Stochastic dominance is apartial order betweenrandom variables.^[1][2] It is a form ofstochastic ordering. The concept is motivated indecision theory anddecision analysis as follows. By standard decision theory, a decision-maker has a utility function${\displaystyle U(x)}$ that encodes theirpreferences, and if the decision-maker needs to pick between several gambles, each gamble's outcome is aprobability distribution over possible outcomes (also known as prospects), and can be written as${\displaystyle X_0,X_1,\dots }$. Then, the decision maker should rationally pick the${\displaystyle X_i}$ that maximizes${\displaystyle \mathbb{E}[U(X_i)]}$.

In more general cases, however, the decision-maker's utility function may be not exactly known, so the above procedure cannot take place. Nevertheless, if we know some partial details about the utility function, then this may be enough to conclude something of the form "any utility function${\displaystyle U}$ that satisfies the given constraint must satisfy${\displaystyle \mathbb{E}[U(X_i)]\ge \mathbb{E}[U(X_j)]}$". In this case, we say that${\displaystyle X_i}$ "stochastically dominates"${\displaystyle X_j}$.Risk aversion is a factor only in second order stochastic dominance.^[3]

Stochastic dominance does not give atotal order, but rather only apartial order. 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,${\displaystyle \rho ,\nu }$ stand for probability distributions on${\displaystyle \mathbb{R}}$, while${\displaystyle A,B,X,Y,Z}$ stand for particular random variables on${\displaystyle \mathbb{R}}$. The notation${\displaystyle X\sim \rho }$ means that${\displaystyle X}$ has distribution${\displaystyle \rho }$.

There are a sequence of stochastic dominance orderings, from zeroth${\displaystyle {⪰}_0}$, to first${\displaystyle {⪰}_1}$, to second${\displaystyle {⪰}_2}$, to higher orders${\displaystyle {⪰}_n}$, each one strictly more inclusive than the previous one. That is, if${\displaystyle \rho {⪰}_n\nu }$, then${\displaystyle \rho {⪰}_k\nu }$ for all${\displaystyle k\ge n}$. Further, there exists${\displaystyle \rho ,\nu }$ such that${\displaystyle \rho {⪰}_{n+1}\nu }$ but not${\displaystyle \rho {⪰}_n\nu }$. Each level of stochastic dominance corresponds to stronger assumptions about the decision-maker's utility function. The stronger these assumptions, the more pairs of gambles can be ranked.

Stochastic dominance could trace back to (Blackwell, 1953),^[4] but it was not developed until 1969–1970.^[3]

The simplest case of stochastic dominance isstatewise dominance (also known asstate-by-state dominance). The idea is that anyone who prefers more to less i.e. hasmonotonically increasing preferences) will always (weakly) prefer a statewise dominant gamble. It is defined as

Random variable${\displaystyle A}$ is statewise dominant over random variable${\displaystyle B}$ if${\displaystyle A}$ gives at least as good a result in every state (every possible set of outcomes).

In symbols,${\displaystyle P(A\ge B)=1}$. This is written as${\displaystyle A{⪰}_{0}B}$.

For strict statewise dominance, an extra condition is needed:${\displaystyle A}$ gives a strictly better result in at least one state. In symbols,${\displaystyle P(A\ge B)=1}$ and${\displaystyle P(A>B)>0}$. This is written as${\displaystyle A{\succ }_{0}B}$.

Similarly, if${\displaystyle P(A=B)=1}$, then both${\displaystyle A{⪰}_{0}B}$ and${\displaystyle B{⪰}_{0}A}$, so${\displaystyle A{\sim }_{0}B}$, and they are equivalent in the sense of statewise dominance.

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.

Statewise dominance impliesfirst-order stochastic dominance (FSD),^[5] which is defined as:

Random variable A has first-order stochastic dominance over random variable B if for any outcomex, A gives at least as high a probability of receiving at leastx as does B, and for somex, A gives a higher probability of receiving at leastx. In notation form,${\displaystyle P[A\ge x]\ge P[B\ge x]}$ for allx.

This is written as${\displaystyle A{⪰}_{1}B}$. Similarly to the case of zeroth-order,${\displaystyle A{\succ }_{1}B}$ requires the extra condition: for somex,${\displaystyle P[A\ge x]>P[B\ge x]}$.

In terms of thecumulative distribution functions,${\displaystyle A{⪰}_{1}B}$ means that${\displaystyle F_A(x)\le F_B(x)}$ for allx.${\displaystyle A{\succ }_{1}B}$ requires the extra condition: for some${\displaystyle x}$.

If${\displaystyle A,B}$ are comparable according to first-order dominance, then${\displaystyle \mathrm{\forall }x,F_A(x)\le F_B(x)}$ or${\displaystyle \mathrm{\forall }x,F_A(x)\ge F_B(x)}$. In both cases, we say that "the distributions of${\displaystyle A}$ and${\displaystyle B}$ do not intersect".

When${\displaystyle A,B}$ are comparable according to${\displaystyle {⪰}_1}$,Wilcoxon rank-sum test tests for first-order stochastic dominance.^[6]

Let${\displaystyle \rho ,\nu }$ be two probability distributions on${\displaystyle \mathbb{R}}$, such that${\displaystyle {\mathbb{E}}_{X\sim \rho }[|X|],{\mathbb{E}}_{X\sim \nu }[|X|]}$ are both finite, then the following conditions are equivalent, thus they may all serve as the definition of first-order (weak) stochastic dominance:^[7]

• For any${\displaystyle u:\mathbb{R}\to \mathbb{R}}$ that is non-decreasing,${\displaystyle {\mathbb{E}}_{X\sim \rho }[u(X)]\ge {\mathbb{E}}_{X\sim \nu }[u(X)]}$.
• ${\displaystyle F_{\rho }(t)\le F_{\nu }(t),\mathrm{\forall }t\in \mathbb{R}.}$
• There exists three random variables${\displaystyle X\sim \rho ,Y\sim \nu ,\delta }$, such that${\displaystyle X=Y+\delta }$, and${\displaystyle \delta \ge 0}$.

For strict first-order stochastic dominance, add a non-equality in the definition, as usual.

The first definition states that a gamble${\displaystyle \rho }$ first-order stochastically dominates gamble${\displaystyle \nu }$if and only if everyexpected utility maximizer with an increasing utility function prefers gamble${\displaystyle \rho }$ over gamble${\displaystyle \nu }$.

The third definition is equivalent to the second. The third definition states that we can construct a pair of gambles${\displaystyle X,Y}$ with distributions${\displaystyle \rho ,\nu }$, such that gamble${\displaystyle X}$ always pays at least as much as gamble${\displaystyle Y}$. More concretely, construct first a uniformly distributed${\displaystyle Z\sim \mathrm{U}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}(0,1)}$, then use theinverse transform sampling to get${\displaystyle X=F_X^{-1}(Z),Y=F_Y^{-1}(Z)}$, then${\displaystyle X\ge Y}$ for any${\displaystyle Z}$. This then implies the second definition. The argument can be run backwards to show that the second definition implies the third.

Consider three gambles over a single toss of a fair six-sided die:

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 first-order dominates B.

Gamble C does not statewise dominate B because B gives a better yield in states 4 through 6, but C first-order 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 first-order 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 first-order stochastically dominates a second gamble, theexpected 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 first-order dominate A.

The other commonly used type of stochastic dominance issecond-order stochastic dominance.^[1][8][9] Roughly speaking, for two gambles${\displaystyle \rho }$ and${\displaystyle \nu }$, gamble${\displaystyle \rho }$ has second-order stochastic dominance over gamble${\displaystyle \nu }$ if the former is more predictable (i.e. involves less risk) and has at least as high a mean. Allrisk-averseexpected-utility maximizers (that is, those with increasing and concave utility functions) prefer a second-order stochastically dominant gamble to a dominated one. Second-order dominance describes the shared preferences of a smaller class of decision-makers (those for whom more is betterand who are averse to risk, rather thanall those for whom more is better) than does first-order dominance.

In terms of cumulative distribution functions${\displaystyle F_{\rho }}$ and${\displaystyle F_{\nu }}$,${\displaystyle \rho }$ is second-order stochastically dominant over${\displaystyle \nu }$ if and only if${\displaystyle {\int }_{-\mathrm{\infty }}^x[F_{\nu }(t)-F_{\rho }(t)]dt\ge 0}$ for all${\displaystyle x}$, with strict inequality at some${\displaystyle x}$. Equivalently,${\displaystyle \rho }$ dominates${\displaystyle \nu }$ in the second order if and only if${\displaystyle {\mathbb{E}}_{X\sim \rho }[u(X)]\ge {\mathbb{E}}_{X\sim \nu }[u(X)]}$ for all nondecreasing andconcave utility functions${\displaystyle u(x)}$.

Second-order stochastic dominance can also be expressed as follows: Gamble${\displaystyle \rho }$ second-order stochastically dominates${\displaystyle \nu }$ if and only if there exist some gambles${\displaystyle y}$ and${\displaystyle z}$ such that${\displaystyle x_{\nu }\stackrel{d}{=}(x_{\rho }+y+z)}$, with${\displaystyle y}$ always less than or equal to zero, and with${\displaystyle \mathbb{E}(z\mid x_{\rho }+y)=0}$ for all values of${\displaystyle x_{\rho }+y}$. Here the introduction of random variable${\displaystyle y}$ makes${\displaystyle \nu }$ first-order stochastically dominated by${\displaystyle \rho }$ (making${\displaystyle \nu }$ disliked by those with an increasing utility function), and the introduction of random variable${\displaystyle z}$ introduces amean-preserving spread in${\displaystyle \nu }$ which is disliked by those with concave utility. Note that if${\displaystyle \rho }$ and${\displaystyle \nu }$ have the same mean (so that the random variable${\displaystyle y}$ degenerates to the fixed number 0), then${\displaystyle \nu }$ is a mean-preserving spread of${\displaystyle \rho }$.

Let${\displaystyle \rho ,\nu }$ be two probability distributions on${\displaystyle \mathbb{R}}$, such that${\displaystyle {\mathbb{E}}_{X\sim \rho }[|X|],{\mathbb{E}}_{X\sim \nu }[|X|]}$ are both finite, then the following conditions are equivalent, thus they may all serve as the definition of second-order stochastic dominance:^[7]

• For any${\displaystyle u:\mathbb{R}\to \mathbb{R}}$ that is non-decreasing, and (not necessarily strictly) concave,${\displaystyle {\mathbb{E}}_{X\sim \rho }[u(X)]\ge {\mathbb{E}}_{X\sim \nu }[u(X)]}$

• ${\displaystyle {\int }_{-\mathrm{\infty }}^t{F}_{\rho }(x)dx\le {\int }_{-\mathrm{\infty }}^t{F}_{\nu }(x)dx,\mathrm{\forall }t\in \mathbb{R}.}$
• There exists two random variables${\displaystyle X\sim \rho ,Y\sim \nu }$, such that${\displaystyle Y=X-\delta +ϵ}$, where${\displaystyle \delta \ge 0}$ and${\displaystyle \mathbb{E}[ϵ|X-\delta ]=0}$.

These are analogous with the equivalent definitions of first-order stochastic dominance, given above.

• First-order stochastic dominance ofA overB is a sufficient condition for second-order dominance ofA overB.
• IfB is a mean-preserving spread ofA, thenA second-order stochastically dominatesB.

• ${\displaystyle {\mathbb{E}}_{\rho }(x)\ge {\mathbb{E}}_{\nu }(x)}$ is a necessary condition for${\displaystyle \rho }$ to second-order stochastically dominate${\displaystyle \nu }$.
• ${\displaystyle \underset{\rho }{min}(x)\ge \underset{\nu }{min}(x)}$ is a necessary condition for${\displaystyle \rho }$ to second-order dominate${\displaystyle \nu }$. The condition implies that the left tail of${\displaystyle F_{\nu }}$ must be thicker than the left tail of${\displaystyle F_{\rho }}$.

Let${\displaystyle F_{\rho }}$ and${\displaystyle F_{\nu }}$ be the cumulative distribution functions of two distinct investments${\displaystyle \rho }$ and${\displaystyle \nu }$.${\displaystyle \rho }$ dominates${\displaystyle \nu }$ inthe third order if and only if both

• ${\displaystyle {\int }_{-\mathrm{\infty }}^x\left({\int }_{-\mathrm{\infty }}^z[F_{\nu }(t)-F_{\rho }(t)]dt\right)dz\ge 0\text{ for all }x,}$
• ${\displaystyle {\mathbb{E}}_{\rho }(x)\ge {\mathbb{E}}_{\nu }(x)}$.

Equivalently,${\displaystyle \rho }$ dominates${\displaystyle \nu }$ in the third order if and only if${\displaystyle {\mathbb{E}}_{\rho }U(x)\ge {\mathbb{E}}_{\nu }U(x)}$ for all${\displaystyle U\in D_3}$.

The set${\displaystyle D_3}$ has two equivalent definitions:

• the set of nondecreasing, concave utility functions that arepositively skewed (that is, have a nonnegative third derivative throughout).^[10]
• the set of nondecreasing, concave utility functions, such that for any random variable${\displaystyle Z}$, therisk-premium function${\displaystyle {\pi }_u(x,Z)}$ is a monotonically nonincreasing function of${\displaystyle x}$.^[11]

Here,${\displaystyle {\pi }_u(x,Z)}$ is defined as the solution to the problem$${\displaystyle u(x+\mathbb{E}[Z]-\pi )=\mathbb{E}[u(x+Z)].}$$See more details atrisk premium page.

• Second-order dominance is a sufficient condition.

• ${\displaystyle {\mathbb{E}}_{\rho }(\log (x))\ge {\mathbb{E}}_{\nu }(\log (x))}$ is a necessary condition. The condition implies that thegeometric mean of${\displaystyle \rho }$ must be greater than or equal to the geometric mean of${\displaystyle \nu }$.
• ${\displaystyle \underset{\rho }{min}(x)\ge \underset{\nu }{min}(x)}$ is a necessary condition. The condition implies that the left tail of${\displaystyle F_{\nu }}$ must be thicker than the left tail of${\displaystyle F_{\rho }}$.

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.^[12]Arguably the most powerful dominance criterion relies on the accepted economic assumption ofdecreasing absolute risk aversion.^[13][14]This involves several analytical challenges and a research effort is on its way to address those.^[15]

Formally, the n-th-order stochastic dominance is defined as^[16]

• For any probability distribution${\displaystyle \rho }$ on${\displaystyle [0,\mathrm{\infty })}$, define the functions inductively:

$${\displaystyle F_{\rho }^1(t)=F_{\rho }(t),F_{\rho }^2(t)={\int }_0^t{F}_{\rho }^1(x)dx,\cdots }$$

• For any two probability distributions${\displaystyle \rho ,\nu }$ on${\displaystyle [0,\mathrm{\infty })}$, non-strict and strict n-th-order stochastic dominance is defined as$${\displaystyle \rho {⪰}_n\nu \text{ iff }{F}_{\rho }^n\le F_{\nu }^n\text{ on }[0,\mathrm{\infty })}$$$${\displaystyle \rho {\succ }_n\nu \text{ iff }\rho {⪰}_n\nu \text{ and }\rho \ne \nu }$$

These relations are transitive and increasingly more inclusive. That is, if${\displaystyle \rho {⪰}_n\nu }$, then${\displaystyle \rho {⪰}_k\nu }$ for all${\displaystyle k\ge n}$. Further, there exists${\displaystyle \rho ,\nu }$ such that${\displaystyle \rho {⪰}_{n+1}\nu }$ but not${\displaystyle \rho {⪰}_n\nu }$.

Define the n-th moment by${\displaystyle {\mu }_k(\rho )={\mathbb{E}}_{X\sim \rho }[X^k]=\int x^{k}d{F}_{\rho }(x)}$, then

Stochastic dominance relations may be used as constraints in problems ofmathematical optimization, in particularstochastic programming.^[17][18][19] In a problem of maximizing a real functional${\displaystyle f(X)}$ over random variables${\displaystyle X}$ in a set${\displaystyle X_0}$ we may additionally require that${\displaystyle X}$ stochastically dominates a fixed randombenchmark${\displaystyle B}$. In these problems,utility functions play the role ofLagrange 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${\displaystyle f(X)+\mathbb{E}[u(X)-u(B)]}$ over${\displaystyle X}$ in${\displaystyle X_0}$, where${\displaystyle u(x)}$ is a certain utility function. If thefirst order stochastic dominance constraint is employed, the utility function${\displaystyle u(x)}$ isnondecreasing; if the second order stochastic dominance constraint is used,${\displaystyle u(x)}$ isnondecreasing andconcave. Asystem of linear equations can test whether a given solution if efficient for any such utility function.^[20]Third-order stochastic dominance constraints can be dealt with using convex quadratically constrained programming (QCP).^[21]

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