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{\large {\bf Defining the Mean-Preserving Spread: 3-pt versus 4-pt}\\[0pt]
}

\bigskip  Eric Rasmusen and Emmanuel Petrakis\\[0pt]

Published in: {\it Decision Making Under Risk and Uncertainty: New Models
and Empirical Findings ,} edited by John Geweke. Amsterdam: Kluwer, 1992
(with Emmanuel Petrakis ). ISBN: 0-7923-1904-4.\\[0pt]

\bigskip {\it Abstract}
\end{center}

\noindent  The standard way to define a mean-preserving spread is in terms
of changes in the probability at four points of a distribution (Rothschild
and Stiglitz [1970]). Our alternative definition is in terms of changes in
the probability at just three points. Any 4-pt mean-preserving spread can be
constructed from two 3-pt mean-preserving spreads, and any 3-pt
mean-preserving spread can be constructed from two 4-pt mean-preserving
spreads. The 3-pt definition is simpler and more often applicable. It also
permits easy rectification of a mistake in the Rothschild-Stiglitz proof
that adding a mean-preserving spread is equivalent to other measures of
increasing risk.

\bigskip

Draft: 5.10 (Draft 1.1, October 1988)

{\small \noindent  2001: Eric Rasmusen, Professor of Business Economics and
Public Policy and Sanjay Subhedar Faculty Fellow, Indiana University, Kelley
School of Business, BU 456,  1309 E 10th Street,  Bloomington, Indiana,
47405-1701.  Office: (812) 855-9219. Fax: 812-855-3354.
Erasmuse@indiana.edu. Php.indiana.edu/$\sim$erasmuse. }

We would like to thank Peter Fishburn and John Pratt for their comments.

%---------------------------------------------------------------

\newpage

The question of what ``risky'' means is central to information economics.
One way to define risk is to say that asset X is riskier than asset Y if
every individual with a strictly concave utility function prefers Y to X. By
another definition, X is riskier if it is distributed like Y plus an
additional asset with zero mean and positive variance. Still another
definition says that X is riskier if the distribution of Y has the same mean
as X but dominates it in the sense of second-order stochastic dominance or
adding a mean-preserving spread. It turns out that all three definitions are
equivalent. This is best known to economists from the classic article of
Rothschild \& Stiglitz (1970), although as the authors themselves pointed
out two years later, various components of their key theorem could have been
drawn from existing mathematics (Blackwell and Girshick, 1954; Hardy,
Littlewood, and Polya, 1953).

The standard way to define a mean-preserving spread is in terms of changes
in the probability at four points of a distribution (Rothschild and Stiglitz
[1970]). Our alternative definition is in terms of changes in the
probability at just three points. Any 4-pt mean-preserving spread can be
constructed from two 3-pt mean-preserving spreads, and any 3-pt
mean-preserving spread can be constructed from two 4-pt mean-preserving
spreads. The 3-pt definition is simpler and more often applicable. It also
permits easy rectification of a mistake in the Rothschild-Stiglitz proof
that adding a mean-preserving spread is equivalent to other measures of
increasing risk.

Let $F$ and $G$ be cumulative density functions of the discrete random
variables $X$ and $Y$, where $Pr (X=a_i) = f_i$, $Pr (Y=a_i) = g_i$, and $%
\gamma_i = g_i -f_i.$ A mean-preserving spread (MPS) is a set of $\gamma$'s
such that if $Y$ differs from $X$ by a single MPS then $Y$ has the same mean
as $X$, but more weight in the tails. Rothschild and Stiglitz use the
following 4-pt definition, illustrated in Figure 1a:

A {\bf 4pt MPS} is a set of four locations $a_1 <a_2 <a_3 <a_4$ and four
probabilities $\gamma_1 \geq 0,\gamma_2 \leq 0, \gamma_3 \leq 0, \gamma_4
\geq 0$ such that $-\gamma_1= \gamma_2$, $\gamma_3= -\gamma_4$, and $\sum_i
\gamma_i a_i =0$.

As an alternative, we suggest the 3-pt MPS, illustrated in Figure 1b.

A {\bf 3pt MPS} is a set of three locations $a_1 <a_2 <a_3$ and three
probabilities $\gamma_1 \geq 0,\gamma_2 \leq 0, \gamma_3 \geq 0$ such that $%
\sum_i \gamma_i =0$ and $\sum_i \gamma_i a_i =0$.

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Compared to the 4-pt MPS, the 3-pt MPS is simpler, more intuitive, and more
often applicable (it requires one fewer point of positive probability).
Mathematically, the two definitions are equivalent. Proof of this
equivalence exists implicitly but not very accessibly in the mathematics
literature (the Fishburn [1982] theorem on the properties of the convex
cones of certain sets of signed measures in multiple dimensions). We will
prove the equivalence more simply here by construction.

%---------------------------------------------------------------

\noindent  THEOREM 1a: {\it Any 4-pt MPS can be constructed from two 3-pt
MPS's.}

{\it Proof.} Begin with a 4-pt MPS: 
\[
MPS^0= \{a_1,a_2, a_3, a_4; \gamma_1, -\gamma_1, -\gamma_4, \gamma_4\}. 
\]
We claim that $MPS^0$ is the sum of the 3-pt MPS's 
\[
MPS^1= \{a_1, a_2, a_3; \gamma_1, -(\gamma_1+y), y\} 
\]
and 
\[
MPS^2= \{a_2, a_3, a_4; y, -(y+\gamma_4), \gamma_4\}, 
\]
where $y>0$, and 
\begin{equation}  \label{e11aa}
\gamma_1 a_1 - (\gamma_1+y)a_2 + y a_3 =0
\end{equation}
These definitions make $MPS^1$ and $MPS^2$ spreads, and condition (\ref
{e11aa}) makes $MPS^1$ mean-preserving. When added together, spreads $MPS^1$
and $MPS^2$ equal 
\[
\{a_1,a_2, a_3, a_4; \gamma_1+0, -(\gamma_1+y)+y, y-(y+\gamma_4),
0+\gamma_4\}, 
\]
which is $MPS^0$. It must be shown that $MPS^2$ is mean-preserving. The fact
that $MPS^0$ is mean-preserving implies that 
\begin{equation}  \label{e11a}
\gamma_1 a_1 -\gamma_1a_2 -\gamma_4a_3 + \gamma_4a_4 = 0
\end{equation}
Equating (\ref{e11aa}) and (\ref{e11a}) gives 
\begin{equation}  \label{e11b}
\gamma_1 a_1 -\gamma_1a_2 -\gamma_4a_3 + \gamma_4a_4 = \gamma_1 a_1 -
(\gamma_1+y)a_2 + y a_3,
\end{equation}
which is equivalent to 
\begin{equation}  \label{e11c}
y a_2 +(-y-\gamma_4)a_3 + \gamma_4a_4 = 0.
\end{equation}
Equation (\ref{e11c}) is the condition that $MPS^2$ be mean-preserving.
Thus, all three spreads are mean-preserving. (Note that the construction can
use any of a broad set of different values for $y$.) \newline
Q.E.D.

\noindent  THEOREM 1b: {\it Any 3-pt MPS can be constructed from two 4-pt
MPS's.}

{\it Proof.} Begin with the 3-pt MPS 
\[
MPS^0= (a_1, a_3, a_5; \gamma_1, -(\gamma_1+\gamma_3), \gamma_3). 
\]
We claim $MPS^0$ is the sum of the 4-pt MPS's 
\[
MPS^1= (a_1, a_2, a_3, a_4; \gamma_1, -\gamma_1, -\gamma_3, \gamma_3) 
\]
and 
\[
MPS^2= (a_2, a_3, a_4, a_5; \gamma_1, -\gamma_1, -\gamma_3, \gamma_3), 
\]
where $a_2$ and $a_4$ are chosen to satisfy 
\begin{equation}  \label{e91}
a_1\gamma_1 - a_2 \gamma_1 - a_3\gamma_3 + a_4\gamma_2 = 0
\end{equation}
and 
\begin{equation}  \label{e92}
a_2\gamma_1 - a_3 \gamma_1 - a_4\gamma_3 + a_5\gamma_3 = 0.
\end{equation}
When added together, spreads $MPS^1$ and $MPS^2$ equal 
\[
\{a_1, a_2, a_3, a_4, a_5; \gamma_1+0, -\gamma_1+ \gamma_1, -\gamma_3-
\gamma_1, \gamma_3- \gamma_3, 0+\gamma_3\}, 
\]
or 
\[
\{a_1, a_2, a_3, a_4, a_5; \gamma_1, 0, -(\gamma_3+ \gamma_1), 0,
\gamma_3\}, 
\]
which is $MPS^0$. (Note that any of a large number of values of $a_2$ and $%
a_4$ satisfy (\ref{e91}) and (\ref{e92}).) \newline
Q.E.D.

Although the two definitions are mathematically equivalent, we suggest that
the 3-pt MPS is superior. A good definition does two things: it defines a
useful idea, and it does so in a way that is simple and convenient to use.
The 3-pt and 4-pt MPS define the same useful idea. But the 3-pt MPS has a
distinct, if modest, advantage in simplicity and convenience. It is the
simplest possible definition, because two points cannot spread probability
while preserving the mean. It matches the intuition of the spread
exactly---to take probability from a point and move it to each side of that
point in such a way that the mean stays the same, whereas the 4-pt MPS takes
probability away from two points. Finally, the 3-pt MPS is more often
applicable, since it uses fewer points of the support. Two distributions $F$
and $G$, each with three points of positive probability, might differ by a
single 3-pt MPS, but to move between them using 4-pt MPS's would require
negative probabilities in the intermediate step.

\bigskip  Risk can also be analyzed using cumulative distributions. Figure 2
shows that if cumulative distribution $G$ equals cumulative distribution $F$
plus a MPS, then the difference $G-F$ looks like Figure 2a for a 4-pt MPS,
and like Figure 2b for a 3-pt MPS.

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Rothschild and Stiglitz use the ``integral conditions'' to look at spreads
using cumulative distributions. The first integral condition preserves the
mean: 
\begin{equation}  \label{int6}
\int_0^1 [G(x)- F(x)]dx = 0;
\end{equation}
and the second integral condition makes the change a spread: 
\begin{equation}  \label{int7}
\int_0^y [G(x)- F(x)]dx \geq 0, \;\;\; 0 \leq y \leq 1.
\end{equation}
Condition (\ref{int7}) ensures that $F$ dominates $G$ in the sense of
2nd-order stochastic dominance. If $F$ and $G$ differ by either a 3-pt or a
4-pt MPS, then condition (\ref{int7}) is satisfied. That the implication
runs the other way too can be seen from p. 630 of Fishburn (1982) or by
combining Lemma 1 of Rothschild and Stiglitz (1970) with Theorems 1a and 1b
of the present article. Lemma 1 of Rothschild and Stiglitz (1970) contains a
mistake, but one which can be rectified by using the 3-pt MPS. The first
paragraph of its proof says ``By (7), $a_2 < a_3$,'' where it should say,
``By (7), $a_2 \leq a_3$.'' The false step would rule out the $G-F$ shown in
Figure 2b, which is clearly an example of stochastic dominance. The proof is
thus invalid for the 4-pt MPS, but it can easily be made valid for the 3-pt
MPS. Rather than writing a new proof for the 4-pt MPS, it is easier to use
the equivalence of the 3-pt and 4-pt MPS to establish that the subsequent
propositions in Rothschild and Stiglitz are correct.

The usefulness of the mean-preserving spread lies in its equivalence to
other definitions of risk. Since the 3-pt and 4-pt definitions are
equivalent, either of them can be used in the theorem below, which says that
different ways of comparing distributions of wealth $F(x)$ and $G(x)$ are
equivalent.

\bigskip \newpage

THEOREM (Rothschild and Stiglitz [1970] Theorem 2): The following three
statements are equivalent:

(A) {\it Risk Aversion}.  For every bounded concave function $U$, $\int U(z)
dF(z) \geq \int U(z) dG(z)$. (Every risk averter prefers $F$ to $G$.)

(B) {\it Noise.}  There exists a random variable $Z$ such that $E(Z|X =x)=0$
for all $x$, and $Y \stackrel{=}{d} X+Z$. ($G$ is distributed as $F$ plus
noise.)

(C) {\it MPS/Stochastic Dominance.}  The difference $G-F$ satisfies the
integral conditions (\ref{int6}) and (\ref{int7}). ($F$ has the same mean as 
$G$, but 2nd-order stochastically dominates it.)  ($G$ equals $F$ plus a
sequence of 3-pt or 4-pt MPS's.)  ($G$ has more weight in the tails than $F$%
.)

\bigskip

David Hirshleifer points out that it is easy to see the equivalence of (B)
and (C) using the 3-pt MPS. The 3-pt MPS takes probability away from point $%
a_2$ and moves it to points $a_1$ and $a_3$. This is like waiting for the
realization specified by the original distribution, and then, if the
realization is $a_2$, adding a new gamble that either (a) leaves the outcome
as $a_2$, or (b) moves it to $a_1$ or $a_3$.

To conclude: risk is so central to information economics that it is
important to have its definition be as clear and convenient as possible.
There are a number of different ways to define risk that can be shown to be
mathematically equivalent, of which the 4-pt MPS is perhaps the best-known.
We have suggested that the 4-pt MPS be replaced by the 3-pt MPS, which
combines the attractive properties of the 4-pt MPS with additional
simplicity and intuitiveness.

%---------------------------------------------------------------

\titlepage
\noindent {\bf References.}

Blackwell, David and Meyer Girshick (1954) {\it Theory of Games and
Statistical Decisions} New York: Wiley.

Fishburn, Peter (1982) ``Moment-Preserving Shifts and Stochastic
Dominance,'' {\it Mathematics of Operations Research}, 7, 629-634.

Hardy, G., J. Littlewood, and G. Polya (1953) {\it Inequalities}, 2nd
Edition London: Cambridge University Press (1st Edition, 1934).

Rothschild, Michael and Joseph Stiglitz (1970) ``Increasing Risk I,'' {\it %
Journal of Economic Theory}, 2: 225-243.

Rothschild, Michael and Joseph Stiglitz (1972) ``Addendum to `Increasing
Risk: I. A Definition','' {\it Journal of Economic Theory}, 5, 306.

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