Stochastic dominance is a
partial order
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
between
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s.
It is a form of
stochastic ordering. The concept arises in
decision theory and
decision analysis in situations where one gamble (a
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
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
preference
In psychology, economics and philosophy, preference is a technical term usually used in relation to choosing between alternatives. For example, someone prefers A over B if they would rather choose A than B. Preferences are central to decision theo ...
s regarding sets of possible outcomes and their associated probabilities. Only limited knowledge of preferences is required for determining dominance.
Risk aversion
In economics and finance, risk aversion is the tendency of people to prefer outcomes with low uncertainty to those outcomes with high uncertainty, even if the average outcome of the latter is equal to or higher in monetary value than the more ce ...
is a factor only in second order stochastic dominance.
Stochastic dominance does not give a
total order
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexive) ...
, but rather only a
partial order
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
: 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.
Throughout the article,
stand for probability distributions on
, while
stand for particular random variables on
. The notation
means that
has distribution
.
There are a sequence of stochastic dominance orderings, from first
, to second
, to higher orders
. The sequence is increasingly more inclusive. That is, if
, then
for all
. Further, there exists
such that
but not
.
Stochastic dominance could trace back to (Blackwell, 1953), but it was not developed until 1969–1970.
Statewise dominance
The simplest case of stochastic dominance is statewise dominance (also known as state-by-state dominance), defined as follows:
: Random variable A is statewise dominant over random variable 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
monotonic
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...
ally increasing preferences) will always prefer a statewise dominant gamble.
First-order
Statewise dominance is implied by first-order stochastic dominance (FSD), which is defined as:
: Random variable A has first-order stochastic dominance over random variable 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,