Stochastic dominance

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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 U ( x ) {\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 X 0 , X 1 , … {\displaystyle X_{0},X_{1},\dots } . Then, the decision maker should rationally pick the X i {\displaystyle X_{i}} that maximizes E [ U ( X i ) ] {\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 U {\displaystyle U} that satisfies the given constraint must satisfy E [ U ( X i ) ] ≥ E [ U ( X j ) ] {\displaystyle \mathbb {E} [U(X_{i})]\geq \mathbb {E} [U(X_{j})]} ". In this case, we say that X i {\displaystyle X_{i}} "stochastically dominates" X j {\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.