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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 $a_1 0$, 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.
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\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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