Stochastic dominance#Second-order

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.

File:Normaldistdominance2.svg

File:Normaldistnondominance2.svg

Statewise dominance implies first-order stochastic dominance (FSD),{{cite journal | last1 = Quirk | first1 = J. P. | last2 = Saposnik | first2 = R. | year = 1962 | title = Admissibility and Measurable Utility Functions | journal = Review of Economic Studies | volume = 29 | issue = 2 | pages = 140–146 | doi=10.2307/2295819| jstor = 2295819 }} 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, $P\; [A\; \backslash ge\; x]\backslash ge\; P\; [B\; \backslash ge\; x]$ for all x, and for some x, $P[A\; \backslash ge\; x]>P[B\; \backslash ge\; x]$.

In terms of the cumulative distribution functions of the two random variables, A dominating B means that $F\_A(x)\; \backslash le\; F\_B(x)$ for all x, with strict inequality at some x.

In the case of non-intersecting{{Clarification needed|date=March 2024}} distribution functions, the Wilcoxon rank-sum test tests for first-order stochastic dominance.Seifert, S. (2006). Posted Price Offers in Internet Auction Markets. Deutschland: Physica-Verlag. Page 85, ISBN 9783540352686, https://books.google.com/books?id=a-ngTxeSLakC&pg=PA85

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

• For any $u:\; \backslash R\; \backslash to\; \backslash R$ that is non-decreasing, $\backslash mathbb\; E\_\{X\backslash sim\; \backslash rho\}[u(X)]\; \backslash geq\; \backslash mathbb\; E\_\{X\backslash sim\; \backslash nu\}[u(X)]$

• $F\_\backslash rho(t)\; \backslash leq\; F\_\backslash nu(t),\; \backslash quad\; \backslash forall\; t\; \backslash in\; \backslash R.$
• There exists two random variables $X\backslash sim\; \backslash rho,\; Y\; \backslash sim\; \backslash nu$, such that $X\; =\; Y\; +\; \backslash delta$, where $\backslash delta\; \backslash geq\; 0$.

The first definition states that a gamble $\backslash rho$ first-order stochastically dominates gamble $\backslash nu$ if and only if every expected utility maximizer with an increasing utility function prefers gamble $\backslash rho$ over gamble $\backslash nu$.

The third definition states that we can construct a pair of gambles $X,\; Y$ with distributions $\backslash rho,\; \backslash nu$, such that gamble $X$ always pays at least as much as gamble $Y$. More concretely, construct first a uniformly distributed $Z\backslash sim\backslash mathrm\{Uniform\}(0,\; 1)$, then use the inverse transform sampling to get $X\; =\; F\_X^\{-1\}(Z),\; Y\; =\; F\_Y^\{-1\}(Z)$, then $X\; \backslash geq\; Y$ for any $Z$.

Pictorially, the second and third definition are equivalent, because we can go from the graphed density function of A to that of B both by pushing it upwards and pushing it leftwards.

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

: $$

\begin{array}{rcccccc}

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

\hline

\text{gamble A wins }\$ & 1 & 1 & 2 & 2 & 2 & 2 \\

\text{gamble B wins }\$ & 1 & 1 & 1 & 2 & 2 & 2 \\

\text{gamble C wins }\$ & 3 & 3 & 3 & 1 & 1 & 1 \\

\hline

\end{array}

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, 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 first-order dominate A.

The other commonly used type of stochastic dominance is second-order stochastic dominance.{{cite journal | last1 = Hanoch | first1 = G. | last2 = Levy | first2 = H. | year = 1969 | title = The Efficiency Analysis of Choices Involving Risk | journal = Review of Economic Studies | volume = 36 | issue = 3| pages = 335–346 | doi=10.2307/2296431| jstor = 2296431 }}{{cite journal | last1 = Rothschild | first1 = M. |author-link=Michael Rothschild | last2 = Stiglitz | first2 = J. E. |author-link2=Joseph Stiglitz | year = 1970 | title = Increasing Risk: I. A Definition | journal = Journal of Economic Theory | volume = 2 | issue = 3 | pages = 225–243 |doi=10.1016/0022-0531(70)90038-4 }} Roughly speaking, for two gambles $\backslash rho$ and $\backslash nu$, gamble $\backslash rho$ has second-order stochastic dominance over gamble $\backslash nu$ 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 $F\_\backslash rho$ and $F\_\backslash nu$, $\backslash rho$ is second-order stochastically dominant over $\backslash nu$ if and only if $\backslash int\_\{-\backslash infty\}^x\; [F\_\backslash nu(t)\; -\; F\_\backslash rho(t)]\; \backslash ,\; dt\; \backslash geq\; 0$ for all $x$, with strict inequality at some $x$. Equivalently, $\backslash rho$ dominates $\backslash nu$ in the second order if and only if $\backslash mathbb\; E\_\{X\backslash sim\; \backslash rho\}[u(X)]\; \backslash geq\; \backslash mathbb\; E\_\{X\backslash sim\; \backslash nu\}[u(X)]$ for all nondecreasing and concave utility functions $u(x)$.

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

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

| last1=Mas-Colell | first1=Andreu

| last2=Whinston | first2=Michael Dennis

| last3=Green | first3=Jerry R.

| url=https://www.worldcat.org/oclc/32430901

| title=Microeconomic theory

| date=1995

| isbn=0-19-507340-1

| location=New York

| at=Proposition 6.D.1

| oclc=32430901}}

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

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

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

• First-order stochastic dominance of A over B is a sufficient condition for second-order dominance of A over B.
• If B is a mean-preserving spread of A, then A second-order stochastically dominates B.

• $\backslash mathbb\; E\_\backslash rho(x)\; \backslash geq\; \backslash mathbb\; E\_\backslash nu(x)$ is a necessary condition for $\backslash rho$ to second-order stochastically dominate $\backslash nu$.
• $\backslash min\_\backslash rho(x)\backslash geq\; \backslash min\_\backslash nu(x)$ is a necessary condition for $\backslash rho$ to second-order dominate $\backslash nu$. The condition implies that the left tail of $F\_\backslash nu$ must be thicker than the left tail of $F\_\backslash rho$.

Let $F\_\backslash rho$ and $F\_\backslash nu$ be the cumulative distribution functions of two distinct investments $\backslash rho$ and $\backslash nu$. $\backslash rho$ dominates $\backslash nu$ in the third order if and only if both

• $\backslash int\_\{-\backslash infty\}^x\; \backslash left(\backslash int\_\{-\backslash infty\}^z\; [F\_\backslash nu(t)\; -\; F\_\backslash rho(t)]\; \backslash ,\; dt\backslash right)\; dz\; \backslash geq\; 0\; \backslash text\{\; for\; all\; \}\; x,$

• $\backslash mathbb\; E\_\backslash rho(x)\; \backslash geq\; \backslash mathbb\; E\_\backslash nu(x)$.

Equivalently, $\backslash rho$ dominates $\backslash nu$ in the third order if and only if $\backslash mathbb\; E\_\backslash rho\; U(x)\; \backslash geq\; \backslash mathbb\; E\_\backslash nu\; U(x)$ for all $U\backslash in\; D\_3$.

The set $D\_3$ has two equivalent definitions:

• the set of nondecreasing, concave utility functions that are positively skewed (that is, have a nonnegative third derivative throughout).{{Cite web |last1=Chan |first1=Raymond H. |last2=Clark |first2=Ephraim |last3=Wong |first3=Wing-Keung |date=2012-11-16 |title=On the Third Order Stochastic Dominance for Risk-Averse and Risk-Seeking Investors |url=https://mpra.ub.uni-muenchen.de/42676/ |access-date=2022-12-25 |website=mpra.ub.uni-muenchen.de |language=en}}
• the set of nondecreasing, concave utility functions, such that for any random variable $Z$, the risk-premium function $\backslash pi\_u(x,\; Z)$ is a monotonically nonincreasing function of $x$.{{Cite journal |last=Whitmore |first=G. A. |date=1970 |title=Third-Degree Stochastic Dominance |url=https://www.jstor.org/stable/1817999 |journal=The American Economic Review |volume=60 |issue=3 |pages=457–459 |jstor=1817999 |issn=0002-8282}}

Here, $\backslash pi\_u(x,\; Z)$ is defined as the solution to the problem$$u(x\; +\; \backslash mathbb\; E[Z]\; -\; \backslash pi)\; =\; \backslash mathbb\; E\; [u(x\; +\; Z)].$$See more details at risk premium page.

• Second-order dominance is a sufficient condition.

• $\backslash mathbb\; E\_\backslash rho(\backslash log(x))\backslash geq\; \backslash mathbb\; E\_\backslash nu(\backslash log(x))$ is a necessary condition. The condition implies that the geometric mean of $\backslash rho$ must be greater than or equal to the geometric mean of $\backslash nu$.
• $\backslash min\_\backslash rho(x)\backslash geq\backslash min\_\backslash nu(x)$ is a necessary condition. The condition implies that the left tail of $F\_\backslash nu$ must be thicker than the left tail of $F\_\backslash rho$.

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.{{cite journal |last1=Ekern |first1=Steinar |title=Increasing Nth Degree Risk |journal=Economics Letters |date=1980 |volume=6 |issue=4 |pages=329–333 |doi=10.1016/0165-1765(80)90005-1}}

Arguably the most powerful dominance criterion relies on the accepted economic assumption of decreasing absolute risk aversion.{{cite journal | last1 = Vickson | first1 = R.G. | year = 1975 | title = Stochastic Dominance Tests for Decreasing Absolute Risk Aversion. I. Discrete Random Variables | journal = Management Science | volume = 21 | issue = 12| pages = 1438–1446 | doi=10.1287/mnsc.21.12.1438}}{{cite journal | last1 = Vickson | first1 = R.G. | year = 1977 | title = Stochastic Dominance Tests for Decreasing Absolute Risk Aversion. II. General random Variables | journal = Management Science | volume = 23 | issue = 5| pages = 478–489 | doi=10.1287/mnsc.23.5.478}}

This involves several analytical challenges and a research effort is on its way to address those.

See, e.g. {{cite journal | last1 = Post | first1 = Th. | last2 = Fang | first2 = Y. | last3 = Kopa | first3 = M. | year = 2015 | title = Linear Tests for DARA Stochastic Dominance | journal = Management Science | volume = 61 | issue = 7| pages = 1615–1629 | doi=10.1287/mnsc.2014.1960}}

Formally, the n-th-order stochastic dominance is defined as {{Cite journal |last=Fishburn |first=Peter C. |date=1980-02-01 |title=Stochastic Dominance and Moments of Distributions |url=https://pubsonline.informs.org/doi/abs/10.1287/moor.5.1.94 |journal=Mathematics of Operations Research |volume=5 |issue=1 |pages=94–100 |doi=10.1287/moor.5.1.94 |issn=0364-765X|url-access=subscription }}

• For any probability distribution $\backslash rho$ on $[0,\; \backslash infty)$, define the functions inductively:

$$F^1\_\backslash rho(t)\; =\; F\_\backslash rho(t),\; \backslash quad\; F^2\_\backslash rho(t)\; =\; \backslash int\_0^t\; F^1\_\backslash rho(x)dx,\; \backslash quad\; \backslash cdots$$

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

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

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

{{Math theorem|name=Theorem|note=|math_statement=

If $\backslash rho\; \backslash succ\_n\; \backslash nu$ are on $[0,\; \backslash infty)$ with finite moments $\backslash mu\_k(\backslash rho),\; \backslash mu\_k(\backslash nu)$ for all $k\; =\; 1,\; 2,\; ...,\; n$, then $(\backslash mu\_1(\backslash rho),\; \backslash ldots,\; \backslash mu\_n(\backslash rho))\backslash succ^*\_n\; (\backslash mu\_1(\backslash nu),\; \backslash ldots,\; \backslash mu\_n(\backslash nu))$.

Here, the partial ordering $\backslash succ^*\_n$ is defined on $\backslash mathbb\; R^n$ by $v\; \backslash succ^*\_n\; w$ iff $v\; \backslash neq\; w$, and, letting $k$ be the smallest such that $v\_k\; \backslash neq\; w\_k$, we have $(-1)^\{k-1\}\; v\_k\; >\; (-1)^\{k-1\}\; w\_k$

}}

Stochastic dominance relations may be used as constraints in problems of mathematical optimization, in particular stochastic programming.{{cite journal |author-link=Darinka Dentcheva |last1=Dentcheva |first1=D. |author-link2=Andrzej Piotr Ruszczyński |last2=Ruszczyński |first2=A. |title=Optimization with Stochastic Dominance Constraints |journal=SIAM Journal on Optimization |volume=14 |issue=2 |year=2003 |pages=548–566 |doi=10.1137/S1052623402420528 |citeseerx=10.1.1.201.7815 |s2cid=12502544 }}{{cite journal | last1 = Kuosmanen | first1 = T | year = 2004 | title = Efficient diversification according to stochastic dominance criteria | journal = Management Science | volume = 50 | issue = 10| pages = 1390–1406 | doi=10.1287/mnsc.1040.0284}}{{cite journal |author-link=Darinka Dentcheva |last1=Dentcheva |first1=D. |author-link2=Andrzej Piotr Ruszczyński |last2=Ruszczyński |first2=A. |title=Semi-Infinite Probabilistic Optimization: First Order Stochastic Dominance Constraints |journal=Optimization |volume=53 |issue=5–6 |year=2004 |pages=583–601 |doi=10.1080/02331930412331327148 |s2cid=122168294 }} 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

$f(X)\; +\; \backslash mathbb\; E[u(X)\; -\; u(B)]$ 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. A system of linear equations can test whether a given solution if efficient for any such utility function.{{cite journal | last1 = Post | first1 = Th | year = 2003 | title = Empirical tests for stochastic dominance efficiency | journal = Journal of Finance | volume = 58 | issue = 5| pages = 1905–1932 | doi=10.1111/1540-6261.00592}}

Third-order stochastic dominance constraints can be dealt with using convex quadratically constrained programming (QCP).{{cite journal |last1=Post |first1=Thierry |last2=Kopa |first2=Milos |title=Portfolio Choice Based on Third-Degree Stochastic Dominance |year=2016 |journal=Management Science |volume=63 |issue=10 |pages=3381–3392 |doi=10.1287/mnsc.2016.2506 |ssrn=2687104 }}

• 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

{{Reflist|30em}}

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Category:Random variable ordering