What are some interesting examples of stochastic preferences for random variables?

Question

Let $X$ and $Y$ be stochastic variables. We say that $X$ is preferred over $Y$ when $X \succeq Y$. There are many types of preference relations. For instance, we may have:

1. $X \succeq Y$ iff $E(X) \geq E(Y)$. Here, $E(X)$ is the expectation of $X$.
2. Let $u^{X}_{\alpha} := \sup \{ t \in \mathbb{R} \, | \, P(X \leq t) \leq \alpha \} $. Furthermore, let $U(X) := u^{X}_{\alpha}$ Then we could define $X \succeq Y$ iff $U(X) \geq U(Y)$.
3. Define $(XY)_{\alpha} := \{ \alpha \, | \, F_{X}^{-1} (\alpha) \geq F_{Y}^{-1} (\alpha) \} $. Then $X \succeq Y$ iff $\lambda((XY)_{\alpha}) \geq \frac{1}{2}$. Here, $\lambda$ denotes the Lebesgue measure.

I think all of these preference relations each in their own way meaningfully compare the stochastic variables $X$ and $Y$ with one another. I'm wondering whether there are other important or interesting ways to compare stochastic variables. Can you give me a good example?

Answer 1

Well, this is a big area. One thing to consider is what properties you want in your preference relation, must it be complete, should it be transitive, and so forth. That aside, here are some examples, drawn from my area of familiarity, which is economics and finance.

1. Variance: $X \succeq Y \ \rm{if} \ E(X -E(X))^2 \leq E(Y-E(Y))^2$
2. more generally, choose a utility function $u()$, and $X \succeq Y \ \rm{if}\ E(u(X))\geq E(u(Y)),$ where $u()$ is increasing, continuous, and concave. For example, for positive random variables you might choose $\ln(x)$ as your utility function.
3. First Order Stochastic Dominance: $X \succeq Y \ \rm{if} \ F(z)\leq G(z)$ for all $z$, where $F$ is the Cumulative Distribution Function of $X$ and $G$ is the CDF of $Y$. $Y$ has more weight in the tails, and thus is riskier.
4. Second Order Stochastic Dominance, $X \succeq Y \ \rm{if} \ \int_0^z[F(z)-G(z)]dz\geq 0.$

There is also third order stochastic dominance. Some of these definitions are equivalent, for example second order stochastic dominance holds if and only if $E(u(x))\geq E(u(y))$ for every increasing concave utility function. Ingersoll, Theory of Financial Decision Making, is good on this.
All this is very heavily centered on expected utility, but of course there are alternatives, and papers by Mark Machina have explored the issue.