Stochastic dominance^{[1]}^{[2]} 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 decisionmakers. 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 decisionmakers will differ regarding which gamble is preferable without them generally being considered to be equally attractive.
A related concept not included under stochastic dominance is deterministic dominance, which occurs when the least preferable outcome of gamble A is more valuable than the most highly preferred outcome of gamble B.
Contents

Statewise dominance 1

Firstorder stochastic dominance 2

Secondorder stochastic dominance 3

Sufficient conditions for secondorder stochastic dominance 3.1

Necessary conditions for secondorder stochastic dominance 3.2

Thirdorder stochastic dominance 4

Sufficient condition for thirdorder stochastic dominance 4.1

Necessary conditions for thirdorder stochastic dominance 4.2

Higherorder stochastic dominance 5

Stochastic dominance constraints 6

See also 7

References 8
Statewise dominance
The simplest case of stochastic dominance is statewise dominance (also known as statebystate dominance), defined as follows: gamble A is statewise dominant over gamble B if A gives at least as good a result in every state (every possible set of outcomes), and a strictly better result 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 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.
Firstorder stochastic dominance
Statewise dominance is a special case of the canonical firstorder stochastic dominance, defined as follows: Gamble A has firstorder stochastic dominance over gamble B if for any 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, P [A \ge x]\ge P [B \ge x] for all x, and for some x, P[A \ge x]>P[B \ge x].
In terms of the cumulative distribution functions of the two gambles, A dominating B means that F_A(x) \le F_B(x) for all x, with strict inequality at some x.
Gamble A firstorder stochastically dominates gamble B if and only if every expected utility maximizer with an increasing utility function prefers gamble A over gamble B.
Firstorder stochastic dominance can also be expressed as follows: If and only if A firstorder stochastically dominates B, there exists some gamble y such that x_B \overset {d}{=} (x_A+y) where y\le 0 in all possible states (and strictly negative in at least one state); here \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.
For example, consider a single toss of a fair die with the six possible outcomes (states) summarized in this table along with the amount won in each state by each of three alternative gambles:

\text{State (die result)} \quad 1 \quad 2 \quad 3 \quad 4 \quad 5 \quad 6

\text{Gamble A wins}\ $ \quad 1 \quad 1 \quad 2 \quad 2 \quad 2 \quad 2

\text{Gamble B wins}\ $ \quad 1 \quad 1 \quad 1 \quad 2 \quad 2 \quad 2

\text{Gamble C wins}\ $ \quad 3 \quad 3 \quad 3 \quad 1 \quad 1 \quad 1
Here gamble A statewise dominates gamble B because A gives at least as good a yield in all possible states (outcomes of the die roll) and gives a strictly better yield in one of them (state 3). Since A statewise dominates B, it also firstorder dominates B. Gamble C does not statewise dominate B because B gives a better yield in states 4 through 6, but C firstorder stochastically dominates B because Pr(B ≥ 1) = Pr(C ≥ 1) = 1, Pr(B ≥ 2) = Pr(C ≥ 2) = 3/6, and Pr(B ≥ 3) = 0 while Pr(C ≥ 3) = 3/6 > Pr(B ≥ 3). Gambles A and C cannot be ordered relative to each other on the basis of firstorder stochastic dominance because Pr(A ≥ 2) = 4/6 > Pr(C ≥ 2) = 3/6 while on the other hand Pr(C ≥ 3) = 3/6 > Pr(A ≥ 3) = 0.
In general, although when one gamble firstorder stochastically dominates a second gamble, the expected value of the payoff under the first will be greater than the expected value of the payoff under the second, the converse is not true: one cannot order lotteries with regard to stochastic dominance simply by comparing the means of their probability distributions. For instance, in the above example C has a higher mean (2) than does A (5/3), yet C does not firstorder dominate A.
Secondorder stochastic dominance
The other commonly used type of stochastic dominance is secondorder stochastic dominance. Roughly speaking, for two gambles A and B, gamble A has secondorder stochastic dominance over gamble B if the former is more predictable (i.e. involves less risk) and has at least as high a mean. All riskaverse expectedutility maximizers (that is, those with increasing and concave utility functions) prefer a secondorder stochastically dominant gamble to a dominated one. Secondorder dominance describes the shared preferences of a smaller class of decisionmakers (those for whom more is better and who are averse to risk, rather than all those for whom more is better) than does firstorder dominance.
In terms of cumulative distribution functions F_A and F_B, A is secondorder stochastically dominant over B if and only if the area under F_A from minus infinity to x is less than or equal to that under F_B from minus infinity to x for all real numbers x, with strict inequality at some x; that is, \int_{\infty}^x [F_B(t)  F_A(t)]dt \geq 0 for all x, with strict inequality at some x. Equivalently, A dominates B in the second order if and only if E[u(A)] \geq E[u(B)] for all nondecreasing and concave utility functions u(x).
Secondorder stochastic dominance can also be expressed as follows: Gamble A secondorder stochastically dominates B if and only if there exist some gambles y and z such that x_B \overset {d}{=} (x_A + y + z), with y always less than or equal to zero, and with E(zx_A+y)=0 for all values of x_A+y. Here the introduction of random variable y makes B firstorder stochastically dominated by A (making B disliked by those with an increasing utility function), and the introduction of random variable z introduces a meanpreserving spread in B which is disliked by those with concave utility. Note that if A and B have the same mean (so that the random variable y degenerates to the fixed number 0), then B is a meanpreserving spread of A.
Sufficient conditions for secondorder stochastic dominance

Firstorder stochastic dominance of A over B is a sufficient condition for secondorder dominance of A over B.

If B is a meanpreserving spread of A, then A secondorder stochastically dominates B.
Necessary conditions for secondorder stochastic dominance

E_A(x) \geq E_B(x) is a necessary condition for A to secondorder stochastically dominate B.

\min_A(x)\leq\min_B(x) is a necessary condition for A to secondorder dominate B. The condition implies that the left tail of F_B must be thicker than the left tail of F_A.
Thirdorder stochastic dominance
Let F_A and F_B be the cumulative distribution functions of two distinct investments A and B. A dominates B in the third order if and only if

\int_{\infty}^x \int_{\infty}^z [F_B(t)  F_A(t)] \, dt \, dz \geq 0 for all x,
and there is at least one strict inequality. Equivalently, A dominates B in the third order if and only if E_AU(x) \geq E_BU(x) for all nondecreasing, concave utility functions U that are positively skewed (that is, have a positive third derivative throughout).
Sufficient condition for thirdorder stochastic dominance

Secondorder stochastic dominance is a sufficient condition.
Necessary conditions for thirdorder stochastic dominance

E_A(\log(x))\geq E_B(\log(x)) is a necessary condition. The condition implies that the geometric mean of A must be greater than or equal to the geometric mean of B.

\min_A(x)\leq\min_B(x) is a necessary condition. The condition implies that the left tail of F_B must be thicker than the left tail of F_A.
Higherorder stochastic dominance
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.
Stochastic dominance constraints
Stochastic dominance relations may be used as constraints ^{[3]} ^{[4]} in problems of mathematical optimization, in particular stochastic programming. In a problem of maximizing a real functional f(X) over random variables X in a set X_0 we may additionally require that X stochastically dominates a fixed random benchmark B . In these problems, utility functions play the role of Lagrange multipliers associated with stochastic dominance constraints. Under appropriate conditions, the solution of the problem is also a (possibly local) solution of the problem to maximize f(X) + E[u(X)  u(B)] over X in X_0 , where u(x) is a certain utility function. If the first order stochastic dominance constraint is employed, the utility function u(x) is nondecreasing; if the second order stochastic dominance constraint is used, u(x) is nondecreasing and concave.
See also
References

^ Hadar, J., and Russell, W.,"Rules for Ordering Uncertain Prospects", American Economic Review 59, March 1969, 2534.

^ Bawa, Vijay S., "Optimal Rules for Ordering Uncertain Prospects," Journal of Financial Economics 2, 1975, 95121.

^ Dentcheva, D., and Ruszczyński, A., "Optimization with Stochastic Dominance Constraints," SIAM Journal on Optimization 14, 2003, 548566.

^ Dentcheva, D., and Ruszczyński, A., "SemiInfinite Probabilistic Optimization: First Order Stochastic Dominance Constraints," Optimization 53, 2004, 583601.
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