Stochastic dominance is a partial order between random variables.[1][2] It is a form of stochastic ordering. The concept arises in decision theory and decision analysis in situations where one gamble (a probability distribution over possible outcomes, also known as prospects) can be ranked as superior to another gamble for a broad class of decision-makers. It is based on shared preferences regarding sets of possible outcomes and their associated probabilities. Only limited knowledge of preferences is required for determining dominance. Risk aversion is a factor only in second order stochastic dominance.

Stochastic dominance does not give a total order, but rather only a partial 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, stand for probability distributions on , while stand for particular random variables on . The notation means that has distribution .

There are a sequence of stochastic dominance orderings, from first , to second , to higher orders . The sequence is increasingly more inclusive. That is, if , then for all . Further, there exists such that but not .

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

Statewise dominance (Zeroth-Order)

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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.

First-order

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  und  , X and Y are not comparable through first-order stochastic dominance.

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,   for all x, and for some x,  .

In terms of the cumulative distribution functions of the two random variables, A dominating B means that   for all x, with strict inequality at some x.

In the case of non-intersecting[clarification needed] distribution functions, the Wilcoxon rank-sum test tests for first-order stochastic dominance.[6]

Equivalent definitions

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Let   be two probability distributions on  , such that   are both finite, then the following conditions are equivalent, thus they may all serve as the definition of first-order stochastic dominance:[7]

  • For any   that is non-decreasing,  
  •  
  • There exists two random variables  , such that  , where  .

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

The third definition states that we can construct a pair of gambles   with distributions  , such that gamble   always pays at least as much as gamble  . More concretely, construct first a uniformly distributed  , then use the inverse transform sampling to get  , then   for any  .

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.

Extended example

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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, 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.

Second-order

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The other commonly used type of stochastic dominance is second-order stochastic dominance.[1][8][9] Roughly speaking, for two gambles   and  , gamble   has second-order stochastic dominance over gamble   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   and  ,   is second-order stochastically dominant over   if and only if   for all  , with strict inequality at some  . Equivalently,   dominates   in the second order if and only if   for all nondecreasing and concave utility functions  .

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

Equivalent definitions

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Let   be two probability distributions on  , such that   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   that is non-decreasing, and (not necessarily strictly) concave, 
  •  
  • There exists two random variables  , such that  , where   and  .

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

Sufficient conditions

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  • 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.

Necessary conditions

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  •   is a necessary condition for A to second-order stochastically dominate B.
  •   is a necessary condition for A to second-order dominate B. The condition implies that the left tail of   must be thicker than the left tail of  .

Third-order

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Let   and   be the cumulative distribution functions of two distinct investments   and  .   dominates   in the third order if and only if both

  •  
  •  .

Equivalently,   dominates   in the third order if and only if   for all  .

The set   has two equivalent definitions:

  • the set of nondecreasing, concave utility functions that are positively skewed (that is, have a nonnegative third derivative throughout).[10]
  • the set of nondecreasing, concave utility functions, such that for any random variable  , the risk-premium function   is a monotonically nonincreasing function of  .[11]

Here,   is defined as the solution to the problem See more details at risk premium page.

Sufficient condition

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  • Second-order dominance is a sufficient condition.

Necessary conditions[citation needed]

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  •   is a necessary condition. The condition implies that the geometric mean of   must be greater than or equal to the geometric mean of  .
  •   is a necessary condition. The condition implies that the left tail of   must be thicker than the left tail of  .

Higher-order

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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 of decreasing 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   on  , define the functions inductively:

 

  • For any two probability distributions   on  , non-strict and strict n-th-order stochastic dominance is defined as  

These relations are transitive and increasingly more inclusive. That is, if  , then   for all  . Further, there exists   such that   but not  .

Define the n-th moment by  , then

Theorem — If   are on   with finite moments   for all  , then  .

Here, the partial ordering   is defined on   by   iff  , and, letting   be the smallest such that  , we have  

Constraints

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Stochastic dominance relations may be used as constraints in problems of mathematical optimization, in particular stochastic programming.[17][18][19] In a problem of maximizing a real functional   over random variables   in a set   we may additionally require that   stochastically dominates a fixed random benchmark  . 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   over   in  , where   is a certain utility function. If the first order stochastic dominance constraint is employed, the utility function   is nondecreasing; if the second order stochastic dominance constraint is used,   is nondecreasing and concave. A system 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]

See also

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References

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  1. ^ a b Hadar, J.; Russell, W. (1969). "Rules for Ordering Uncertain Prospects". American Economic Review. 59 (1): 25–34. JSTOR 1811090.
  2. ^ 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.
  3. ^ 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.
  4. ^ 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
  5. ^ 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.
  6. ^ 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
  7. ^ a b 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)
  8. ^ 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.
  9. ^ 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.
  10. ^ 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.
  11. ^ Whitmore, G. A. (1970). "Third-Degree Stochastic Dominance". The American Economic Review. 60 (3): 457–459. ISSN 0002-8282. JSTOR 1817999.
  12. ^ Ekern, Steinar (1980). "Increasing Nth Degree Risk". Economics Letters. 6 (4): 329–333. doi:10.1016/0165-1765(80)90005-1.
  13. ^ 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.
  14. ^ 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.
  15. ^ 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.
  16. ^ 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.
  17. ^ 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.
  18. ^ Kuosmanen, T (2004). "Efficient diversification according to stochastic dominance criteria". Management Science. 50 (10): 1390–1406. doi:10.1287/mnsc.1040.0284.
  19. ^ 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.
  20. ^ Post, Th (2003). "Empirical tests for stochastic dominance efficiency". Journal of Finance. 58 (5): 1905–1932. doi:10.1111/1540-6261.00592.
  21. ^ 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.