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# Stochastic dominance

Stochastic dominance is a form of stochastic ordering. The term 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" under B. In terms of the cumulative distribution functions of the two lotteries, A dominating B means that $A\left(x\right) le B\left(x\right)$ for all x. For example, consider a coin-toss where heads and tails give returns 1 and 3 respectively for A, and 2 and 1 respectively for B. In this example there is no statewise dominance, but A has first-order stochastic dominance over B. 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, and so one cannot order lotteries by comparing the means of their probability distributions.

The other commonly used case of stochastic dominance is second-order stochastic dominance. Roughly speaking, lottery A has second-order stochastic dominance over lottery B if the former is more predictable, ie. involves less risk. 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.

References

* Bawa, Vijay S., "Optimal Rules for Ordering Uncertain Prospects," "Journal of Financial Economics" 2, 1975, 95-121.
** This paper defines first-order, second-order, and third-order stochastic dominance and the sets of utility functions over which dominance implies maximal expected utility for all functions in the set.
** For example, first-order stochastic dominance (FSD) occurs if and only if the dominant distribution has a higher expected utility for all increasing and continuously differentiable utility functions.

*http://mathworld.wolfram.com/Dominance.html
* [http://cepa.newschool.edu/het/essays/uncert/increase.htm A good discussion of first and second order stochastic dominance]

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