What are some natural ways to compare random variables? Suppose that two random variables $X$ and $Y$ represent the probability that one makes a certain positive amount of money. We say that person $i$ prefers $X$ to $Y$ when $X \succsim_{i} Y$. People can, however, have different criteria for their preferences.
For instance, a person $a$ might say: I prefer $X$ to $Y$ when the expectation of $X$ is bigger than the expectation of $Y$. So in this case: $X \succsim_{a} Y \iff E(X) > E(Y)$. Let's call this the expectational preference.$^{*}$
Another type of preferences involves quantiles of random variables. Let $$u_{\beta}^{X} := \sup \{ t \in \mathbb{R} \mbox{  } | \mbox{  } P(X \leq t) \leq \beta\}.$$
Then person $d$ might choose a quantile 
$\beta_{d}$
and have the following preference: $X \succsim_{d} Y \iff u_{\beta_{d}}^{X} \geq u_{\beta_{d}}^{Y}$. Let us call this type of preference the quantile preference. 
I am interested in other natural ways to compare random variables with one another, besides expectational and quantile preferences. What are some other good reasons why a person $i$ may prefer random variable $X$ to $Y$? Why are these good reasons? 
$^{*}$Note: I know the expectational preference is a special case of the collection of von Neumann-Morgenstern functions, as it is the preference associated with case when the utility function $u(x) = ax + b$, when $a >0$. So please don't mention von Neumann-Morgenstern preferences as an answer, as I am aware of these already, too!
 A: In contrast to real numbers there exists no all-purpose total order on random variables. This is why it is so hard to say something in general and might be the reason why your question has received so little votes/answers. 
To say something meaningful I will focus on actuarial/economical applications. Furthermore, in applications you have the additional challenge that three legitimate goals overlap. You might want to order risk for mathematical reasons (e.g. as a tool in a proof), you might want to describe how rational people should behave (e.g. the axioms leading to von-Neumann utility) or you would like to describe how real humans behave in real life (e.g. prospect theory ). 
So it is not surprising that a standard reference lists (in Chapter 3) several different ways to compare risks, most of them partial not total orders. Among them integral orders (your "expectational preference" is a special case) and various orders based on risk measures. Again your quantile based measure being a special case.
A well-known example of a quantile based stochastic ordering and arguably the "best approximation" to the order between real numbers is stochastic dominance:
$$ X\leq_{SD}Y \text{ iff } u_\beta^X \leq u_\beta^Y \text{ for all $\beta\in[0,1]$}.$$
It can be shown that $X\leq_{SD}Y$ is equivalent to the existence of $\tilde{X}$ and $\tilde{Y}$ with the same distribution as $X$ and $Y$ respectively and the property that $$ P[\tilde{X}\leq\tilde{Y}]=1.$$ See Chapter 3.3 of the reference above to see more characterizations and properties.
Of course stochastic dominance is a partial, not a total order. For optimisation in general and in finance in particular you need total orders. From my experience expected utility is by far the most dominant approach to portfolio optimisation. The only other approach with any practical relevance is the one based on return-risk ratios. For those you need a risk measure $\rho$ to define $X$ as better than $Y$ iff $$ \frac{E[X]}{\rho(X)} > \frac{E[Y]}{\rho(Y)}. $$
These measures are popular since they have a straightforward interpretation as return per unit of risk. They often come under the moniker of ROC (Return on capital) or RORAC (Return on risk adjusted capital). Of course each choice of $\rho$ (and each choice of return definition) will define a different order. Accordingly, there is a gazillion of them around. That said, again by far the most prominent of these return-risk ratios is the Sharpe-Ratio where total return of a portfolio above the risk free rate is divided by its standard deviation. 
