# Stochastic dominance

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The term **stochastic dominance** is used in decision theory to refer to situations where one lottery (a probability distribution over outcomes) can be ranked as superior to another. It is based on preferences regarding outcomes (e.g., if each outcome is expressed as a number, e.g. gain or utility, a higher value is preferred), but requires only limited knowledge of preferences with regard to distributions of outcomes, which depend e.g. on risk aversion.

The simplest case is **statewise dominance** (also known as **state-by-state dominance**), defined as follows: lottery A is statewise dominant over lottery B if A gives a better outcome than B in every possible state of nature (more precisely, at least as good an outcome in every state, with strict inequality in at least one state). For example, if a dollar is added to one or more prizes in a lottery, the new lottery statewise dominates the old one. 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 monotonic preferences) will always prefer a statewise dominant lottery.

The canonical case of stochastic dominance is referred to as **first-order stochastic dominance**, defined as follows: lottery A has first-order stochastic dominance over lottery B if for any outcome *x*, A gives a higher probability of receiving an outcome equal to or better than *x* than B. For example, throwing a coin, head and tail give 1 and 2, respectively, with A, and 3 and 1, respectively, with B. In this example there is no statewise dominance.

The other commonly used case of stochastic dominance is **second-order stochastic dominance**. All risk-averse expected-utility maximizers prefer a second-order stochastically dominant lottery to a dominated lottery. The same is true for non-expected utility maximizers with concave local utility functions.

Higher orders of stochastic dominance have also been analyzed, as have generalizations of the dual relationship between stochastic dominance orderings and classes of preference functions.

## External linksEdit

- There's a good discussion of first and second order stochastic dominance at http://cepa.newschool.edu/het/essays/uncert/increase.htm .

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