<p>Let <span class="math-container">$F$</span> be the cdf of the uniform distribution on <span class="math-container">$[-1,1]$</span>. Since <span class="math-container">$F$</span> and <span class="math-container">$G$</span> have the same mean, <span class="math-container">$G$</span> is a mean-preserving spread of <span class="math-container">$F$</span> if and only if
<span class="math-container">\begin{equation}
\int_{-\infty}^x F(t)\mathrm dt \le \int_{-\infty}^x G(t)\mathrm dt\, \qquad \text{for all }x\in[-1,1].
\end{equation}</span>
Proving the inequality should be straightforward.</p>
<p>The key condition here is <span class="math-container">$F$</span> and <span class="math-container">$G$</span> have the same mean. But if <span class="math-container">$F$</span> is <em>any</em> cdf on <span class="math-container">$[-1,1]$</span>, the &quot;mean-preserving&quot; part will not always hold. For example, suppose <span class="math-container">$F$</span> is a distribution that gives <span class="math-container">$-1$</span> a probability of <span class="math-container">$0.3$</span> and <span class="math-container">$1$</span> a probability of <span class="math-container">$0.7$</span>. Then <span class="math-container">$G$</span> cannot be a &quot;mean-preserving&quot; spread of <span class="math-container">$F$</span> because the two have different means. (However, you can still say that <span class="math-container">$F$</span> second order stochastically dominates <span class="math-container">$G$</span>, which is a slightly more general concept than MPS.)</p>

CC BY-SA 4.0
2023-12-07T05:36:31.100Z
2023-12-07T05:36:31.100Z
Herr K.
<p>Suppose G is a lottery where the payoff is equal to -1 with 0.5 probability and 1 with 0.5 probability. I'm trying to show that G is a mean preserving spread of a lottery with uniform distribution on [-1,1] <em>and</em> (stronger) that given any c.d.f. on [-1,1], G is a mean preserving spread of that c.d.f. Any help on either problem would be greatly appreciated.</p>
<p>I understand intuitively how this is true, but I'm having a hard time formalizing this into a proof. I haven't seen an example of <em>how</em> to show a lottery is a mean preserving spread of another.</p>

<microeconomics><mathematical-economics><risk-aversion>
Mean preserving spread of a lottery
Justin Cross
# Mean preserving spread of a lottery

(Asked by: username_0 on 2023-12-07)

Suppose G is a lottery where the payoff is equal to -1 with 0.5 probability and 1 with 0.5 probability. I'm trying to show that G is a mean preserving spread of a lottery with uniform distribution on [-1,1] *and* (stronger) that given any c.d.f. on [-1,1], G is a mean preserving spread of that c.d.f. Any help on either problem would be greatly appreciated.

I understand intuitively how this is true, but I'm having a hard time formalizing this into a proof. I haven't seen an example of *how* to show a lottery is a mean preserving spread of another.

**Tags:** microeconomics, mathematical-economics, risk-aversion

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## 1 answers

### Answer by username_1 on 2023-12-07. Score: 2

Let $F$ be the cdf of the uniform distribution on $[-1,1]$. Since $F$ and $G$ have the same mean, $G$ is a mean-preserving spread of $F$ if and only if
\begin{equation}
\int\_{-\infty}^x F(t)\mathrm dt \le \int\_{-\infty}^x G(t)\mathrm dt\, \qquad \text{for all }x\in[-1,1].
\end{equation}
Proving the inequality should be straightforward.

The key condition here is $F$ and $G$ have the same mean. But if $F$ is *any* cdf on $[-1,1]$, the "mean-preserving" part will not always hold. For example, suppose $F$ is a distribution that gives $-1$ a probability of $0.3$ and $1$ a probability of $0.7$. Then $G$ cannot be a "mean-preserving" spread of $F$ because the two have different means. (However, you can still say that $F$ second order stochastically dominates $G$, which is a slightly more general concept than MPS.)


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