# Mean preserving spread of a lottery

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.

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 $$$$\int_{-\infty}^x F(t)\mathrm dt \le \int_{-\infty}^x G(t)\mathrm dt\, \qquad \text{for all }x\in[-1,1].$$$$ 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.)