Let's say we're in a discrete probability space so that $f(x) \in \mathcal{R}^n$. Intuitively, you need some function $U: \mathcal{R}^n \rightarrow \mathcal{R}$ so you can optimize $U(f(x))$. You can only optimize a single objective!
Optimizing a single objective function may sound quite constraining, but it is not! Rather a single objective can represent incredibly diverse preferences you may have over what is a better or worse solution.
Skipping ahead, a simple place to start may be choosing a random variable $\lambda$ then solving:
$$
\begin{array}{*2{>{\displaystyle}r}}
\mbox{minimize (over $x$)} & E\left[\lambda f(x) \right] \\
\mbox{subject to} & x \in X
\end{array}
$$
This is a simple linear re-weighting of $E[f(x)]$. Anyway, here's an argument for why collapsing multiple objectives to a single objective is typically ok.
Basic setup:
- You have a choice variable $x$ and a feasible set $X$.
- Your choice of $x$ leads a random outcome $\tilde{y} = f(x)$
- You have rational preferences $\prec$ over the random outcome. (Basically, you can say whether you prefer one random outcome $\tilde{y}$ to another.)
Your problem is to choose $x^*\in X$ such that:
$$ \nexists_{x \in X} \quad f(x^*) \prec f(x) $$
In English, you wan to choose $x^*$ so that no feasible choice $x$ leads to an outcome preferred to $f(x^*)$.
Equivalence to maximizing utility (under certain technical conditions)
For technical simplicity, I'll say we're in a discrete probability space with $n$ outcomes so I can represent random outcome $\tilde{y}$ with a vector $\mathbf{y} \in \mathcal{R}^n$.
Under certain technical conditions (that aren't limiting in a practical sense), the above problem is equivalent to maximizing a utility function $U(\mathbf{y})$. (The utility function assigns more preferred outcomes a higher number.)
This logic would apply to any problem where your choice leads to multiple outcome variables.
$$
\begin{array}{*2{>{\displaystyle}r}}
\mbox{maximize (over $x$)} & U(f(x)) \\
\mbox{subject to} & x \in X
\end{array}
$$
Giving more structure to utility function $U$: Expected Utility hypothesis:
If we're in a probabilistic setting and we accept the Neumann-Morgernstern axioms, the overall utility function $U$ has to take a special form:
$$U(\mathbf{y}) = E[u(y_i)] = \sum_i p_i u(y_i) $$
Where $p_i$ is the probability of state $i$ and $u$ is a concave utility function. The curvature of $u$ measures risk aversion. Simply substituting this specialized form of $U$ you get:
$$
\begin{array}{*2{>{\displaystyle}r}}
\mbox{maximize (over $x$)} & \sum_i p_i u(y_i) \\
\mbox{subject to} & x \in X \\
& \mathbf{y} = f(x)
\end{array}
$$
Observe that the simple case $u(y_i) = y_i$ is maximizing the expected value (i.e. no risk aversion).
Another approach: $\lambda$ weights
Another thing to do is:
$$
\begin{array}{*2{>{\displaystyle}r}}
\mbox{maximize (over $x$)} & \sum_i \lambda_i y_i \\
\mbox{subject to} & x \in X \\
& \mathbf{y} = f(x)
\end{array}
$$
Intuitively, you can choose weights $\lambda_i$ that are larger or smaller than the probability $p_i$ of a state occurring, and this captures the importance of a state.
The deeper justification of this approach is that under certain technical conditions, there exists lambda weights $\boldsymbol{\lambda}$ such that the above problem and the earlier problems (eg. maximizing $U(f(x))$) have the same solution.
