Stochastic dominance

Stochastic dominance is a partial order between random variables. It is a form of stochastic ordering. The concept arises in decision theory and decision analysis in situations where one gamble (a probability distribution over possible outcomes, also known as prospects) can be ranked as superior to another gamble for a broad class of decision-makers. It is based on shared preferences regarding sets of possible outcomes and their associated probabilities. Only limited knowledge of preferences is required for determining dominance. Risk aversion is a factor only in second order stochastic dominance. Stochastic dominance is a partial order between random variables. It is a form of stochastic ordering. The concept arises in decision theory and decision analysis in situations where one gamble (a probability distribution over possible outcomes, also known as prospects) can be ranked as superior to another gamble for a broad class of decision-makers. It is based on shared preferences regarding sets of possible outcomes and their associated probabilities. Only limited knowledge of preferences is required for determining dominance. 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. The simplest case of stochastic dominance is statewise dominance (also known as state-by-state dominance), defined as follows: For example, if a dollar is added to one or more prizes in a lottery, the new lottery statewise dominates the old one because it yields a better payout regardless of the specific numbers realized by the lottery. Similarly, if a risk insurance policy has a lower premium and a better coverage than another policy, then with or without damage, the outcome is better. Anyone who prefers more to less (in the standard terminology, anyone who has monotonically increasing preferences) will always prefer a statewise dominant gamble. Statewise dominance is a special case of the canonical first-order stochastic dominance (FSD), which is defined as: In terms of the cumulative distribution functions of the two random variables, A dominating B means that F A ( x ) ≤ F B ( x ) {displaystyle F_{A}(x)leq F_{B}(x)} for all x, with strict inequality at some x. Gamble A first-order stochastically dominates gamble B if and only if every expected utility maximizer with an increasing utility function prefers gamble A over gamble B. First-order stochastic dominance can also be expressed as follows: If and only if A first-order stochastically dominates B, there exists some gamble y {displaystyle y} such that x B = d ( x A + y ) {displaystyle x_{B}{overset {d}{=}}(x_{A}+y)} where y ≤ 0 {displaystyle yleq 0} in all possible states (and strictly negative in at least one state); here = d {displaystyle {overset {d}{=}}} means 'is equal in distribution to' (that is, 'has the same distribution as'). Thus, we can go from the graphed density function of A to that of B by, roughly speaking, pushing some of the probability mass to the left.