Stochastic ordering

A real random variable ${\displaystyle A}$ is less than a random variable ${\displaystyle B}$ in the "usual stochastic order" if

${\displaystyle \Pr(A>x)\leq \Pr(B>x){\text{ for all }}x\in (-\infty ,\infty ),}$

where ${\displaystyle \Pr(\cdot )}$ denotes the probability of an event. This is sometimes denoted ${\displaystyle A\preceq B}$ or ${\displaystyle A\leq _{st}B}$. If additionally ${\displaystyle \Pr(A>x)<\Pr(B>x)}$ for some ${\displaystyle x}$, then ${\displaystyle A}$ is stochastically strictly less than ${\displaystyle B}$, sometimes denoted ${\displaystyle A\prec B}$. In decision theory, under this circumstance B is said to be first-order stochastically dominant over A.

Stochastic dominance relations are a family of stochastic orderings used in decision theory:[1]

• Zeroth-order stochastic dominance: ${\displaystyle A\prec _{(0)}B}$ if and only if ${\displaystyle A\leq B}$ for all realizations of these random variables and ${\displaystyle A<B}$ for at least one realization.
• First-order stochastic dominance: ${\displaystyle A\prec _{(1)}B}$ if and only if ${\displaystyle \Pr(A>x)\leq \Pr(B>x)}$ for all ${\displaystyle x}$ and there exists ${\displaystyle x}$ such that ${\displaystyle \Pr(A>x)<\Pr(B>x)}$.
• Second-order stochastic dominance: ${\displaystyle A\prec _{(2)}B}$ if and only if ${\displaystyle \int _{-\infty }^{x}[\Pr(B>t)-\Pr(A>t)]\,dt\geq 0}$ for all ${\displaystyle x}$, with strict inequality at some ${\displaystyle x}$.

There also exist higher-order notions of stochastic dominance. With the definitions above, we have ${\displaystyle A\prec _{(i)}B\implies A\prec _{(i+1)}B}$.

An ${\displaystyle \mathbb {R} ^{d}}$-valued random variable ${\displaystyle A}$ is less than an ${\displaystyle \mathbb {R} ^{d}}$-valued random variable ${\displaystyle B}$ in the "usual stochastic order" if

${\displaystyle {\rm {E}}[f(A)]\leq {\rm {E}}[f(B)]{\text{ for all bounded, increasing functions }}f\colon \mathbb {R} ^{d}\longrightarrow \mathbb {R} }$

Other types of multivariate stochastic orders exist. For instance the upper and lower orthant order which are similar to the usual one-dimensional stochastic order. ${\displaystyle A}$ is said to be smaller than ${\displaystyle B}$ in upper orthant order if

${\displaystyle \Pr(A>\mathbf {x} )\leq \Pr(B>\mathbf {x} ){\text{ for all }}\mathbf {x} \in \mathbb {R} ^{d}}$

and ${\displaystyle A}$ is smaller than ${\displaystyle B}$ in lower orthant order if[2]

${\displaystyle \Pr(A\leq \mathbf {x} )\leq \Pr(B\leq \mathbf {x} ){\text{ for all }}\mathbf {x} \in \mathbb {R} ^{d}}$

All three order types also have integral representations, that is for a particular order ${\displaystyle A}$ is smaller than ${\displaystyle B}$ if and only if ${\displaystyle {\rm {E}}[f(A)]\leq {\rm {E}}[f(B)]}$ for all ${\displaystyle f\colon \mathbb {R} ^{d}\longrightarrow \mathbb {R} }$ in a class of functions ${\displaystyle {\mathcal {G}}}$.[3] ${\displaystyle {\mathcal {G}}}$ is then called generator of the respective order.

The following stochastic orders are useful in the theory of random social choice. They are used to compare the outcomes of random social choice functions, in order to check them for efficiency or other desirable criteria.[4] The dominance orders below are ordered from the most conservative to the least conservative. They are exemplified on random variables over the finite support {30,20,10}.

Deterministic dominance, denoted ${\displaystyle A\succeq _{dd}B}$, means that every possible outcome of ${\displaystyle A}$ is at least as good as every possible outcome of ${\displaystyle B}$: for all x<y, ${\displaystyle \Pr[A=x]\cdot \Pr[B=y]=0}$. In other words: ${\displaystyle \Pr[A\geq B]=1}$. For example, ${\displaystyle 0.6*30+0.4*20\succeq _{dd}0.5*20+0.5*10}$.

Bilinear dominance, denoted ${\displaystyle A\succeq _{bd}B}$, means that, for every possible outcome, the probability that ${\displaystyle A}$ yields the better one and ${\displaystyle B}$ yields the worse one is at least as large as the probability the other way around: for all x<y, ${\displaystyle \Pr[A=x]\cdot \Pr[B=y]\leq \Pr[A=y]\cdot \Pr[B=x]}$ For example, ${\displaystyle 0.5*30+0.5*20\succeq _{bd}0.33*30+0.33*20+0.34*10}$.

Stochastic dominance (already mentioned above), denoted ${\displaystyle A\succeq _{sd}B}$, means that, for every possible outcome x, the probability that ${\displaystyle A}$ yields at least x is at least as large as the probability that ${\displaystyle B}$ yields at least x: for all x, ${\displaystyle \Pr[A\geq x]\geq \Pr[B\geq x]}$. For example, ${\displaystyle 0.5*30+0.5*10\succeq _{sd}0.5*20+0.5*10}$.

Pairwise-comparison dominance, denoted ${\displaystyle A\succeq _{pc}B}$, means that the probability that that ${\displaystyle A}$ yields a better outcome than ${\displaystyle B}$ is larger than the other way around: ${\displaystyle \Pr[A\geq B]\geq \Pr[B\geq A]}$. For example, ${\displaystyle 0.67*30+0.33*10\succeq _{pc}1.0*20}$.

Downward-lexicographic dominance, denoted ${\displaystyle A\succeq _{dl}B}$, means that ${\displaystyle A}$ has a larger probability than ${\displaystyle B}$ of returning the best outcome, or both ${\displaystyle A}$ and ${\displaystyle B}$ have the same probability to return the best outcome but ${\displaystyle A}$ has a larger probability than ${\displaystyle B}$ of returning the second-best best outcome, etc. Upward-lexicographic dominance is defined analogously based on the probability to return the worst outcomes. See lexicographic dominance.

• Stochastic dominance
• Stochastic - meaning of the term

• M. Shaked and J. G. Shanthikumar, Stochastic Orders and their Applications, Associated Press, 1994.
• E. L. Lehmann. Ordered families of distributions. The Annals of Mathematical Statistics, 26:399–419, 1955.