Stochastic dominance – Wikipedia
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[edit]
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[edit]
Statewise dominance is implied by 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, for all x, with strict inequality at some x.
Equivalent definitions[edit]
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:[6]
- 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[edit]
Consider three gambles over a single toss of a fair six-sided die:
- Second-order[edit]
The other commonly used type of stochastic dominance is second-order stochastic dominance.[1][7][8] 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[edit]
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:[6]
- For any that is non-decreasing, and (not necessarily strictly) concave,
These are analogous with the equivalent definitions of first-order stochastic dominance, given above.
Sufficient conditions[edit]
- 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[edit]
- 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[edit]
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).[9]
- the set of nondecreasing, concave utility functions, such that for any random variable , the risk-premium function is a monotonically nonincreasing function of .[10]
Here,
is defined as the solution to the problem
See more details at risk premium page.
Sufficient condition[edit]
- Second-order dominance is a sufficient condition.
Necessary conditions[citation needed][edit]
Higher-order[edit]
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 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[edit]
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
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 functionis 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.[19]
Third-order stochastic dominance constraints can be dealt with using convex quadratically constrained programming (QCP).[20]See also[edit]
References[edit]
- ^ a b Hadar, J.; Russell, W. (1969). “Rules for Ordering Uncertain Prospects”. American Economic Review. 59 (1): 25–34. JSTOR 1811090.
- ^ 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.
- ^ 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.
- ^ 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
- ^ 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 b Mas-Colell, Andreu (1995). Microeconomic theory. Michael Dennis Whinston, Jerry R. Green. New York. Proposition 6.D.1. ISBN 0-19-507340-1. OCLC 32430901.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ Whitmore, G. A. (1970). “Third-Degree Stochastic Dominance”. The American Economic Review. 60 (3): 457–459. ISSN 0002-8282. JSTOR 1817999.
- ^ Ekern, Steinar (1980). “Increasing Nth Degree Risk”. Economics Letters. 6 (4): 329–333. doi:10.1016/0165-1765(80)90005-1.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ Kuosmanen, T (2004). “Efficient diversification according to stochastic dominance criteria”. Management Science. 50 (10): 1390–1406. doi:10.1287/mnsc.1040.0284.
- ^ 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.
- ^ Post, Th (2003). “Empirical tests for stochastic dominance efficiency”. Journal of Finance. 58 (5): 1905–1932. doi:10.1111/1540-6261.00592.
- ^ 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.
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