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Stochastic Dominance and Utility

Two strategies with different return distributions. Which is better? The honest answer depends on preferences, and stochastic dominance identifies the cases where it does not.

First-order dominance

First-order dominance
FX(x)≤FY(x)for all xF_X(x) \leq F_Y(x) \quad \text{for all } x

One distribution beats another at every threshold, so every increasing utility prefers it and no risk preference need be assumed.

XX's CDF lies below YY's everywhere, meaning XX gives a higher probability of exceeding every threshold.

Anyone who prefers more to less prefers XX, regardless of risk attitude. This is close to unanimous agreement, and consequently it is rare: real alternatives usually cross somewhere.

Second-order dominance

∫−∞xFX(t) dt≤∫−∞xFY(t) dtfor all x\int_{-\infty}^x F_X(t)\,dt \leq \int_{-\infty}^x F_Y(t)\,dt \quad \text{for all } x

A weaker condition, and a more useful one. Every risk-averse decision maker prefers XX.

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