Second-Order Stochastic Dominance¶

Definition¶

For nonnegative integrable gain or wealth variables $X,Y$ , $Y$ has second-order stochastic dominance over $X$ if $\int_0^tF_X(s)\,\mathrm{d}s\ge\int_0^tF_Y(s)\,\mathrm{d}s$ for every $t\ge0$ $0$ . Equivalently, every increasing concave utility weakly prefers $Y$ to $X$ , with expectations defined.

Interpretation¶

This comparison is stated for favourable outcomes. When the variables denote losses, put the minus sign inside wealth utility. A larger mean alone does not establish second-order dominance.

Integrated CDFs of a fair draw X of 0 or 2 and a certain gain Y = 1; the X curve lies above the Y curve up to t = 2, and the two curves coincide from there on.

The figure reads the Examples case as gains: $X$ is the fair draw of 0 or 2 and $Y=1$ is certain. Then $\int_0^tF_X(s)\,\mathrm{d}s=t/2$ for $0\le t\le2$ $2$ , while $\int_0^tF_Y(s)\,\mathrm{d}s=(t-1)_+$ there; the shaded gap shows that the first integral is never below the second, which is the definition of $Y$ dominating $X$ in second order. From $t=2$ on both integrals equal $t-1$ $1$ , because once $t$ exceeds every outcome the integral equals $t$ minus the mean, and the means are equal.

Move the certain gain $y$ and the spread of $X$ : while $y\ge\mathbb E[X]=1$ the integrated CDF of $X$ is never below that of $Y=y$ $y$ , so the certain gain dominates. Push $y$ below 1 and the curves cross.

Convex order and more dangerous risks compares dispersion at equal means. In that case $Y\le_{\mathrm{cx}}X$ makes every concave utility prefer the gain $Y$ , explaining the direction of the equal-mean second-order comparison.

Key Results¶

First-order dominance of $Y$ over $X$ implies second-order dominance.
Dominance requires $\mathbb{E}[Y]\ge\mathbb{E}[X]$ for integrable variables.
At equal means, $Y$ dominating $X$ in second order corresponds to $Y\le_{\mathrm{cx}}X$ $X$ .
For losses, $X\le_{\mathrm{sl}}Y$ corresponds to preference for wealth $w-X$ over $w-Y$ by increasing concave utility.

Examples¶

As gains, a certain 1 second-order dominates a fair draw of 0 or 2. As losses, the same certain 1 is the less dangerous risk. The preference direction is the same after the outcome meaning is specified.

Prerequisites¶

Sources¶

Risk Insurance · Lecture 10

Aliases: increasing concave order, second-order dominance of wealth, SSD, second-order dominance, second-degree stochastic dominance Keywords: integrated CDF, increasing concave utility, ∫_0^t F_X(s) ds ≥ ∫_0^t F_Y(s) ds, risk-averse preference, gains versus losses Symbols: integral

Last updated: 17 October 2026 21:53