%Paper: ewp-game/9405002
%From: wilhelm@neuefeind.wustl.edu (Wilhelm Neuefeind)
%Date: Tue, 17 May 94 10:22:33 CDT

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\begin{document}

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\newtheorem{defi}{Definition}
\begin{titlepage}
\quad
\vskip2cm
\begin{center}
  Continuous Linear Representability of\\
  Binary Relations*\\ \quad \\
 

  {\sc Wilhelm Neuefeind} \\
           {\it    Washington University in
               Saint Louis}\\
  {\sc Walter Trockel**}\\
             {\it  IMW, 
               Universit\"{a}t Bielefeld}\\ \quad \\
\end{center}\vskip2cm

\noindent
{\bf Running title:} Representability of Relations

\vskip1cm

\noindent Send correspondence to:\\
Professor Wilhelm Neuefeind \\
Department of Economics, Box 1208\\
Washington University\\
St. Louis, MO 63130-4899
\vskip1cm
\vfill


* Thanks are due for helpful comments to Manfred Nermuth, David Schmeidler,
Antonio Villar, and anonymous referees.

** Financial support from the DFG under Tr 120/4-2 is gratefully acknowledged.
\end{titlepage}
\newpage

\noindent {\bf Summary:} A very general result on continuous
linear representability of binary relations on topological
vector spaces is presented. Applications of this result
include individual decision making under uncertainty, i.e.
expected utility theory and collective decision making, 
in particular, utilitaristic social welfare functions.
\vskip.5cm

\noindent {\it Journal of Economic Literature} Classification Numbers:  D71,
D81
\vskip.5cm 
\section{Introduction}
\pagestyle{plain}

This paper presents a very general result on continuous
linear representability of binary relations on topological
vector spaces. 

Important applications of this result
include individual decision making under uncertainty
(expected utility theory) and collective decision making
(utilitaristic social welfare functions).  In the latter
area, the paper adds  to known results on the
existence of social welfare functionals; see e.g. Maskin 
[1978], Roberts [1980], or d'Aspremont [1985].  In addition
to allowing for infinitely many individuals, it demonstrates
the power of a translation-invariance assumption which allows
to considerably weaken other assumptions, which are usually made in this
context, as completeness,
transitivity, and continuity of the social planner's
preferences (or, in the expected utility case, the
individual's).



The result is in the spirit of Schmeidler's [1971] who derives
completeness from strong continuity properties of partial
transitive preferences. We use an assumption, to be called
indifference-invariance, which is  strictly weaker than 
transitivity.\footnote{The relation of that assumption to transitivity has been
studied earlier by Rader [1963] and Sonnenschein [1965] for the case of
complete preferences.} This assumption, together with the
translation-invariance
postulate, allows us to get completeness, transitivity, and continuous
representability from much weaker assumptions on the relation.


For the sake of clarity and ease of exposition, we postpone a more detailed
discussion until we have presented the formal set-up and the
Proposition.



\section{Notation and basic definitions}

Let $V$ be a topological vector space and $R$ be a binary relation on
$V$. By $I$
and $P$ we denote the strict and the indifference relations, respectively, 
derived
from $R$:
\[\begin{array}{lcccl}
  \forall\ x,y\in V & : & x \ P\  y & :\Leftrightarrow &
                            x \ R\  y \mbox{\ and not\ }
                            y \ R\  x;\\
                    &   & x \ I\  y & :\Leftrightarrow &
                            x \ R\  y \mbox{\ and\ } y \ R\  x.
\end{array}\]
\newpage
We shall employ the following notation for all $x\in V\!$:



\[\begin{array}{lcll}
  R(x) & \defeq & \{y\in V\mid x\ R\ y\} &
  \mbox{\ lower contour set of $x$} ;\\
  R^{-1}(x) & \defeq & \{ y\in V \mid y \ R\  x\} &
  \mbox{\ upper contour
  set of $x$} ;\\
  P(x) & \defeq & \{ y\in V \mid x\ P\ y\} &
  \mbox{\ worse set of $x$} ;\\
  P^{-1}(x) & \defeq & \{ y\in V \mid y\ P\ x\} &
  \mbox{\ better set of $x$}          ;\\
  I(x) & \defeq & \{ y \in V \mid x\ I\ y\} &
  \mbox{\ indifference set of $x$} .
\end{array}\]

By definition,  $I$ is {\em symmetric}, i.e. $x\ I\ y$ implies $ y\ I\ x$
 for all $x,y\in V$, and $P$ is {\em irreflexive}, i.e. $x\not\in 
P(x)$ for all $x \in V\!$.

\begin{defi}
  A binary relation $R$ on $V$ is called {\em translation-invariant} 
iff $\, \forall\ y,z\in V\!: R(y) + z \subseteq R(y+z)$.
\end{defi}

\begin{defi}
  A  binary relation $R$ on $V$ is called {\em indifference-invariant} 
  iff $ \forall x,y \in V\!: x\ P\ y \Rightarrow I(x) \times I(y) \subseteq
P^{-1}(y) \times P(x).$ 
  \end{defi}

 Clearly, indifference-invariance of a relation is strictly weaker than its
transitivity.  Rader [1963] and Sonnenschein [1965]  give conditions under
which transitivity   and indifference-invariance of $R$ are in fact equivalent.
 They  use completeness, continuity, and connectedness assumptions on 
  $R$ and one half of indifference-invariance, i.e.\quad
 $x\ P\ y  \Rightarrow I(x) \subseteq P^{-1}(y)$ for all $ x,y \in V,\;$ to
obtain that equivalence. In our set-up, 
 a local continuity assumption on and translation-invariance of
  $R$  allow us to get an even stronger result. 
 
  Note that, for translation-invariant relations $R$, indifference-invariance
of $P$ is characterized by $P(0) + I(0) \subset P(0)$. We will use this fact
later on without explicitly mentioning it.

\begin{defi}
  A binary relation $R$ on $V$ is called {\em lower closed}
  (resp.~{\em lower open}) {\em at} $x\in V$ iff $R(x)$ is closed
  (resp.~$P(x)$ is open). It is {\em lower continuous at} $x\in V$ iff
  it is lower closed and lower open at $x\in V\!$.
\end{defi}

Upper continuity is defined symmetrically. Note that a relation is called {\it
continuous} if it is lower continuous and upper continuous at all $x \in V\!$.


\section{Result and Discussion}

\noindent{\bf Proposition:}
 {\it Let $R$ be a 
 translation- and indifference-invariant binary relation on $V$ which is
   lower continuous at 
  some point 		of $V\!$.
  If $P \not= \phi$, then $R$ is a continuous complete preordering,
  representable by a continuous linear utility function.}
\vskip.2cm


In the case of expected utility theory, $V$ contains the set
of lotteries available to an individual decision maker with
a preference $R$.  Here, our result asserts the existence of
a continuous linear utility function for an {\em a priori}
partial, locally continuous, and non-ordered preference on
the set of lotteries.  In contrast, earlier work as  in Theorem 4.2.2 of
Blackwell-Girshick [1954], in Theorem 2 of Grandmont [1972], and the results in
Einy [1989], Trockel [1992], and Candeal-Indurain [1994]  all require complete,
transitive, and globally continuous preferences.

Our result does not imply (nor is it implied by) Theorem 8
in Herstein-Milnor [1953]. They assume completeness and transitivity of the
relation  while we do not. Our set of alternatives is a linear topological
space while theirs is a mixture set and may not have a topology. Our local
continuity assumption plays the same r\^ole as their Archimedian axiom which
amounts to a continuity
assumption when the mixture set equals $\reals^n$. While, in general, the
question of continuity of the utility may be meaningless in their setup,   the
utility may not be continuous even if the mixture set can be
embedded in or is a linear topological space.\footnote{  To verify this and,
thus,  that our result does not imply theirs, observe that any linear
topological space is a mixture set, in particular, any ${\cal{L}}^p$.  It is
well known (see e.g. Rudin [1991]) that, for $p \in (0,1)$, the latter spaces
have only a trivial topological dual.  Therefore, linear utility
representations exist due to their result, but continuous do not.}



In the case of the application of our Proposition to a social choice problem,
the vector space $V$ is the space of utility allocations to the individuals
making up society.  The relation $R$ on $V$ is a social welfare ordering, i.e.
a planner's preference on utility allocations.  Our Proposition  gives a very
general utilitarianism result for the social welfare ordering $R$.  In contrast
to the set-up usually considered, it allows for infinitely many individuals in
a society as well as for {\em a priori} partial, non-ordered preferences of the
social planner.  In particular, our indifference-invariant social relations do
{\em a priori} not exclude cycles.\footnote{Indeed, indifference-invariance of
$R$ is consistent with cycles. It would be quite a natural property of a
non-transitive social preference relation because the ranking of two social
alternatives could be influenced by a replacement of two socially indifferent
substitutes.}  They only require that, in choice situations between two utility
alloca

tions, either one of them can be replaced by an indifferent one without
influencing the choice.  Thus, no rationality assumptions are required for the
planner's preference.



The translation-invariance here represents cardinality and unit comparability
with a fixed unit.  It is the power of this assumption which allows us to
deduce continuity from local semi-continuity and indifference-invariance, and
even implies $R$ to be a linearly representable (thus transitive and complete)
social welfare ordering.



In this literature, a distinction is made between weak and strong
utilitarianism results.  The strong notion requires {\it representation} of the
planner's preference $R$ by a linear functional, while the weak notion only
requires $R$ to be {\it included}  in the relation generated by the functional.
 The ``weak notion'' results (see e.g. d'Aspremont-Gevers [1977] and Roberts
[1980]), usually rely on Theorem 4.3.1 in Blackwell-Girshick [1954] which does
not require continuity assumptions (and does not yield continuity of the
representation).  Our Proposition, thus, does not strengthen these results.
It, however, generalizes the version of this theorem, which includes
semi-continuity as an assumption,  on page 120 in Blackwell-Girshick [1954],
and extends it to infinite dimensions.  Therefore, our result allows the
derivation of a more desirable strong utilitarianism result with a continuous
social welfare functional.



It is natural, to compare this result with that of Maskin [1978], whose
approach is quite different.  He starts with a product space of utility
functions and assumptions on the social welfare functional and derives, with
the help of the welfarism theorem (see d'Aspremont-Gevers [1977]), a social
welfare ordering on the (finite-dimensional) space of utility allocations and
proves its representability.  We apply our Proposition directly to the latter
space (allowing for infinite dimension).  Maskin's assumption of {\em full
comparability} is replaced by our assumption of {\em translation-invariance}.
Neither one of these assumptions is implied by the other.  This substitution,
however, has the interesting and pleasant consequence that the strong
separation property needed in Maskin's paper can be dispensed with and that (as
already pointed out) continuity can be derived from local semi-continuity.

In closing we note
that, in view of Trockel's [1989] classification of Cobb-Douglas
representable utility functions, our present result may be translated
into one giving minimal requirements for a binary relation on
$\reals^n_{++}$ to be representable by a Cobb-Douglas utility
function.


\section{ Proof of the Proposition}

\noindent{\bf Proof:} 
(a) First we show that $I$ is transitive.\footnote{
Note that completeness of an indifference-invariant relation trivially implies
this result. Here we show it without first showing the completeness of the 
relation.}
\vskip.1cm
\noindent By translation-invariance,  $R(0) = -R^{-1}(0)$, $P(0) = -P^{-1}(0)$,
$I(0)= -
I(0)$,  and $R(x) = x + R(0)$,  $P(x) = x + P(0)$, $I(x) = x +I(0)$ for
  any $x\in V$. Hence, lower continuity  at one 
point implies (global) continuity. 


There exists $\bar{x} \in P(0)$. We will show\footnote{This argument borrows
from Schmeidler [1971].} that
\begin{equation}
V=P(0) \cup P^{-1}(\bar{x}) = R(0) \cup R^{-1}(\bar{x}). 
\end{equation}
Because the middle term is open and non-empty  and the last term is closed, 
equality of the two in fact implies that both terms equal the connected set
$V$. To 
prove (1), let $y \in I(0) \cup I(\bar{x})$. If $y \in I(0)$, then $-y 
\in I(0)$ and
indifference-invariance implies $\bar{x} - y \in P(0)$, i.e. $\bar{x} \in
P(y)$, 
thus $y \in P^{-1}(\bar{x})$. If $y \in I(\bar{x})$, then $y- \bar{x} \in I(0)$
and, 
again by indifference-invariance, $y=  \bar{x} +(y - \bar{x}) \in P(0)$. 

We will now show that $I$ is transitive, i.e. $I(0) + I(0) \subseteq I(0)$.

\noindent Because of (1), $\frac{1}{2}\bar{x} \in P(0) \cup P^{-1}(\bar{x})$.
We will 
show that $ \frac{1}{2}\bar{x} \in  P(0)$, which is clear if
$\frac{1}{2}\bar{x} \not\in P^{-1}(\bar{x})$. If $\frac{1}{2}\bar{x} \in
P^{-1}(\bar{x}),$ i.e. 
$\bar{x} \in P(\frac{1}{2}\bar{x})$, then, by translation-invariance, 
$\frac{1}{2}\bar{x} \in P(0)$.  Since (1) is true for any $z \in P(0)$, 
iterations of this argument show that $0 \in clP(0)$, where $cl$ 
denotes the closure operator in the topology of $V$.

Next we will show that $R(0) = cl P(0)$. Because $R(0)$ is closed, we only 
have to show that $I(0) \subseteq cl P(0).$ To this end, let $z \in I(0)$. Then

$z = 0 + z \in cl P(0) + I(0) \subseteq cl( P(0) + I(0)) \subseteq cl P(0)$,
where 
the last inclusion holds because of indifference-invariance. 

This implies 
that $R(0) +I(0) = cl P(0) + I(0) \subseteq cl(P(0) +I(0)) \subseteq cl P(0) = 
R(0)$, hence, $I(0) + I(0) \subseteq R(0)$. By translation-invariance, we also
have $R^{-1}(0)  +I(0) \subseteq R^{-1}(0)$, whence    $I(0) + I(0) \subseteq
R^{-1}(0)$  and we are done.


\vskip.4cm
(b)  Secondly, we show that $R$ is complete.
\vskip.1cm
\noindent Assume 
that this is not the case, i.e. there exists $x\in V$ not comparable to 0.
By (1), this implies $x \in P^{-1}(\bar{x})$. This will allow us to prove\
\begin{equation} 
 R(x) \cap R(0) = P(x) \cap P(0). 
\end{equation}
In fact, if $z \in I(x) \cap P(0)$ or $z \in I(0) \cap P(x)$, then, by
indifference-invariance, $x \in P(0)$ or $x \in P^{-1}(0)$, respectively -- a
contradiction to our assumption.  If $z \in I(x) \cap I(0)$, then transitivity
of $I$ implies $x \in I(0)$ -- again a contradiction.

Because  the set described  in (2) does not own $x$, it is closed, open, and
not the whole space. Therefore it is empty, a contradiction to $\bar{x} \in
P(0) \cap P(x)$.

\vskip.4cm
(c)  Next we show that $I(0)$ is a closed linear
  subspace of $V$.
 \vskip.1cm 
\noindent In fact, let $x\ I\ 0$, $y \ I\  0$. Translation-invariance of $R$
and transitivity of $I$ give us $0\ I\ -x$
  and thus $-x+y \ I\ y$. Transitiviy    of $I$ implies $-x+y\ I\ 0$, and
$I(0)$ is a group. 
 
  It is left to show that $ x \in I(0)$ implies $\lambda x \in I(0)$  for any
$\lambda \in 
\reals_+$. 
  We will show that $x \ I\ 0$ implies $\frac{1}{2} x \ I \ 0$. Assume not. 
Then by completeness, $\frac{1}{2} x \in P (0)$ or $\frac{1}{2} x\in
P^{-1}(0)$; say 
$\frac{1}{2} x \in P(0)$. Translation-invariance yields $-x\ I\ 0$ , and
indifference-invariance $-\frac{1}{2} x =\frac{1}{2} x -x \in P(0)$, i.e.
$\frac{1}{2} x\in P^{-1}(0)$, a contradiction to the fact that $P$ is 
asymmetric. Iterations of this argument yield $\frac{1}{2^k} x\ I\ 0$ for all 
$k \in \nat$. The group property of $I(0)$ allows us to infer that
$\frac{n}{2^k} 
x \in I(0)$ for all $n,k \in \nat $. Because the set of these points is dense
in 
$\reals_+\cdot x$ and because $I(0)$ is closed  as the intersection of the
closed sets $R^{-1}(0)$ and 
$R(0)$, we are done. 
 
 \vskip.4cm 
(d)  Finally, we prove linear continuous representability.
\vskip.1cm
  \noindent Let $W$ be an algebraic complement of $I(0)$, i.e. $W$ is a linear
subspace of  $V$ with $I(0) \cap W = \{0\}$, and $V = I(0) + W$. By
non-triviality of $R$,  $W \not = \left\{ 0 \right\}$. Then, $C \defeq W
\setminus \left\{ 0 \right\}$ clearly is a subset of $P(0) \cup P^{-1}(0)$,
and, thus, cannot be connected. However, if dim $W \geq 2$, then $C$ is
connected as the union of finitely many connected sets with mutually non-empty
intersections. Hence dim $W = 1$ and $I(0)$ is a hyperplane. Therefore, there
exists  a continuous linear functional $p$
 with $p(\bar{x}) < 0$ and $H^p \defeq \mbox{Ker\ } p = I(0)$ (Robertson and
Robertson, 1964, Chapter II).    Let $H_-^p \defeq \{ y\in V \mid p(y) < 0 \}$
\break and
  $H_+^p  \defeq  -H_-^p$.  Either one of these open halfspaces is
  non-empty and connected.  Because $R$ is complete, $H_-^p = (H_-^p \cap 
P(0)) \cup (H_-^p \cap P^{-1}(0))$. 
   Because $\bar{x} \in H_-^p \cap P(0)$ and the sets $P(0)$ and $P^{-1}(0)$ 
are  open,  $H_-^p\cap P^{-1}(0) = \emptyset$. Thus  $H_-^p \subset P(0)$. By
  translation-invariance, this gives $H_+^p = -H_-^p \subset -P(0)= P^{-1}(0)$.
  Hence $( H_{\rule{0mm}{2mm}}^p,H_+^p,H_-^p)$ and $(I(0) , P^{-1}(0),P(0))$ 
both are partitions of $V$ with $H_+^p \subset P^{-1}(0)$ and $H_-^p \subset 
P(0)$, and $I(0) =H^p$. Therefore, these partitions are identical and  $p$ is
(up to multiplication by a positive
  scalar) the uniquely determined continuous linear 
  functional
  representing $R$. In particular, $R$ must be transitive and
  continuous  in addition to being complete. \qed



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%\art author | date | title | journal | vol | pp. 
 \def\art#1 | #2 | #3 | #4 | #5 | #6.
 {#1: #3. #4 {\bf #5,} #6 (#2)} 
 
  %\book author | date | title | city | press. 
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  {#1: #3 #4: #5 #2} 
 
  %vol author | date | title | in | eds | city | press. 
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\end{description}

\end{document}

