Stochastic Dominance - First-order Stochastic Dominance

Statewise dominance is a special case of the canonical first-order stochastic dominance, defined as follows: gamble A has first-order stochastic dominance over gamble B if for any good outcome x, A gives at least as high a probability of receiving at least x as does B, and for some x, A gives a higher probability of receiving at least x. In notation form, for all x, and for some x, . In terms of the cumulative distribution functions of the two gambles, A dominating B means that for all x, with strict inequality at some x. For example, consider a die-toss where 1 through 3 wins $1 and 4 through 6 wins $2 in gamble B. This is dominated by a gamble A that yields $3 for 1 through 3 and $1 for 4 through 6, and it is also dominated by a gamble C that gives $1 for 1 and 2 and $2 for 3 through 6. Gamble A would have statewise dominance over B if we re-ordered the die toss outcome by value won, but gamble C has first-order stochastic dominance over B without statewise dominance no matter how we order the prospects . Further, although when A dominates B, the expected value of the payoff under A will be greater than the expected value of the payoff under B, this is not a sufficient condition for dominance, and so one cannot order lotteries with regard to stochastic dominance simply by comparing the means of their probability distributions.

Every expected utility maximizer with an increasing utility function will prefer gamble A over gamble B if A first-order stochastically dominates 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 such that where in all possible states (and strictly negative in at least one state); here 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.