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Stochastic dominance
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<h2 id="statewise-dominance-zeroth-order">Statewise dominance (Zeroth-Order)</h2> <p>The simplest case of stochastic dominance is statewise dominance (also known as state-by-state dominance), defined as follows: </p> 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. <p>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 <a href="/facts/Monotonic/MjQK0YL2">monotonically</a> increasing preferences) will always prefer a statewise dominant gamble. </p> <h2 id="first-order">First-order</h2> <p>Statewise dominance implies first-order stochastic dominance (FSD),<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> which is defined as: </p> Random variable A has first-order stochastic dominance over random variable B if for any outcome <i>x</i>, A gives at least as high a probability of receiving at least <i>x</i> as does B, and for some <i>x</i>, A gives a higher probability of receiving at least <i>x</i>. In notation form, P [ A ≥ x ] ≥ P [ B ≥ x ] {\displaystyle P[A\geq x]\geq P[B\geq x]} for all <i>x</i>, and for some <i>x</i>, P [ A ≥ x ] > P [ B ≥ x ] {\displaystyle P[A\geq x]>P[B\geq x]} . <p>In terms of the <a href="/facts/Cumulative_distribution_function/WaKU8tp4">cumulative distribution functions</a> of the two random variables, A dominating B means that F A ( x ) ≤ F B ( x ) {\displaystyle F_{A}(x)\leq F_{B}(x)} for all <i>x</i>, with strict inequality at some <i>x</i>. </p><p>In the case of non-intersecting distribution functions, the <a href="/facts/Wilcoxon_rank-sum_test/VHGPpw8m">Wilcoxon rank-sum test</a> tests for first-order stochastic dominance.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p> <h3>Equivalent definitions</h3> <p>Let ρ , ν {\displaystyle \rho ,\nu } be two probability distributions on R {\displaystyle \mathbb {R} } , such that E X ∼ ρ [ | X | ] , E X ∼ ν [ | X | ] {\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 stochastic dominance:<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p> <ul><li>For any u : R → R {\displaystyle u:\mathbb {R} \to \mathbb {R} } that is non-decreasing, E X ∼ ρ [ u ( X ) ] ≥ E X ∼ ν [ u ( X ) ] {\displaystyle \mathbb {E} _{X\sim \rho }[u(X)]\geq \mathbb {E} _{X\sim \nu }[u(X)]} </li></ul> <ul><li> F ρ ( t ) ≤ F ν ( t ) , ∀ t ∈ R . {\displaystyle F_{\rho }(t)\leq F_{\nu }(t),\quad \forall t\in \mathbb {R} .} </li> <li>There exists two random variables X ∼ ρ , Y ∼ ν {\displaystyle X\sim \rho ,Y\sim \nu } , such that X = Y + δ {\displaystyle X=Y+\delta } , where δ ≥ 0 {\displaystyle \delta \geq 0} .</li></ul> <p>The first definition states that a gamble ρ {\displaystyle \rho } first-order stochastically dominates gamble ν {\displaystyle \nu } <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> every <a href="/facts/Expected_utility_hypothesis/EboBlw1D">expected utility</a> maximizer with an increasing <a href="/facts/Utility/wlMDG74L">utility function</a> prefers gamble ρ {\displaystyle \rho } over gamble ν {\displaystyle \nu } . </p><p>The third definition states that we can construct a pair of gambles X , Y {\displaystyle X,Y} with distributions ρ , ν {\displaystyle \rho ,\nu } , such that gamble X {\displaystyle X} always pays at least as much as gamble Y {\displaystyle Y} . More concretely, construct first a uniformly distributed Z ∼ U n i f o r m ( 0 , 1 ) {\displaystyle Z\sim \mathrm {Uniform} (0,1)} , then use the <a href="/facts/Inverse_transform_sampling/fIDxBxT6">inverse transform sampling</a> to get X = F X − 1 ( Z ) , Y = F Y − 1 ( Z ) {\displaystyle X=F_{X}^{-1}(Z),Y=F_{Y}^{-1}(Z)} , then X ≥ Y {\displaystyle X\geq Y} for any Z {\displaystyle Z} . </p><p>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. </p> <h3>Extended example</h3> <p>Consider three gambles over a single toss of a fair six-sided die: </p> State (die result) 1 2 3 4 5 6 gamble A wins $ 1 1 2 2 2 2 gamble B wins $ 1 1 1 2 2 2 gamble C wins $ 3 3 3 1 1 1 {\displaystyle {\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}}} <p>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. </p><p>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). </p><p>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. </p><p>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. </p> <h2 id="second-order">Second-order</h2> <p>The other commonly used type of stochastic dominance is second-order stochastic dominance.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a><a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a><a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> Roughly speaking, for two gambles ρ {\displaystyle \rho } and ν {\displaystyle \nu } , gamble ρ {\displaystyle \rho } has second-order stochastic dominance over gamble ν {\displaystyle \nu } if the former is more predictable (i.e. involves less risk) and has at least as high a mean. All <a href="/facts/Risk_aversion/erarEwDf">risk-averse</a> <a href="/facts/Expected_utility_hypothesis/EboBlw1D">expected-utility maximizers</a> (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 <i>and</i> who are averse to risk, rather than <i>all</i> those for whom more is better) than does first-order dominance. </p><p>In terms of cumulative distribution functions F ρ {\displaystyle F_{\rho }} and F ν {\displaystyle F_{\nu }} , ρ {\displaystyle \rho } is second-order stochastically dominant over ν {\displaystyle \nu } if and only if ∫ − ∞ x [ F ν ( t ) − F ρ ( t ) ] d t ≥ 0 {\displaystyle \int _{-\infty }^{x}[F_{\nu }(t)-F_{\rho }(t)]\,dt\geq 0} for all x {\displaystyle x} , with strict inequality at some x {\displaystyle x} . Equivalently, ρ {\displaystyle \rho } dominates ν {\displaystyle \nu } in the second order if and only if E X ∼ ρ [ u ( X ) ] ≥ E X ∼ ν [ u ( X ) ] {\displaystyle \mathbb {E} _{X\sim \rho }[u(X)]\geq \mathbb {E} _{X\sim \nu }[u(X)]} for all nondecreasing and <a href="/facts/Concave_function/HxVEBkce">concave</a> utility functions u ( x ) {\displaystyle u(x)} . </p><p>Second-order stochastic dominance can also be expressed as follows: Gamble ρ {\displaystyle \rho } second-order stochastically dominates ν {\displaystyle \nu } if and only if there exist some gambles y {\displaystyle y} and z {\displaystyle z} such that x ν = d ( x ρ + y + z ) {\displaystyle x_{\nu }{\overset {d}{=}}(x_{\rho }+y+z)} , with y {\displaystyle y} always less than or equal to zero, and with E ( z ∣ x ρ + y ) = 0 {\displaystyle \mathbb {E} (z\mid x_{\rho }+y)=0} for all values of x ρ + y {\displaystyle x_{\rho }+y} . Here the introduction of random variable y {\displaystyle y} makes ν {\displaystyle \nu } first-order stochastically dominated by ρ {\displaystyle \rho } (making ν {\displaystyle \nu } disliked by those with an increasing utility function), and the introduction of random variable z {\displaystyle z} introduces a <a href="/facts/Mean-preserving_spread/Wm0Twogl">mean-preserving spread</a> in ν {\displaystyle \nu } which is disliked by those with concave utility. Note that if ρ {\displaystyle \rho } and ν {\displaystyle \nu } have the same mean (so that the random variable y {\displaystyle y} degenerates to the fixed number 0), then ν {\displaystyle \nu } is a mean-preserving spread of ρ {\displaystyle \rho } . </p> <h3>Equivalent definitions</h3> <p>Let ρ , ν {\displaystyle \rho ,\nu } be two probability distributions on R {\displaystyle \mathbb {R} } , such that E X ∼ ρ [ | X | ] , E X ∼ ν [ | X | ] {\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:<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> </p> <ul><li>For any u : R → R {\displaystyle u:\mathbb {R} \to \mathbb {R} } that is non-decreasing, and (not necessarily strictly) concave, E X ∼ ρ [ u ( X ) ] ≥ E X ∼ ν [ u ( X ) ] {\displaystyle \mathbb {E} _{X\sim \rho }[u(X)]\geq \mathbb {E} _{X\sim \nu }[u(X)]} </li></ul> <ul><li> ∫ − ∞ t F ρ ( x ) d x ≤ ∫ − ∞ t F ν ( x ) d x , ∀ t ∈ R . {\displaystyle \int _{-\infty }^{t}F_{\rho }(x)dx\leq \int _{-\infty }^{t}F_{\nu }(x)dx,\quad \forall t\in \mathbb {R} .} </li> <li>There exists two random variables X ∼ ρ , Y ∼ ν {\displaystyle X\sim \rho ,Y\sim \nu } , such that Y = X − δ + ϵ {\displaystyle Y=X-\delta +\epsilon } , where δ ≥ 0 {\displaystyle \delta \geq 0} and E [ ϵ | X − δ ] = 0 {\displaystyle \mathbb {E} [\epsilon |X-\delta ]=0} .</li></ul> <p>These are analogous with the equivalent definitions of first-order stochastic dominance, given above. </p> <h3>Sufficient conditions</h3> <ul><li>First-order stochastic dominance of <i>A</i> over <i>B</i> is a sufficient condition for second-order dominance of <i>A</i> over <i>B</i>.</li> <li>If <i>B</i> is a mean-preserving spread of <i>A</i>, then <i>A</i> second-order stochastically dominates <i>B</i>.</li></ul> <h3>Necessary conditions</h3> <ul><li> E ρ ( x ) ≥ E ν ( x ) {\displaystyle \mathbb {E} _{\rho }(x)\geq \mathbb {E} _{\nu }(x)} is a necessary condition for <i>A</i> to second-order stochastically dominate <i>B</i>.</li> <li> min ρ ( x ) ≥ min ν ( x ) {\displaystyle \min _{\rho }(x)\geq \min _{\nu }(x)} is a necessary condition for <i>A</i> to second-order dominate <i>B</i>. The condition implies that the left tail of F ν {\displaystyle F_{\nu }} must be thicker than the left tail of F ρ {\displaystyle F_{\rho }} .</li></ul> <h2 id="third-order">Third-order</h2> <p>Let F ρ {\displaystyle F_{\rho }} and F ν {\displaystyle F_{\nu }} be the cumulative distribution functions of two distinct investments ρ {\displaystyle \rho } and ν {\displaystyle \nu } . ρ {\displaystyle \rho } dominates ν {\displaystyle \nu } in the third order if and only if both </p> <ul><li> ∫ − ∞ x ( ∫ − ∞ z [ F ν ( t ) − F ρ ( t ) ] d t ) d z ≥ 0 for all x , {\displaystyle \int _{-\infty }^{x}\left(\int _{-\infty }^{z}[F_{\nu }(t)-F_{\rho }(t)]\,dt\right)dz\geq 0{\text{ for all }}x,} </li> <li> E ρ ( x ) ≥ E ν ( x ) {\displaystyle \mathbb {E} _{\rho }(x)\geq \mathbb {E} _{\nu }(x)} .</li></ul> <p>Equivalently, ρ {\displaystyle \rho } dominates ν {\displaystyle \nu } in the third order if and only if E ρ U ( x ) ≥ E ν U ( x ) {\displaystyle \mathbb {E} _{\rho }U(x)\geq \mathbb {E} _{\nu }U(x)} for all U ∈ D 3 {\displaystyle U\in D_{3}} . </p><p>The set D 3 {\displaystyle D_{3}} has two equivalent definitions: </p> <ul><li>the set of nondecreasing, concave utility functions that are <a href="/facts/Skewness/rdS8luFa">positively skewed</a> (that is, have a nonnegative third derivative throughout).<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a></li> <li>the set of nondecreasing, concave utility functions, such that for any random variable Z {\displaystyle Z} , the <a href="/facts/Risk_premium/ngjVLPpF">risk-premium</a> function π u ( x , Z ) {\displaystyle \pi _{u}(x,Z)} is a monotonically nonincreasing function of x {\displaystyle x} .<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a></li></ul> <p>Here, π u ( x , Z ) {\displaystyle \pi _{u}(x,Z)} is defined as the solution to the problem u ( x + E [ Z ] − π ) = E [ u ( x + Z ) ] . {\displaystyle u(x+\mathbb {E} [Z]-\pi )=\mathbb {E} [u(x+Z)].} See more details at <a href="/facts/Risk_premium/ngjVLPpF">risk premium</a> page. </p> <h3>Sufficient condition</h3> <ul><li>Second-order dominance is a sufficient condition.</li></ul> <h3>Necessary conditions</h3> <ul><li> E ρ ( log ( x ) ) ≥ E ν ( log ( x ) ) {\displaystyle \mathbb {E} _{\rho }(\log(x))\geq \mathbb {E} _{\nu }(\log(x))} is a necessary condition. The condition implies that the geometric mean of ρ {\displaystyle \rho } must be greater than or equal to the geometric mean of ν {\displaystyle \nu } .</li> <li> min ρ ( x ) ≥ min ν ( x ) {\displaystyle \min _{\rho }(x)\geq \min _{\nu }(x)} is a necessary condition. The condition implies that the left tail of F ν {\displaystyle F_{\nu }} must be thicker than the left tail of F ρ {\displaystyle F_{\rho }} .</li></ul> <h2 id="higher-order">Higher-order</h2> <p>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.<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> Arguably the most powerful dominance criterion relies on the accepted economic assumption of <a href="/facts/Risk_aversion/erarEwDf">decreasing absolute risk aversion</a>.<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a><a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> This involves several analytical challenges and a research effort is on its way to address those. <a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> </p><p>Formally, the n-th-order stochastic dominance is defined as <a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a> </p> <ul><li>For any probability distribution ρ {\displaystyle \rho } on [ 0 , ∞ ) {\displaystyle [0,\infty )} , define the functions inductively:</li></ul> <p> F ρ 1 ( t ) = F ρ ( t ) , F ρ 2 ( t ) = ∫ 0 t F ρ 1 ( x ) d x , ⋯ {\displaystyle F_{\rho }^{1}(t)=F_{\rho }(t),\quad F_{\rho }^{2}(t)=\int _{0}^{t}F_{\rho }^{1}(x)dx,\quad \cdots } </p> <ul><li>For any two probability distributions ρ , ν {\displaystyle \rho ,\nu } on [ 0 , ∞ ) {\displaystyle [0,\infty )} , non-strict and strict n-th-order stochastic dominance is defined as ρ ⪰ n ν iff F ρ n ≤ F ν n on [ 0 , ∞ ) {\displaystyle \rho \succeq _{n}\nu \quad {\text{ iff }}\quad F_{\rho }^{n}\leq F_{\nu }^{n}{\text{ on }}[0,\infty )} ρ ≻ n ν iff ρ ⪰ n ν and ρ ≠ ν {\displaystyle \rho \succ _{n}\nu \quad {\text{ iff }}\quad \rho \succeq _{n}\nu {\text{ and }}\rho \neq \nu } </li></ul> <p>These relations are transitive and increasingly more inclusive. That is, if ρ ⪰ n ν {\displaystyle \rho \succeq _{n}\nu } , then ρ ⪰ k ν {\displaystyle \rho \succeq _{k}\nu } for all k ≥ n {\displaystyle k\geq n} . Further, there exists ρ , ν {\displaystyle \rho ,\nu } such that ρ ⪰ n + 1 ν {\displaystyle \rho \succeq _{n+1}\nu } but not ρ ⪰ n ν {\displaystyle \rho \succeq _{n}\nu } . </p><p>Define the n-th moment by μ k ( ρ ) = E X ∼ ρ [ X k ] = ∫ x k d F ρ ( x ) {\displaystyle \mu _{k}(\rho )=\mathbb {E} _{X\sim \rho }[X^{k}]=\int x^{k}dF_{\rho }(x)} , then </p> <p>Theorem—If ρ ≻ n ν {\displaystyle \rho \succ _{n}\nu } are on [ 0 , ∞ ) {\displaystyle [0,\infty )} with finite moments μ k ( ρ ) , μ k ( ν ) {\displaystyle \mu _{k}(\rho ),\mu _{k}(\nu )} for all k = 1 , 2 , . . . , n {\displaystyle k=1,2,...,n} , then ( μ 1 ( ρ ) , … , μ n ( ρ ) ) ≻ n ∗ ( μ 1 ( ν ) , … , μ n ( ν ) ) {\displaystyle (\mu _{1}(\rho ),\ldots ,\mu _{n}(\rho ))\succ _{n}^{*}(\mu _{1}(\nu ),\ldots ,\mu _{n}(\nu ))} . </p><p>Here, the partial ordering ≻ n ∗ {\displaystyle \succ _{n}^{*}} is defined on R n {\displaystyle \mathbb {R} ^{n}} by v ≻ n ∗ w {\displaystyle v\succ _{n}^{*}w} iff v ≠ w {\displaystyle v\neq w} , and, letting k {\displaystyle k} be the smallest such that v k ≠ w k {\displaystyle v_{k}\neq w_{k}} , we have ( − 1 ) k − 1 v k > ( − 1 ) k − 1 w k {\displaystyle (-1)^{k-1}v_{k}>(-1)^{k-1}w_{k}} </p> <h2 id="constraints">Constraints</h2> <p>Stochastic dominance relations may be used as constraints in problems of <a href="/facts/Mathematical_optimization/oRn8Iv5I">mathematical optimization</a>, in particular <a href="/facts/Stochastic_programming/Nr1S4vA4">stochastic programming</a>.<a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a><a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a><a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a> In a problem of maximizing a real functional f ( X ) {\displaystyle f(X)} over random variables X {\displaystyle X} in a set X 0 {\displaystyle X_{0}} we may additionally require that X {\displaystyle X} stochastically dominates a fixed random <i>benchmark</i> B {\displaystyle B} . In these problems, <a href="/facts/Utility/wlMDG74L">utility</a> functions play the role of <a href="/facts/Lagrange_multiplier/U5mSI5YP">Lagrange multipliers</a> 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 ) + E [ u ( X ) − u ( B ) ] {\displaystyle f(X)+\mathbb {E} [u(X)-u(B)]} over X {\displaystyle X} in X 0 {\displaystyle X_{0}} , where u ( x ) {\displaystyle u(x)} is a certain utility function. If the first order stochastic dominance constraint is employed, the utility function u ( x ) {\displaystyle u(x)} is <a href="/facts/Monotonic_function/MjQK0YL2">nondecreasing</a>; if the second order stochastic dominance constraint is used, u ( x ) {\displaystyle u(x)} is <a href="/facts/Monotonic_function/MjQK0YL2">nondecreasing</a> and <a href="/facts/Concave_function/HxVEBkce">concave</a>. A system of linear equations can test whether a given solution if efficient for any such utility function.<a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a> Third-order stochastic dominance constraints can be dealt with using convex quadratically constrained programming (QCP).<a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Modern_portfolio_theory/VzQqKnuK">Modern portfolio theory</a></li> <li><a href="/facts/Marginal_conditional_stochastic_dominance/7wGRzINb">Marginal conditional stochastic dominance</a></li> <li><a href="/facts/Responsive_set_extension/AnKYJzbD">Responsive set extension</a> - equivalent to stochastic dominance in the context of preference relations.</li> <li><a href="/facts/Quantum_catalyst/n99TYdR7">Quantum catalyst</a></li> <li><a href="/facts/Ordinal_Pareto_efficiency/qKs2yTkw">Ordinal Pareto efficiency</a></li> <li><a href="/facts/Lexicographic_dominance/9MqXh92K">Lexicographic dominance</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Hadar, J.; Russell, W. (1969). "Rules for Ordering Uncertain Prospects". American Economic Review. 59 (1): 25–34. JSTOR 1811090. <a href="/wiki/American_Economic_Review" target="_blank">/wiki/American_Economic_Review</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Bawa, Vijay S. (1975). "Optimal Rules for Ordering Uncertain Prospects". Journal of Financial Economics. 2 (1): 95–121. doi:10.1016/0304-405X(75)90025-2. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Blackwell, David (June 1953). "Equivalent Comparisons of Experiments". The Annals of Mathematical Statistics. 24 (2): 265–272. doi:10.1214/aoms/1177729032. ISSN 0003-4851. <a href="https://doi.org/10.1214%2Faoms%2F1177729032" target="_blank">https://doi.org/10.1214%2Faoms%2F1177729032</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Levy, Haim (1990), Eatwell, John; Milgate, Murray; Newman, Peter (eds.), "Stochastic Dominance", Utility and Probability, London: Palgrave Macmillan UK, pp. 251–254, doi:10.1007/978-1-349-20568-4_34, ISBN 978-1-349-20568-4, retrieved 2022-12-23 <a href="978-1-349-20568-4" target="_blank">978-1-349-20568-4</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Quirk, J. P.; Saposnik, R. (1962). "Admissibility and Measurable Utility Functions". Review of Economic Studies. 29 (2): 140–146. doi:10.2307/2295819. JSTOR 2295819. <a href="/wiki/Review_of_Economic_Studies" target="_blank">/wiki/Review_of_Economic_Studies</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>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 <a href="https://books.google.com/books?id=a-ngTxeSLakC&pg=PA85" target="_blank">https://books.google.com/books?id=a-ngTxeSLakC&pg=PA85</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Mas-Colell, Andreu; Whinston, Michael Dennis; Green, Jerry R. (1995). Microeconomic theory. New York. Proposition 6.D.1. ISBN 0-19-507340-1. OCLC 32430901.{{cite book}}: CS1 maint: location missing publisher (link) <a href="0-19-507340-1" target="_blank">0-19-507340-1</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Hadar, J.; Russell, W. (1969). "Rules for Ordering Uncertain Prospects". American Economic Review. 59 (1): 25–34. JSTOR 1811090. <a href="/wiki/American_Economic_Review" target="_blank">/wiki/American_Economic_Review</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Hanoch, G.; Levy, H. (1969). "The Efficiency Analysis of Choices Involving Risk". Review of Economic Studies. 36 (3): 335–346. doi:10.2307/2296431. JSTOR 2296431. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Rothschild, M.; Stiglitz, J. E. (1970). "Increasing Risk: I. A Definition". Journal of Economic Theory. 2 (3): 225–243. doi:10.1016/0022-0531(70)90038-4. <a href="/wiki/Michael_Rothschild" target="_blank">/wiki/Michael_Rothschild</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Mas-Colell, Andreu; Whinston, Michael Dennis; Green, Jerry R. (1995). Microeconomic theory. New York. Proposition 6.D.1. ISBN 0-19-507340-1. OCLC 32430901.{{cite book}}: CS1 maint: location missing publisher (link) <a href="0-19-507340-1" target="_blank">0-19-507340-1</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Chan, Raymond H.; Clark, Ephraim; Wong, Wing-Keung (2012-11-16). "On the Third Order Stochastic Dominance for Risk-Averse and Risk-Seeking Investors". mpra.ub.uni-muenchen.de. Retrieved 2022-12-25. <a href="https://mpra.ub.uni-muenchen.de/42676/" target="_blank">https://mpra.ub.uni-muenchen.de/42676/</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Whitmore, G. A. (1970). "Third-Degree Stochastic Dominance". The American Economic Review. 60 (3): 457–459. ISSN 0002-8282. JSTOR 1817999. <a href="https://www.jstor.org/stable/1817999" target="_blank">https://www.jstor.org/stable/1817999</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Ekern, Steinar (1980). "Increasing Nth Degree Risk". Economics Letters. 6 (4): 329–333. doi:10.1016/0165-1765(80)90005-1. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Vickson, R.G. (1975). "Stochastic Dominance Tests for Decreasing Absolute Risk Aversion. I. Discrete Random Variables". Management Science. 21 (12): 1438–1446. doi:10.1287/mnsc.21.12.1438. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>Vickson, R.G. (1977). "Stochastic Dominance Tests for Decreasing Absolute Risk Aversion. II. General random Variables". Management Science. 23 (5): 478–489. doi:10.1287/mnsc.23.5.478. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>See, e.g. Post, Th.; Fang, Y.; Kopa, M. (2015). "Linear Tests for DARA Stochastic Dominance". Management Science. 61 (7): 1615–1629. doi:10.1287/mnsc.2014.1960. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>Fishburn, Peter C. (1980-02-01). "Stochastic Dominance and Moments of Distributions". Mathematics of Operations Research. 5 (1): 94–100. doi:10.1287/moor.5.1.94. ISSN 0364-765X. <a href="https://pubsonline.informs.org/doi/abs/10.1287/moor.5.1.94" target="_blank">https://pubsonline.informs.org/doi/abs/10.1287/moor.5.1.94</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> <li id="fn:19"><p>Dentcheva, D.; Ruszczyński, A. (2003). "Optimization with Stochastic Dominance Constraints". SIAM Journal on Optimization. 14 (2): 548–566. CiteSeerX 10.1.1.201.7815. doi:10.1137/S1052623402420528. S2CID 12502544. <a href="/wiki/Darinka_Dentcheva" target="_blank">/wiki/Darinka_Dentcheva</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li> <li id="fn:20"><p>Kuosmanen, T (2004). "Efficient diversification according to stochastic dominance criteria". Management Science. 50 (10): 1390–1406. doi:10.1287/mnsc.1040.0284. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li> <li id="fn:21"><p>Dentcheva, D.; Ruszczyński, A. (2004). "Semi-Infinite Probabilistic Optimization: First Order Stochastic Dominance Constraints". Optimization. 53 (5–6): 583–601. doi:10.1080/02331930412331327148. S2CID 122168294. <a href="/wiki/Darinka_Dentcheva" target="_blank">/wiki/Darinka_Dentcheva</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></p></li> <li id="fn:22"><p>Post, Th (2003). "Empirical tests for stochastic dominance efficiency". Journal of Finance. 58 (5): 1905–1932. doi:10.1111/1540-6261.00592. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></p></li> <li id="fn:23"><p>Post, Thierry; Kopa, Milos (2016). "Portfolio Choice Based on Third-Degree Stochastic Dominance". Management Science. 63 (10): 3381–3392. doi:10.1287/mnsc.2016.2506. SSRN 2687104. <a href="/wiki/Management_Science_(journal)" target="_blank">/wiki/Management_Science_(journal)</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></p></li> </ol>