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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.

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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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