Stochastic Dominance - Second-order Stochastic Dominance

The other commonly used type of stochastic dominance is second-order stochastic dominance. Roughly speaking, for two gambles A and B, gamble A has second-order stochastic dominance over gamble B if the former is more predictable (i.e. involves less risk) and has at least as high a mean. All risk-averse expected-utility maximizers (that is, those with increasing and concave utility functions) prefer a second-order stochastically dominant gamble to a dominated gamble. The same is true for non-expected utility maximizers with utility functions that are locally concave.

In terms of cumulative distribution functions and, A is second-order stochastically dominant over B if and only if the area under from minus infinity to is less than or equal to that under from minus infinity to for all real numbers, with strict inequality at some ; that is, for all, with strict inequality at some . Equivalently, dominates in the second order if and only if for all nondecreasing and concave utility functions .

Second-order stochastic dominance can also be expressed as follows: If and only if A second-order stochastically dominates B, there exist some gambles and such that, with always less than or equal to zero, and with for all values of . Here the introduction of random variable makes B first-order stochastically dominated by A (making B disliked by those with an increasing utility function), and the introduction of random variable introduces a mean-preserving spread in B which is disliked by those with concave utility. Note that if A and B have the same mean (so that the random variable degenerates to the fixed number 0), then B is a mean-preserving spread of A.