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\ctrline{\bf WILSON'S THEORY OF SYNDICATES}
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\ctrline{\bf REVISITED}
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\ctrline{by}
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\ctrline{Amin H.\ Amershi (Stanford University)}
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\ctrline{Jan Stoeckenius (Stanford University)}
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\noindent February 1981 \par
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\noindent$\null↑*$Comments welcome.  Please do not quote.\par
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\noindent{\bf I. INTRODUCTION AND COUNTEREXAMPLE}\par
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In a path-breaking paper, Arrow [1953] presented the first effective
analysis of exchange markets under uncertainty.  Borch [1962]
derived general
results for \po\ risk sharing. In a seminal paper, Wilson [1968]
addressed the following question:
Suppose a group is comprised of $N$ individuals who obey the Savage
axioms, and suppose the group \po ly shares risk.  When can the group's
joint decision be modeled {\sl as if} the group were a single
individual (we will make precise the {\sl as if} part of this
statement below).  Wilson demonstrated that that this condition was
equivalent to a separability condition on the Lagrange multiplier in
Borch's first order conditions for \po ity.  We will illustrate this by
two examples:\par 
\yyskip
\noindent {\bf Example 1: } Suppose two individuals with logarithimic
utilities are risk sharing over gambles defined on the open interval
$(0,1)$ (\ie, $U↓1(x) = U↓2(x) = \ln(x)$).  The first individual has
beliefs corresponding to the uniform distribution over the interval,
and the beliefs of the other correspond to the function $2s$ (\ie,
$f↓1(s) = 1$, $f↓2(s) = 2s$).  There are many possible \po\ risk
sharing contracts in this setting, each corresponding to a set of
Borch first order conditions. One of these sets of first order
conditions is:
$$U↓1↑\prime (z↓1(x,s))f↓1(s) = U↓2↑\prime(z↓2(x,s))f↓2(s) =
\mu↓0(x,s)$$ \par
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$${1 \over {z↓1(x,s)}} = {{2s} \over {z↓2(x,s)}} = \mu↓0(x,s),$$
where $z↓i$ is the \ith individual's share in the event that the
(group's) payoff is $x$ and the state is $s$ (note that $\Twosum
z↓i(x,s) = x$) and $\mu↓0$ is the Lagrange multiplier of
this set of first order conditions.  A solution to this problem is
$$z↓1(x,s)={x \over {1+2s}},\ \ z↓2(x,s)={{2xs} \over {1+2s}},\ \
\mu↓0(x,s) = {{1+2s} \over x}.$$
Note that $\mu↓0$ is separable (\ie, $\mu↓0(x,s) =
\theta(x)\phi(s)$), thus in this case the group acts as if it were a
single individual, with utility $2\ln(x)$ and beliefs ${1 \over 2} +
s$.\par
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\noindent {\bf Example 2: } Let the situation be as above but suppose
that the second individual has a square root utility function (\ie,
$U↓2(x) = \sqrt{x}$).  The first order condition then becomes
$${1 \over {z↓1(x,s)}} = {s \over {\sqrt{z↓2(x,s)}}} =
\mu↓0(x,s).$$
A solution to this expression is
$$\eqalignn{z↓1(x,s) ⊗= {{-1+\sqrt{1+4xs↑2}} \over {2s↑2}},\cr
           z↓2(x,s) ⊗= {{2xs↑2+1-\sqrt{1+4xs↑2}} \over 2s↑2},\cr
         \mu↓0(x,s) ⊗= {{2s↑2} \over {-1+\sqrt{1+4xs↑2}}}.\cr}$$
If $\mu↓0$ is separable, then
$${{\dbyd{\mu↓0}{x}} \over {\mu↓0}} = {{\theta↑\prime(x)\phi(s)}
\over {\theta(x)\phi(s)}} = {{\theta↑\prime(x)} \over {\theta(x)}}$$
is independent of $s$.  In the above case, however,
$${{\dbyd{\mu↓0}{x}} \over {\mu↓0}} = {{2s↑2} \over
{1+4xs↑2-\sqrt{1+4xs↑2}}}$$
is clearly not independent of $s$, so the group can not be regarded as
a single individual.\par
\yyskip
The above example makes clear that, at least {\it a priori}, the
question of the separability of $\mu↓0$ is non-trivial.  Wilson
[1968] put forth a general result of great importance in the study of
linear financial instruments, which we have reproduced below.\par
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\noindent{\bf THEOREM} (Wilson [1968], Theorem 6):\quad If the members
of a group have heterogeneous beliefs, then the group acts jointly
{\sl as if} it were an individual Savage rational decision maker if
and only if for each state the group's \rsa\ is linear in wealth.
(equivalently, the Lagrange multiplier $\mu↓0$ corresponding to a
\po\ contract involving wealth sharing rules $z↓i$ is separable iff
for each $s$ the \sr s $z↓i(\cdt,s)$ are linear in the group payoff $x$).\par
\yyskip
The {\sl if} part of this theorem is correct, but, as will be shown in the
following counterexample, the {\sl only if} part is not correct in its
present form; \ie, linear sharing rules are sufficient but not
necessary for the separability of $\mu↓0$.\par
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\noindent {\bf Counterexample to Theorem:\quad}Consider the three member
group $G = \{ 1,2,3 \} $ composed of Savage rational decision makers
with utilities $U↓1$, $U↓2$, and $U↓3$, respectively.  Let the
state space of interest be the open interval from zero to one, \ie\ $S =
(0,1)$.  Let the beliefs of the \ith decision maker be denoted by the
probability density function $f↓i$, and suppose \par
\yyskip
\ctrline{\vbox{\baselineskip 15pt \lineskip 3pt \lineskiplimit 3pt
          \halign{$\lft{#},$\qquad \qquad ⊗ $\lft{#}$\cr
          U↓1(x) = {1 \over 2} \ln (x) ⊗ f↓1(s)=1,\cr
          U↓2(x) = {1 \over 2} \ln (x) ⊗ f↓2(s)=2s,\cr
      U↓3(x) = 2\sqrt{x}           ⊗ f↓3(s)={1 \over 2} + s.\cr}}}
\yyskip
\noindent Set
$$\eqalignn{z↓1(x,s) ⊗= {x \over {(1+2s)(1 + \sqrt{1 + 4x})}},\cr
           z↓2(x,s) ⊗= {{2xs} \over {(1+2s)(1 + \sqrt{1+4x})}},\cr
           z↓3(x,s) ⊗= {{2x↑2} \over {1 + 2x + \sqrt{1+4x}}}.,\cr}$$
We note that
$$\eqalignn{U↓i↑\prime(z↓i(x,s))f↓3(s) ⊗= {{1+2s} \over 2} \
                                             {{1+\sqrt{1+4x}}\over x} \cr
                                         ⊗= \mu↓0(x,s)\ 
                                                 \hbox{for i = 1,2,3,}\cr}$$
hence the specified $z↓i$'s form a \po\ \sr, and as $\mu↓0$ is
separable the group acts jointly as if it were an individual.
However, the \rsa\ is clearly non-linear.\par
Before discussing the significance of this result, it is necessary to
consider more closely the meaning of the statement that the group acts
(or can be modeled as acting) {\sl as if} it were an individual.  The
aspect of the group that we wish to model is its choice over joint
actions (projects).  However, it is well known (see Krantz {\it et.\
al.\ }[1971]) that a group need not possess a complete order over all
projects (maps from states to outcomes) in spite of the fact that each
member of the group possesses such an order.  Nevertheless, the group
does possess at least a partial order over projects, this being the
Pareto order (defined in the usual way; for details see section II
below).  Any ordering the group may possess, if it is to be
economically reasonable (recall we are ignoring externalities) must
``agree with'' this basic partial order; \ie, if $a$ (a project) is
Pareto-superior to $a↑\prime$, then $a$ must be preferred by the group
to $a↑\prime$.  We call such orders {\bf
Pareto-inclusive}.\par
We may, therefore, ask either one of two reasonable questions:\par
\yyskip
\noindent 1.  For a given group, when does there exist a complete order
on projects which is Pareto-inclusive and may be represented via a
group utility function and a group probability distribution.\par
\yyskip
\noindent 2.  If we suppose the group possesses a complete order over
projects, then when can this order be represented via a group utility
function and a group probability distribution.\footnote{More
generally, one could suppose the group possesses a Pareto-inclusive
partial order and ask when there exists a decomposable total order which is
inclusive of this partial order.  Little seems to be gained, however,
by going to this generality.}\par
\yyskip
These two questions are different for all groups of more than one
member.  The first question can be rephrased in the context of a
centrally planned economy.  Its answer determines when a (Savage
rational) dictatorial decision making mechanism can be employed by the
group without violating \po ity.  There may be several different
dictators who fulfill the required role, each with a different
complete ordering.  Furthermore, the group may possess a complete
ordering that is different from any of these orderings.  However, for
those situations in the answer set to question one, it is possible to
say that the group behaves as if it was a Savage rational individual
over certain sets of projects.  We will show below that this is
sufficient to derive results concerning aggregation and separation
properties of the economy.\par
     The answer set of the second question is clearly a subset of the
first.  The interpretation given to the term {\sl as if} by the second
question is the standard one employed in axiomatic decision theory.
For situations within its answer set one can say that the group may in
all decisions be replaced by a composite individual.  However, the
economic analysis of the two questions is essentially identical.
Therefore, which interpretation one takes in understanding what
follows is a matter of taste and not of any immediate economic
significance for our results.\par
Returning to the counter-example, its significance stems from its implications
about the manner in which syndicates form.  As will be shown
below, syndicates can be built up from smaller syndicates, and each of
these subsyndicates appears to be a single individual to all other
members of the large syndicate.  If Wilson's result were correct,
syndicates could form in a arbitrary manner, \ie, any partition of the group
could be regarded as dividing it into subsyndicates.  In effect, to
each member the syndicate would appear to be a two-member
organization, comprised of himself and the subsyndicate formed by all
the other members.\par
What the counter-example demonstrates is that there exist syndicates
which are not arbitrarily decomposable.  In the example, individuals 2
and 3 could not, by themselves, form a syndicate.  Thus individual 1
could not regard the group as being composed of two members.  When
negotiating the optimal sharing rule he must consider the utilities
and beliefs of {\sl both} the other individuals.\par
An alternate way to consider this problem is to imagine the difficulty
involved in soliciting participation in a syndicate.  If all
syndicates were arbitrarily decomposable, then any {m-member} subgroup could
solicit additional members by issuing a prospectus containing its
joint beliefs and joint utility function.  Any individual who would
seek to join this syndicate would need no additional information.
However, as the example shows, syndicates are not always arbitrarily
decomposable.  If individuals 2 and 3 were to form a group, the
prospectus they issue to 1 would have to list more information, in
spite of the fact that once individual 1 joined, the three-member
group would form a syndicate.\par
In the following section we present a formal development of the theory
of syndicates, covering many of the points made above.  We will also
demonstrate a class of nondecomposable syndicates.  In addition we will prove
a weaker form of Wilson's theorem, to the effect that a group can form
a syndicate under an arbitrary set of beliefs iff its sharing rule
is always linear.\footnote{Wilson has suggested to us that one could
make a stronger statement, to the effect that if a group forms a set
under any set of beliefs that is {\sl not} nowhere dense (in some
appropriate sense) then the group has linear sharing rules.  At an
intuitive level this statement appears correct, but for the moment it
must remain a conjecture.}\par
In Section III below we investigate an issue that has been studied in
a separate branch of the literature, namely separation and aggregation
phenomena of state-contingent securities markets (see, \eg, Rubinstein
[1974], Brennan and Kraus [1978]).  Although the links between these
phenomena and the syndicate results have been alluded to, a formal
investigation of these links has never been carried out.  We show that
there are indeed such links, and demonstrate that the study of (strong
form) {\sl unanimity} in syndicates (see, \eg, Wilson [1968], Rossing
[1974], Ekern and Wilson [1974]) is equivalent to the aggregation
results.  We also extend the aggregation results in several directions
and pose some new questions.\par