Mean-preserving spread

In probability and statistics, a mean-preserving spread (MPS)^[1] is a change from one probability distribution A to another probability distribution B, where B is formed by spreading out one or more portions of A's probability density function while leaving the mean (the expected value) unchanged. As such, the concept of mean-preserving spreads provides a stochastic ordering of equal-mean gambles (probability distributions) according to their degree of risk; this ordering is partial, meaning that of two equal-mean gambles, it is not necessarily true that either is a mean-preserving spread of the other. Definitionally, if B is a mean-preserving spread of A then A is said to be a mean-preserving contraction of B.

Ranking gambles by mean-preserving spreads is a special case of ranking gambles by second-order stochastic dominance – namely, the special case of equal means: If B is a mean-preserving spread of A, then A is second-order stochastically dominant over B; and the converse holds if A and B have equal means.

If B is a mean-preserving spread of A, then B has a higher variance than A; but the converse is not in general true, because the variance is a complete ordering while ordering by mean-preserving spreads is only partial.

Example

This example from ^[2] shows that to have a mean-preserving spread does not require that all or most of the probability mass move away from the mean. Let A have equal probabilities 1 / 100 on each outcome x_Ai , with x_Ai = 198 for $i=1,\dots, 50$ and x_Ai = 202 for $i=51,\dots,100$; and let B have equal probabilities 1 / 100 on each outcome x_Bi, with x_B1 = 100, x_Bi = 200 for $i=2,\dots,99$, and 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).

Mathematical definitions

Let x_A and 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 $x_B \overset {d}{=} (x_A + z)$ for some random variable z having E(z | x_A) = 0 for all values of x_A. Here $\overset{d}{=}$ 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 F_A and 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 F_A from minus infinity to x is less than or equal to that under F_B from minus infinity to x for all real numbers x, with strict inequality at some x.

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

Relation to expected utility theory

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.

See also

• Stochastic ordering

References

1. ^ Rothschild, Michael, and Stiglitz, Joseph, "Increasing risk I: A definition," Journal of Economic Theory, 1970, 225–243.
2. ^ Landsberger, M., and Meilijson, I., "Mean-preserving portfolio dominance," Review of Economic Studies 60, April 1993, 479–485.