Mean-preserving spread

This example shows that to have a mean-preserving spread does not require that all or most of the probability mass move away from the mean.[2] Let A have equal probabilities ${\displaystyle 1/100}$ on each outcome ${\displaystyle x_{Ai}}$ , with ${\displaystyle x_{Ai}=198}$ for ${\displaystyle i=1,\dots ,50}$ and ${\displaystyle x_{Ai}=202}$ for ${\displaystyle i=51,\dots ,100}$; and let B have equal probabilities ${\displaystyle 1/100}$ on each outcome ${\displaystyle x_{Bi}}$, with ${\displaystyle x_{B1}=100}$, ${\displaystyle x_{Bi}=200}$ for ${\displaystyle i=2,\dots ,99}$, and ${\displaystyle x_{B100}=300}$. Here B has been constructed from A by moving one chunk of 1% probability from 198 to 100 and moving 49 probability chunks from 198 to 200, and then moving one probability chunk from 202 to 300 and moving 49 probability chunks from 202 to 200. This sequence of two mean-preserving spreads is itself a mean-preserving spread, despite the fact that 98% of the probability mass has moved to the mean (200).

Let ${\displaystyle x_{A}}$ and ${\displaystyle x_{B}}$ be the random variables associated with gambles A and B. Then B is a mean-preserving spread of A if and only if ${\displaystyle x_{B}{\overset {d}{=}}(x_{A}+z)}$ for some random variable ${\displaystyle z}$ having ${\displaystyle E(z\mid x_{A})=0}$ for all values of ${\displaystyle x_{A}}$. Here ${\displaystyle {\overset {d}{=}}}$ means "is equal in distribution to" (that is, "has the same distribution as").

Mean-preserving spreads can also be defined in terms of the cumulative distribution functions ${\displaystyle F_{A}}$ and ${\displaystyle F_{B}}$ of A and B. If A and B have equal means, B is a mean-preserving spread of A if and only if the area under ${\displaystyle F_{A}}$ from minus infinity to ${\displaystyle x}$ is less than or equal to that under ${\displaystyle F_{B}}$ from minus infinity to ${\displaystyle x}$ for all real numbers ${\displaystyle x}$, with strict inequality at some ${\displaystyle x}$.

Both of these mathematical definitions replicate those of second-order stochastic dominance for the case of equal means.

If B is a mean-preserving spread of A then A will be preferred by all expected utility maximizers having concave utility. The converse also holds: if A and B have equal means and A is preferred by all expected utility maximizers having concave utility, then B is a mean-preserving spread of A.

• Stochastic ordering
• Risk (statistics)
• Scale parameter

• Mas-Colell, A.; Whinston, M. D.; Green, J. R. (1995). Microeconomic Theory. New York: Oxford University Press. pp. 197–199. ISBN 0-19-510268-1 – via Google Books.