DP2014-23

Partial Stochastic Dom inanceTakashi KAMIHIGASHIJohn STACHURSKI

May 19, 2014

Partial Stochastic Dominance1

Takashi Kamihigashia,b and John Stachurskic
a Research Institute for Economics and Business Administration, Kobe University
b IPAG Business School, Paris, Francec Research School of Economics, Australian National University

May 29, 2014
A BSTRACT. The stochastic dominance ordering over probability distributions is
one of the most familiar concepts in economic and financial analysis. One difficulty with stochastic dominance is that many distributions are not ranked at all,even when arbitrarily close to other distributions that are. Because of this, several
measures of ”partial” or ”near” stochastic dominance have been introduced into
the literature—albeit on a somewhat ad hoc basis. This paper argues that there is asingle measure of extent of stochastic dominance that can be regarded as the mostnatural default measure from the perspective of economic analysis.
JEL Classifications: D81, G11
Keywords: Stochastic dominance, stochastic order

1. I NTRODUCTION
Stochastic dominance is a fundamental concept in economics, finance and operations research, which permits comparison of outcomes in environments involving
1

This paper has benefited from helpful discussions with Kieron Meagher and Rabee Tourky,

and financial support from the Japan Society for the Promotion of Science and Australian ResearchCouncil Discovery Grant DP120100321.
Email addresses: tkamihig@rieb.kobe-u.ac.jp, john.stachurski@anu.edu.au1

2

uncertainty. The basic properties of first order stochastic dominance were initiallyprovided by Lehmann (1955), and introduced to economics by Quirk and Saposnik(1962). As stated by Border (1992), “the concept has been independently discovered too many times for an exhaustive listing.” (Here and below, stochastic dominance always refers to first order stochastic dominance unless otherwise specified.)
For probability distributions µ and ν on some set S, stochastic domination of µby ν means that if X and Y are S-valued random elements with distributions µand ν respectively, then E h( X ) ≤ E h(Y ) for any increasing bounded function

h : S → R. In the one dimensional setting, stochastic domination is equivalent topointwise ordering of the cumulative distribution functions (cdfs). In particular,
cdf F is (weakly) stochastically dominated by cdf G if G ( x ) ≤ F ( x ) for all x ∈ R.
See, for example, section 6.D of Mas-Colell et al. (1995).

Stochastic dominance is one of the cornerstones of the theory of choice under uncertainty. One immediate consequence of the definition of stochastic dominance isthat any von-Neumann–Morgenstern expected utility maximizer with monotone
preferences will weakly prefer stochastically dominant distributions. In addition,
extensions and alternatives to expected utility theory typically treat the notion ofstochastic dominance as fundamental, either as an axiom or as a test of the implications of axiomatic treatments (see, e.g., Machina (1982), Tversky and Kahneman(1992), Halevy and Feltkamp (2005) or Safra and Segal (2008)).
The notion of stochastic dominance also plays a central role in the theory of finance. For example, stochastic dominance is closely related to existence of arbitrage opportunities. In standard settings, existence of arbitrage opportunities
implies stochastic dominance and that the converse is true under relatively weak
conditions (Jarrow (1986)). In addition, stochastic dominance is fundamental to thestudy of portfolio diversification, partly because many of the theorems of classicalmean-variance portfolio theory apply only to certain classes of utility functions
and return distributions (Hadar and Russell (1971)).

3

In social welfare studies, stochastic dominance is again fundamental, as policies
or decision rules that shift outcomes up in terms of stochastic dominance can be
regarded as universally preferred under very weak assumptions on individual
choices. In fact Blackorby and Donaldson (1980) show that all social welfare functions satisfying three basic axioms are equivalent in the sense that they induce thesame order on the set of distributions as first order stochastic dominance. Related
analysis can be found in Atkinson (1987) and Foster and Shorrocks (1988).
Stochastic dominance also serves as a standard ordering for considering parametric monotonicity in settings where distributions are compared. These distributions
can take the form of distributions over actions, beliefs, valuations, preferences, information or other related quantities (see, e.g., Maskin and Riley (2000), Brock andDurlauf (2001), Rosenberg et al. (2007), or Ambrus et al. (2013)). The most common notion of monotonicity in stochastic dynamic models is defined in terms ofstochastic dominance (e.g., Stokey and Lucas (1989)). There is a large literature on
econometric tests for stochastic dominance. See, for example, Linton et al. (2005)
and Lee et al. (2009).
Further discussion of economic and financial applications can be found in standard textbooks such as Stokey and Lucas (1989) or Mas-Colell et al. (1995). Moredetailed treatments can be found in Levy (2006) and Föllmer and Schied (2011).
One difficulty with stochastic dominance is that the measure is fragile, in the sensethat it typically fails to be invariant to small changes in the distributions underconsideration. For example, it is well known that two normal distribution are
ranked by stochastic dominance if and only if their variances are identical. As
a consequence, any arbitrarily small change in variance for either of two ranked
distributions will break the ranking. Given these facts, it is not surprising that normal distributions estimate from data are almost never ranked. Similar issues afflict
estimation with other kinds of distributions, a point we return to below.

4

0.9
0.8
0.7

µ = N (0, 1)
ν = N (0.2, 1)
ν0 = N (5, 0.5)

ν0

0.6
0.5
0.4

ν

µ

0.3
0.2
0.1
0.0

−2

0

2

4

6

F IGURE 1.
To help illustrate ideas, consider the normal distributions µ, ν and ν0 shown in figure 1. Let sd indicate stochastic dominance. Since ordering normal distributions

requires identical variance, the ranking µ sd ν holds while µ sd ν0 fails. Intu-

itively, µ sd ν0 fails because ν0 has smaller variance, and hence holds less massin its tails. These thin tails mean that, for very large c, the probability that a draw
from µ exceeds c is larger than the probability that a draw from ν0 exceeds c. Thisviolates stochastic dominance. It should be noted, however, that the smallest suchc is larger than 10, and the amount of mass that either of these distributions puts
on any of these sets is numerically indistinguishable from zero, even with 64 bit
precision.
These issues translate into various difficulties in applied settings. While stochastic
dominance is a popular tool in the theory of portfolio choice, in practice it often
fails to provide effective analysis. As phrased by Levy (1992), “the difficulty withstochastic dominance rules is that, in general, [they result] in a large efficient setof investments, i.e. in many cases this framework is unable to rank the two risky
options under consideration.” On the empirical side, Davidson and Duclos (2013)5show that, as a consequence of the sensitivity of the ordering, it is in fact not possible to reject non-dominance in favor of dominance over the entire supports of two
distributions when these distributions are continuous. This ”necessitates a recasting of the usual theoretical links between stochastic dominance relationships andorderings in terms of poverty, social welfare, and expected utility. It also highlights better why a failure to reject the usual null hypothesis of unrestricted dominance cannot be interpreted as an acceptance of unrestricted dominance, since unrestricted dominance can never be inferred from continuous data” (Davidson and
Duclos, 2013, p. 87).
Because of the relative fragility of stochastic dominance, several notions of partial
stochastic dominance have been introduced into the literature. An early one is”restricted” first order stochastic dominance up to some point c, which is defined(Atkinson, 1987, p. 751) as holding for one-dimensional cumulative distributions
F and G whenever(1)

G(x) ≤ F(x)for allx ≤ c.

(Recall that, for cdfs F and G on the real line, stochastic domination of F by G canequivalently be defined as G ( x ) ≤ F ( x ) at every x.) Restricted first order stochas-tic dominance is a specialized measure intended for situations where interest is
focused on what happens below some threshold, such as a poverty line. It is notsuitable as a general measure of approximate stochastic dominance.2
Another measure of partial stochastic dominance is the notion of “almost stochastic dominance” from the finance literature (see, e.g., Leshno and Levy (2002); Levy
(2009)). This measure also compares cdfs, and is described in detail below. Several
2One issue is that in general the appropriate choice of c might not be clear. A second issue isillustrated by the following example. Let F = N (0, 0.1) and G = N (10, 1). The distribution G”approximately” dominates F in the sense that the probability that a draw from F exceeds a draw
from G is essentially indistinguishable from zero. At the same time, condition (1) fails for any
c ∈ R.

6other measures are considered in Stoyanov et al. (2012), who analyze degrees ofviolation of stochastic dominance using quasi-semidistances.
The aim of this paper is to make the case that there is a single measure of extent of
stochastic dominance, denoted below by σ, that can arguably be considered as thedefault measure from the perspective of economic analysis (without claiming to be
the most suitable in every individual setting). For one dimensional distributions
with cdfs F and G, the measure can be expressed by the simple representation(2)

σ ( F, G ) = 1 − sup{ G ( x ) − F ( x )}.
xThis value measures the extent to which F is dominated by G. It is not difficult toverify that 0 ≤ σ( F, G ) ≤ 1 and σ ( F, G ) = 1 if and only if F is stochastic dominatedby G.

While the measure σ( F, G ) seems almost naı̈ve, we show that it has a number ofdifferent and useful representations, can be motivated from several entirely different perspectives, possesses attractive continuity properties, and extends naturally
to multidimensional or more abstract settings. As an example of an alternative
representation presented in the paper, we show that σ ( F, G ) can also be expressedasσ( F, G ) = max P{ X ≤ Y }X,Y

where the supremum is over all pairs of random variables ( X, Y ) such that X ∼ F

and Y ∼ G. One way to think of this expression is as providing the “maximal

regret probability” for a decision maker with monotone preferences who chooses
F over G. Other interpretations and alternative expressions are discussed below.

Applying the measure σ to the distributions in figure 1, we calculate that σ(µ, ν) =1 as expected (since µ is dominated by ν). Comparing µ and ν0 , it can be shown
analytically that σ(µ, ν0 ) is strictly less than one. However, from the perspective of
64 bit floating point arithmetic, σ(µ, ν0 ) is indistinguishable from 1.

7

The main part of our paper proceeds as follows. First we present several alternative measures of extent of stochastic dominance, all motivated from different
perspectives, and then show that all are equivalent to σ (sections 2 and 3). Next weshow that σ has desirable continuity and robustness properties (section 4). All ofthese results are stated and proved in a general setting, for probability measures onan arbitrary space with arbitrary closed partial order. Hence our analysis accommodates a variety of different state spaces and state space orderings, including
multivariate and vector-valued state spaces with standard partial orders, infinite
dimensional state spaces (e.g., sets of distributions) and so on. Section 5 discusses
two applications. Proofs are given in section 7.
2. A M EASURE OF S TOCHASTIC D OMINANCE
We begin with some preliminaries and then define a measure of degree of stochastic dominance.
2.1. Definitions. Let S be a Polish space with closed partial order  and Borel sets

B.3 As usual, we say that h : S → R is an increasing function if x  y impliesh( x ) ≤ h(y). We say that h is bounded if there exists a K ∈ R with |h( x )| ≤ K

for all x ∈ S. We say that a set B ⊂ S is an increasing set if x ∈ B and x  yimplies y ∈ B. Equivalently, B is an increasing set if its indicator function 1B is anincreasing function. We write• M for the finite measures4 on (S, B )• P ⊂ M for the probability measures• isB for the increasing sets in B

• ibB for the increasing, bounded, Borel measurable functions from S to R.
3A Polish space is a completely metrizable topological space. A closed partial order on S is abinary relation  on S × S that is reflexive, transitive and antisymmetric, and such that its graph{( x, y) ∈ S × S : x  y} is closed in the product topology.

4A finite measure on (S, B ) is a countably additive function µ : B → R+ satisfying µ ( S ) < ∞.

8

Two natural partial orders on M are defined as follows: Given µ and ν in M , wesay that µ is stochastically dominated by ν and write µ sd ν ifµ(S) = ν(S)In our setting, the condition

and

R

Z

h dµ ≤

h dµ ≤RZh dν

∀ h ∈ ibB.

h dν for all h ∈ ibB can be replaced byµ( I ) ≤ ν( I ) for all I ∈ isB, or even µ( I ) ≤ ν( I ) for all closed I ∈ isB (Kamae

et al., 1977, theorem 1). We say that µ is setwise dominated by ν and write µ ≤ ν ifµ( B) ≤ ν( B)∀ B ∈ B.

For each µ, ν ∈ P, we defineΦ(µ, ν) := {(µ0 , ν0 ) ∈ M × M : µ0 ≤ µ, ν0 ≤ ν, µ0 sd ν0 }.
If µ and µ0 are measures with µ0 ≤ µ, then µ0 is sometimes called a componentof µ. In this terminology, we can think of Φ(µ, ν) as the set of “ordered component pairs” corresponding to (µ, ν), where “ordered” means ordered by stochastic
dominance. Note that (µ0 , ν0 ) ∈ Φ(µ, ν) implies that µ0 (S) = ν0 (S).
It is clear that if (µ0 , ν0 ) is any ordered component pair, then the total mass µ0 (S) =ν0 (S) must be no more than one; and that if µ is stochastically dominated by ν, thenthe original pair (µ, ν) is itself an ordered component pair, with each element of thepair having total mass that attains the maximum of unity. It is also intuitive that
if µ is “almost” dominated by ν, then we can choose relatively large components,
so that the mass of each element of the ordered component pair approaches one.
These ideas suggest regarding the mass of the largest ordered component pair as ameasure of the extent to which µ is dominated by ν. Thus, for each µ, ν in P, wedefine σ (µ, ν) by(3)

σ(µ, ν) := sup{µ0 (S) : (µ0 , ν0 ) ∈ Φ(µ, ν)},and regard it as a measure of degree of first order stochastic domination of µ by ν.
Its basic properties are as follows:
Proposition 2.1. For any µ, ν ∈ P,

9

1. σ(µ, ν) is well defined and contained in [0, 1].
2. µ sd ν if and only if σ(µ, ν) = 1 .

3. σ(µ, ν) = σ (ν, µ) = 1 if and only if µ = ν.
In particular, 0 ≤ σ(µ, ν) ≤ 1, with higher values representing ”greater” domination of µ by ν, and with σ(µ, ν) = 1 attained if and only if stochastic dominance iscomplete.
Proposition 2.2. For any µ, ν ∈ P, there exists a pair (µ0 , ν0 ) ∈ Φ(µ, ν) such that
µ0 (S) = ν0 (S) = σ(µ, ν).
In other words, the sup in (3) can be replaced by max. Like most of our results, theproofs of propositions 2.1 and 2.2 are deferred to section 7.
Two particularly important properties of σ are as follows. First, this measure ofextent of stochastic dominance turns out to be equivalent to several other natural measures of the same quantity, each of which can be motivated from considerations related to choice under uncertainty. Second, σ exhibits “continuity” or“robustness,” in the sense that, when µ is “almost” dominated by ν, the value ofσ (µ, ν) is almost 1. These topics are treated below, beginning with equivalences.
3. E QUIVALENCES
In this section we introduce several other measures of extent of stochastic dominance, all of which have some claim to be natural measures of this concept. We
then show that each of these measures is equivalent to σ.
3.1. A Coupling Measure. For agents with monotone preferences, stochastic dominance ranks distributions. Let us now also consider choosing between arbitrary
distributions that may or may not be ordered. We will consider a setting where
preferences are monotone but make no assumptions beyond that. For example, we
can consider a decision maker evaluating distributions on behalf of agents whose10

preferences are known only to be monotone (consider, for example, a fund manager or a policy maker).
Given that the agents have monotone preferences, one way for the decision makerto think about distribution µ being less desirable than distribution ν is to consider
the probability that choosing prospect µ turns out to be worse than ν ex post—that is, the probability that a draw X from µ is dominated by a draw Y from ν.
This probability depends on the joint distribution of X and Y. In the current setting we have no information on the joint distribution. Absent such information,
one approach is to make the best decision under the worst case scenario, whichis the joint distribution making X  Y most likely. This leads us to the value

supX,Y P{ X  Y }, where the supremum is over all pairs ( X, Y ) with marginals µand ν. The larger this value, the more µ can be regarded as ”dominated” by ν.

To state this idea more formally, recall that, given (µ, ν) ∈ P × P, a pair of S-

valued random variables ( X, Y ) defined on some probability space (Ω, F , P) iscalled a coupling of (µ, ν) if µ( B) = P{ X ∈ B} and ν( B) = P{Y ∈ B} for all
B ∈ B. In other words, X and Y live on the same probability space and havemarginals µ and ν. The set of all couplings of (µ, ν) ∈ P × P will be denoted by

C (µ, ν). Given this notation, the reasoning in the preceding paragraph suggests(4)s(µ, ν) :=sup( X,Y )∈C (µ,ν)

P{ X  Y }.

as a measure of the extent to which µ is stochastically dominated by ν.
To help understand this idea better, let’s consider the case where µ sd ν. For sucha pair we can draw on a well known result due to Strassen (1965), which, in thecurrent setting, tells us that(5)µ sd ν⇐⇒∃ ( X, Y ) ∈ C (µ, ν) s.t. P{ X  Y } = 1.

It follows that the measure s(µ, ν) defined in (4) attains its maximum (i.e., equals
one) when µ is stochastically dominated by ν. In fact, as we will see, the twostatements are equivalent, in the sense that if µ is not stochastically dominated by11

ν, then s(µ, ν) will be strictly less than one. Moreover, as will also be shown below,
when µ is “almost” dominated by ν, the value will be close to one.
Before going further, it is worth noting that we can replace sup in (4) with max
because, somewhat surprisingly, the maximum always exists:
Proposition 3.1. For all (µ, ν) ∈ P × P, there exists a pair ( X, Y ) ∈ C (µ, ν) such that

P{ X  Y } =

(6)

sup
( X 0 ,Y 0 )∈C (µ,ν)

P{ X 0  Y 0 }.

While it might at first seem that s defined in (4) is a new and entirely different
measure of degree of stochastic dominance, we show below that it coincides with
σ.
3.2. Deviation Measures. There is another class of functions on P × P that havea strong claim to providing a measure of extent of stochastic dominance. To construct them, let µ and ν be any two probability measures, and observe that, bydefinition, µ sd ν if and only if
(7)

sup
h∈ibB

Z

h dµ −

Z

h dν



≤ 0.

From a social welfare or choice-under-uncertainty perspective, this inequality statesthat every agent with monotone preferences will achieve at least the same expectedutility under ν as under µ.
Note that zero can always be achieved in (7) by choosing h = 0 or h = 1. Moreover,
if µ is not completely dominated by ν, then a strictly positive value can be obtainedby the supremum. This value gives some indication of the extent to which µ fails
to be dominated by ν. Hence we can consider
ZZ(8) θ (µ, ν) := suph dµ − h dν
where
h∈ H

H := {h ∈ ibB : 0 ≤ h ≤ 1}

as an inverse measure of the degree to which µ is dominated by ν, taking smaller
values when domination by ν is greater, and zero when domination is complete. A

12measure similar in spirit to θ is mentioned in Stoyanov et al. (2012), with the only
difference being the class of functions over which the supremum is taken.
In the definition of θ, the restriction of ibB to functions with range in [0, 1] is aconvenient normalization. In fact we can go a step further and restrict to indicatorfunctions, which leads to the alternative measureθ I (µ, ν) := sup {µ( B) − ν( B)} .
B∈isB

3.3. Equivalence. It turns out that all of the preceding measures are equivalent in
the sense made precise by the following theorem:
Theorem 3.1. For any (µ, ν) ∈ P × P we have
σ (µ, ν) = s(µ, ν) = 1 − θ (µ, ν) = 1 − θ I (µ, ν).
As with other results, the proof of theorem 3.1 can be found in section 7.
For some calculations it can be convenient to consider the variation(9)θO
I ( µ, ν ) : = sup { µ ( B ) − ν ( B )}
B∈iO

where

iO := {all open B ∈ isB }.

O
Evidently θ O
I ≤ θ I . In fact, as the proof of theorem 3.1 shows, we have θ I = θ I on

all of P × P, and hence θ O
I ( µ, ν ) = 1 − σ ( µ, ν ) is also valid. Moreover, it is trivial

to confirm thatsup {µ( B) − ν( B)} = sup {ν( D ) − µ( D )} ,

B∈iO

D ∈dC

where dC is the closed decreasing sets in B. This is useful, particularly in the onedimensional case, where dC is the set of all intervals (−∞, x ] over x ∈ R. In thissetting it implies that for cdfs F and G on R, we have σ ( F, G ) = 1 − supx { G ( x ) −F ( x )}. This is the expression for σ given in the introduction.

Remark 3.1. One benefit of the last expression is that the measure σ is straightforward to compute in one dimension. Aside from numerical optimization, if G and F

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are both absolutely continuous, then the first order condition for maxx∈R { G ( x ) −F ( x )} is just f ( x ) = g( x ), where f and g are the densities of F and G respectively.

Remark 3.2. For other calculations, alternative expressions for σ can be easier towork with. To illustrate, let S be a general space—not necessarily one dimensional—
and suppose that µ is degenerate, putting all mass on x̄ ∈ S. Let ν be arbitrary. If

we wish to compute σ (µ, ν), it is enough to observe that if ( X, Y ) is any couplingof (µ, ν), then, since X puts all mass on x̄, we must have P{ X  Y } = P{ x̄  Y }.
We conclude that σ(µ, ν) = ν( Ix̄ ) where Ix̄ := { x ∈ S : x̄  x }.
4. C ONTINUITY
Since we wish to use σ (µ, ν) as a measure of the extent to which µ is dominated by
ν, one desirable characteristic is that σ(µ, ν) is “almost” 1 whenever µ is “almost”
dominated by ν. This turns out to be valid in many cases, although some care isurequired when defining ”almost.” To state a precise result, let’s write µn → µ tomean that kµn − µk → 0 as n → ∞, where k · k is the total variation normkµn − µk := sup |µn ( B) − µ( B)|.
B ∈B

Now, given sequences of probability measures {µn } and {νn }, the following result

holds:uu

Proposition 4.1. If µn → µ and νn → ν, then σ (µn , νn ) → σ(µ, ν).
Put differently, σ : P × P → [0, 1] is continuous in the product topology inducedby the total vairation norm. In fact the proof of proposition 4.1 gives the bound|σ(µn , νn ) − σ(µ, ν)| ≤ kµn − µk + kνn − νk.
Specializing this bound to the case where µ sd ν and νn = ν, we get σ (µn , ν) ≥1 − kµn − µk. This quantifies the idea that σ (µn , ν) is almost 1 when µn almost
equal to a distribution dominated by ν.

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The total variation convergence used in proposition 4.1 can be checked relatively
easily when the probability measures in question have densities, since pointwise
convergence of densities implies total variation convergence by Scheffé’s lemma(Williams, 1991, p. 55). Also, in view of Pinsker’s inequality, proposition 4.1 remains valid when comparison is made based on measures of relative entropy.
For example, if µ sd ν and either the Kullback-Leibler divergence or the Jensen-

Shannon distance from µn to µ and νn to ν goes to zero, then σ (µn , νn ) → σ (µ, ν).
See Devroye and Lugosi (2001).

By comparison, consider the concept of “almost stochastic dominance” mentioned
in the introduction. Let S be a bounded interval of R and let F and G be cdfs
on S. Let e ≥ 0. Leshno and Levy (2002) define F to be e-almost stochasticallydominated by G if

Z

(10)

{ F<G}( G ( x ) − F ( x ))dx ≤ e

Z| G ( x ) − F ( x )|dx.

Here { F < G } := { x ∈ S : F ( x ) < G ( x )}. We can convert this to a measure ofextent of stochastic dominance by considering the smallest e such that (10) is valid.

That is,a( F, G ) = inf e ≥ 0 :Z{ F<G}( G ( x ) − F ( x ))dx ≤ eZ| G ( x ) − F ( x )|dx .

Smaller values of a( F, G ) are associated with a greater degree of dominance of G
over F. Moreover a( F, G ) = 0 if and only if F sd G.
Unfortunately, the measure a fails to have the form of continuity discussed inproposition 4.1. For example, let F = G = Gn = N (0, 1), the standard normaluu

distribution, and let Fn = N (1/n, 1). Here Gn → G is trivial and Fn → F is valid
because the densities converge pointwise everywhere on R (which is sufficient for
total variation convergence as discussed above). At the same since Fn ( x ) < Gn ( x )for all x ∈ R and n ∈ N, we have a( Fn , Gn ) = 1 for all n. This compares toa( F, G ) = 0 (because F = G).

15

Note that the claim in proposition 4.1 may fail if we replace total variation converRR
gence with weak convergence. (Recall that µn → µ weakly if gdµn → gdµ forw

all continuous bounded g : S → R. Symbolically we write µn → µ.) The reason
the proposition is not generally valid in this case is that weak convergence is a
topological notion, while the stochastic dominance measure σ is defined only by
the ordering on S.
Note however that total variation convergence can be weakened while still maintaining the conclusions of proposition 4.1. For example, in the one-dimensionalsetting, it is sufficient that convergence takes place in the Kolmogorov metric. Thatis, if F, G, Fn and Gn are cdfs on R withlim sup | Fn ( x ) − F ( x )| = lim sup | Gn ( x ) − G ( x )| = 0,

n→∞

n→∞

x

x

then σ ( Fn , Gn ) → σ ( F, G ). A sufficient condition for convergence of cdfs { Hn } to
H in the Kolmogorov metric is that Hn → H weakly and H is continuous (see, e.g.,

Dudley (2002), p. 389). This is obviously the case when H has a density. This leads
to the following corollary:
Corollary 4.1. Let Fn , Gn , F and G be cdfs on R. The following statements are true:w

1. If Fn → F and F is continuous, then σ ( Fn , G ) → σ( F, G ).
w

2. If Gn → G and G is continuous, then σ( F, Gn ) → σ( F, G ).
w

w

3. If Fn → F, Gn → G and both F and G are continuous, then σ( Fn , Gn ) → σ( F, G ).

In other words, weak convergence is enough when the limiting distributions are
sufficiently regular.
5. A PPLICATIONS
The concepts discussed in this paper can be applied to any setting where distributions are compared, from industrial organization (e.g., distributions of firm size or
productivity) to labor economics (wage distributions), development and growth16(wealth distributions), macroeconomics (parameter uncertainty represented via
distributions), network analysis (degree distributions), social welfare (income distributions) and so on. Here we consider a two problems, one related to wealthdistribution and the other to portfolio choice. Our discussion illustrates the pointthat some properties only hold ”in the limit.” The measure σ allows us to makethis idea precise in the context of stochastic dominance.
5.1. Wealth Distributions. Let W0 be an initial wealth or asset distribution acrosssome set of agents, and suppose that individual wealth grows according to somespecified dynamics. Interesting examples can be found in Piketty (1997), Benhabib
et al. (2011) and many other places. Since the main points change little, here we
simply suppose that wealth grows geometrically and deterministically via Wt+1 =aWt for some a ≥ 1. As a result, Wt = at W0 , or, with lower case representing logs,

wt = t ln a + w0 .

Now consider an alternative economy with wealth distribution w00 = ln W00 andgrowth rate a0 > a. Despite this higher growth rate, one cannot in general claimthat wt0 = t ln a0 + w00 will stochastically dominate wt for any finite t. An obvious
counterexample is as follows: Suppose that W0 and W00 are lognormal, so that wt
and wt0 are normal. If w0 and w00 do not have identical variance, then the same willbe true of wt and wt0 , and hence they will fail to be ordered for any finite t.
On the other hand, it seems intiutive that wt0 will ”almost” stochastically dominate
wt when t is large enough. We can formalize this using σ. In particular, we canshow that σ ( Ft , Ft0 ) → 1 as t → ∞, where Ft and Ft0 are the distributions of wt

and wt0 respectively. To do this, observe that, by (4) and theorem 3.1, the statement
σ ( Ft , Ft0 ) ≥ P{wt ≤ wt0 } is always valid. Henceσ( Ft , Ft0 ) ≥ P{t ln a + w0 ≤ t ln a0 + w00 } = P{w0 − w00 ≤ t(ln a0 − ln a)}.

Letting G be the cdf of w0 − w00 and κ := ln a0 − ln a, we can express this asσ ( Ft , Ft0 ) ≥ G (tκ ) for all t. Since G is a cdf and κ > 0, we have shown thatσ ( Ft , Ft0 ) → 1 as claimed.

17

5.2. A Portfolio Problem. Next we turn to the portfolio problem. To begin, consider a set of securities indexed by j ∈ N. Let x j = g + e j be the payoff of the j-th
security. Here g is a common factor with distribution function G, and {e j } j∈N is asequence of uncorrelated shocks with zero mean and finite variance σ2 . A portfolio
is a sequence θ = {θ1 , θ2 . . .} such that ∑∞j=1 θ j = 1. Each portfolio is identified witha payoff ∑∞j=1 θ j x j . A variance minimizing finite diversified portfolio is a portfoliothat puts weight 1/n on n securities. More formally, it is a portfolio {θ j } of theform θ j = n−1 1{ j ∈ In } where In is a finite set containing n elements of N. Since

it changes nothing substantial, we always take In := {1, . . . , n}. Let θn∗ denote thisportfolio. The payoff of θn∗ is g + n1 ∑nj=1 e j .

The diversified portfolio θn∗ uses the law of large numbers to reduce idiosyncraticrisk. Each increase in n decreases the variance of the payoff. In the limit only factor
risk remains. However, the limit of the sequence of diversified portfolios is nottrivial to describe, since the share θ j = 1/n converges to zero, while the constraint∑ j θ j = 1 must be respected. Werner (1997) places the limiting portfolio on a firm
theoretical foundation by representing it as a density charge (i.e., a purely finitely
additive measure) on the power set of N. Here we investigate limiting behavior
from a different angle, at least as it pertains to stochastic dominance.
Suppose we are interested in comparing the diversified portfolio against some alternative portfolio (e.g., a portfolio containing only a risk free asset). For the sakeof generality we will consider an arbitrary alternative, with payoff distribution F.
Let Gn be the distribution function for the payoff g + n1 ∑nj=1 e j of θn∗ . Regarding Gn
and F we have the following result:
Proposition 5.1. If G is continuous, then σ( F, Gn ) → σ( F, G ).
The proof is a simple consequence of the results in section 4. A useful implicationof proposition 5.1 is that if the distribution G of the factor g stochastically dominates F, then σ( F, Gn ) → 1. In other words, the payoff of the diversified portfolio
Gn ”stochastically dominates F in the limit,” as continued diversification reduces

18

idiosyncratic risk towards zero. This is true even if F fails to be dominated by Gn
for all n, as would be the case if the shocks e j have sufficiently large support.5

6. C ONCLUSION
This paper shows that there is a single measure of extent of stochastic dominancethat can arguably be treated as the default measure of this property from the perspective of economic analysis. The main results of the paper demonstrate that thismeasure has a number of different and informative representations that are in fact
equivalent. As a consequence, the measure can be motivated from several very
different perspectives. In addition, the measure is shown to possess important
continuity properties, and extends naturally to multidimensional or more abstract
settings. Two applications are given.
As a caveat we reiterate that, as with many problems, the existence of a default
measure does not preclude the optimality of other measures in certain specific settings. On the other hand, it does provide a common frame of reference for theoretical work. Moreover, since the measure can be estimated, the preceding analysis
also opens up new avenues for econometric investigation of stochastic dominance.
Finally, the question of whether analogous measures and results exist for the caseof higher order stochastic dominance (second order, third order, etc.) is left to future research.
5To see why continuity of G matters, suppose that F and G are both degenerate distributions

concentrated on some risk free rate R. Let e j be standard normal, so that Gn = N ( R, 1/n). Thenσ( F, G ) = 1 but σ ( F, Gn ) = 1/2 for all n (see remark 3.2 on how to obtain the last equality). The
intuitive reason for failure of ”stochastic dominance in the limit” here is as follows: Suppose wehave an agent with preferences defined by von Neumann-Morgenstern utility function u( x ) =1{ R ≤ x}. This agent cares only about equaling or beating the risk free rate. Despite monotone

increasing preferences, the expected payoff of Gn for this agent is always half of that of F, regardlessof how large n is.

19

7. P ROOFS

7.1. Preliminaries. Throughout the proofs we let O and C be the open and closed
subsets of S respectively. In addition, let dC be the decreasing sets in C , and let iO
be the increasing sets in O. As in (8), let H := {h ∈ ibB : 0 ≤ h ≤ 1}. For ourpartial order  on S, let G be the graph. That is

G := {( x, y) ∈ S × S : x  y}.
Let S × S be the Cartesian product of S with itself, and let B ⊗ B be the product σalgebra. Let πi be the i-th coordinate projection, so that π1 ( x, y) = x and π2 ( x, y) =y for any ( x, y) ∈ S × S. As usual, given Q ⊂ S × S, we let π1 ( G ) be all x ∈ S suchthat ( x, y) ∈ Q, and similarly for π2 . A probability measure λ on (S × S, B ⊗ B )is said to have marginals µ and ν ifµ( B) = λ( B × S)andν( B) = λ(S × B)

∀ B ∈ B.

Equivalently, µ = λ ◦ π1−1 and ν = λ ◦ π2−1 . We will apply the following deep
theorem due to Strassen (1965), slightly specialized to the current setting:

Theorem 7.1 (Strassen’s theorem). Let µ, ν ∈ P. For any e ≥ 0 and any closed set
K ⊂ S × S, there exists a probability measure λ on (S × S, B ⊗ B ) with marginals µ andν such that λ(K ) ≥ 1 − e if and only if

ν( F ) ≤ µ(π1 (K ∩ (S × F ))) + e,

∀F ∈ C.

20

By applying Strassen’s theorem and some other more elementary results, we will
now show that the following chain of inequalities holds:
ZZh dµ − h dν ≤ 1 − σ(µ, ν)
(11)
sup
h∈ H

(12)

≤ 1−

(13)

sup
( X,Y )∈C (µ,ν)

P{ X  Y }≤ sup {µ( I ) − ν( I )}
I ∈iO

(14)

≤ sup

Z

h∈ H

h dµ −

Zh dν .

Fix µ, ν ∈ P. Regarding inequality (11), fix h ∈ H. Let (µ0 , ν0 ) be an element of

Φ(µ, ν) with µ0 (S) = ν0 (S) = σ (µ, ν), the existence of which follows from propos-

tion 2.2. Let µr := µ − µ0 and νr := ν − ν0 . We have
Z

hdµ −

Z

hdν =

=
≤
≤

Z

hdµr +

Z

hdµ0 −

Z

Z

r

hdµ −

Z

r

hdν +

Z

Z

r

hdµ −

Z

hdνr

Z

hdµr ≤ µr (S) = 1 − σ(µ, ν).

hdνr −

Z

0

hdµ −

hdν0
Z

hdν

0



Next we prove inequality (12). To prove this inequality it is enough to show that if( X, Y ) is a coupling of (µ, ν), then P{ X  Y } ≤ σ( X, Y ). To see that this is so, let( X, Y ) be a coupling of (µ, ν), and defineµ0 ( B) := P{ X ∈ B, X  Y }andν0 ( B) := P{Y ∈ B, X  Y }.

Clearly µ0 ≤ µ, ν0 ≤ ν and µ0 (S) = P{ X  Y } = ν0 (S). Moreover, for any
increasing set I ∈ B we haveµ0 ( I ) = P{ X ∈ I, X  Y } ≤ P{Y ∈ I, X  Y } = ν0 ( I ).
Hence (µ0 , ν0 ) ∈ Φ(µ, ν) and we obtain P{ X  Y } = µ0 (S) ≤ σ(µ, ν).

21

We now turn to inequality (13). For any F ⊂ S, let F d be the smallest decreasing

set containing F. In other words, F d is all x ∈ S such that x  y for some y ∈ F.

Note that if F is closed then, since  is a closed partial order, so is the set F d (i.e.,
F ∈ C =⇒ F d ∈ dC ). Let

e := sup {ν( D ) − µ( D )}.
D ∈dC

Evidentlye ≥ sup{ν( F d ) − µ( F d )} ≥ sup{ν( F ) − µ( F d )}.
F ∈C

F ∈C

Noting that F d can be expressed as π1 (G ∩ (S × F )), it follows that, for any F ∈ C ,ν( F ) ≤ µ(π1 (G ∩ (S × F ))) + e.
Since  is a closed partial order, the set G is closed, and Strassen’s theorem (theorem 7.1) applies. From this theorem we obtain a probability measure λ on the
product space S × S such that (a) λ(G) ≥ 1 − e and (b) λ has marginals µ and ν.
Because complements of increasing sets are decreasing and vice versa, we have(15)

sup {µ( I ) − ν( I )} = sup {ν( D ) − µ( D )} = e ≥ 1 − λ(G).
I ∈iO

D ∈dC

Now consider the probability space (Ω, F , P) = (S × S, B ⊗ B, λ), and let X = π1
and Y = π2 . We then haveλ(G) = λ{( x, y) ∈ S × S : x  y} = P{ X  Y }.
Combining this equality with (15), we have shown the existence of a ( X, Y ) ∈C (µ, ν) withsup {µ( I ) − ν( I )} ≥ 1 − P{ X  Y }.
I ∈iOInequality (13) now follows.
Since inequality (14) is trivial we have now established the chain (11)–(14). It follows that all the terms in the chain are equal.

22

7.2. Remaining Proofs. We now turn to the remaining proofs from the paper. We
prove propostion 2.2 first, since it is independent of other claims and will be usedbelow.
Proof of propostion 2.2. Let {µ̃i }i∈N and {ν̃i }i∈N be sequences in M such that(16)µ̃i (S) ↑ σ(µ, ν),∀i ∈ N, (µ̃i , ν̃i ) ∈ Φ(µ, ν).

Since µ̃i ≤ µ and ν̃i ≤ ν for all i ∈ N, each of these sequences has a weakly
convergent subsequence by Prohorov’s theorem (Dudley, 2002, theorem 11.5.4).
By taking further subsequences, we can assume that µ̃i → µ̃ and ν̃i → ν̃ weakly forsome µ̃, ν̃ ∈ M . Since µ̃i ≤ µ and ν̃i ≤ ν for all i ∈ N, it follows from HernándezLerma and Lasserre (2003), theorem 1.5.5, that for all B ∈ B, µ̃i ( B) → µ̃( B) andν̃i ( B) → ν̂( B). Hence, for any B ∈ B, we have µ̃( B) ≤ µ( B) and ν̃( B) ≤ ν( B)

since µ̃i ( B) ≤ µ( B) and ν̃i ( B) ≤ ν( B) for all i ∈ N. Thus µ̃ ≤ µ and ν̃ ≤ ν. Sinceµ̃i (S) = ν̃i (S) for all i ∈ N, we also have µ̃(S) = ν̃(S). Moreover, σ(µ, ν) = µ̃(S),since µ̃i (S) ↑ σ (µ, ν). For any increasing I ∈ B, the fact that µ̃i ( I ) ≤ ν̃i ( I ) for alli ∈ N gives us µ̃( I ) ≤ ν̃( I ). Thus µ̃ sd ν̃. We have shown that the pair (µ̃, ν̃)satisfies the claim in proposition 2.2.



Now we can return to the proof of proposition 2.1.
Proof of proposition 2.1. The first claim is trivial. Regarding the second, it is clearthat if µ sd ν, then (µ, ν) ∈ Φ(µ, ν), and hence σ(µ, ν) = 1. To see the converse,

suppose that σ (µ, ν) = 1. Then, in view of proposition 2.2, there exists an ordered
component pair (µ0 , ν0 ) ∈ Φ(µ, ν) with µ0 (S) = ν0 (S) = 1. Since µ0 ≤ µ andµ0 (S) = 1, it must be that µ0 = µ. (Otherwise, there exists a B ∈ B with µ0 ( B) <µ( B), from which we obtain µ0 ( Bc ) > µ( Bc ). This contradicts µ0 ≤ µ.) For thesame reason we have, ν0 = ν. Because µ0 sd ν0 , we have shown that µ sd ν.

Finally, for the third claim, suppose first that µ = ν. In that case we have µ sd νand hence σ (µ, ν) = 1. Similarly, ν sd µ, and hence σ (ν, µ) = 1. Conversely,

23

suppose that σ(µ, ν) = σ(ν, µ) = 1. In that case both µ sd ν and ν sd µ. Since sd

is a partial order (Kamae and Krengel (1978)) it is antisymmetric. Hence µ = ν. 
Proof of theorem 3.1. The theorem follows directly from inequalities (11)–(14).

Proof of proposition 3.1. Let µ and ν be given. It view of theorem 3.1, it suffices toconstruct a coupling ( X, Y ) ∈ C (µ, ν) such that P{ X  Y } = σ (µ, ν). In constructing the coupling, we can and do assume that σ := σ (µ, ν) > 0.6

To begin, observe that, by proposition 2.2, there exists a pair (µ0 , ν0 ) ∈ Φ(µ, ν) with
µ0 (S) = ν0 (S) = σ. Letµr : =µ − µ01−σandνr : =

ν − ν0.
1−σ

It is easy to check that both µr and νr are probability measures, as are µ0 /σ andν0 /σ. Moreover, µ and ν can be expressed as the convex combinationsµ = (1 − σ ) µr + σ

µ0
σ

and

ν = (1 − σ ) νr + σ

ν0.
σ

Thus, to draw from µ, one can draw from µr with probability (1 − σ) and µ0 /σwith probability σ. A similar statement holds for ν. We construct a coupling ( X, Y )as follows. Let U, X 0 , Y 0 , X r and Y r be random variables on a common probability
space such that
D

D

D

D

(a) X 0 = µ0 /σ, Y 0 = ν0 /σ, X r = µr and Y r = νr(b) U is uniform on [0, 1](c) U is independent of ( X 0 , Y 0 , X r , Y r ) and(d) P{ X 0  Y 0 } = 1.
The only nontrivial existence problem here is (d), but such a pair can be constructed via Strassen’s theorem since µ0 /σ sd ν0 /σ. Now let
X : = 1 {U ≤ σ } X 0 + 1 {U > σ } X r

and Y := 1{U ≤ σ }Y 0 + 1{U > σ}Y r .

6If not, then for any ( X, Y ) ∈ C (µ, ν) we have 0 ≤ P{ X  Y } ≤ σ (µ, ν) = 0.

24

Evidently ( X, Y ) is a coupling of (µ, ν). Moreover, for this pair, we have

P{ X  Y } ≥ P{ X  Y, U ≤ σ} = P{ X 0  Y 0 , U ≤ σ}.
In view of independence the right hand side is equal to P{ X 0  Y 0 }P{U ≤ σ}.
But this is just σ = σ (µ, ν), so P{ X  Y } ≥ σ(µ, ν). Theorem 3.1 tells us that thereverse is also true, so P{ X  Y } = σ(µ, ν) as claimed.

Proof of proposition 4.1. Let {µn }, {νn }, µ and ν be as in the statement of the proposition. Applying theorem 3.1, we have|σ(µn , νn ) − σ(µ, ν)| = |1 − σ(µn , νn ) − (1 − σ(µ, ν))|= sup {µn ( I ) − νn ( I )} − sup {µ( I ) − ν( I )}I ∈isBI ∈isB≤ sup |µn ( I ) − νn ( I ) − {µ( I ) − ν( I )}|I ∈isB≤ sup |µn ( B) − µ( B)| + sup |νn ( B) − ν( B)| .
u

u

B ∈B

B ∈B

Since µn → µ and νn → ν, we now have σ(µn , νn ) → σ(µ, ν).



Proof of proposition 5.1. Under our assumptions on {e j }, the weak law of large numw

bers gives g + n1 ∑nj=1 e j → g in probability as n → ∞. Hence Gn → G. It follows

from corollary 4.1 that σ( F, Gn ) → σ ( F, G ).



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