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\begin{document}
\title{Toeholds and Takeovers }
\author{Jeremy Bulow\\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
\begin{itemize}
\item {\large A slightly revised version is published in: \textit{Journal of
Political Economy} 1999, 107(3), 427-454.}
\end{itemize}
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\noindent\textbf{Abstract} \vskip20pt \noindent 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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\textsl{Keywords}: Toeholds, Takeovers, Auctions, Mergers, Corporate
Acquisitions, Footholds, Winner's Curse, Common Value Auctions.
\noindent%
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\textsl{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\textit{, }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.\textbf{\ }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 \textit{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.\textbf{\ }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\textbf{\ }%
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
\[
\limsup_{\Delta t_{i}\to0}{{{{{{{{{{\frac{ b_{i}(t_{i}+\Delta t_{i}%
)-b_{i}(t_{i})}{\Delta t_{i}}}}}}}}}}}} \leq{{{{{{{{{{\frac{1}{\theta_{j}}}}}%
}}}}}}}\left( {{{{{{{{{{\frac{1}{1-t_{i}}}}}}}}}}}}\right) \left[
b_{i}(t_{i})-v\bigl(t_{i},\phi_{j}(t_{i})\bigr)\right] .
\]
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
%
$k=\int_{0}^{1}v(t,\phi_{j}(t))(1-t)^{{{{{{{{{{\frac{1}{\theta_{j}}}}}}}}}}%
}-1}dt$. So we have
\begin{equation}
b_{i}(t_{i})={{{{{{{{{{\frac{\int_{t_{i}}^{1}v(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\textbf{Proposition 1.} \textit{\ There exists a unique Nash
equilibrium. In it bidder }$i$\textit{\ remains in the bidding until the price
reaches }
\begin{equation}
b_{i}(t_{i})={{{{{{{{{{\frac{\int_{t_{i}}^{1}v\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}
\textit{and bidder }$j$\textit{'s strategy can be expressed symmetrically.}\newline
Note that our equilibrium is unique, in stark contrast to the case without
toeholds in which \textit{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}%
\begin{array}
[c]{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_{i}b_{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_{j}dt_{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}^{1}v(t_{i},t_{j})dt_{j}dt_{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
\[%
\begin{array}
[c]{ll}%
\displaystyle\Pi_{0} & =\int_{t_{i}=0}^{1}\int_{t_{j}=0}^{1}\left[ \left(
v(t_{i},t_{j})-(1-t_{i}){{{{{{{{{{\frac{\partial v}{\partial t_{i}}}}}}}}}}}%
}(t_{i},t_{j})\right) \right. p_{i}(t_{i},t_{j})\\
\displaystyle & \quad+\left. \left( v(t_{i},t_{j})-(1-t_{j}){{{{{{{{{{\frac
{\partial v}{\partial t_{j}}}}}}}}}}}}(t_{i},t_{j})\right) p_{j}(t_{i}%
,t_{j})\right] dt_{j}dt_{i}-\theta_{i}b_{i}(0)-\theta_{j}b_{j}(0),
\end{array}
\]
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\textbf{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.
\vskip20pt
\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 $t_{j}=1,$ \textit{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
\textit{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 \textit{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, \textit{all
types of bidder }$j$\textit{\ with sufficiently pessimistic information also
bid more aggressively if }$i$\textit{'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 --- \textit{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\textbf{Proposition 4.} \textit{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 \textit{less} on the grounds that his marginal profit when
he wins is lower, but \textit{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\textbf{Proposition 5.} \textit{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,\textbf{\ }implies first-price auctions are likely to
perform better.\newline
\noindent\textbf{Proposition 6.} \textit{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 }$\lambda\neq
1$, \textit{the first-price auction yields a higher expected price for all
}$\theta_{i},\theta_{j}$\textit{\ such that }$\theta_{j}=\lambda\theta_{i}%
\leq$\textit{\ }$\overline{\theta}$\textit{, 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 \textit{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\textbf{Example.} \textit{In the linear example }$v=t_{i}+t_{j}%
,$\textit{\ a sufficient condition for the expected price to be higher in a
first-price auction than an ascending auction is } $\theta_{i}<{{{{{{{{{{\frac
{1}{8}}}}}}}}}}}\,\theta_{j}$\textit{\ or }$\theta_{i}>8\,\theta_{j}%
.$\textit{\ If }$\theta_{k}<0.1,k=i,j,$\textit{\ a sufficient condition is
}$\theta_{i}<{{{{{{{{{{\frac{1}{4}}}}}}}}}}}\,\theta_{j}$\textit{\ or }%
$\theta_{i}>4\,\theta_{j}$.\newline
%\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\textbf{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\textbf{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
$\lambda_{2}$, $\displaystyle\lim_{\theta_{k}\rightarrow0}\pi_{k}(0)=0, k=i,
j,$ so the result follows straightforwardly from (\ref{jj}).\quad$\Box.$\newline
\medskip\noindent\textbf{Proof of Proposition 5: } Using the argument leading
up to (\ref{j}), the expected sale price in the second-price auction is
\[
{{{{{{{{{{\frac{1}{(1-\theta_{i}-\theta_{j})}}}}}}}}}}}\left\{ \int_{t_{i}%
=0}^{1}\int_{t_{j}=0}^{1}v(t_{i},t_{j})dt_{j}dt_{i}-\left[ \pi_{i}%
(0)+\int_{t_{i}=0}^{1}\int_{t_{j}=0}^{\phi_{j}(t_{i})}(1-t_{i}){{{{{{{{{{\frac
{dv}{dt_{i}}}}}}}}}}}}(t_{i},t_{j})dt_{j}dt_{i}\right] \right.
\]
\[
\left. -\left[ \pi_{j}(0)+\int_{t_{j}=0}^{1}\int_{t_{i}=0}^{\phi_{i}(t_{j}%
)}(1-t_{j}) {{{{{{{{{{\frac{dv}{dt_{j}}}}}}}}}}}}(t_{i},t_{j})dt_{i}%
dt_{j}\right] \right\}
\]
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 $\pi_{i}(0)=\theta_{i}b_{i}(0)$ 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
\[
{{{{{{{{{{\frac{1}{(1-2\theta)}}}}}}}}}}}\int_{t=0}^{1}2v(t,t)\left\{ \left[
(1-t)-(1-t) ^{{{{{{{{{{\frac{1-\theta}{\theta}}}}}}}}}} }\right] -\left[ t
^{{{{{{{{{{\frac{\theta}{1-\theta}}}}}}}}}} }-t\right] \right\} dt.
\]
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\textbf{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}\rightarrow0}\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 $\displaystyle\lim_{\theta_{i}\rightarrow0}\widetilde{
\phi}_{j}(t_{i})=t_{i}$ for every $t_{i}$. Furthermore, by our assumption that
$t_{i}>t_{j} \Longrightarrow MR_{i} >MR_{j}$, $E(1)>E(\lambda)$ for all
$\lambda\neq1$. Finally it is straightforward that $\displaystyle\lim
_{\theta_{k}\rightarrow0}\pi_{k}(0)=\lim_{\theta_{k}\rightarrow0}%
\widetilde{\pi} _{k}(0)=0,k=i,j,$ for all $\lambda$, so the result follows
easily from (\ref{jj}).\quad$\Box.$\newline \
\newpage
%\setcounter{page}{1}
%
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\markboth{References}{}
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%\item
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%
%\item
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%
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%
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%
%
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\end{enumerate}
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. However, the assumption is a stronger one for common-value auctions
than for private-value auctions. See 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. [WAS FOOTNOTE 16]%
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this effect is small.\footnote{In particular, there is only a small reduction,
as a proportion, in the discount by which a bidder underbids the value
conditional on his being marginal. It is the relative sizes of these discounts
that determines bidders' relative hazard rate.} So the relative sizes of the
winners' curses of winning against bidders with large toeholds and with small
toeholds are not too different in first price auctions. Therefore%
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These were after Figure 2
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