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\begin{document}

\title{Rescheduling with many lenders: Relying on the information of others}
\author{Claude Fluet\thanks{%
Email: fluet.claude-denys@uqam.ca. Financing from FQRSC and SSHRC is
gratefully acknowledged.} \\
%EndAName
Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al and CIRPEE \and Paolo G.
Garella\thanks{%
Email: garella@spbo.unibo.it, financing from Italian Ministry of Education
under the Cofin 2003 program is gratefully acknowledged. } \\
%EndAName
University of Bologna \\
}
\date{25 August 2005 \emph{(Preliminary and incomplete)}\\
}
\maketitle

\begin{abstract}
{\small Can inertia in terminating unsuccessful projects be due to the
presence of multiple lenders in loan arrangements? Is it possible that a
lender reschedules, apparently betting against his odds? We show that the
fear of being last in a liquidation prevents the aggregation of the lenders'
private information about the value of continuation. Then, in a loan with
multiple lenders, privately observed bad news that would prompt liquidation
if information was aggregated or if the lender acted myopically, may instead
lead to rescheduling.. The rational gamble is that other lenders may have
more precise information. At the resulting equilibrium, it turns out that
rescheduling may occur even if all lenders received bad news. This is a
source of inefficiency (increasing the cost of capital) compared to the
ideal of perfect information sharing. However, from a social point of view,
given that information cannot be shared, the equilibrium does not exhibit
excessive reliance on the information of others. } \noindent {\small %
Keywords: Debt contracts, asymmetric information, rescheduling, bankruptcy,
Bayesian games. \noindent }\noindent {\small JEL: G32, G33 }
\end{abstract}

\baselineskip18pt \thispagestyle{empty} \newpage \setcounter{page}{1}

\section{Introduction}

\textbf{Large and dramatic defaults give rise to debates, more or less
technical in kind, wether banks have properly done their work in processing
information. Large defaults by firms in Europe and the US, as well as by
sovereigns around the world, have often raised questions if some banks knew
that continuation was unlikely to be profitable, and if yes , why they did
not act. Often, events making the news originate from large loans, like
loans to large firms or sovereigns, with enormous amounts of money at stake,
and multiple creditors involved. Collusion \textquotedblright behind the
doors\textquotedblright ,between subgroups of lenders and the borrower,
trying to conceal information and to protect reciprocally at the expenses of
other lenders, can sometimes be invoked to explain matters, but one does not
expect misbehavior of this sort to be too frequent in developed financial
markets, where public institutions and the markets operate to discipline
managers in both, banks and firms. Also, skepticism towards big banks may
simply arise out of political prejudice against sophisticated financial
institutions and finance in general. However, to dismiss the issue may also
be a mistake. Indeed, is it trivially true that banks, and lenders in
general, because they protect their own interests, make the socially optimal
use of the information they collect? Must a lender that receives news about
a borrower's financial health going to act accordingly? What if other
lenders are involved in the same loan arrangement?}

\textbf{To note, when a borrower gets in trouble, if a lender does not renew
a loan it will likely trigger liquidation, which is quite an irrevocable
act. In the presence of a foggy view of the borrower's current profile, a
lender can "think twice", and consider rescheduling, which can be "undone"
at a later period by refusing further delays, may. This can be relevant, in
particular, if the loan arrangement is with multiple lenders. Then, the
single lender knows that the others also receive information about the
prospects of a trouble borrower. In this context, information precision
should be an important matter for taking a decision. A lender receiving a
bad but imprecise signal about the borrower, if other lenders are involved,
may be reluctant to liquidate. The gamble is that other lenders may have
obtained more precise information and be in a better position to decide. Can
this be rational? And, suppose then that a loan arrangement is with two
lenders, what happens if both decide to reschedule if they are ill informed?
Is there too much relying on the possibility of other lenders being better
informed? Does this mean that there is excessive rescheduling at
equilibrium? And what is meant by this?}

\textbf{To answer these questions we shall make use of a simple two
lenders-one borrower model, where agents are involved in a multi-stage game,
under imperfect information. }

The inefficiencies arising from resorting to multiple lenders in financing
firms are well known and the literature has been trying to explain multiple
lending arrangements as a response to problems of asymmetric information or
other imperfections in the loan market. Papers in this vein include Bolton
and Scharfstein (1996), Rajan (1992), Detragiache, Garella and Guiso (2000)
and, more related to the present framework, Dewatripont and Maskin (1995).
The presence of multiple lenders in recontracting of loan arrangements may
lead to a lack of coordination, or to free riding, that prevents socially
optimal debt restructuring (Detragiache (1994), Hart and Moore (1995),
Bolton and Scharfstein (1996), Detragiache and Garella (1996). In
particular, making renegotiation difficult may act as a device to correct
for borrowers' incentives to use funds in suboptimal ways, like for instance
to undertake socially inefficient projects (Hart and Moore 1995). On the
other hand, having more than one lender dilutes the incentives to monitor
the borrower, counterbalancing the discipline effect. As Dewatripont and
Maskin (1995) argue, in a setup where (i) initial debt is signed with one
lender only, (ii) due to limited fund availability, refinancing involves a
second lender, the incentives to monitor by the first lender are reduced,
because part of the gains are appropriated by the second. Then, less
refinancing will occur at equilibrium, mitigating the inefficiencies from a
\textquotedblright soft budget constraint\textquotedblright\ problem that
would arise if credit were to be provided by a centralized agency. While,
however, insufficient incentive to gather information, or also to monitor,
has been a concern in the case of multiple lenders, so far the literature
seems to have neglected the question whether lenders, and banks more
specifically, make the socially optimal use of the information \emph{in
their possession}, so as to provide a framework for addressing the issues
here recalled at the outset. Minetti and Guiso (2004) and Minetti (2203)
study a setup that shares some assumptions with the present paper: multiple
lenders arrangements involve banks that are either relationship banks, or
transaction banks, where relationship banks have better information about
the borrower than transaction banks do. Further, they assume that
relationship banks, thanks to their superior information, have the advantage
that they can selectively choose which assets to repossess in a liquidation.
Related to this issue, Berger, Klapper and Udell (2001) empirically test
various hypotheses about the role of relationship vs. arms-length loans in
Argentina. Our paper shares with Minetti and Guiso (2004) the idea that
lenders receive private information that can differ in precision, although,
in our framework, in order to build our argument, bad news necessarily come
bundled into coarse signals. We analyze a simple recontracting game
involving one (large) borrower and two creditors. The project financed at
date 0 may terminate successfully at date 1, with full payments of all debt
obligations. However, with some probability, the project does not deliver
any return at date 1 but it does so at date 2, with uncertain prospects, but
with no need for fresh funds. By assumption, according to the common priors,
the expected value of returns at date 2 is such that if the project does not
delivers at date 1, liquidation is better than rescheduling. Returns,
however, can also end up to be high enough to make rescheduling worthwhile.
The date 1 liquidation value of the loan is common knowledge and
deterministic and liquidation follows if at least one lender goes for it. At
date 1, each lender decides wether to accept rescheduling or file for
bankruptcy, after observing a private signal about the value of the loan
returns at date 2. The signal that banks receive can be of two kinds:
\textquotedblright precise\textquotedblright\ or \textquotedblright
coarse\textquotedblright , according to a random draw by Nature. The \emph{%
coarse} signal is also assumed to be \textquotedblright
bad\textquotedblright\ in that it brings no more favorable information than
it was contained in the common priors, so that a bank should file for
liquidation if based its decision only upon it. The \emph{precise} signal
reveals the true state of Nature and can be, therefore, good or bad. \textbf{%
One could also enlarge the spectrum of possible signals so as to allow for a
coarse but good signal in addition to the two here considered. As it will be
clear from the analysis below, this change could be accommodated, without
essentially altering the equilibrium strategies of lenders when they receive
the bad and coarse signal. Hence, without altering the basic results. }%
Information received at the rescheduling stage could be merged, but, as we
shall show, a communication game will in general fail to provide equilibria
where truthful revelation of information occurs, and therefore information
is not merged. Accordingly, decisions wether to reschedule or not must be
taken based only upon the received signal.\footnote{%
Other reasons may prevent information to be merged. Guiso and Minetti (2004)
argue that the borrower can selectively control the information flows. We
can also think of cases where lenders have private interests with large
firms, for instance when they also finance suppliers of these firms, or when
they can buy or sell outstanding bonds issued by the borrower. In both cases
the lenders have incentives to manipulate the revelation of information.}
The idea that information received by lenders can differ not only in value,
but also in precision is central to the developments below. Strikingly, we
show that if a bank receives the coarse signal, that would unambiguously
trigger liquidation if information was merged, \emph{or if there was not a
second lender}, will not do so at equilibrium and will adopt a softer
attitude. If, at one extreme, the (exogenous) likelihood that the other
lender is informed is large, indeed, rescheduling by a coarsely informed
bank is a dominant strategy and occurs with probability equal to one. When
both banks receive the bad, but coarse, signal, they both ignore it and
reschedule instead of liquidating. This behavior, that an external observer
would address as \textquotedblright irresponsible\textquotedblright ,
because it amounts to act as if the information was favorable, arises out of
equilibrium strategies enacted by rational players and its economics is
fully understandable when the analysis of the game is fully developed. 
\textbf{Of course, the use of the information, is not socially optimal if
compared to a situation where information is merged. However, we prove that
a social planner, were it able to direct the banks' actions, given the same
information structure, could not improve upon the equilibrium solution.
Hence, roughly speaking, there need not be excessive rescheduling at
equilibrium.} Lenders in our game have interests that are not fully
conflicting: if some \emph{good} information is around, everybody is better
off by rescheduling rather than liquidating. Key to the decision to
reschedule by an ill informed lender is the possibility that the other bank
is informed and triggers liquidation when it is optimal to do so. Of course,
and paradoxically, when both lenders are ill informed they both rely on this
likelihood and inefficient rescheduling occurs. Finally, suppose that if
liquidation occurs the two banks are treated asymmetrically, in the sense
that if one triggers liquidation while the other reschedules, the first gets
some higher payoff than the second---if both liquidate, one can simply
assume that they are treated equally. The analysis of the game played by the
two banks, then, reveals that a lender receiving the precise signal will
liquidate also for some values of the date 2 returns in the range where
rescheduling is optimal. This excessive liquidation is chosen because there
is a chance that the other bank is not fully informed and does not
reschedule with probability equal to one.\textbf{\ Therefore, our model also
predicts that equilibrium runs on the borrower's assets can occur, or
excessive rescheduling can occur, the effective path of play depending upon
how Nature distributes information to the lenders}. The paper is organized
as follows. Section 2 provides the model under the assumption that if banks
take different decisions at the rescheduling stage, the one that triggers
liquidation has a higher payoff than the other. There, it is shown that
information cannot be merged as an equilibrium of a communication game. In
section 3 we simplify our analysis by assuming that the advantage from
unique liquidation tends to zero, a merely technical device that grants that
a well informed bank always takes the socially efficient decision. This
allows to focus on the strategic aspects related to the ill informed bank
\textquotedblright relying on the information of the other\textquotedblright
. Section 4 deals with the efficiency property of the equilibria and
contains comments on the economics of over-rescheduling. Section 5 gives the
results stemming from the analysis of the game where the advantage from
unique liquidation does not vanish, so that a well informed lender may not
reschedule for values of the observed date 2 returns that exceed the
liquidation value. Section 6 provides the conclusions.

\section{The model}

There are three periods, an entrepreneur with no endowment, and lenders. At
date 0, the entrepreneur seeks financing for a project. The amount to be
raised is normalized to 1 and two lenders, henceforth the ``banks'',
participate in equal measure to the provision of funds by means of a debt
contract. Banks are small or the project is large, so that financing must be
obtained from two lenders. The credit market is competitive and lenders earn
zero expected profit at equilibrium. For simplicity, their opportunity cost
of funds is zero. If successful, which occurs with probability $\gamma $,
the project is completed by date 1 and yields the return $R>1$, where $R$ is
sufficiently large for the net expected value of the project to be positive.
With probability $1~-~\gamma $, the project runs into problems and is not
completed by date 1. It can then be liquidated, yielding $L<1$. If allowed
to continue, it delivers the random return $\widetilde{X}$ at date~2. We
take $\widetilde{X}$ to be distributed over the interval $[0,1]$ with
cumulative distribution function $F(X)$. The expected return from
continuation is denoted by $\overline{X}$. An unsuccessful project continues
up to date 2 only if both banks reschedule their loan. Banks do not
coordinate for reasons to be explained below. The payoffs from rescheduling
or filing to liquidate are as follows. When each bank reschedules, they each
get $\widetilde{X}/2$ at date 2. When each bank seeks to liquidate, they
each get $L/2$. However, when one bank reschedules and the other refuses to
do so, the latter obtains $\beta L$ where $\beta \in (\frac{1}{2},1)$ and
the former $(1-\beta )L$. Non rescheduling by one bank ultimately entails
liquidation and $\beta >\frac{1}{2}$ reflects the first-mover advantage in a
run on the firm's assets. At date 1, prior to the rescheduling decision,
each bank obtains independent information about the value of continuation.
This information is either perfect or it is unfavorable but imprecise.
Perfect information means that the bank learns $X$. The imprecise
information is denoted by $\phi $. This information is unfavorable in the
sense that $E(\widetilde{X}\mid \phi )<L$, where $E$ is the expectation
operator; i.e., taken on its own, $\phi $ means that the socially
appropriate decision is to liquidate the project. A bank does not know how
well informed the other bank is. Moreover, the information received by the
banks is non verifiable or ``soft''. Specifically, a bank may announce that
it learned that the value of continuation was some$~X$, but it can offer no
proof even if the announcement is true. Thus, any exchange of information at
date 1 is cheap talk. The consequence is that information will not be shared
because of the first-mover advantage in a liquidation run. To show this, we
expand the set of actions at date 1 to allow for a communication game. One
can imagine that, prior to the play of the rescheduling decisions, banks
simultaneously make announcements, $m_{i}$, of the form $m_{i}=\hat{X}$ or $%
m_{i}=$ $\phi $, after which they simultaneously play ``reschedule'' or
``liquidate'' as described above. The following result justifies\textbf{\ }%
the analysis of the full game under the assumption that no communication
between banks is possible.

\begin{lemma}
If a communication game is played at date 1, it is a dominant strategy to
announce $m_{i}=\hat{X}$ such that $\frac{1}{2}\hat{X}\geq \beta L$.
\end{lemma}

The result is obvious. If a bank indeed learned $X$ satisfying the
inequality, it would like to convince the other bank that continuation is
profitable for both of them. However, a bank learning $X$ such that $\frac{1%
}{2}X<\beta L$ would also gain from making the same announcement if it were
believed (which requires that the other is ill informed), since it would be
the first mover in liquidation. Similarly, a bank learning $\phi $ cannot
gain by announcing that it received a poor but unfavorable signal. For
instance,\textbf{\ }suppose the two banks are ill informed and each
truthfully announces $\phi $, prompting liquidation at the next stage. Then
one bank would have been better off announcing a favorable $X$ and be the
first mover should the announcement be believed. Thus, in equilibrium,
favorable announcements will never be believed and will be equivalent to
being told nothing. We now return to the original game tree and take it that
banks do not communicate at all. To simplify the exposition, we henceforth
assume that $\phi $ is in fact ``no information'', so that $E(\widetilde{X}%
\mid \phi )=\overline{X}$. With probability $\theta $ a bank learns the true 
$X$, with probability $1-\theta $ it learns nothing. To preserve that $\phi $
is unfavorable, we assume 
\begin{equation}
\overline{X}<L<1.  \label{ass}
\end{equation}
Thus, taken on its own, ``no information'' means that from a collective
standpoint it would be best to liquidate. The next section analyzes the
rescheduling game. Figure 1 summarizes the set-up.

\section{Rescheduling decisions}

To simplify the exposition further, we now assume that $\beta $ is close to
one half. In effect, we will write all payoffs as if $\beta =\frac{1}{2}$.
By continuity, the actual equilibrium will be close to what we compute, but
encumbered by ``second order'' terms. With $\beta $ arbitrarily close to one
half, when one bank reschedules and the other liquidates, the payoffs
simplify to $L/2$ for each bank. In section~5, we derive the banks'
strategies when $\beta $\ differs from one half and discuss how this affects
the equilibrium outcome. Let \textit{liq }and \textit{res} refer to
``liquidate'' and ``reschedule''. Strategies are denoted by $\alpha $, the
probability that a bank plays \textit{liq}. We write $\alpha (X)$ for the
strategy played by a well informed bank which learns $X$; similarly, $\alpha
(\phi )$ is the strategy played when ill informed. A bank's expected payoff
is denoted by$~u$.\footnote{%
While this section describes a simultaneous game, we introduce a sequence in
section~6. The equilibrium concept is Perfect Bayesian Equilibrium, namely,
a belief system and a strategy profile such that beliefs are based upon the
equilibrium strategies and are constructed using Bayes' rule whenever
possible, and strategies obey the criterion of sequential rationality.}
Given the simplification that $\beta $ is arbitrarily close to one half,
bank $i$'s expected payoff from playing \textit{liq }does not depend on its
information nor on bank $j$'s strategy ($i,j=1,2$; $i\neq j$). Thus, 
\begin{equation}
u_{i}(liq,\alpha _{j}\mid \phi )=u_{i}(liq,\alpha _{j}\mid X)=\frac{L}{2}%
,\quad \text{for all }X.  \label{liq}
\end{equation}
This is not so for the expected payoff from playing \textit{res}. If bank $i$
is informed, its expected payoff is 
\begin{eqnarray}
&&u_{i}(res,\alpha _{j}\!\mid \!X)=  \nonumber \\
&&\frac{1}{2}\left\{ (1\!-\!\theta )\left[ \alpha _{j}(\phi )L+\left(
1\!-\!\alpha _{j}(\phi )\right) X\right] +\theta \left[ \alpha
_{j}(X)L+\left( 1\!-\!\alpha _{j}(X)\right) X\right] \right\} .  \label{resX}
\end{eqnarray}
If it is ill-informed, the expected payoff from \textit{res }is 
\begin{eqnarray}
&&u_{i}(res,\alpha _{j}\!\mid \!\phi )=  \nonumber \\
&&\frac{1}{2}E\left\{ (1\!-\!\theta )\!\left[ \alpha _{j}(\phi )L+\left(
1\!-\!\alpha _{j}(\phi )\right) X\right] \!+\!\theta \!\left[ \alpha
_{j}(X)L+\left( 1\!-\!\alpha _{j}(X)\right) X\right] \right\} \!.
\label{resPhi}
\end{eqnarray}
In the last two equations, $\theta $ is the probability that bank $j$ is
well informed. The difference between (\ref{resX}) and (\ref{resPhi}) is the
expectation operator in the latter, since the bank whose payoff is
represented in (\ref{resPhi}) does not know$~X$. Comparing (\ref{liq}) and (%
\ref{resX}), the best strategy of an informed bank is obviously \textbf{(it
may not be so obvious to the reader )} to choose \textit{liq }if $L>X$ and 
\textit{res }otherwise. At equilibrium, therefore, for either bank, 
\begin{equation}
\alpha _{i}(X)=\left\{ 
\begin{array}{ll}
1 & \text{if }X<L, \\ 
0 & \text{if }X\geq L.%
\end{array}
\right.  \label{a(X)}
\end{equation}
Note that this coincides with the socially optimal decision. To derive the
best response of an ill informed bank, define\textbf{\ }$Z=\max [X,L]$. This
is the total return that would accrue from an unsuccessful project if banks
were perfectly informed at date 1 and took the appropriate decision. The
expected value is 
\begin{equation}
\overline{Z}\equiv E(Z)=F(L)L+(1-F(L))E\left( X\mid X\geq L\right) .
\label{Zbar}
\end{equation}
Substituting in (\ref{resPhi}) and using (\ref{a(X)}), the payoff from 
\textit{res} for an ill informed bank can then be rewritten as 
\begin{eqnarray}
u_{i}(res,\alpha _{j}\! &\mid &\!\phi )=  \nonumber \\
&&\frac{1}{2}\left\{ (1-\theta )\left[ \alpha _{j}(\phi )L+\left( 1-\alpha
(\phi )\right) \overline{X}\right] +\theta \overline{Z}\right\} .
\label{resPhi2}
\end{eqnarray}
Comparing (\ref{liq}) and (\ref{resPhi2}), bank $i$'s best response, when
ill informed, depends\textbf{\ }on the other bank's strategy when the latter
is also ill informed and on the likelihood of poor information. The
expression in\textbf{\ }(\ref{resPhi2}) is increasing in $\theta $ since $%
\overline{Z}>L>\overline{X}$. Accordingly,\textbf{\ }if $\theta $ is
sufficiently large, it is best to play \textit{res} and rely on the other
bank to make the appropriate decision. Conversely, if $\theta $ is close to
zero, the best move is to play \textit{liq}. The expression in (\ref{resPhi2}%
) is also increasing in $\alpha _{j}(\phi )$. The greater the probability
that the other bank plays \textit{liq }when ill informed, the safer it is to
play \textit{res }when one is also ill informed since the probability of a
``wrong'' rescheduling decision is smaller.

\begin{proposition}
At equilibrium, for $\beta =1/2$, well informed banks take the
sociallyoptimal decisions. Ill informed ones always reschedule their loan if 
\begin{equation}
\theta \geq \widehat{\theta }\equiv \frac{L-\overline{X}}{\overline{Z}-%
\overline{X}}\,.  \label{thetaHat}
\end{equation}
When $\theta <\widehat{\theta }$, there are two equilibria. In equilibrium $%
M $, ill informed banks play a mixed strategy, rescheduling with the
strictly positive probability 
\begin{equation}
1-\alpha _{i}(\phi )=\frac{\theta }{1-\theta }\left( \frac{\overline{Z}-L}{L-%
\overline{X}}\right) ,\quad i=1,2.  \label{alphi}
\end{equation}
In equilibrium $P$, the banks play pure strategies: one always reschedules,
the other always liquidates.
\end{proposition}

The essence of the result is that banks sometimes rely on others to take the
appropriate decision (proofs are henceforth in the Appendix). The gamble,
from the prespective of an ill informed bank, is that the other lender may
have more precise information. When this is sufficiently likely (i.e., $%
\theta ~\geq ~\widehat{\theta }$), each bank, when ill informed, completely
disregards its own unfavorable information and relies fully on the other
bank to be better informed. In fact, ``temporizing'' through a rescheduling
decision is then a dominant strategy---see the proof. When the likelihood of
the other bank being well informed is small ($\theta ~<~\widehat{\theta }$),
there are two possibilities. In the pure strategy equilibrium $P$, one bank
is passive and ``delegates'' to the other lender the liquidation versus
rescheduling decision. In turn, the lender in charge always liquidates when
ill informed, completely discounting the possibility that the other bank may
have obtained favorable information. In the symmetric strategy equilibrium $%
M $, an ill informed bank is indifferent between rescheduling or filing to
liquidate. The greater the likelihood that the other bank is well informed,
the larger the probability of rescheduling, i.e., of delegating the decision
to the other lender. The next section discusses the extent of the
inefficiencies characterizing these equilibria.

\section{Inefficiency}

Inefficiency compared to the first-best with shared information is not
surprising. Still,\textbf{\ }it is of interest to explore how the nature and
extent of the inefficiency is affected by the amount of information in the
system. Moreover, while the perfect sharing of information represents an
obvious benchmark, we also ask whether the equilibria can be improved upon
even though information is not shared. Let $B$ denote the facial value of
the loan (i.e., $B=1+\rho $ where $\rho $ is the rate of interest on the
debt contract). By assumption, $R$ is sufficiently large for $R>B$ to
obtain. Let $\overline{Y}$ denote the total amount that the lenders\textbf{\ 
}(both banks together) expect to recuperate from an unsuccessful
project---i.e., one that will not be\textbf{\ }completed at date 1. Recall
that a project is successful with probability $\gamma $. In a perfectly
competitive credit market, banks earn zero ex ante expected profits so that $%
B$ satisfies 
\[
1=\gamma B+(1-\gamma )\overline{Y}. 
\]
The larger $\overline{Y}$, the smaller $B$ (equivalently, the smaller\textbf{%
\ }$\rho $). We measure inefficiency by how small $\overline{Y}$ is. In
terms of the lenders' strategies, the amount recuperated on average is 
\begin{eqnarray}
\overline{Y}(\theta )\! &=&\!\theta ^{2}\overline{Z}+\theta (1-\theta )\left[
\widehat{\alpha }_{2}L+\left( 1-\widehat{\alpha }_{2}\right) \overline{Z}%
\right] +(1-\theta )\theta \left[ \widehat{\alpha }_{1}L+\left( 1-\widehat{%
\alpha }_{1}\right) \overline{Z}\right]  \nonumber \\
&&+\,\left( 1-\theta \right) ^{2}\left[ \left( 1-(1-\widehat{\alpha }_{1})(1-%
\widehat{\alpha }_{2})\right) L+(1-\widehat{\alpha }_{1})(1-\widehat{\alpha }%
_{2})\overline{X}\right] ,  \label{y_bar}
\end{eqnarray}
where $\widehat{\alpha }_{i}$ is the strategy $\alpha _{i}(\phi )$\ when ill
informed. Each bank is assumed to play the socially optimal---and
equilibrium---strategy when well informed. The derivation of (\ref{y_bar})
is straightforward. The total expected return $\overline{Z}$ is obtained
when both banks are well informed or when only one is, which occurs with
probability $2\theta (1-\theta )$, and the other reschedules. When both
banks are ill informed, the total return from an unsuccessful project is $L$
if at least one bank liquidates, otherwise it is on average $\overline{X}$.
In a first best with perfectly\textbf{\ }aggregated information, the amount
expected to be recuperated is\textbf{\ } 
\begin{equation}
\overline{Y}^{*}(\theta )=\left( 1-\theta \right) ^{2}L+\left( 1-\left(
1-\theta \right) ^{2}\right) \overline{Z}.  \label{optY}
\end{equation}
When information is shared, $\overline{Z}$ is expected to be recuperated if
at least one of the lenders is informed, hence with probability $%
1-\left(1-\theta \right) ^{2}$. When none is informed, the project is
appropriately liquidated. Denote the equilibrium outcome by $\overline{Y}%
^{e}(\theta )$. From proposition 1, when $\theta \geq \widehat{\theta }$, 
\begin{equation}
\overline{Y}^{e}(\theta )=\overline{Y}_{II}(\theta )\equiv \left( 1-\theta
\right) ^{2}\overline{X}+\left( 1-\left( 1-\theta \right) ^{2}\right) 
\overline{Z}.  \label{YII}
\end{equation}
The subscript in the middle expression emphasizes that in equilibrium both
lenders always reschedule; that is, we set $\widehat{\alpha }_{1}=\widehat{%
\alpha }_{2}=0$ in (\ref{y_bar}). When $\theta <\widehat{\theta }$, the
outcome depends on which equilibrium we pick. In the pure strategy
equilibrium, one banks always reschedules when ill informed and the other
always liquidates. Hence, 
\begin{equation}
\overline{Y}^{e}(\theta )=\overline{Y}_{P}(\theta )\equiv (1-\theta
)L+\theta \overline{Z}.  \label{YP}
\end{equation}
In the mixed strategy equilibrium, both lenders play the same strategy and 
\begin{eqnarray}
\overline{Y}^{e}(\theta ) &=&\overline{Y}_{M}(\theta )\equiv \theta ^{2}%
\overline{Z}+2\theta (1-\theta )\left[ \widehat{\alpha }L+\left( 1-\widehat{%
\alpha }\right) \overline{Z}\right]  \nonumber \\
&&+\,\left( 1-\theta \right) ^{2}\left[ \left( 1-\left( 1-\widehat{\alpha }%
\right) ^{2}\right) L+\left( 1-\widehat{\alpha }\right) ^{2}\overline{X}%
\right] ,  \label{YM}
\end{eqnarray}
where $\widehat{\alpha }$ is the optimal strategy $\alpha (\phi )$\ defined
in proposition 1. Comparing with the first-best amount in (\ref{optY}), $%
\overline{Y}^{e}(\theta )\leq \overline{Y}^{*}(\theta )$ with strict
inequality when $\theta \in (0,1)$. There is maximum waste of information
when the probability that individual banks are well informed is neither too
large nor too small. When the information is on average either very good ($%
\theta $ close to unity) or very bad ($\theta $ close to zero), the social
loss from the non sharing of information is negligible. Relying on the other
bank to be well informed has no social cost if indeed the other bank is very
likely to be informed. Conversely, when the likelihood is small, at least
one bank will almost be certain to liquidate---this is the bank ``in
charge'' in equilibrium $P$ or both banks in equilibrium $M$, since $\alpha
(\phi )$ in (\ref{alphi}) then approaches unity. Consider now the nature of
the inefficiency. When $\theta \geq \widehat{\theta }$, ill informed banks
always reschedule. Rescheduling may therefore occur even though both banks
have unfavorable---albeit imprecise---information. Compared to the first
best, the problem is therefore too much rescheduling. By contrast, when $%
\theta <\widehat{\theta } $, inefficient rescheduling never occurs in the
pure strategy equilibrium, but there is inefficient liquidation. The bank
``in charge'' may then liquidate even though the other lender observed $%
X\geq L$. In the mixed strategy equilibrium, both types of inefficiencies
occur. Which of these two equilibria is socially preferable? In the proof of
the next proposition, we show that the pure strategy equilibrium is socially
preferable, i.e., $\overline{Y}_{P}(\theta )>\overline{Y}_{M}(\theta )$ for $%
\theta \in (0,\widehat{\theta })$. A more general question is whether a
Pareto improvement could be achieved by imposing rescheduling strategies on
banks, subject to the constraint that they are consistent with\textbf{\ }the
banks'private information. Second-best optimal strategies potentially differ
from the equilibrium ones only in the event that banks are ill informed. To
characterize the socially optimal strategies, it is therefore sufficient to
choose $\widehat{\alpha }_{1}$ and $\widehat{\alpha }_{2}$ in (\ref{y_bar})
so as to maximize $\overline{Y}(\theta )$.

\begin{proposition}
The following strategies are second-best optimal, subject to the constraint
that lenders cannot credibly share information: if $\theta \geq \widehat{%
\theta }$, $\alpha _{1}=\alpha _{2}=0$; if $\theta <\widehat{\theta }$, $%
\alpha _{1}=1$ and $\alpha _{2}=0$ or $\alpha _{1}=0$ and $\alpha _{2}=1$.
\end{proposition}

%TCIMACRO{\TeXButton{Proof}{\proof}}%
%BeginExpansion
\proof%
%EndExpansion
See the Appendix.\medskip

The result is surprising. When lenders obtain information that cannot be
shared and if the likelihood of information is sufficiently high ($\theta
\geq \widehat{\theta }$), there is indeed excessive rescheduling compared to
the ideal of perfectly aggregated information. However, given the constraint
that information cannot be shared, rescheduling decisions are socially
optimal in a second-best sense. Put differently, in equilibrium, there is
not excessive reliance on others being well informed. For instance, the
outcome would be worse if ill informed banks always acted myopically,
liquidating on the basis of the imprecise but unfavorable information $\phi $%
.\footnote{%
We would then have $\overline{Y}(\theta )=(1-\theta ^{2})L+\theta ^{2}%
\overline{Z}$, which is easily seen to be less than $\overline{Y}%
_{II}(\theta )$ when $\theta \geq \widehat{\theta }$.}\textbf{\ }When $%
\theta <\widehat{\theta }$, the pure strategy equilibrium is second-best
efficient. It is socially efficient for the bank ``in charge'' to act
myopically---at the risk of inefficient liquidation---and for one bank to
fully rely on the other lender's decision. Inefficient rescheduling in a
second-best sense therefore only arises in the mixed strategy equilibrium,
which requires a sufficiently small probability of banks being well informed.

\section{Herding}

Our results bear some similarity with so-called herding phenomena. However,
the mechanism leading to a ``wrong'' outcome with herding is of a very
different nature. Herding (see for instance Banerjee 1992) occurs when
players who receive private signals \textit{observe} the actions taken by
other players and update accordingly. In such a setup, the first player, so
to speak, can determine a cascade of optimal deviations from the actions
dictated by the private signals, even when the information owned by all
players, if merged, would point towards a superior solution. In our case,
observing other players' actions is inessential. An ill informed bank knows
that, if it reschedules, its mistake can be corrected by the better informed
lender, when there is one, but not if it liquidates. This suffices to gamble
against information that is unfavorable but poor. Indeed, modifying our
set-up so as to introduce a sequence of moves has no effect on the results.
Suppose one bank, say bank 1, moves first. Denote its strategy by $\widehat{%
\alpha }_{1}(\cdot )$ where the dot refers to the bank's private
information. Bank 2 moves after observing the action of bank 1. Its strategy
is described by $\widehat{\alpha }_{2}(liq,\cdot )$ and $\widehat{\alpha }%
_{2}(res,\cdot )$, where again the dot is the bank's private information and
where \textit{liq} and \textit{res} refer to bank 1's action. We have the
following result.

\begin{proposition}
Let $\alpha _{1}(\cdot )$ and $\alpha _{2}(\cdot )$ be equilibrium
strategies of the simultaneous game, where the argument is $\phi $ or the
realization of $\widetilde{X}$. Then $\widehat{\alpha }_{1}(\cdot )=\alpha
_{1}(\cdot )$ and $\widehat{\alpha }_{2}(liq,\cdot )=1$, $\widehat{\alpha }%
_{2}(res,\cdot )=\alpha _{2}(\cdot )$ are equilibrium strategies of the
sequential game where bank 1's play of res or liq is observed by bank 2.
\end{proposition}

%TCIMACRO{\TeXButton{Proof}{\proof }}%
%BeginExpansion
\proof %
%EndExpansion
See the Appendix.\medskip Bank 2 is now better informed than when moves are
simultaneous, but the outcome is not improved. The intuition is that bank
2's action matters only when bank 1 plays \textit{res}, as in the
simultaneous game. If bank 2 is well informed, it chooses the socially
efficient action, again as in the simultaneous game. In one class of
equilibria, an ill informed bank 2 replicates \textit{res }because\textit{\ }%
the play of \textit{res }by bank 1 represents ``good news''.\footnote{%
Rescheduling by bank 1 is good news if $\theta $ is large, since the
possibility that the bank rescheduled even though ill informed is then
small. It is also good news if $\theta $ is small and bank 1 plays \textit{%
res }only when well informed.} Thus, an ill informed bank 2 plays \textit{res%
} when its decision matters. For bank 1, the play of \textit{res} therefore%
\textit{\ }has the same expected payoff as in the simultaneous game.

\section{Extensions}

To complete the analysis, we briefly explore the case where the first-mover
advantage in liquidation is non negligible (we restrict the discussion to
the simultaneous game). Recall that, if it is the only one to liquidate, a
bank gets the payoff $L\beta $ while the other lender gets $(1-\beta )L$,
where $\beta \in (\frac{1}{2},1)$. Compared to the equilibrium where $\beta $
is arbitrarily close to one half, a larger value has two effects. First,
well informed banks will now inefficiently liquidate unless their
information is sufficiently favorable. Secondly, ill informed banks
liquidate more often. Both effects reinforce one another. To see this, let $%
X_{\beta }^{\prime }$ denote the equilibrium cutoff such that an informed
bank reschedules when observing $X\geq X_{\beta }^{\prime }$.. From a social
point of view, rescheduling should take place when $X>L$. However, with $%
\beta $ greater than one half, an informed bank will now liquidate if $X$ is
sufficiently close to $L$. One reason is the possibility of gaining the
first-mover advantage should the other lender reschedule. Another is that
rescheduling is now dangerous as the other lender might liquidate because he
is ill informed. The payoff then would be $(1-\beta )L$ rather than $L/2$ as
in section 3. At the same time, an ill informed bank will anticipate that
the other lender, if informed, will be less prone to rescheduling. The
strategy of an ill informed bank will therefore also change and lean more
towards liquidation. At the extreme, if $\beta $ is large enough, the
advantage from liquidating is so large that it becomes a dominant strategy,
whatever the signal received. Otherwise, when the value of $\beta $ is not
too large, equilibrium strategies are similar to those already studied,
except for the inefficient liquidation when at least one bank is well
informed and the fact that inefficient rescheduling takes place less often
when both banks are ill informed. In other words, the first-mover advantage
in liquidation remedies some of the excess rescheduling, but at the cost of
excessive liquidations. The frequency of inefficient liquidation increases
with $X_{\beta }^{\prime }$. Let $\alpha _{\beta }(\phi )$ denote the
probability of liquidation by an ill informed lender, as a function of $%
\beta $. The threshold triggering rescheduling for a well informed bank is
then (all derivations are in the Appendix) 
\begin{equation}
X_{\beta }^{\prime }=L\frac{2\beta -\alpha _{\beta }(\phi )(1-\theta )}{%
1-\alpha _{\beta }(\phi )(1-\theta )}\text{.}  \label{ERprime}
\end{equation}
Clearly, $X_{\beta }^{\prime }>L$ for $\beta >1/2$ with the threshold
tending to $L$, the socially efficient cutoff, when $\beta $ tends to one
half.. As in section 3, an ill informed bank always reschedules if the
probability of the other bank being well informed is sufficiently large.
Specifically, $\alpha _{\beta }(\phi )$ equals zero if 
\begin{equation}
\theta \geq \widehat{\theta }_{\beta }\equiv \frac{2\beta L-\overline{X}}{%
\left[ F(2\beta L)L+\left( 1-F(2\beta L)\right) E\left( X\mid X\geq 2\beta
L\right) -\overline{X}\right] },  \label{newalph}
\end{equation}
where it is easily verified that $\widehat{\theta }_{\beta }$ is increasing
in $\beta $. When $\theta <\widehat{\theta }_{\beta }$ but assuming it is
not too small, an ill informed bank randomizes, rescheduling with
probability 
\begin{equation}
1-\alpha _{\beta }(\phi )=\frac{\theta \left[ E(X\mid X\geq X_{\beta
}^{\prime })-L\right] (1-F(X_{\beta }^{\prime }))-L(2\beta -1)}{(L-\bar{X}%
)(1-\theta )}.  \label{newtheta}
\end{equation}
It is easily shown that $\alpha _{\beta }(\phi )$ is larger than the
corresponding value in proposition 1. Note also that (\ref{newtheta}) holds
only if $\theta $ is not too small, otherwise the expression becomes
negative since $\beta >1/2$. Therefore, if $\theta $ is sufficiently small,
the equilibrium is in pure strategies with $\alpha _{\beta }(\phi )=1$. This
is a further contrast with the equilibrium in section 3.

\section{Concluding comments}

The notion that excessive lending may result in loan arrangements is not
new, where excessive lending may take different meanings according to the
context. Financing projects that have negative expected present value may
result in adverse selection setups (De Meza and Webb (1987), or providing
loans that allow managers to pursue inefficient projects in the future (Hart
and Moore (1995), are just two examples. but overlending may also result
from imperfect bankruptcy procedures that give too much protection to
borrowers. Similarly, inefficient monitoring may hamper the ability to make
the efficient financing decisions. Our idea is that inefficiencies may arise
at the rescheduling stage because lenders may rationally decide to disregard
a bad signal received. We stress how excessive rescheduling may result so
that \emph{ex-post}, if the information possessed by all lenders was
revealed, it could be verified that all lenders had bad news and all lenders
decided to reschedule. However, as we have shown, in the same game, if some
or all lenders receive the good and precise signal, too much liquidation
obtains. Recall, indeed, that rescheduling by an ill informed lender is not
always a pure strategy, and randomization may occur: then, because a well
informed lender knows that the other can be ill informed and liquidate
instead of refinancing a good project, he will also liquidate, avoiding the
risk of being last in a liquidation procedure. Therefore, \emph{ex-post}, it
may appear that too much or too little liquidation has occurred with respect
to the social optimum, according to how the information has been
disseminated in the system. \textbf{[[[ this paragraph was used in the
herding section }Our argument bears some similarity with studies of
so-called herding phenomena, but the mechanism leading to a ''wrong''
outcome with herding is of a very different nature. Herding (see for
instance Banerjee 1992) occurs when players that receive private signals 
\emph{observe} the actions taken by other players and accordingly revise
their priors. In that setup, the first player, so to speak, can determine a
cascade of optimal deviations from the actions dictated by the private
signals, even when the information owned by all players, if merged, would
point towards a superior solution. In our case, although introducing a
sequence of moves would not change the result, observing other payers'
actions is inessential. An ill informed bank, \underline{due to the
liquidation institute}, knows that its mistake can be corrected by the
better informed one, if there is one. This suffices to gamble against the
information received....\textbf{there is much imbedded in the
underlined...we should comment...In particular, why can't liquidation by one
bank be renegotiated when the other bank observed }$X>L$\textbf{?? ]]]}
Another instance of players not making use of the information in their
possession can be found in Brandenburger and Polak (1996), where firms'
managers maximize the value of the firms' share by taking actions that they
would take if they had no different information from the market
participants. There, managers that take suboptimal choices have superior
information. The finance literature also deals with various kinds of
inefficient management that tries to please analysts (see for instance
Degeorge, Patel and Zeckhauser 1999 and 2005 and the references
therein).\bigskip \appendix

\begin{center}
{\Large \textbf{Appendix}}
\end{center}

{\Large \bigskip } \noindent \textbf{Proof of proposition 1: }From (\ref{liq}%
) and (\ref{resPhi2})\textbf{,} when ill informed, bank $i$ plays \textit{%
res }if 
\begin{equation}
(1-\theta )\left[ \alpha _{j}(\phi )L+\left( 1-\alpha _{j}(\phi )\right) 
\overline{X}\right] +\theta \overline{Z}\geq L.  \label{inq1}
\end{equation}
Consider first the case where $\theta \geq \widehat{\theta }$ as defined in
the proposition. The above condition is then satisfied with $\alpha
_{j}(\phi )=0$. Moreover, the left-hand side of (\ref{inq1}) is increasing
in $\alpha _{j}(\phi )$ because $L>\overline{X}$. Hence, the condition holds
for all $\alpha _{j}(\phi )$, which means that \textit{res }is a dominant
strategy \textbf{for bank }$i$. This proves the first part of the
proposition. When $\theta <\widehat{\theta }$, condition (\ref{inq1}) does
not hold if $\alpha _{j}(\phi )=0$. The best response to the pure strategy 
\textit{res }is therefore the pure strategy \textit{liq.} Now, (\ref{inq1})
is satisfied as a strict inequality if $\alpha _{j}(\phi )=1$ (since $%
\overline{Z}>L$ and given $\theta >0$). Thus, \textit{res }is itself the
best response to \textit{liq}, proving equilibrium $P$. From \textbf{this} 
\textbf{last} argument, when $0<\theta <\widehat{\theta }$, there exists $%
\alpha _{j}(\phi )\in (0,1)$ such that (\ref{inq1}) holds as an equality.
Solving for $\alpha _{j}(\phi )$ yields (\ref{alphi}) and proves equilibrium 
$M$. 
%TCIMACRO{\TeXButton{End Proof}{\endproof}}%
%BeginExpansion
\endproof%
%EndExpansion
\medskip

\noindent \textbf{Proof of proposition 2: }We first show that $\overline{Y}%
_{P}(\theta )\geq \overline{Y}_{M}(\theta )$ with strict inequality for $%
\theta \in (0,\widehat{\theta })$, where the expressions are as defined in (%
\ref{YP}) and (\ref{YM}). From (\ref{alphi}), substitute for $\widehat{%
\alpha }=\alpha (\phi )$ in (\ref{YM}), yielding 
\[
\overline{Y}_{M}(\theta )=L+\frac{\theta ^{2}(\overline{Z}-L)(\overline{Z}-%
\overline{X})}{(L-\overline{X})}. 
\]
Define 
\[
h(\theta )\equiv \overline{Y}^{e}(\widehat{\theta })-\overline{Y}_{I}(\theta
)=\left[ L+\frac{\theta ^{2}(\overline{Z}-L)(\overline{Z}-\overline{X})}{(L-%
\overline{X})}\right] -\left[ (1-\theta )L+\theta \overline{Z}\right] . 
\]
The function $h(\theta )$ is a quadratic, with roots at $\theta =0$ and $%
\theta =\widehat{\theta }=(L-\overline{X})/(\overline{Z}-\overline{X})$, and
it is strictly convex, hence $h(\theta )<0$ when $\theta \in (0,\widehat{%
\theta })$. The optimal $\alpha _{1}$ and $\alpha _{2}$ maximize $\overline{Y%
}(\theta )$ in (\ref{Ybar}). Let $\mu _{i}$ be the multiplier associated
with the constrant $\alpha _{i}\leq 1$ and $\nu _{i}$ the multiplier
associated with $\alpha _{i}\geq 0$, $i=1,2$. The Lagrangean is 
\[
\mathcal{L}=\overline{Y}(\theta )+\mu _{1}(1-\alpha _{1})+\nu _{1}\alpha
_{1}+\mu _{2}(1-\alpha _{2})+\nu _{2}\alpha _{2}. 
\]
The necessary conditions for a maximum are the first-order conditions 
\begin{eqnarray}
\partial \mathcal{L}/\partial \alpha _{1}\!\! &=&\!\!(1\!-\!\theta
)[(1\!-\!\theta )(L\!-\!\overline{X})(1\!-\!\alpha _{2})-\theta (\overline{Z}%
\!-\!L)]-\mu _{1}+\nu _{1}\!=\!0,  \label{foca} \\
\partial \mathcal{L}/\partial \alpha _{2}\!\! &=&\!\!(1\!-\!\theta
)[(1\!-\!\theta )(L\!-\!\overline{X})(1\!-\!\alpha _{1})-\theta (\overline{Z}%
\!-\!L)]-\mu _{2}+\nu _{2}\!=\!0,  \label{focb}
\end{eqnarray}
together with complentary slackness and non-negativity of the multipliers, 
\begin{equation}
\mu _{i}(1-\alpha _{i})=\nu _{i}\alpha _{i}=0\text{, }\mu _{i}\geq 0\text{, }%
\nu _{i}\geq 0\text{, }i=1,2.  \label{cs}
\end{equation}
When $\theta \geq \widehat{\theta }$, $(1-\theta )(L-\overline{X})(1-\alpha
_{i})-\theta (\overline{Z}-L)<0$ for all $\alpha _{i}$. Hence, there is
onlyone solution to (\ref{foca}), (\ref{focb}) and (\ref{cs}) and it
involves $\nu _{i}>0$, implying $\alpha _{i}=0$, $i=1,2$. When $\theta <%
\widehat{\theta }$, the first term on the right-hand side of (\ref{foca})
and (\ref{focb}) is zero if $\alpha _{1}=\alpha _{2}=\alpha (\phi )$, where
the latter is as defined in (\ref{alphi}). Together with $\mu _{i}=\nu
_{i}=0 $, $i=1,2$, this therefore constitutes one solution to the set of
necessary conditions. It corresponds to equilibrium $M$. Another possibility
is the corner solution defined by $\alpha _{1}=1$ and $\alpha _{2}=0$. Since 
$(1-\theta )(L-\overline{X})-\theta (\overline{Z}-L)>0$ for $\theta <%
\widehat{\theta }$, $\alpha _{2}=0$ in (\ref{foca}) implies $\mu _{1}>0$ and
therefore $\alpha _{1}=1$. In (\ref{focb}), $\alpha _{1}=1$ implies $\nu
_{2}>0$, hence $\alpha _{2}=0$. The corner solution corresponds to
equilibrium $P$. From the above result, $\theta \in (0,\widehat{\theta })$
implies $\overline{Y}_{P}(\theta )>\overline{Y}_{M}(\theta )$. The
maximization problem is therefore solved by the corner solution. 
%TCIMACRO{\TeXButton{End Proof}{\endproof}}%
%BeginExpansion
\endproof%
%EndExpansion
\medskip

\noindent \textbf{Proof of proposition 3: }When bank 1 chooses \textit{liq},
bank 2 cannot affect the outcome and it is therefore a best response to also
choose \textit{liq}, hence\textit{\ }$\widehat{\alpha }_{2}(liq,\cdot )=1$.
Obviously, $\widehat{\alpha }_{1}(X)=\widehat{\alpha }_{2}(res,X)=\alpha (X)$%
, i.e., well informed banks choose the socially efficient action (when their
action matters). Thus, we need only discuss $\widehat{\alpha }_{1}(\phi )$
and $\widehat{\alpha }_{2}(res,\phi )$. In the simultaneous game (and given
that the other bank behaves efficiently when well informed), bank 2 weakly
prefers \textit{res }to \textit{liq }if 
\begin{equation}
(1-\theta )\left[ \alpha _{1}L+\left( 1-\alpha _{1})\right) \overline{X}%
\right] +\theta \overline{Z}\geq L,  \label{cond1}
\end{equation}
where $\alpha _{1}$ is short-hand for $\alpha _{1}(\phi )$. In the
sequential game, after the play of \textit{res }by bank 1, \textit{res }is
weakly preferred to \textit{liq }by bank 2 if 
\begin{equation}
\frac{\theta (1-F(L))E(X\mid X\geq L)+(1-\theta )(1-\alpha _{1})\overline{X}%
}{\theta (1-F(L))+(1-\theta )(1-\alpha _{1})}\geq L,  \label{cond2}
\end{equation}
where $\alpha _{1}$ is short-hand for $\widehat{\alpha }_{1}(\phi )$. The
left-hand side is (twice) the expected payoff from playing \textit{res},
given that bank 2 observed $\phi $ and the play of \textit{res }by bank 1
(the probability of the latter is the numerator of the expression). Now,
from (\ref{Zbar}), substitute for $\overline{Z}$ in (\ref{cond1}), which
then writes as 
\begin{equation}
(1-\theta )  \label{cond1b}
\end{equation}
It is easily checked that (\ref{cond2}) and (\ref{cond1b}) are in fact
equivalent. Thus, $\widehat{\alpha }_{1}(\phi )=\alpha _{1}(\phi )$ implies $%
\widehat{\alpha }_{2}(res,\phi )=\alpha _{2}(\phi )$ as best response. We
now show the converse. In the sequential game, when ill informed, bank 1
weakly prefers \textit{res }to \textit{liq }if 
\begin{equation}
(1-\theta )\left[ \alpha _{2}L+\left( 1-\alpha _{2})\right) \overline{X}%
\right] +\theta \overline{Z}\geq L,  \label{cond3}
\end{equation}
where $\alpha _{2}$ is short-hand for $\widehat{\alpha }_{2}(res,\phi )$.
The condition is the same in the simultaneous game, but with $\alpha _{2}$
short-hand for $\alpha _{2}(\phi )$. It follows that $\widehat{\alpha }%
_{2}(res,\phi )=\alpha _{2}(\phi )$ implies $\widehat{\alpha }_{1}(\phi
)=\alpha _{1}(\phi )$, thereby concluding the proof. 
%TCIMACRO{\TeXButton{End Proof}{\endproof}}%
%BeginExpansion
\endproof%
%EndExpansion
\medskip

\noindent \textbf{Proof of the statements in section 6: }We first derive the
formula for $X_{\beta }^{\prime }$ in (\ref{ERprime}). Write $\alpha _{\beta
}=\alpha _{\beta }(\phi )$ for short. If a well informed bank plays \textit{%
\theta \left( \frac{X}{2}\right) +(1-\theta )\left[ \alpha _{\beta }(1-\beta
)L+(1-\alpha _{\beta })\frac{X}{2}\right] $, given that the other lender
will also play \mathit{res }if informed. If the bank plays \mathit{liq}, the
expected payoff is $\left( L\beta \right) \left( (1-\theta )(1-\alpha
_{\beta })+\theta \right) +\frac{L}{2}(1-\theta )\alpha _{\beta }$. For $%
X_{\beta }^{\prime }$to define a cutoff point, the two payoffs must be equal
at $X=X_{\beta }^{\prime }$, which leads to (\ref{ERprime}). To prove (\ref%
{newalph}), set $\alpha _{\beta }=0$in (\ref{ERprime}) so that $X_{\beta
}^{\prime }=2\beta L$. If it plays \mathit{res, }an ill informed bank gets 
\begin{eqnarray*}
&&u(res,\alpha \!\mid \!\phi )= \\
&&\theta \left\{ \left( 1-F(2\beta L)\right) \frac{E(X\mid X\geq 2\beta L)}{2%
}+F(2\beta L)(1-\beta )L\right\} +(1-\theta )\frac{\overline{X}}{2}.
\end{eqnarray*}
If it plays \mathit{liq}, the expected payoff is 
\[
u(liq,\alpha \!\mid \!\phi )=\theta \left\{ \left( 1-F(2\beta L)\right)
\beta L+F(2\beta L)\frac{L}{2}\right\} +(1-\theta )\beta L. 
\]
One easily checks that $u(res,\alpha \!\mid \!\phi )\geq u(liq,\alpha \!\mid
\!\phi )$ is equivalent to $\theta \geq \widehat{\theta }_{\beta }$ as
defined in (\ref{newalph}). Note that $\widehat{\theta }_{\beta }<1$ only if 
$\beta $ is not too large, which is implicitly assumed here. To prove (\ref%
{newtheta}), assume $\theta <\widehat{\theta }_{\beta }$ and write $\pi $
for the belief that an ill informed bank assigns to the event that the other
lender plays \mathit{res}. At equilibrium, $\pi =\theta (1-F(X_{\beta
}^{\prime }))+(1-\theta )(1-\alpha _{\beta })$. For an ill informed bank,
the payoff from \mathit{liq} is therefore 
\[
\pi \left( \beta L\right) +(1-\pi )\frac{L}{2}. 
\]
The payoff from \mathit{res }is 
\[
\pi \left( \frac{\theta (1-F(X_{\beta }^{\prime }))}{\pi }\frac{E(X\mid
X\geq X_{\beta }^{\prime })}{2}+\frac{(1-\theta )(1-\alpha _{\beta })}{\pi }%
\frac{\bar{X}}{2}\right) +(1-\pi )(1-\beta )L. 
\]
Substituting for $\pi $ and setting the two payoffs equal to one another
leads to~(\ref{newtheta}). Note that the equality is consistent with $\alpha
_{\beta }(\phi )<1$ only if $\theta $ is not too small. Finally, we prove
that $\alpha _{\beta }(\phi )>\alpha (\phi )$, where the latter is as
defined in proposition 1. From (\ref{ERprime}) and (\ref{newtheta}), $%
X_{\beta }^{\prime }=L$ and $\alpha _{\beta }(\phi )=\alpha (\phi )$ when $%
\beta =1/2$. Now, from (\ref{newtheta}), $\alpha _{\beta }(\phi )$ is
increasing in $\beta $. It is also increasing in $X_{\beta }^{\prime }$
since 
\[
\frac{\partial \alpha _{\beta }(\phi )}{\partial X_{\beta }^{\prime }}=\frac{%
f(X_{\beta }^{\prime })\left( X_{\beta }^{\prime }-L\right) }{(1-\theta )(L-%
\overline{X})} 
\]
and $X_{\beta }^{\prime }>L$ when $\beta >1/2$. The two results together
imply $\alpha _{\beta }(\phi )>\alpha (\phi )$ as claimed.\bigskip }

\end{document}

