{t=0}^{t_i}\int_{t_j=0}^{\phi
_j(t)} {{{{{{{{{{\frac{\partial v}{\partial t}}}}}}}}}}}(t,t_j)\, dt_j\, dt,
\end{array}
\end{equation}
since a bidder with $t_i=0$ always sells at $b_i(0)$.
Obviously, bidder $i$'s expected profits (after averaging across all
possible values of his information and simplifying) are
\begin{equation}
{\Pi _i=\int_{t_i=0}^1\pi _i(t_i)dt_i=\theta
_ib_i(0)+\int_{t_i=0}^1\int_{t_j=0}^{\phi _j(t_i)}(1-t_i) {{{{{{{{{{\frac{%
\partial v}{\partial t_i}}}}}}}}}}}(t_i,t_j)dt_jdt_i}. \label{i}
\end{equation}
The expected surplus accruing to all shareholders except the bidders is
\begin{equation}
\Pi _0=\int_{t_i=0}^1\int_{t_j=0}^1v(t_i,t_j)dt_jdt_i-\Pi _i-\Pi _j,
\label{j}
\end{equation}
and the average sale price is $\Pi_0/(1-\theta _i-\theta _j)$.
It is also useful to note that (\ref{i}) can be written as
\begin{equation}
\Pi _{i}=\theta
_{i}b_{i}(0)+\int_{t_{i}=0}^{1}\int_{t_{j}=0}^{1}p_{i}(t_{i},t_{j})(1-t_{i}){%
{{{{{{{{{\frac{\partial v}{\partial t_{i}}}}}}}}}}}}%
(t_{i},t_{j})dt_{j}dt_{i}, \label{ii}
\end{equation}
in which $p_{i}(t_{i},t_{j})$ is the probability with which $i$ wins the
company if the bidders' signals are $t_{i}$ and $t_{j}$. So substituting $%
(p_{i}(t_{i},t_{j})+p_{j}(t_{i},t_{j}))v(t_{i},t_{j})$ for $v(t_{i},t_{j})$,
we can collect terms to rewrite (\ref{j}) as
\[
\begin{array}{ll}
\displaystyle \Pi _{0} & =\int_{t_{i}=0}^{1}\int_{t_{j}=0}^{1}\left[ \left(
v(t_{i},t_{j})-(1-t_{i}){{{{{{{{{{\frac{\partial v}{\partial t_{i}}}}}}}}}}}}%
(t_{i},t_{j})\right) \right. p_{i}(t_{i},t_{j}) \\
\displaystyle & \quad +\left. \left( v(t_{i},t_{j})-(1-t_{j}){{{{{{{{{{\frac{%
\partial v}{\partial t_{j}}}}}}}}}}}}(t_{i},t_{j})\right) p_{j}(t_{i},t_{j})%
\right] dt_{j}dt_{i}-\theta _{i}b_{i}(0)-\theta _{j}b_{j}(0),
\end{array}
\]
or
\begin{equation}
\Pi _{0}=E_{t_{i},t_{j}}(MR_{\hbox{winning bidder}})-\theta
_{i}b_{i}(0)-\theta _{j}b_{j}(0), \label{jj}
\end{equation}
in which $MR_{i}$ is $i$'s ``marginal revenue'' as defined in Section~2.%
%TCIMACRO{
%\TeXButton{newline}{\newline%
%}}%
%BeginExpansion
\newline%
%
%EndExpansion
\noindent {\bf Linear Example}
\nobreak
As an example we explicitly compute the case in which the company's value is
just the sum of the bidders' signals, $v=t_i+t_j $. Performing the
integration in (\ref{g}) we have
\begin{equation}
b_i(t_i)=2 - {{{{{{{{{{\frac{1}{1+\theta _j}}}}}}}}}}}(1-t_i) -{{{{{{{{{{%
\frac{1}{1+\theta _i}}}}}}}}}}}(1-t_i)^{\theta _i/\theta _j}. \label{Lbid}
\end{equation}
Hence
\begin{equation}
\pi _i(t_i)=\theta _i\left( {{{{{{{{{{\frac{\theta _i}{\theta _i+1}}}}}}}}}}}%
+ {{{{{{{{{{\frac{\theta _j}{\theta _j+1}}}}}}}}}}}\right) +t_i-\left( {{{{{{%
{{{{\frac{\theta _j}{\theta _i+\theta _j}}}}}}}}}}}\right) \left( 1-(1-t_i)
^{{{{{{{{{{\frac{\theta _i+\theta _j}{\theta _j}}}}}}}}}}}\right) .
\label{lprofit}
\end{equation}
So also
\begin{equation}
\Pi _i=\theta _i\left( {{{{{{{{{{\frac{\theta _i}{\theta _i+1}}}}}}}}}}}+ {{{%
{{{{{{{\frac{ \theta _j}{\theta _j+1}}}}}}}}}}}+ {{{{{{{{{{\frac{1}{%
2\theta_i+4\theta _j}}}}}}}}}}}\right) , \label{Lprofit}
\end{equation}
\begin{equation}
\Pi _0=1-(\theta _i+\theta _j)\left( {{{{{{{{{{\frac{\theta _i}{\theta _i+1}}%
}}}}}}}}}+ {{{{{{{{{{\frac{\theta _j}{\theta _j+1}}}}}}}}}}}\right) -\left( {%
{{{{{{{{{\frac{\theta _i}{2\theta _i+4\theta _j}}}}}}}}}}}\right) -\left( {{{%
{{{{{{{\frac{\theta _j}{4\theta _i+2\theta _j}}}}}}}}}}}\right)
\label{Lrevenue}
\end{equation}
and the average sale price is
\[
\left[ {{{{{{{{{{\frac{\theta _j(2\theta _j+\theta _i+1)}{(\theta
_j+1)(2\theta _j+\theta _i)}}}}}}}}}}}\right] +\left[ {{{{{{{{{{\frac{\theta
_i(2\theta _i+\theta _j+1)}{(\theta _i+1)(2\theta _i+\theta _j)}}}}}}}}}}} %
\right] .
\]
The bidding functions for this example are illustrated in Figure~1 for the
case in which the toeholds are $\theta_1=0.05$ and $\theta_2=0.01$. Observe
that the bidder with the larger toehold always bids more than in the
symmetric equilibrium without toeholds, while the bidder with the smaller
toehold bids less than if neither bidder had a toehold except for very low
values of his signal. Figure~2 also shows the bidding functions when the
toeholds are $\theta_1=0.10$ and $\theta_2=0.01$; increasing bidder~1's
toehold makes that bidder bid more aggressively (and increases his expected
profits) for all values of his signal.
\vspace{.3in} \centerline{\underline{Figure 1 goes here.}} \vspace{.3in}
\begin{center}
Figure 1: Equilibrium Bidding Functions With and Without Toeholds\\[0pt]
for Linear Example $v=t_1+t_2$ with Toeholds of 5\% and 1\%.
\end{center}
\smallskip
\vspace{.3in} \nobreak
\centerline{\underline{Figure 2 goes here.}} \nobreak
\vspace{.3in} \nobreak
\begin{center}
Figure 2: Equilibrium Bidding Functions With Different Size Toeholds \\[0pt]
\nobreak for Linear Example $v=t_{1}+t_{2}$. \\[0pt]
\nobreak Dashed lines: bidding functions with toeholds of 10\% and 1\%; \\[%
0pt]
\nobreak Solid lines: bidding functions with toeholds of 5\% and 1\%.
\end{center}
\bigskip
The next section describes properties of the equilibrium, including those
illustrated in the figures, that apply in the general case.
\vskip 20pt
\section{Properties of the Equilibrium}
If there were no toeholds, type $t_{i}$ would bid up to the price $v\bigl(%
t_{i},\phi _{j}(t_{i})\bigr)$ at which he would just be indifferent about
winning the auction, but it is immediate from equation (\ref{c}) that every
bidder except the highest possible type, $t_{i}=1,$ bids beyond this price.%
\footnote{%
Of course, this does not mean bidders necessarily bid more than if there
were no toeholds, since the functions $\phi _k(\cdot )$ are different.} So
except for types $t_{i}=1$ and $t_{j}=1,$ {\it any bidder who narrowly
``wins'' the auction loses money}.
From equation (\ref{d}), bidder $i$ always quits at a rate $\theta
_{j}/\theta _{i}$ times as fast as bidder $j$, so it follows immediately
that $i$ ``wins'' the auction, i.e. buys the company, with probability $%
\left( {{{{{{{{{{\frac{\theta _{i}}{\theta _{i}+\theta _{j}}}}}}}}}}}}%
\right) $. Thus {\it probabilities of winning the auction are highly
sensitive to the relative sizes of bidders' stakes}, and a bidder's
probability of winning is increasing in his stake.
It also follows that {\it increasing a bidder's stake increases his
probability of winning, conditional on whatever information he has} (i.e. $%
\phi _j(t_i)$ is strictly increasing in $\theta_i$ for all $0\phi
_j(t_i)$ for all $00}$
for all $t_{i}<1$. This is what we expect --- a higher stake makes a bidder
more like a seller who wants to set a high price, than like a pure buyer who
wants to buy low.
Since $b_i(0)=b_j(0)$ and bidding strategies are continuous, {\it all types
of bidder }$j${\it \ with sufficiently pessimistic information also bid more
aggressively if }$i${\it 's stake is increased}. The intuition is that
because $i$ is bidding more aggressively, low types of bidder $j$ should
take the opportunity to bid the price up under him.
However, for higher types of bidder $j$ it is not clear whether increasing $%
i $'s stake should make $j$ more or less aggressive: bidder $j$ also has to
take account of the larger winner's curse of winning against a
more-aggressive bidder $i$. In fact, there is no general result about
whether raising $i$'s stake raises or lowers $j$'s bid.\footnote{%
It is easy to check for the linear case that
\[
{{{{{{{{{{\frac{\partial b_j(t_j)}{\partial\theta _i}}}}}}}}}}} = {{{{{{{{{{%
\frac{(1-t_j)}{(1+\theta _i)^2}}}}}}}}}}}\left[ 1+ {{{{{{{{{{\frac{(1+\theta
_i)^2}{(1+\theta _j)}}}}}}}}}}} {{{{{{{{{{\frac{\theta _j}{\theta _i^2}}}}}}}%
}}}} (1-t_j) ^{{{{{{{{{{\frac{\theta _j-\theta _i}{\theta _i}}}}}}}}}} }\log
(1-t_j)\right].
\]
So an increase in the share of the bidder with the larger toehold leads to
the opponent bidding more/less aggressively according to whether his type is
below/above some cutoff level. An increase in the share of the bidder with
the smaller toehold always results in both weak and strong types of the
opponent bidding more aggressively while intermediate types bid less
aggressively.}
%[The result of the last sentence is claimed by Ming (I haven't
%checked it). I wonder whether one could have a case (in a different example)
%in which every type of $j$ bids more aggressively.]
Even though raising a bidder's stake makes some types of his opponent more
aggressive --- so results in lower ex-post profits for some types of the
bidder --- {\it increasing a bidder's stake always increases his expected
profits, whatever his signal}. In fact, increasing a bidder's toehold
increases his expected profits in two ways; it both raises the price $%
b_{i}(0)$ at which the bidding starts and at which the bidder can sell out
if he has the lowest possible signal, and also increases the incremental
surplus that he earns from any higher signal (since $%\TeXButton{bg}
\displaystyle {{{{{{{{{{\frac{d\phi _{j}}{d\theta _{i}}}}}}}}}}}}(t_{i})>0$
for all $0\theta _j,$ this is smaller than the probability $%
\displaystyle \left( {{{{{{{{{{\frac{\theta _i}{\theta _i+\theta j}}}}}}}}}}}
\right) $ with which $i$ would win the ascending auction, so it also follows
that\newline
\noindent {\bf Proposition 4.} {\it The probability that the bidder with the
higher signal wins the auction is greater in the first-price auction than in
the ascending auction.}\newline
Thus the outcomes of first-price auctions are less sensitive to toeholds
than are the outcomes of ascending auctions, %\footnote{
%[We would like to interpret this as meaning first-price auctions are more
%efficient. Of course we can't in a pure common-value model but we can maybe
%extrapolate in this direction in our conclusion.]}
%\footnote{
%[We might want to use the term second-price auction somewhere or sometimes.]}
although it remains true that the bidder with the larger toehold has a
higher probability of winning.
The intuition is that a bidder with a toehold still has an incentive to bid
higher than otherwise: bidding more aggressively is less costly when winning
the auction means buying only fraction $(1-\theta )$ rather than all of the
company. However, this effect is generally small unless $\theta $ is close
to 1 (in which case the bidder has control anyway\footnote{%
Our model therefore assumes $\theta <{{{{{{{{{{\frac{1}{2}}}}}}}}}}}.$}).
Furthermore, and more importantly, the indirect or ``strategic'' effect due
to the winner's curse on the opponent is much smaller in first-price than in
ascending auctions.\footnote{%
In an ascending auction, when bidder $i$\ bids more aggressively, bidder $j$%
\ must bid less, because conditional on winning at any price his revenue is
lower. (That is, bidding strategies are ``strategic substitutes'' in the
terminology introduced by Bulow, Geanakoplos, and Klemperer (1985).) In a
first-price auction, by contrast, bidder $j$'s response to bidder $i$\
bidding more is ambiguous: when $i$\ bids more, $j$\ wants to bid {\it less}
on the grounds that his marginal profit when he wins is lower, but {\it more}
on the grounds that his probability of winning is lower so increasing his
bid is less costly. So the ascending-auction logic that when $i$\ bids a
little more, $j$\ bids a similar amount less, so $i$\ bids a similar amount
more, so $j$\ bids a similar amount less, etc., does not apply in
first-price auctions.} So the extreme outcome of the ascending auction, that
a bidder with a relatively small toehold is almost completely driven out of
the bidding, does not arise in the first-price auction.
Because toeholds provide greater incentives for bidding aggressively in
ascending auctions than in first-price auctions, ascending auctions yield
higher prices on average when toeholds are symmetric:\newline
\noindent {\bf Proposition 5.} {\it With symmetric toeholds, the expected
sale price is higher in an ascending auction than in a first-price auction.}
\footnote{%
See Appendix for proof. This result does not depend on the assumption of
pure common values. Singh (1995) obtains this result for the pure
private-values case.}\newline
However, when toeholds are very asymmetric, the winner's curse effect that
the bidder with the smaller toehold is forced to quit at a very low value in
an ascending auction,{\bf \ }implies first-price auctions are likely to
perform better.\newline
\noindent {\bf Proposition 6.} {\it With asymmetric toeholds, the expected
sale price is higher in a first-price auction than in an ascending auction,
if the toeholds are sufficiently small. (I.e. for any }$\lambda \neq 1$,
{\it the first-price auction yields a higher expected price for all }$\theta
_{i},\theta _{j}${\it \ such that }$\theta _{j}=\lambda \theta _{i}\leq $%
{\it \ }$\overline{\theta }${\it , for some }$\overline{\theta }$.)\footnote{%
See Appendix for proof.
%This result may not be robust to relaxing the pure common values assumption.
}\newline
A more formal way to understand Propositions~5 and 6 is to recall that the
expected sale price equals $\Pi _0/(1-\theta _i-\theta _j)$ and $\Pi _0$ can
be written as in (\ref{jj}) for the ascending auction. By a exactly similar
logic, $\Pi _0$ for the first-price auction can also be written as in (\ref
{jj}) except that the term $b_i(0)$ is replaced by the expected price
received by bidder $i$ in a first-price auction if $i$ has the lowest
possible signal, that is $\int_{t_j=0}^1\widetilde{b}_j(t_j)dt_j,$ and the
term $b_j(0)$ is replaced similarly. There are therefore two differences
between a first-price auction and an ascending auction:
First, the price received by bidder $i$ with signal zero in a first-price
auction $\left( \int_{t_j=0}^1\widetilde{b}_j(t_j)dt_j\right) $ is the
average bid of a bidder $j$ who does not know $i$'s signal, whereas in an
ascending auction bidder $i$ must drop out immediately at $b_i(0)$. When
toeholds are symmetric this is the only distinction between the expressions
for $\Pi _0$ for the two types of auction, so the ascending auction yields
higher prices for symmetric toeholds (Proposition~5).
Second, as Proposition~4 demonstrates, the first-price auction is won by the
bidder with the higher signal in more cases than in the ascending auction,
so the first-price auction is more often won by the bidder with the higher
marginal revenue and so is likely to have the higher expected marginal
revenue of the winning bidder.\footnote{%
However this need not be the case, even under our assumption that the bidder
with the higher signal has the higher marginal revenue, because it is {\it %
not} true that the higher signal wins in the first-price auction in every
case in which it wins in the ascending auction.} In the limit as toeholds
became arbitrarily tiny, this is the only distinction between the
expressions for $\Pi _{0}$ for the two types of auction, so we expect the
first-price auction to yield higher prices for asymmetric toeholds if the
toeholds are not too large (Proposition~6).\footnote{%
An example which shows that if the bidder with the higher signal does not
always have the higher marginal revenue, then an ascending auction may
always yield a higher expected price than a first-price auction is $%
v=t_{i}^{3}+t_{j}^{3}$.} %\footnote{
%An example which shows that if the bidder with the higher signal
%does not always have the higher marginal revenue, then an
%ascending auction may always yield a higher expected price than a
%first-price auction is $v=t_i^3+t_j^3.$
%}
If bidders' toeholds are neither small nor symmetric, the sale-price
comparison between the two auction forms is ambiguous, but our leading
example suggests that first-price auctions are likely to be better in
practice if there is much asymmetry in the relative sizes of the toeholds.
\noindent {\bf Example.} {\it In the linear example }$v=t_{i}+t_{j},${\it \
a sufficient condition for the expected price to be higher in a first-price
auction than an ascending auction is } $\theta _{i}<{{{{{{{{{{\frac{1}{8}}}}}%
}}}}}}\,\theta _{j}${\it \ or }$\theta _{i}>8\,\theta _{j}.${\it \ If }$%
\theta _{k}<0.1,k=i,j,${\it \ a sufficient condition is }$\theta _{i}<{{{{{{{%
{{{\frac{1}{4}}}}}}}}}}}\,\theta _{j}${\it \ or }$\theta _{i}>4\,\theta _{j}$%
.%\noindent {\bf Example~2.} {\it If toeholds are sufficiently asymmetric,
%the expected sale price is higher in a first-price auction than
%in an ascending auction if }
%$v=\alpha (t_i^\beta +t_j^\beta )${\it \ for any }
%$\beta <\frac{3}{2}${\it \ and any }$\alpha$,
%{\it or if }
%$v(0,t_j)=v(t_i,0)=v(0,0)$ {\it for any }
%$t_i, t_j$.\footnote{If $\theta_k < 1/2$
%for $k=i, j$, and if $\theta_i/\theta_j\to 0$
%or $\theta_i/\theta_j \to \infty$, Proposition~1 shows that the expected
%price in the ascending auction goes to 0 while Proposition~3 shows that
%the expected price in the first-price auction is finite.}\\
\section{Changing the Game: (B) Selling a Second Toehold}
An alternative approach to compensating for the advantage that a bidder with
a toehold has is to ``level the playing field'' by selling shares (or
equivalently options) to the second bidder so that he has an equal stake.%
\footnote{%
Selling shares at price $p$ is equivalent in this context to giving options
for the same number of shares at exercise price $p$.} Even if these shares
are sold very cheap (so that all types of the second bidder will wish to buy
them) the likely higher price from a fairer contest may more than outweigh
the cost to the remaining shareholders of diluting their stake.
For example, with the linear value function $v=t_i+t_j,$ if just one of the
two bidders has a toehold, say $\theta$, the expected profits of the
non-bidding shareholders are ${{{{{{{{{{\frac{1}{2}}}}}}}}}}}- {{{{{{{{{{%
\frac{\theta ^2}{1+\theta}}}}}}}}}}} $ (from (\ref{Lrevenue})).
%$\left(=\int_{t_j=0}^1v(0,t_j)dt_j-\theta b_i(0)\right)$.
%\footnote{
%[This formula is nice and simple, but I didn't get nice simple formulae for
%the general case for the rest of this example, so we should probably leave
%this out unless Ming can do better. {\bf REPLY:} we can write down
%simple enough formula for all other quantities involved in
%this paragraph, but we cannot claim that symmetric toeholds generate
%more revenue, unless we restrict our discussion to a certain class
%of valuation functions, such as ones with $v(0,t_j)=v(t_i,0)=0$.]}.
The bidder without the toehold makes zero expected profit (whatever his
signal) so, even if he had the lowest possible signal, he would be prepared
to pay ${{{{{{{{{{\frac{2\theta ^2}{1+\theta }}}}}}}}}}},$ that is, $\theta
b(0)$ when both bidders have a stake of $\theta ,$ for a stake of equal
size. The expected profits of the non-bidding shareholders would then be ${{{%
{{{{{{{\frac{2\theta ^2}{1+\theta }}}}}}}}}}}$ plus the expected profits
from the bidding, ${{{{{{{{{{\frac{2}{3}}}}}}}}}}}- {{{{{{{{{{\frac{4\theta
^2}{1+\theta }}}}}}}}}}}$, which equals ${{{{{{{{{{\frac{2}{3}}}}}}}}}}}-{{{{%
{{{{{{\frac{2\theta ^2}{1+\theta }}}}}}}}}}}$ in all. This exceeds the
expected profits if there were no such sale, $( {{{{{{{{{{\frac{1}{2}}}}}}}}}%
}}-{{{{{{{{{{\frac{\theta ^2}{1+\theta }}}}}}}}}}})$, for all $\theta\leq {{{%
{{{{{{{\frac{1}{2}}}}}}}}}}}$.
In fact, even if the stake could only be given away free,\footnote{%
Note that we have set the base price of the stock to be zero if both bidders
observe the lowest possible signal. So ``given away free'' here means
selling them at the base price of the stock.} giving away the stake would
dominate not doing so for all $\theta \leq {{{{{{{{{{\frac{1}{4}}}}}}}}}}}$.
\footnote{%
Thus selling shares, or giving options, at a price close to the lowest
possible value of the company may be acceptable management behavior in a
context in which the value function is hard to assess.} \footnote{%
In fact selling, or giving, a second toehold is even more desirable than
this if it is done through e.g. issuing new shares that dilute the size of
the first bidder's stake, rather than by just selling a fraction of the
non-bidding shareholders' shares. Dilution is probably more realistic but it
was not needed for our result.}
%Of course the non-bidders might do better still by selling the second bidder
%a different size stake, or offering it at a higher price so that possibly
%only some types of second bidder would accept. We next analyze the optimal
%price per share $c$ and the optimal number of shares $\theta_j$ that the
%non-bidders should offer to bidder $j$ (who starts with no toehold) when
%bidder $i$ has a toehold $\theta_i>0$.
%
%Facing the offer of $(c,\theta_j)$, bidder $j$ either accepts
%the offer to buy $\theta_j$ shares at $c$ per share and then participates
%in the auction, or refuses the offer in which case we assume that
%he still participates in the auction.\footnote{This assumption is
%made such that our analysis is robust to assuming that bidder $j$ comes
%into the auction with a tiny toehold.}
%Since bidder $j$ expects no profit in the subsequent auction if he
%refuses the offer,
%only bidders $j$ with
%signals exceeding $\hat{t}$ accept the offer, with $\hat t$ determined by
%the condition that the starting bid in the subsequent auction must
%be equal to $c$.
%
%The information that the offer is accepted or refused affects the auction.
%In the linear symmetric example $v=t_i+t_j$,
%if bidder $j$ accepts the offer, the
%subsequent auction is for a company worth $\widetilde{v}=\hat{t}+t_i+(1-%
%\hat{t})\widetilde t_j,$ with $\widetilde{t}_j\equiv (t_j-\hat{t})/(1-\hat{t%
%})$ uniformly distributed on [0,1]; this is just the asymmetric linear
%example we solved in Section 3. If bidder $j$ refuses the offer,
%the subsequent auction is between bidder $i$ with
%$\theta_i>0$ and bidder $j$ with $\theta_j=0$,
%for a company that is now worth
%$\widetilde v= t_i + \hat t\cdot \bar t_j$, with $\bar t_j = t_j/{\hat t}$
%uniformly distributed on $[0,1]$; this again is the asymmetric linear example
%that we solved in Section~3.
%
%Taking into account of bidder $j$'s decision to accept or refuse
%the offer and the expected revenue in the subsequent auctions,
%the management can maximize non-bidders' expected
%profit over $\theta _j$ and $c$. In this
%example, numerical calculations show that maximum non-bidder profits are
%achieved by choosing $c$ such that all types of bidder $j$ are just willing
%to participate, and choosing $\theta_j$ a little less than $\theta_i$.
%For $\theta_i=1\%$, optimal $\theta _j=0.999\%$ (reflecting the advantage of a
%symmetric auction); for $\theta _i=20\%,$ optimal $\theta _j=14.7\%$
%(reflecting a tradeoff between the advantage of symmetry and the cost of
%selling stock cheap).
%
%Our example here suggests that it is sometimes in the interest of
%the majority of the shareholders for the management to help a ``white knight''
%to obtain a stake in the company
%when facing a raider with a large toehold.
%
%\item I'm not sure how far we should go down the route of optimising over $c
%$ and $\theta _j$ (see footnote) since it raises issues (such as offering a
%schedule) we can't deal with.
%
%\item One question that a referee might ask is ``What exactly is the
%problem we're solving?'' We seem to have assumed it is ``$\max \Pi _0$
%across simple and natural schemes''. We may need to explain why you should $%
%\max \Pi _0$? (?fiduciary duty is to max sale price) and what are the
%constraints on how? (Without constraints you would just expropriate the
%toehold and run any simple auction. The solutions to $\max (\Pi _0+\Pi _1)$
%and $\max (\Pi _0+\Pi _1+\Pi _2)$ are also trivial.)
%
%\item ???We could solve some explicit mechanism design problem properly.
%
%\item Jeremy wanted to talk about the difficulty an outsider has in
%entering (including investment banking fees) and the fact he has to give up
%information to enter, so insider wins vast percentage of time.
%\end{itemize}
\section{Asymmetric Value Functions}
Our analysis thus far has assumed that the value function is symmetric in
bidders' signals, that is, that the bidders have equally valuable private
information about the value of the company. In fact, none of our analysis
depends on this assumption. However, if the value function is not symmetric,
it is implausible that the bidder with the higher signal will always have
the higher marginal revenue, and dropping this assumption requires dropping
Propositions~2 and 6. (Propositions 1, 3, 4, and 5 are unaffected; they
depend neither on the value function being symmetric, nor on any assumption
about marginal revenues.)
If the bidders' information is not equally valuable, then the bidder to
whose information the value is less sensitive (the bidder, $k$, with the
lower ${{{{{{{{{{\frac{\partial v}{\partial t_{k}}}}}}}}}}}}(\cdot ,\cdot )$%
) will typically have a higher marginal revenue when $t_{i}=t_{j}$, that is,
when each bidder receives a signal that is the same fraction of the way down
the distribution of signals that he could have received. Therefore, by
contrast with Proposition~2, an auction in which the low-information bidder
has the larger toehold and so sometimes wins when he has the lower signal
may yield higher expected revenue than an auction with symmetric toeholds.%
\footnote{%
For example, if the value function is linear but twice as sensitive to $i$'s
signal as to $j$'s signal (i.e., $v=2t_i+t_j$), then in the limit as all
toeholds become tiny, the expected sale price is maximized when $j$'s
toehold is approximately three times as large as $i$'s toehold.} Similarly,
by contrast with Proposition~6, if the low-information bidder has the larger
toehold, an ascending auction may be preferred to a first-price auction,
since the ascending auction gives a greater bias in favour of the larger
toeholder's probability of winning. Of course, if the low-information bidder
also has the smaller toehold, then an ascending auction will be particularly
disastrous.
\section{Conclusion}
Toeholds can dramatically influence takeover battles. A bidder with a large
toehold will have an incentive to bid aggressively, essentially because
every price she quotes is both a bid for the rest of the company and an ask
for her own shares. This increased aggressiveness will cause a competitor to
alter his strategy as well. A competitor with a smaller toehold who is
relatively pessimistic about the value of the company will become more
aggressive, counting on the large toeholder to buy him out at a higher
price. If the competitor has an optimistic assessment of the company's
prospects, though, the large toeholder's aggressive strategy will cause the
competitor to become more conservative, because of an exacerbated winner's
curse.
Because toeholds make a bidder more aggressive, which can make a competitor
more conservative, which can make the bidder still more aggressive, and so
on, even small toeholds can have large effects. A toehold can sharply
improve a bidder's chance of winning an auction, and raise the bidder's
expected profits at the expense of both other bidders and stockholders.
The strategic consequences that so benefit the toeholder create a problem
for a board of directors interested in attaining the highest possible sales
price for their investors. The board of a target company may therefore wish
to ``level the playing field'' by selling a toehold to a new bidder, or by
changing the rules of the auction.
%- In general, asymmetric toeholds may lead to socially very inefficient
%outcomes - contests that are won by the largest toeholder rather than the
%highest value. We expect first-price auctions to improve this. And we expect
%``levelling the playing field'' to improve this.\\
%- Something about the general case (WHICH OF OUR RESULTS ARE ROBUST?) (We
%could have one day have an Appendix solving an example of the general case
%and showing how robust our results are)\\
%- Burkart interprets existing empirical results. Should we?\\
%
%- ? Discuss our policy proposal (or is it too hard to address properly in
%our model)\\
%
%- ? More about ``keeping bidders evenly matched''. One take-home message
%might be that there are different kinds of advantage e.g. (a) having a
%(larger) toehold (b) having an informational advantage (c) having a
%private-value advantage (i.e. it's common knowledge the object is worth more
%to one person that to the other). You don't want one bidder to have an
%advantage unless the other has a compensating advantage (possibly of a
%different kind). If one bidder {\it does} have an advantage you should try
%to cancel it or compensate for it. If you can't do that, then you should
%probably try a first-price auction rather than an ascending auction.\\
%
%- Anything more about the issue that a small toehold may give you effective
%control.\\
%
%This is not exhaustive; just some thoughts (and obviously we may restate the
%crucial points of the introduction).\\ \ \
\newpage %\setcounter{page}{1}
\section*{Appendix: Proofs}
\markboth{Appendix}{}
\noindent{\bf Proof of Lemma 1: } Let $\bar B$ be the lowest price level at
or below which, with probability 1, at least one bidder has dropped out. It
is easy to see that if a low type gets the same expected surplus from two
different quitting prices and the lower price is below $\bar B$, then a
higher type always strictly prefers the higher quitting price. So at least
up to $\bar B$, higher types quit (weakly) after lower types.
Define the common bidding range as price levels below $\bar B$.
Now if $i$ has an ``atom'' (that is, an interval of his types drops out at a
single price) within the common range, then $j$ cannot have an atom at the
same price, since an interval of $j$'s types cannot all prefer to quit
simultaneously with $i$'s atom rather than leave either just before or just
after.
We next argue that the equilibrium bidding functions $b_{i}(t_{i})$ and $%
b_{j}(t_{j})$ are single-valued and continuous on the common range, that is,
there are no ``gaps'' (no intervals of prices within the common range within
which a bidder drops out with probability zero). The reason is that if $i$
has a gap, then $j$ would do better to raise the price to the top of the gap
(thus raising the price $j$ receives for his share) than to drop out during
the gap. So $j$ must have a gap that starts no higher than the start of $i$%
's gap. Furthermore, unless $i$ has an atom at the start of the gap, $j$
would do better to raise the price to the top of the gap than to drop out
just below the start of $i$'s gap, that is, $j$'s gap starts lower than $i$%
's. So, since we have already shown that $i$ and $j$ cannot both have atoms
at the same price, we obtain a contradiction.\footnote{%
Note that without toeholds, gaps would be feasible, since a bidder who knows
he will be the next to drop out is indifferent about the price at which he
does so.}
Similarly it follows that $b_i(0)=b_j(0)$, since if $b_i(0)>b_j(0)$, then
type 0 of bidder~$j$ would do strictly better to increase his bid a little.
Now, observe that if $i$ has an atom in the common range, there cannot be a $%
t_j$ that is willing to drop out just after the atom quits; $t_j$ would
either prefer to quit just before the atom (if $t_j$'s value conditional on $%
i$ being among the types within the atom is less than the current price) or
prefer to quit a finite distance later (since $t_j$'s lowest possible value
conditional on $i$ being above the atom must otherwise strictly exceed the
current price). So, since we have already shown there are no gaps, any atom
must be at the top of the common bidding range.
It now follows that $b_i(0)=b_j(0)>v(0,0)$, since if not then type 0 of
bidder~$j$ would do better to raise his bid slightly; raising his bid by $%
\varepsilon$ gains $\varepsilon \theta _j$ when he still sells (with
probability close to 1) and loses less than $\varepsilon (1-\theta _j)$ when
he ends up buying (which happens with a probability that can be made
arbitrarily small by reducing $\varepsilon $).
At the top of the common range, assume, without loss of generality, that $j$
is the player who quits with probability 1 by or at price $\bar{B}$. Then,
for some $\hat{t}_{i}$, the types (and only the types) $t_{i}\geq \hat{t}%
_{i} $ of $i$ quit at or above $\bar{B}$ (by the argument in the first
paragraph of this proof). Then $\bar{B}\geq v(\hat{t}_{i},1)$ (so that it is
always rational for $j$ to sell at $\bar{B}$). But also $\bar{B}\leq v(\hat{t%
}_{i},1)$ (because either type $t_{i}=\hat{t}_{i}$ is willing to buy at $%
\bar{B}$ with probability 1; or if type $\hat{t}_{i}$ is not buying with
probability 1, then $j$ must have an atom at $\bar{B}$ and $\hat{t}_{i}$ is
bidding $\bar{B}$, so $\bar{B}\leq v(\hat{t}_{i},1)$ otherwise $\hat{t}_{i}$
will quit just before $j$'s atom). So $\bar{B}=v(\hat{t}_{i},1)$. Now we
can't have $\hat{t}_{i}<1$ or $j$'s types just below 1 would prefer quitting
just after $\bar{B}$ to just before $\bar{B}$; either $i$ has an atom at $%
\bar{B}$ so buying just above $\bar{B}=v(\hat{t}_{i},1)$ is profitable, or $%
i $ does not have an atom so raising $t_{j}$'s bid by $\varepsilon $ gains $%
\varepsilon \theta _{j}$ when he still sells (with probability close to 1,
conditional on having reached price $\bar{B}=v(\hat{t}_{i},1)$) and loses
less than $\varepsilon (1-\theta _{j})$ when he ends up buying (which
happens with a probability that can be made arbitrarily small by reducing $%
\varepsilon $). So $\bar{B}=v(1,1)$, and it is straightforward that neither
player can have an atom at this price (no type below 1 would wish to win
with probability 1 at this price).\footnote{%
Note that we have only shown that players quit by $\bar B$ with probability
1. Strictly speaking, in a Nash equilibrium, the (zero-probability) types $%
t_i=1$ and $t_j=1$ can quit above $\bar B$, since it is a zero-probability
event that the price will reach $\bar B$. (In a perfect Bayesian
equilibrium, however, all types including $t_i=1$ and $t_j=1$ must quit by $%
\bar B$.)}
Finally, since we showed that there can be no interval within the bidding
range within which a bidder quits with probability zero, note that bidders
cannot choose mixed strategies. $\quad \Box$.\newline
\medskip \noindent{\bf Proof of Proposition 2: } Since the correspondence
function $\phi _j(t_i)$ is independent of $\theta _i$ for any given ratio $%
\theta _i:\theta _j,E_{t_i,t_j}(MR_{\hbox{winning bidder}})$ is also
independent of $\theta _i $ for any given ratio and is strictly lower for
the ratio $\lambda _1$ than the ratio $\lambda _2$ by our assumption that $%
t_i>t_j \Longrightarrow MR_i>MR_j$. But for any $\lambda _1$ or $\lambda _2$%
, $\displaystyle \lim_{\theta_k\rightarrow 0}\pi_k(0)=0, k=i, j,$ so the
result follows straightforwardly from (\ref{jj}).\quad $\Box .$\newline
\medskip \noindent{\bf Proof of Proposition 5: } Using the argument leading
up to (\ref{j}), the expected sale price in the second-price auction is
\[
{{{{{{{{{{\frac{1}{(1-\theta _i-\theta _j)}}}}}}}}}}}\left\{
\int_{t_i=0}^1\int_{t_j=0}^1v(t_i,t_j)dt_jdt_i-\left[ \pi
_i(0)+\int_{t_i=0}^1\int_{t_j=0}^{\phi _j(t_i)}(1-t_i){{{{{{{{{{\frac{dv}{%
dt_i}}}}}}}}}}}(t_i,t_j)dt_jdt_i\right] \right.
\]
\[
\left. -\left[ \pi _j(0)+\int_{t_j=0}^1\int_{t_i=0}^{\phi _i(t_j)}(1-t_j) {{{%
{{{{{{{\frac{dv}{dt_j}}}}}}}}}}}(t_i,t_j)dt_idt_j\right] \right\}
\]
By the same logic, the expected sale price in the first-price auction is the
same expression but substituting $\widetilde{\phi }_k(\cdot )$ for $\phi
_k(\cdot )$ and $\widetilde{\pi }_k(0)$ for $\pi _k(0),k=i,j,$ in which $%
\widetilde{\pi }_k(0)$ is bidder $k$'s surplus when $k$ has his lowest
possible signal. If $\theta _i=\theta _j=\theta $ then $\phi _j(t_i)=%
\widetilde{\phi }_j(t_i)=t_i,$ so the difference between these expressions
is
\[
{{{{{{{{{{\frac{1}{(1-2\theta )}}}}}}}}}}}\left\{ \widetilde{\pi }_i(0)+%
\widetilde{\pi }_j(0)-\pi _i(0)-\pi _j(0)\right\} .
\]
Substituting $\pi _i(0)=\theta _ib_i(0)$ and $\widetilde{\pi }_i(0)=\theta _i%
\displaystyle \int_{t_j=0}^1\widetilde{b}_j(t_j)dt_j$ (since a bidder with
signal zero always sells) yields (after evaluating $\displaystyle %
\int_{t_j=0}^1\widetilde{b}_j(t_j)dt_j$ by parts) that this difference is
\[
{{{{{{{{{{\frac{1}{(1-2\theta )}}}}}}}}}}}\int_{t=0}^12v(t,t)\left\{ \left[
(1-t)-(1-t) ^{{{{{{{{{{\frac{1-\theta }{\theta }}}}}}}}}} }\right] -\left[ t
^{{{{{{{{{{\frac{\theta }{1-\theta }}}}}}}}}} }-t\right] \right\} dt.
\]
This is positive since $v(t,t)$ is monotonic increasing in $t$ and the
expression in curly brackets has expected value zero and is negative for all
$t\in (0,\widehat{t})$ and positive for all $t\in (\widehat{t},1),$ for some
$\widehat{t}.\quad \Box .$\newline
\medskip \noindent{\bf Proof of Proposition 6: } For a given $\lambda$,
write $E(\lambda )$ and $\widetilde{E}(\lambda )$ for the values of $%
E_{t_i,t_j}(MR_{\hbox{winning bidder}}$) for the ascending auction and
first-price auction, respectively. $E(\lambda )$ is independent of $\theta
_i $ (since $\phi _i(\cdot )$ is independent of $\theta _i),$ while $%
\widetilde{E}(\lambda )$ is monotonic continuous decreasing in $\theta _i$
with $\displaystyle \lim_{\theta_i\rightarrow 0}\widetilde{E}(\lambda
)=E(1), $ since $\widetilde{\phi}_j(t_i)=t_i^{{{{{{{{{{\frac{1-\theta _i}{%
1-\lambda \theta _i}}}}}}}}}} }$ is monotonic and continuous in $\theta _i$
for every $t_i$ and $\displaystyle\lim_{\theta _i\rightarrow 0}\widetilde{
\phi }_j(t_i)=t_i$ for every $t_i$. Furthermore, by our assumption that $%
t_i>t_j \Longrightarrow MR_i >MR_j$, $E(1)>E(\lambda )$ for all $\lambda
\neq 1$. Finally it is straightforward that $\displaystyle\lim_{\theta
_k\rightarrow 0}\pi _k(0)=\lim_{\theta _k\rightarrow 0}\widetilde{\pi }
_k(0)=0,k=i,j,$ for all $\lambda $, so the result follows easily from (\ref
{jj}).\quad $\Box .$\newline
\
\newpage %\setcounter{page}{1}
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%\item
%Michael J. Fishman (1988), ``A Theory of Preemptive Takeover
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%
%\item
%Ronald M. Giammarino and Robert L. Heinkel (1986), ``A Model
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%\item
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%\item
%Bernard Lebrun (1995), ``The Existence of an Equilibrium in First Price
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%\item
%Bernard Lebrun (1994), ``First Price Auction: Properties of the
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%\item
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%\item
%S. A. Ravid and Matthew Spiegel (1992), ``On Toeholds and Bidding
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%
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\end{enumerate}
\end{document}