%Paper: ewp-meet/9506001
%From: ejg@res.mpls.frb.fed.us (Ed Green)
%Date: Wed, 28 Jun 95 13:32:06 CDT

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\begin{document}

\title{Implementing Efficient Allocations in \\ a Model of Financial
Intermediation\thanks{This {\bf preliminary and incomplete draft} is
prepared for the 1995 Society of Economic Dynamics and Control
meetings. The views expressed in this paper are
those of the author, and do not necessarily reflect those of the
Federal Reserve Bank of Minneapolis or the Federal Reserve System.}}
\author{Edward J.~Green 
\thanks{Affiliation beginning July, 1995: Research Department, Federal
Reserve Bank of Minneapolis. Email: ``ejg@res.mpls.frb.fed.us''.}
} \date{26 June, 1995} \maketitle

\abstract{In a finite-trader version of the Diamond-Dybvig (1983)
model, the symmetric, ex-ante efficient allocation is implementable by
a direct mechanism (i.e., each trader announces the type of his own
ex-post preference) in which truthful revelation is the strictly
dominant strategy for each trader.  When the model is modified by
formalizing the sequential-service constraint (cf.~Wallace, 1988), the
truth-telling equilibrium implements the symmetric, ex-ante efficient
allocation with respect to iterated elimination of strictly dominated
strategies.}

\Section a{Introduction}

This paper concerns the welfare analysis of maturity transformation in
financial structure. Maturity transformation is the financing of an
intermediary's assets by liabilities (demand deposits at a bank, in
particular) that are callable, and that some traders do call in
equilibrium, before the assets themselves mature. Bryant (1980) shows
that such a portfolio structure is a means of insuring the depositors
against unobservable risks, and he also identifies a
multiplicity-of-equilibrium problem. He implicitly represents a bank
as a rule or ``allocation mechanism'' that specifies the outcome, in
each state of nature, of each possible profile of traders' decisions
regarding whether or not to exercise the call options on their
deposits. This rule constitutes a framework for strategic interaction
among the traders. Bryant observes that maturity transformation is
necessary in order to implement the symmetric, ex-ante efficient,
allocation as a Bayesian Nash equilibrium. He shows also that some
mechanisms that do implement that efficient allocation---notably the
mechanism that most faithfully reflects the features of a bank-deposit
contract in the context of his model---also can possess other
equilibriums that are strictly Pareto dominated by the ``intended''
equilibrium.

Diamond and Dybvig (1983) address a related set of issues to 
Bryant. They study a model that brings the role of aggregate
risk into sharp focus. They prove four main results.

\begin{enumerate}

\item The phenomenon of Pareto-ranked bank-deposit-contract
equilibriums can occur even in an environment where there is no
aggregate risk.

\item However there is an allocation mechanism, suggested by
historical banking regimes that have permitted suspension of
convertability of deposits when a ``run'' occurs, that implements the
symmetric, ex-ante optimal allocation in strictly dominant strategies.
This is intuitively a particularly compelling notion of implementation
that implies, among other things, that the Bayesian Nash equilibrium
is unique. Obviously, then, there cannot be multiple, Pareto-ranked
equilibriums.

\item In some environments with aggrate risk, a deposit scheme with
suspension of payments cannot implement an ex-ante efficient
allocation. 

\item However, despite the presence of aggregate risk, it is possible
to implement the symmetric, ex-ante efficient allocation in Bayesian
Nash equilibrium. If deposit insurance is feasible, then it can
provide one means to do so. Diamond and Dybvig's analysis does not
establish whether or not there is any allocation mechanism that
implements the symmetric, efficient allocation as its unique Bayesian
Nash equilibrium.

Wallace (1988) provides a formalization of the sequential-service
constraint to which previous researchers had appealed informally. 
He proves the following result that bears on Diamond and Dybvig's
fourth point.

\item If the provision of deposit insurance is genuinely regarded as a
feature of the over-all allocation mechanism, and if it is this
over-all mechanism to which the sequential-service constraint applies,
then deposit insurance is not feasible to provide.

\end{enumerate}

Taken together, the last three of these results raise the possibility
that existence of multiple, Pareto-ranked equilibriums might be an
unavoidable problem for any mechanism that implements the symmetric,
ex-ante efficient allocation as a Bayesian Nash equilibrium in an
environment with aggregate risk. Suppose that that were indeed the
situation, and that one believed that traders' strategic interactions
were much more likely to proceed according to the Pareto-dominated
equilibrium than according to the efficient one. If there were another
mechanism that had a unique, ``mediocre,'' equilibrium that were
situated strictly between the other two according to the Pareto
relation, then one might be inclined to choose mechanism and to
tolerate the inefficiency of its equilibrium rather than to incur the
substantial risk of doing even worse, in order to have any chance of
attaining efficiency. To the contrary, if there were a mechanism
possesing a unique equilibrium, and if the symmetric efficient
allocation were the outcome that would result from that equilibrium
being played, then one would reject without hesitation the mechanism
with the mediocre equilibrium if one were convinced that the Bayesian
Nash equilibriums of both mechanisms would actually be played.

This tension between efficiency of outcome allocations and stability
in the sense of uniqueness of Bayesian Nash equilibrium (and of
characterization of equilibrium in terms of strategic dominance) is
the specific topic of this paper. I review some basic concepts
of implementation theory in \sec c, and in \sec f I use this 
implementation framework to present a version of the Diamond-Dybvig
environment with aggregate risk. (The environment
that I study differs from Diamond and Dybvig's in having only finitely
many traders. I formulate this version both to introduce aggregate
risk in a natural and explicit way, and also to clarify the
formulation of the sequential-service constraint.) A 
naturally-defined mechanism makes it a dominant strategy for each
trader to communicate his type truthfully, and via this
dominant-strategy equilibrium it implements the symmetric, ex-ante
efficient allocation. That is, in sharp contrast to Diamond and Dybvig's
deposit-with-suspension mechanism, for this mechanism the distinction
between environments with and without aggregate risk is immaterial.
Finally, in \sec i, I consider the analogous allocation mechanism in
environments with aggregate risk and also a sequential-service
constraint. I show that, under the assumption that traders' utility
functions exhibit non-increasing absolute risk aversion, for traders
truthfully to communicate their types remains the unique strategy
profile that survives iterated elimination of strictly dominated
strategies. Thus, again, the mechanism has a unique Bayesian Nash
equilibrium that possesses an intuitively compelling stability
property, and the outcome of that equilibrium being played is the
symmetric, ex-ante efficient, equilibrium.

\Section c{Intermediation as an allocation mechanism}

One way of viewing a financial intermediary is as a trading club.  People
want to join such a club because features of the environment (including the
informational features that engender problems of ``adverse selection'' and
``moral hazard'') make arms-length transactions infeasible or
unsatisfactory. Instead, a trading club operates according to a charter that
specifies which trades are to be made as function of information provided by
members according to an explicitly defined protocol of communication and
negotiation.

\Subsection b{The environment of a Bayesian allocation mechanism}

Consider a formal representation of an environment where such an
intermediary would have a rationale. Let $\I = \{1, 2, \ldots, I\}$ be a set
of traders who live in a risky environment. The possible {\it states} of
this environment are the sample points of a probability space $(\Omega, \B,
\Pr)$. There is a measurable space of {\it ex-post allocations} which will
be denoted by $(\A, \cal A)$. A {\it state-contingent allocation} is
simply a $\B$-measurable function from $\Omega$ to $\A$. Denote the
set of such $\B$-measurable functions by $\A^\Omega$. If $\vec a \in
\A^\Omega$ and $\o \in \Omega$, then $\vec a(\o)$ is the ex-post
allocation that the state-contingent allocation $\vec a$ specifies for
state $\o$. (Henceforth a state-contingent allocation or an ex-post
allocation will often be called simply an allocation, when it is clear
from context which type of entity is being discussed.)

There is a set $\F \subseteq \A^\Omega$ of {\it feasible state-contingent
allocations.} (That is, $\F$ is a set of $\B$-measurable functions 
$f \colon \Omega \to \A$.) The specification of $F$ is supposed to
reflect both individual restrictions such as nonnegativity of
consumption and also aggregate restrictions such as materials balance.

This model will incorporate the {\it Harsanyi doctrine} that all traders are
Bayesian utility maximizers, and that moreover $\Pr$ characterizes the prior
beliefs common to all traders at ``birth.'' Typically the model is used to
understand the traders' behavior in ``adulthood'' after they have revised
their beliefs in light of experience. The experience of trader $i$ is called
his {\it type}, and is represented as a sub $\sigma$-algebra $\E_i$ of $\B$.
When an trader's type is described in terms of a $\B$-measurable random
variable which the trader is assumed observe, $\E_i$ will be taken to be the
smallest $\sigma$-algebra with respect to which the random variable observed by
$i$ is measurable. (Typically $\E_i$ is strictly smaller than $\B$ itself.)

In addition, assume that there is a sub $\sigma\/$-algebra $\E_0$ of
$\B$ that represents information that is directly usable for
allocation. That is, an allocation can be made contingent on this
information without traders having to reveal it. 

Each trader $i$ has a state-dependent utility function $u_i \colon \A \times
\Omega \to \Re$, and maximizes the expectation of this function conditional
on his type. Denote this conditional expectation by the function by $U_i
\colon \A^\Omega \times \Omega \to \Re$, which is defined by$\,$\note{See
Breiman (1968) or another graduate-level textbook of probability theory for
the definition of expectation conditional on a $\sigma$-algebra. In the
following definition, an trader's conditional expected utility is written
for notational convenience as depending on the entire ex-post allocation.
Actually, in the model to be studied here, an trader $i$'s own consumption
will be the only aspect of the allocation that matters for the determination
of $i$'s utility.}
%
\display b{U_i(\vec a, \o^*) = \mean \bigl[ u_i(\vec a(\o), \o) | \E_i
\bigr](\o^*).}
%

\Subsection d{Specification of an allocation mechanism}

An allocation mechanism is specified in terms of two structures, a
communication protocol and an allocation rule. The allocation rule is a
function which determines an ex-post allocation on the basis of the data
generated by traders' use of the communication protocol.

A communication protocol is described formally in terms of a finite
{\it message space} $M$. Each trader $i$ chooses, on the basis of his
type, a message $m_i \in M$ to send. As a function of the state of the
environment, then, trader $i$'s message is an $\E_i$-measurable
function $\mu_i \colon \Omega \to M$. This function $\mu_i$ will be
called $i$'s {\it communication strategy}.  When each trader follows
his communication strategy in state $\o$, a profile $\mu(\o) =
(\mu_1(\o), \ldots, \mu_I(\o))$ is generated which can be used as an
informational basis for allocation. 

Thus the {\it allocation rule} of the mechanism is a measurable function
%
\display a{\alpha \colon \Omega \times M^I \to \A}
%
such that
%
\display j{\forall m \in M^\I \enspace \alpha(\o, m) \hbox{ is
$\E_0\/$-measurable \quad and \quad} 
\forall \mu \enspace\exists f \in \F \enspace \forall \o \enspace 
f(\o) = \alpha(\o, \mu(\o)).}
%
(The domain of quantification of $\mu$ is the set of all
communication-strategy profiles. The function $\alpha$ must be
restricted in the way specified by \eqn j, in order to guarantee that the 
mechanism will always determine a feasible allocation regardless of which
communication strategies traders choose.) An allocation rule $\alpha$
and a communication-strategy profile $\mu$ together determine an
allocation $\alpha \circ \mu \in \F$.

An equilibrium (specifically a {\it Bayesian Nash equilibrium}) of the
allocation mechanism $(M, \alpha)$ is a communication-strategy profile
$\mu^*$ such that, for any trader $i$ and any profile $\mu$ that $i$ can
obtain by unilaterally changing his communication strategy while others'
strategies remain the same, $U_i(\alpha\circ\mu, \o) \le
U_i(\alpha\circ\mu^*, \o)$ almost surely.

If $\mu^*$ is an equilibrium of $(M, \alpha)$ and $\vec a =
\alpha\circ\mu^*$, then $\vec a$ will be called an {\it implementable
allocation} of $(M, \alpha)$. Let $\IA$ denote the set of every
state-contingent allocation such that there exists a mechanism that
implements it. An allocation $\vec a \in \IA$ is {\it efficient} if
%
\display c{\begin{array}{r c l} \forall \; \vec c \in \IA && \biggl\{ 
\exists \; i \in \I \enspace \Bigl\{ \mean \bigl[ u_i(\vec c(\o), \o) \bigr] <
\mean \bigl[ u_i(\vec a(\o), \o) \bigr] \Bigr\} \\ && \hbox{\quad or \quad }
\forall \; i \in \I \enspace \Bigl\{ \mean \bigl[ u_i(\vec c(\o), \o) \bigr] =
\mean \bigl[ u_i(\vec a(\o), \o) \bigr] \Bigr\} \biggr\}. \end{array}} 
%
This definition conforms to the usual definition of ex-ante Pareto
efficiency (cf.~Myerson, 1991).

Consider the allocation that, in each state of nature, maximizes the sum of
traders' utilities. Typically this allocation will not be implementable, so
it cannot be efficient. However, this maximizing allocation is necessarily
efficient if it is implementable.

\lemma c{Suppose that $\vec a$ is implementable and satisfies 
%
\display l{\sum_{i \in \I} \mean [u_i(\vec a(\o), \o)] 
= \max_{\vec c \in \F} \sum_{i \in \I} \mean [u_i(\vec c(\o), \o)].}
%
Then $\vec a$ is efficient.}

\proof It follows immediately from \eqn l that $\sum_{i \in \I}
\mean [u_i(\vec a(\o), \o)] \ge \sum_{i \in \I} \mean [u_i(\vec c(\o), \o)]$
for every $\vec c \in \IA$. This means that if $\mean [u_i(\vec a(\o), \o)]
\le \mean [u_i(\vec c(\o), \o)]$ for every $i \in \I$, then $\mean [u_i(\vec
a(\o), \o)] = \mean [u_i(\vec c(\o), \o)]$ for every $i \in \I$. That is,
the condition \eqn c defining efficiency must hold. \endproof

\uppercase \lem c has the following, immediate corollary.

\lemma a{Suppose that $\vec a$ is implementable and satisfies 
%
\display k{\forall \o \in \Omega \quad \sum_{i \in \I} u_i(\vec a(\o),
\o) = \max_{a \in F(\o)} \sum_{i \in \I} u_i(a, \o).}
%
Then $\vec a$ is efficient.}

\Section f{Banking---A schematic model}

Bryant (1980) and Diamond and Dybvig (1983) introduce models of
banking which Jacklin (1987) simplifies further to study
capital-structure issues.\note{Jacklin deliberately neglects the {\it
sequential service constraint}, which Diamond and Dybvig discuss
informally and which Wallace (1988) formalizes and analyzes. Wallace
emphasizes that a serious treatment of this constraint shows the
institutional arrangement of deposit insurance as modelled by Diamond
and Dybvig to be infeasible.} Now I formulate a finite-trader version
of Jacklin's maturity-transformation model.

\Subsection g{An environment where a maturity-transforming intermediary has
a role}

Define $\Omega$ and $\Pr$ by
%
\display d{\Omega = \{0, 1\}^\I \hbox{\quad and\quad}
\forall \o \enspace \Pr(\o) = 2^{-I},}
%
and define $\E_0$ and $\E_i$ by
%
\display e{\E_0 = \bigl\{\emptyset, \Omega \bigr\}
\hbox{\quad and \quad } \forall i \in \I \enspace 
\E_i = \bigl\{\emptyset, \Omega, \{\o | \o_i = 0\}, \{\o |
\o_i = 1\}\bigr\}.}
%
Suppose that there is an aggregate endowment of one unit of a good per
person, which can be transformed into a consumption good available at
either date 1 or date 2. The transformation is simply storage until
date 1, but whatever is not consumed at date 1 is augmented by a gross
factor of $R > 1$ at date 2. Thus feasible ex-post allocations are the
elements of the set 
%
\display h{\A = \biggl\{a \colon \I \to \Re_+^2 \Bigm|
\sum_{i \in \I} \bigl[ a_1(i) + R^{-1} a_2(i) \bigr] \le I \biggr\}.}
%
and 
%
\display i{\F = \A^\Omega.}
%

A trader's utility from allocation $a$ in state $\o$ is given by a
function $v \colon \Re_+ \to \Re$ of a consumption aggregate which includes
consumption at both dates if $i$ is of type 1, but which consists of
consumption at date 1 alone if $i$ is of type 0.\note{This formulation
follows Diamond and Dybvig. Jacklin also considers a utility formulation in
which both types of trader receive positive marginal utility from
consumption of each date, but in which type-0 traders discount consumption
at date 2 more heavily than type-1 traders do.} That is,
%
\display n{\forall i \quad\forall \o \quad u_i(a, \o) = 
v \bigl(a_1(i) + \o_i a_2(i)  \bigr).} 
%
Assume that 
%
\display r{\vbox{
\hbox{$v$ is strictly increasing, continuously twice 
differentiable and strictly concave; $\vphantom{\lim_\gamma}$}
\hbox{$v$ satisfies the Inada conditions $\lim_{\gamma \to 0}
v'(\gamma) = \infty$ and $\lim_{\gamma \to \infty} v'(\gamma) = 0$;}
\hbox{$\forall \gamma \enspace  \gamma v''(\gamma) / v'(\gamma) \le -1$
\qquad (Relative risk aversion $\ge 1$ everywhere).}}} 
%

Consider the problem of choosing $\vec a \in \F$ to maximize the sum of
traders' expected utilities, if the allocation could be made measurable in
the traders' types (that is, the ``fully-informed utilitarian social
planner's problem''). By strict concavity of $v$, Jensen's inequality, and
the fact that $R > 1$ while consumption goods at the two dates are perfect
substitutes for type-1 traders, the following conditions should hold. In
each state $\o$, all type-0 traders  should receive identical consumption
bundles $(c_0(\o), 0)$ and all type-1 traders should receive identical
consumption bundles $(0, c_1(\o))$. Letting  $\theta(\o) =  \sum_{i \in I}
\o_i$ as in the preceding example, each ex-post allocation $a = \vec a(\o)$
should satisfy the following two equations (a first-order condition and a
feasibility condition derived from \eqn h, respectively).
%
\display o{v^\prime (c_0(\o)) = R v^\prime (c_1(\o))}
%
and
%
\display p{\bigl[ I - \theta(\o) \bigr] c_0(\o) + R^{-1} \theta(\o)
c_1(\o) = I.} 
%
These two equations determine $\vec a(\o)$ uniquely. It is evident
that $c_0(\o)$ and $c_1(\o)$ depend on $\o$ only through $\theta(\o)$.
The following lemma explains the significance of the assumption
regarding relative risk aversion in \eqn r.

\lemma d{Suppose that $v$ satisfies the assumptions \eqn r, including 
that \enspace $\forall \gamma \; \gamma v''(\gamma) / v'(\gamma) \le
-1$. (Relative risk aversion $\ge 1$ everywhere.) Then the allocation
$\vec a$ defined from \eqn o and \eqn p by 
%
\display F{\bigl[ \vec a(\o) \bigr]_i = \left( (1 - \o_i) \, c_0(\o), 
\; \o_i \, c_1(\o)\right)}
%
is efficient. The consumption level $c_1(\o)$ of type-one traders is a
nondecreasing function of $\theta(\o)$. More generally, let $\eta$ be
a real variable taking values in $(0,I)$ and consider the problem of
maximizing 
%
\display m{(I-\eta) v\left({\gamma \over I-\eta}\right) + \eta
v\left({R(I-\gamma) \over \eta}\right).}
%
The solution, parametrized by $\eta$, is a function $\Gamma(\eta)$
that satisfies
%
\display y{{d \over d \eta} {R(I-\Gamma(\eta)) \over \eta} \ge 0.}
%
}

\proof By \eqn o and \eqn p and the concavity of $v$, $\vec a$
satisfies the optimality condition \eqn k of \lem a. Therefore $\vec
a$ is efficient by \lem a.

To see that the more general monotonicity assertion of the present lemma
implies the more specific assertion regarding $c_1$, note that if $0 <
\theta(\o) < I$, then $c_0(\o) = \Gamma(\eta) / (I-\eta)$ and $c_1(\o)
= R(I-\Gamma(\eta)) / \eta)$ by \eqn o and \eqn p. This equivalence
can be extended to $\theta(\o) \in \{0, I\}$, in view of the Inada
conditions on $v$. (That is, defining $\Gamma(0) = I$ and $\Gamma(I) =
0$ extends the definition of $\Gamma$ on $(0, I)$ continuously.) 

Corresponding to \eqn o, the first-order condition for \eqn m is 
%
\display w{v'\left({\Gamma(\eta) \over I-\eta}\right) -
Rv'\left({R(I-\Gamma(\eta)) \over \eta}\right) = 0.}
%
Taking the derivative of \eqn w with respect to $\eta$ yields
%
\display x{\left[ \Gamma'(\eta) + {\Gamma(\eta) \over I-\eta} \right]
{v''(\Gamma(\eta) /(I-\eta)) \over I-\eta} + R^2 \left[ \Gamma'(\eta)
+ {I-\Gamma(\eta) \over \eta} \right] {v''(R(I-\Gamma(\eta))/\eta)
\over \eta} = 0.}
%

Now consider the derivative in \eqn y.
%
\display z{{d \over d \eta} {R(I-\Gamma(\eta)) \over \eta} = {-R \over
\eta} \left[ \Gamma'(\eta) + {I-\Gamma(\eta) \over \eta} \right].}
%
In order to prove the lemma by establishing \eqn y, then, it must be
shown that the bracketed expression in \eqn z is negative. This
expression is identical to one of the two bracketed expressions in
\eqn x, and clearly those two expressions must either have opposite 
sign or else both be zero, in order for \eqn x to hold. Thus the inequality 
%
\display A{\Gamma'(\eta) + {I-\Gamma(\eta) \over \eta} \le 0,}
%
which proves the lemma, is equivalent to
%
\display B{{I-\Gamma(\eta) \over \eta} \le {\Gamma(\eta) \over I-\eta}}
%
in view of \eqn x.

Inequality \eqn B follows from the assumption that \enspace $\forall
\gamma \; \gamma v''(\gamma) / v'(\gamma) \le -1$. To see this, note
that the assumption implies that 
%
\display C{{\partial \over \partial r} [r v'(rs)] \le 0.}
%
This inequality and equation \eqn w imply that 
%
\display D{v'\left({I-\Gamma(\eta) \over \eta}\right)
\ge v'\left({\Gamma(\eta) \over I-\eta}\right),}
%
which implies \eqn B by the concavity of $v$. \endproof

\Subsection h{A mechanism with a unique, efficient equilibrium}

Next I will show that conditions \eqn o and \eqn p imply that the
efficient allocation can be implemented by a truth-telling equilibrium
of an allocation mechanism analogous to that studied in the preceding
model of the oasis. The mechanism here possesses a property that the
mechanism in that other model lacks: that truth-telling is the {\it
strictly dominant strategy} for each trader. By definition, this
condition means that whether an trader is of type 0 or of type 1, he
receives a higher utility level from revealing his type truthfully
than from misrepresenting it---regardless of what reports other
traders give. It follows (cf.~Myerson, 1991) that the truth-telling
equilibrium is the unique Bayesian Nash equilibrium of the mechanism.
Therefore no alternative, inefficient, ``run'' equilibrium of this
mechanism can exist.

The mechanism with this dominant-strategy property is constructed
analogously to the mechanism that implements the efficient, symmetric
allocation in the oasis economy. After having characterized the
allocation that maximizes ex-ante expected utility (which has been
done already by deriving conditions \eqn o and \eqn p), the mechanism
is defined by depending on the truthfulness of traders' reports, and
using them as the basis for assigning traders the ex-post consumption
bundles determined by that allocation. Recall that, ordinarily, such a
straightforward approach would be unsuccessful because truth-telling
would not be a trader's equilibrium strategy. However, because of the
particular form \eqn n of the state-contingent utility function and
special features of the efficient, symmetric allocation, the approach
does work in this case.

\theorem b{Let $M = \{0, 1\}$ be the set of signals for each trader. 
Define $x \colon M \times \{0, \ldots, I\} \to \Re$ by 
the conditions (analogous to \eqn o and \eqn p) that 
%
\display t{v'(x(0, \eta)) = R v'(x(1, \eta))}
%
and
%
\display u{[I-\eta] x(0, \eta) + R^{-1}\eta x(1, \eta) = I.}
%
Define $\alpha \colon  \Omega \times M^I \to \A$ by
%
\display v{\bigl[ \alpha(\o, m) \bigr]_i = \left( (1 - m_i) \, x(m_i,
\sum_{j \in \I} m_j), \; m_i \, x(m_i, \sum_{j \in \I} m_j)\right) .}
%
The truthful communication strategy $\hat \mu_i(\o) = \o_i$ is the
strictly dominant strategy for each trader $i$. The mechanism thus
implements the efficient, symmetric allocation in strictly dominant
strategies, and consequently the profile of truthful communication
strategies is its unique Bayesian Nash equilibrium}

\proof If $\eta = \theta(\o)$, then conditions \eqn t and \eqn u on
$(x(0,\eta), x(1,\eta))$ are identical to conditions \eqn o and \eqn p
on $(c_0(\o), c_1(\o))$. \uppercase\lem d therefore implies that the
mechanism implements the efficient, symmetric allocation if the
profile of truthful communication strategies is a Bayesian Nash
equilibrium. 

By Myerson (1991), a profile of strictly dominant strategies for a
mechanism is the unique Bayesian Nash equilibrium of the mechanism. 
Therefore, to prove the lemma, it is sufficient to show that truthful
communicatinon is the strictly dominant strategy for each trader.  To
verify this, consider separately each of the two possible values of
$\o_i$. If $\o_i = 0$, then by \eqn t and \eqn v, $i$ will receive a
positive amount of consumption at date 1 if he sends message 0, but
will receive 0 consumption at date 1 if he sends message 1. Because he
has utility only for consumption at date 1 (by the definition \eqn n
of his utility function), and because his utility is strictly
increasing in the amount of this good that he consumes (by \eqn n and
\eqn r), he strictly prefers to send message 0 rather than message 1
in state $\o$.

Now consider the alternative case that $\o_i = 1$. The strict
concavity of $v$ assumed in \eqn r, together with \eqn t, implies that
%
\display q{x(1, 0 + \sum_{j \neq i} \mu_j(\o)) > 
x(0, 0 + \sum_{j \neq i} \mu_j(\o))}
% 
regardless of which communication strategies $\mu_j$ the other traders use.
By \eqn y of \lem d and the fundamental theorem of calculus,
%
\display E{x(1, 1 + \sum_{j \neq i} \mu_j(\o)) \ge
x(1, 0 + \sum_{j \neq i} \mu_j(\o)).}
%
Therefore, given the functional form of $i\/$'s utility function
(equations \eqn n and \eqn r) and the specification of the mechanism
(equation \eqn v), inequalities \eqn q and \eqn E together imply that
trader $i$ must strictly prefer to send message 1 rather than message
0.
\endproof

\Section i{Banking in an environment with sequential service}

The schematic model of banking studied above abstracts from an
important feature of an actual bank: that traders do not all contract
the bank at the same time, and that the bank must deal promptly with
traders who contact it early. The bank therefore is constrained from
making its treatment of those traders contingent on information yet to
be provided by later traders, especially if the early traders wish to
make withdrawals. This feature plays an important role in Diamond and
Dybvig's (1983) intuitive discussion of their model, and it is
formalized by Wallace (1988) who derives further consequences from it.
In view of the striking discepancy between \thm b and Diamond and
Dybvig's analyisis, and of the closer analogy between the theorem and
Jacklin's (1987) analysis that also abstracts from the
sequential-service constraint, it is a salient question whether or not
\thm b can be extended to an environment with sequential service. Now
I investigate this question and find an answer that is more or less in
the affirmative. Specifically, if $v$ satisfies non-increasing
absolute risk aversion as well as the conditions specified in \eqn r,
then the profile of truthful communication strategies is the unique
profile that survives iterated elimination of strictly dominated
strategies. It follows that, as in \thm b, this is the unique Bayesian
Nash equilibrium of the natural mechanism that implements the
efficient allocation. 

In the present formalization of the sequential-service constraint,
every trader contacts the bank at some time during date 0, these
``arrival times'' for different traders are stochastic and
independently distributed, and each trader's arrival time is in his
own information set. This last detail is crucial, for it implies that
a trader who arrives very late can be almost certain that he is the last
trader to arrive. Conditional on being last, truthful communication is
the trader's unique utility-maximizing action. That is, any strategy
that involves some untruthful communication by a trader when he arrives
very late can be eliminated as being dominated by the strategy that
agrees with it except at very late times, but that specifies truthful
communication at those times. This result can then be ``bootstrapped''
to apply to communication at earlier times as well. One should note
that, in Wallace's formalization of sequential service, a trader's
time of arrival is not in his own information set. Under Wallace's
assumption, it seems that iterated elimination of strictly dominated
strategies may not lead necessarily to truthful communication.

\Subsection j{Formalization of sequential service}

Modelling sequential service requires that the maturity-transformation
model must be modified by enlarging the state space $\Omega$ to
represent information about arrival times, and by making corresponding
changes in the definitions of agents' types and of feasible allocations.

To enlarge the state space, replace the definition \eqn d by
%
\display G{\vbox{
\hbox{$\Omega = \{0,1\}^\I \times [0,1]^\I$, and $p \in (0,1)$;}
\hbox{For all $i \le I$ \enspace $\Pr(\o_i = 1) = p$;}
\hbox{For all $i \le I$ \enspace $\o_{I+i}$ is uniformly distributed;}
\hbox{The projections of $\o$ on its coordinates are independent
r.v.'s.}
}}
%
Replace the definition in \eqn e of agent $i\/$'s type by
%
\display H{\E_i = \bigl\{ \{\o|\o_i = 0 \hbox{ and } \o_{I+i} \in A\} 
\cup \{\o|\o_i = 1 \hbox{ and } \o_{I+i} \in B\} | 
A \in {\cal F} \hbox{ and } B \in {\cal F} \bigr\},}
%
where $\cal F$ is the $\sigma\/$-algebra of Borel sets on $[0,1]$.
That is, in each state a trader knows his own utility function and his
own arrival time at the bank, but he knows nothing about the other
traders.

Also replace the specification in \eqn e that the algebra $\E_0$ is
trivial by the following definition, which intuitively specifies that
information about all traders' arrival times may be used directly
(that is, without having to be revealed by the traders' communication)
as a basis for allocation.
%
\display Q{\E_0 = \bigl\{ {\o | \forall i \le I \enspace \o_{I+i} \in
A_i} | A_1 \in {\cal F},\ldots, A_I \in {\cal F} \bigr\} }
%

In order to formulate the sequential service constraint, define the
arrival-order statistics by $\order \colon \{0, \ldots,I\} \times \Omega
\to \I$. That is, $\order(1,\o), \order(2,\o), \ldots \order(I,\o)$ 
are the first, second, $\ldots I\/$th traders in order of arrival
determined by the coordinates $\o_{I+1}, \o_{I+2}, \ldots \o_{2I}$ of
$\o$. Ties can be assumed to be broken arbitrarily in the 
zero-probability event that several traders arrive simultaneously at
the bank. Define the rank statistics $\rank \colon \I \times
\{0, \ldots,I\} \times \Omega \to \{0, \ldots,I\}$, which are inverse
to the order statistics in each state of nature, by $\rank(\order(i,
\o), \o) = i$.

Suppose that $\vec a = ((X^1_0, X^1_1), \ldots, (X^I_0, X^I_1)) \in
\A^\Omega$ is an allocation.\note{In this section, since $\Omega$ is a
continuum, $\A^\Omega$ denotes the set of Borel-measurable functions
from $\Omega$ to $\A$.} The intuitive content of the sequential
service constraint is that the mechanism represents a financial
intermediary (call it a bank) operating at a specific location that
the trader visits at some time during date 1. When trader $i$ visits,
he communicates a message $m \in M$ determined by a communication
strategy $\mu_i$ that is measurable with respect to $\E_i$, and he
then receives $X^i_0(\o)$ immediately. This quantity thus must not
depend on information from traders who arrive later in state $\o$ than
$i$ does, since those traders have not yet communicated their
information to the bank. Since all traders are envisioned to arrive at
the bank at some time before date 2, when the consumption amounts
$X^j_1(\o)$ are distributed, those date-2 quantities are not
analogously constrained.

That is, the amount $X^{\order(1,\o)}_0(\o)$ of consumption given to
trader $\order(1, \o)$ at date 1 must depend only on the identity of
$\order(1,\o)$ and the time $\o_{I+\order(1,\o)}$, both of which the
bank observes, and on that trader's utility parameter
$\o_{\order(1,\o)}$, which he has the opportunity to communicate to
the bank. (Whether or not he actually does communicate his utility
parameter in equilibrium is irrelevant to the formulation of this
constraint, which expresses the limitation imposed by the exogenous
sequential nature of the {\it opportunities} for the bank to acquire
information.) Next, the information that the bank can use to determine
the date-1 consumption of the second trader to arrives consists of
both this information about the first trader, which the bank
remembers, and also the corresponding information about the second
trader himself. And so forth. Formally, $((X^1_0, X^1_1),
\ldots, (X^I_0, X^I_1))$ satisfies the {\it sequential service
constraint} if 
%
\display I{\forall i \quad X^{\order(i,\o)}_0 = \mean \left[
X^{\order(i,\o)}_0 | \order(1,\o), \ldots, \order(i,\o),
\o_{\order(1,\o)}, \ldots, \o_{\order(i,\o)}, \o_{I+\order(1,\o)},
\ldots, \o_{I+\order(i,\o)} \right].} 
%
In view of this constraint, the definition of $\F$ should be replaced
by 
%
\display J{\F = \left\{ \vec a | \vec a \in \A^\Omega \hbox{ and $\vec
a$ satisfies \eqn I} \right\}.}
%

\Subsection k{The efficient, symmetric, state-contingent allocation}

In this section, I consider the solution of the optimization problem
posed in equation \eqn l, that is,
%
\[\hbox{Maximize } \sum_{i \in \I} \mean [u_i(\vec c(\o), \o)] 
\hbox{ subject to } \vec c \in \F,\]
%
with $\F$ defined by \eqn J. Subsequently I will consider the problem
of implementing this allocation, which is efficient by \lem c.

The key to solving problem \eqn K is the observation, formalized below
in \lem e, that the arrival-order statistics $\order(i,\o)$ provide all of
the relevant information about traders' arrival times. More precise
arrival-time information is relevant neither to traders'
enjoyment of utility nor to the technical feasibility of allocations
in the sequential service environment.\note{Each trader will use his
information about his precise arrival time to make inference about his
probable rank in the arrival queue (which he does not observe
directly) though, so this information is relevant to implementation.}
In view of this observation, define mappings $\s^i \colon \Omega \to
\{0,1\}^i$ for $1 \le i \le I$ by 
%
\display L{\forall j \le i \quad \s^i_j(\o) = [j](\o).}
%
Define the set of 0--1 sequences of length at most $I$, including the
null sequence, as $\S$. For $s \in \S$, let $\l(s)$ denote the length
of $s$. For $i \le I$, define $0^i$ to be the sequence consisting of
$i$ consecutive zeros. Define the weak and strict extension-ordering
relations on $\S$ by 
%
\display K{\vbox{
\hbox{$r \le s \quad \iff \quad \l(r) \le \l(s) \hbox{ and } 
\forall i \le \l(r) \enspace [r_i = s_i]$;}
\hbox{$r < s \quad \iff \quad \l(r) < \l(s) \hbox{ and } 
\forall i \le \l(r) \enspace [r_i = s_i]$.}
}}
%

\lemma e{Suppose that $\vec a = ((X^1_0, X^1_1), \ldots, (X^I_0, X^I_1))$
solves problem \eqn l in the sequential-service environment. Then
there exists a vector $x \in \Re^\S_+$ such that 
%
\display M{\forall s \in \S \quad \left[ s_{\l(s)} = 1 \Longrightarrow
x_s = 0 \right]}
%
and, almost surely for all $i$,
%
\display N{X^{\order(i,\o)}_j = \left\{\begin{array}{l@{\quad :\quad }l}
\case{x_{\s^i(\o)}}{j = 0 \hbox{ and } 
\s^i_i(\o) = \o_{\order(i,\o)} = 0;}\\
\case{{R \over \theta(\o)} \left( I - \sum_{r \le \s^I(\o)} x_r \right)}
{j = 1 \hbox{ and } \s^i_i(\o) = \o_{\order(i,\o)} = 1;}\\
\case{0}{\hbox{otherwise.}}
\end{array} \right. }
%
If $v(0) = 0$ and $\theta^*(s) = \sum_{i<\l(s)} s_i$ and $\pi(s) =
p^{\theta^*(s)}(1-p)^{\l(s) - \theta^*(s)}$, and if $\vec a$
and $x$ are related according to \eqn N, then
%
\display O{\sum_{i \in \I} \mean \left[ u_i(\vec a(\o), \o) \right] 
= \sum_{\hbox{
\begin{tiny}\vbox{\hbox{$\l(s) = I$}\hbox{$\theta^*(s) > 0$}} \end{tiny}}}
\pi(s) \left[ \left( \sum_{r \le s} v(x_r) \right)
+ \theta^*(s) v \left(
{R \over \theta^*(s)} \left( I - \sum_{q \le s} x_q \right)
\right) \right] + \pi(0^I)v(x_{0^I}).}
%
}

\proof One can alternatively characterize $\vec a$ in terms of a
vector of random variables \xxx $((Y^1_0, Y^1_1), \ldots, (Y^I_0, Y^I_1))$,
where $Y^i_j(\o) = X^{\order(i,\o)}_j(\o)$ a.s.~for each $i$ and $j$.
Consider the state-contingent allocation $\vec c = ((Z^1_0, Z^1_1),
\ldots, (Z^I_0, Z^I_1))$, defined by $Z^{\order(i,\o)}_j(\o) = \mean
[Y^i_j(\o) | \sigma^i]$ a.s. It is easily verified that $\vec c \in
\F$, and for every $i$, $\mean [u_i(\vec a(\o), \o)] \le
\mean [u_i(\vec c(\o), \o)]$, with strict inequality for at least one
$i$ if $\vec a \neq \vec c$. (This inequality must hold because $v$ is
strictly concave and $\vec c$ is obtained by taking conditional
expectation with respect to $\vec a$.) That is, $\vec a \neq \vec c$
would contradict the hypothesis of the lemma. By construction, $\vec
c$---that is to say, $\vec a$---can be characterized in terms of a
vector $x \in \Re^\S_+$. This vector must actually satisfy \eqn N, by
the same considerations that prove the efficiency assertion in \lem d. 
(Note that, in the context of \eqn N, condition \eqn M states that
traders of type 1 consume exclusively at date 2.) Condition \eqn O is
verified by straightforward computation. \endproof

By \lem e, a solution to optimization problem \eqn l can be found by
optimizing over a set of vectors in $\Re^\S_+$. Specifically, given
the strict concavity of the right side of \eqn O, a solution is
characterized by a vector that satisfies the first-order conditions
for optimization of \eqn O subject to the constraint \eqn M. That is,
the following lemma holds.

\lemma f{A necessary and sufficient condition for a state-contingent
allocation \xxx $\vec a = ((X^1_0, X^1_1), \ldots, (X^I_0, X^I_1))$ to
solve problem \eqn l in the sequential-service environment is that 
there should exist a vector $x \in \Re^\S_+$ that satisfies \eqn M,
\eqn N, and for all $r \in \S$ such that $r_{\l(r)} = 0$,
%
\display P{\pi(r) v'(x_r) - R \left[ \sum_{\hbox{\begin{tiny}
\vbox{\hbox{$\l(s) = I$}\hbox{$\theta^*(s)>0$}\hbox{$r \le s$}}
\end{tiny}}} \pi(s) v' \left({R \over \theta^*(s)} 
\left( I - \sum_{q \le s} x_q \right) \right) \right] 
+ 0^{\theta^*(r)} \pi(0^I) v'(x_{0^I}) = 0.
}
%
}

\Subsection l{A mechanism with a unique, efficient equilibrium}

The first-order condition \eqn P just derived for the
sequential-service environment has analogous structure to the
first-order condition \eqn o in the simultaneous-communication
environment studied in \sec f. Lemma 1, which provides the key to
establishing dominant-strategy implementability of the symmetric,
efficient allocation in that environment, is proved by examining
condition \eqn o. An analogous result is provable on the basis of
condition \eqn P, and it also leads to an implementability result.

\theorem g{Suppose that $v$ satisfies condition \eqn r and also the
condition that 
%
\display R{\forall \gamma \enspace 
{d \over d \gamma} {v''(\gamma) \over v'(\gamma)} \ge 0 \qquad
\hbox{(Absolute risk aversion non-increasing everywhere).}}
%
Let $M = \{0, 1\}$ be the set of signals for each trader. 
Let $x \colon \S \to \Re_+$ be the vector satisfying the optimality
conditions \eqn M and \eqn O. Define $\alpha \colon \Omega \times M^I
\to \A$ by 
%
\display S{\bigl[ \alpha(\o, m) \bigr]_i = \left( (1 - m_i) \, 
x(m_{\order(1, \o)}, \ldots , m_{\order(\rank(i, \o), \o)}),
\; m_i \, {R \over \sum_{j \le I} m_j} \left( \sum_{j \le I} 
x(m_{\order(1, \o)}, \ldots , m_{\order(j, \o)}) \right) \right) .}
%
Then the profile of truthful-communication strateies $\hat \mu_i(\o) =
\o_i$ is the unique profile that survives iterated elimination of
strictly dominated strategies. The mechanism thus implements the 
symmetric, ex-ante efficient by a unique Bayesian Nash equilibrium.}

\Section Z{References}

\parindent=0pt \parskip=\bigskipamount

Bryant, J., ``A model of reserves, bank runs, and deposit insurance,''
{\sl Journal of Banking and Finance}, 1980.

Diamond, D., and P.~Dybvig, ``Bank Runs, Deposit Insurance, and
Liquidity,'' {\sl Journal of Political Economy}, 1983.

Jacklin, C., ``Demand deposits, trading restrictions, and risk
sharing,'' in E.~Prescott and N.~Wallace, eds., {\sl Contractual
Arrangements for Intertemporal Trade}, 1987.

Myerson, R., {\sl Game Theory: Strategy and Cooperation}, 1991.

Wallace, N., ``The Diamond-Dybvig model with sequential service taken
seriously,'' {\sl Quarterly Review of the Federal Reserve Bank of
Minneapolis}, 1988.

\end{document}
