Stochastic 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 monotonically increasing preferences) will always prefer a statewise dominant gamble.

Statewise dominance implies first-order stochastic dominance (FSD),[5] 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, ${\displaystyle P[A\geq x]\geq P[B\geq x]}$ for all x, and for some x, ${\displaystyle P[A\geq x]>P[B\geq x]}$.

In terms of the cumulative distribution functions of the two random variables, A dominating B means that ${\displaystyle F_{A}(x)\leq F_{B}(x)}$ for all x, with strict inequality at some x.

The other commonly used type of stochastic dominance is second-order stochastic dominance.[1][7][8] 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. All risk-averse expected-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 better and who are averse to risk, rather than all 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 ${\displaystyle \int _{-\infty }^{x}[F_{\nu }(t)-F_{\rho }(t)]\,dt\geq 0}$ 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 ${\displaystyle \mathbb {E} _{X\sim \rho }[u(X)]\geq \mathbb {E} _{X\sim \nu }[u(X)]}$ for all nondecreasing and concave 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 ${\displaystyle x_{\nu }{\overset {d}{=}}(x_{\rho }+y+z)}$, 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 a mean-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 ${\displaystyle F_{\rho }}$ and ${\displaystyle F_{\nu }}$ be the cumulative distribution functions of two distinct investments ${\displaystyle \rho }$ and ${\displaystyle \nu }$. ${\displaystyle \rho }$ dominates ${\displaystyle \nu }$ in the third order if and only if both

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

Equivalently, ${\displaystyle \rho }$ dominates ${\displaystyle \nu }$ in the third order if and only if ${\displaystyle \mathbb {E} _{\rho }U(x)\geq \mathbb {E} _{\nu }U(x)}$ for all ${\displaystyle U\in D_{3}}$.

The set ${\displaystyle D_{3}}$ has two equivalent definitions:

• the set of nondecreasing, concave utility functions that are positively skewed (that is, have a nonnegative third derivative throughout).[9]
• the set of nondecreasing, concave utility functions, such that for any random variable ${\displaystyle Z}$, the risk-premium function ${\displaystyle \pi _{u}(x,Z)}$ is a monotonically nonincreasing function of ${\displaystyle x}$.[10]

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 at risk premium page.

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

Formally, the n-th-order stochastic dominance is defined as [15]

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

$${\displaystyle F_{\rho }^{1}(t)=F_{\rho }(t),\quad F_{\rho }^{2}(t)=\int _{0}^{t}F_{\rho }^{1}(x)dx,\quad \cdots }$$

• For any two probability distributions ${\displaystyle \rho ,\nu }$ on ${\displaystyle [0,\infty )}$, non-strict and strict n-th-order stochastic dominance is defined as
  $${\displaystyle \rho \succeq _{n}\nu \quad {\text{ iff }}\quad F_{\rho }^{n}\leq F_{\nu }^{n}{\text{ on }}[0,\infty )}$$

  
  $${\displaystyle \rho \succ _{n}\nu \quad {\text{ iff }}\quad \rho \succeq _{n}\nu {\text{ and }}\rho \neq \nu }$$

  

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

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

Stochastic dominance relations may be used as constraints in problems of mathematical optimization, in particular stochastic programming.[16][17][18] 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 random benchmark ${\displaystyle B}$. 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 ${\displaystyle f(X)+\mathbb {E} [u(X)-u(B)]}$ over ${\displaystyle X}$ in ${\displaystyle X_{0}}$, where ${\displaystyle u(x)}$ is a certain utility function. If the first order stochastic dominance constraint is employed, the utility function ${\displaystyle u(x)}$ is nondecreasing; if the second order stochastic dominance constraint is used, ${\displaystyle u(x)}$ is nondecreasing and concave. A system of linear equations can test whether a given solution if efficient for any such utility function.[19] Third-order stochastic dominance constraints can be dealt with using convex quadratically constrained programming (QCP).[20]

• Modern portfolio theory
• Marginal conditional stochastic dominance
• Responsive set extension - equivalent to stochastic dominance in the context of preference relations.
• Quantum catalyst
• Ordinal Pareto efficiency
• Lexicographic dominance