Stochastic dominance

Stochastic dominance is a partial order between random variables. It is a form of stochastic ordering. The concept is motivated in decision theory and decision analysis as follows. By standard decision theory, a decision-maker has a utility function ${\displaystyle U(x)}$ that encodes their preferences, and if the decision-maker needs to pick between several gambles, each gamble's outcome is a probability distribution over possible outcomes (also known as prospects), and can be written as ${\displaystyle X_{0},X_{1},\dots }$. Then, the decision maker should rationally pick the ${\displaystyle X_{i}}$ that maximizes ${\displaystyle \mathbb {E} [U(X_{i})]}$.

In more general cases, however, the decision-maker's utility function may be not exactly known, so the above procedure cannot take place. Nevertheless, if we know some partial details about the utility function, then this may be enough to conclude something of the form "any utility function ${\displaystyle U}$ that satisfies the given constraint must satisfy ${\displaystyle \mathbb {E} [U(X_{i})]\geq \mathbb {E} [U(X_{j})]}$". In this case, we say that ${\displaystyle X_{i}}$ "stochastically dominates" ${\displaystyle X_{j}}$. Risk aversion is a factor only in second order stochastic dominance.

Stochastic dominance does not give a total order, but rather only a partial order. For some pairs of gambles, neither one stochastically dominates the other, since different members of the broad class of decision-makers will differ regarding which gamble is preferable without them generally being considered to be equally attractive.

Throughout the article, ${\displaystyle \rho ,\nu }$ stand for probability distributions on ${\displaystyle \mathbb {R} }$, while ${\displaystyle A,B,X,Y,Z}$ stand for particular random variables on ${\displaystyle \mathbb {R} }$. The notation ${\displaystyle X\sim \rho }$ means that ${\displaystyle X}$ has distribution ${\displaystyle \rho }$.

There are a sequence of stochastic dominance orderings, from zeroth ${\displaystyle \succeq _{0}}$, to first ${\displaystyle \succeq _{1}}$, to second ${\displaystyle \succeq _{2}}$, to higher orders ${\displaystyle \succeq _{n}}$, each one strictly more inclusive than the previous one. That is, if ${\displaystyle \rho \succeq _{n}\nu }$, then ${\displaystyle \rho \succeq _{k}\nu }$ for all ${\displaystyle k\geq n}$. Further, there exists ${\displaystyle \rho ,\nu }$ such that ${\displaystyle \rho \succeq _{n+1}\nu }$ but not ${\displaystyle \rho \succeq _{n}\nu }$. Each level of stochastic dominance corresponds to stronger assumptions about the decision-maker's utility function. The stronger these assumptions, the more pairs of gambles can be ranked.

Stochastic dominance could trace back to (Blackwell, 1953), but it was not developed until 1969–1970.