Notes on
Supermodularity and Increasing Differences
in Expected Utility
∗
Alejandro Francetich †‡
Department of Decision Sciences and IGIER
Bocconi University, Italy
March 7, 2014
Abstract
Many choice-theoretic and game-theoretic applications in Economics invoke some
form of supermodularity or increasing differences for objective functions defined on
lattices. These notes provide axiomatic foundations for these properties on expectedutility representations of preferences over lotteries.
Keywords: Expected utility; supermodularity; quasi-supermodularity; increasing
differences; single crossing
JEL Classification Numbers: D01, D81
∗ This
is an updated working-paper version of the paper published in Economics Letters 121 (2013)
206-209. There is a typo on page 208 of the published version paper: On line 5 of the proof of Theorem 2,
the second weak inequality should be a strict inequality.
† Via Röntgen 1, 20136 Milan, Italy. Email address: [email protected]
‡ These notes are based on Chapter 1 of my dissertation, submitted to the Graduate School of Business
at Stanford University. I am grateful to David Kreps, Federico Echenique, Yossi Feinberg, Marco Li Calzi,
Paul Milgrom, John Quah, Andrzej Skrzypacz, Christopher Tyson, and an anonymous referee, for helpful
comments and suggestions to improve these notes. I also thank Rohan Dutta and Peter Troyan for carefully
reading previous drafts. Any remaining errors and omissions are all mine.
1
1
Introduction
These notes revisit the axiomatic foundations of the properties of supermodularity
and of increasing differences for expected-utility representations of preferences defined
over lotteries. While the first of these properties relates to a single preference relation
and the second one involves a family of preferences, mathematically, they are closely
related: A supermodular function on a product lattice has increasing differences.
2
Supermodular Expected Utility
2.1
Lattices and supermodularity
Let ( X, ≥ X ) be lattice, and denote the “join” and “meet” operations by ∨ and ∧,
respectively. For any f : X → R, if f ( x ∨ x 0 ) + f ( x ∧ x 0 ) ≥ f ( x ) + f ( x 0 ) for any x, x 0 ∈ X,
then f is supermodular. Following Li Calzi (1990) and Milgrom and Shannon (1994),
f is quasi-supermodular if, for any x, x 0 ∈ X, we have that f ( x ) ≥ f ( x ∧ x 0 ) implies
f ( x ∨ x 0 ) ≥ f ( x 0 ), with the corresponding implication for strict inequality. In words, for
any x, x 0 ∈ X, if “meeting” x with x 0 “downgrades” x, then “joining” x and x 0 “upgrades”
x 0 , according to f . In other words, if the value of x under f is strictly higher than the
value under f of x ∧ x 0 , then f cannot attain a higher value at x 0 than at x ∨ x 0 .
Supermodularity is a cardinal property, while quasi-supermodularity is an ordinal
implication of supermodularity. Any non-decreasing function satisfies the weak part of
quasi-supermodularity, and any strictly increasing function is quasi-supermodular.
2.2
Mixture spaces and Borel probability measures
Let T be a topology on X such that ( X, T ) is a T1 space; that is, a space in which all
singletons are closed sets. Denote by B( X ) the Borel σ-field on X, and let Δ( X ) denote
the space of Borel probability measures on X. Finally, let Δ0 ( X ) ⊆ Δ( X ) be the subset of
simple Borel probability measures on X, that is, the probability measures in Δ( X ) that
have finite support. In particular, for any x ∈ X, the point mass concentrated at x, δx , is
an element of Δ0 ( X ).
A pair ( Z, ∗), where Z is a set and ∗ is an operation ∗ : [0, 1] × Z × Z → Z, is a
mixture space (Fishburn, 1982) if:
• For all z, z0 ∈ Z, ∗(1, z, z0 ) = z;
• For all z, z0 ∈ Z and for all α ∈ [0, 1], ∗(α, z, z0 ) = ∗(1 − α, z0 , z);
2
• For all z, z0 ∈ Z and for all α, β ∈ [0, 1], ∗(α, ∗( β, z, z0 ), z0 ) = ∗(αβ, z, z0 ).
Both Δ( X ) and Δ0 ( X ), coupled with the operation of taking convex combinations of
probability measures, ∗(α, μ, μ0 ) = αμ + (1 − α)μ0 , form mixture spaces. Henceforth, ∗
will denote this specific mixture operation.
Let D ⊆ Δ( X ) be a subset of probability measures that contains all simple probability
measures and that forms a mixture space on its own; that is to say, Δ0 ( X ) ⊆ D and (D , ∗)
is a mixture space. A function f : D → R is linear in ∗ if for all μ, μ0 ∈ D and all α ∈ [0, 1],
f (∗(α, μ, μ0 )) = α f (μ) + (1 − α) f (μ0 ).
2.3
Preferences over lotteries and supermodular expected utility
Let %⊆ D × D be a complete preorder on D . Since D contains all point masses, %
induces a complete preorder on X, %X , given by x 0 %X x if δx0 % δx for any x, x 0 ∈ X.
The asymmetric and symmetric parts of %X , denoted by X and ∼ X , respectively, are
the relations induced by the asymmetric and symmetric parts of %.
In choice-theoretic and game-theoretic applications, supermodularity is imposed on
Bernoulli utility functions; these represent %X . However, the primitive preference is that
over lotteries, namely %. Hence, the relevant link is the link between supermodularity
of representations of %X and properties of %.
In the Mixture Space Theorem (Herstein and Milnor, 1953), the following three axioms are imposed on %:
( a) % is a complete preorder;
(b) For all μ, μ0 , μ00 ∈ D and for all α ∈ (0, 1): μ0 μ implies ∗ (α, μ0 , μ00 ) ∗ (α, μ, μ00 );
(c) For all μ, μ0 , μ00 ∈ D such that μ μ0 μ00 , there exists some α, β ∈ (0, 1) such that
∗ (α, μ, μ00 ) μ0 ∗ ( β, μ, μ00 ).
Axiom ( a) is a necessary assumption for % to admit a numerical representation.
Axiom (b) is an independence assumption, stating that the presence of a third lottery μ00
does not change the ranking of μ, μ0 when mixed with “equal weight.” Finally, axiom (c)
is a continuity or Archimedean axiom. Following Kreps (2013), this last axiom rules out
the existence of “supergood” or “superbad” lotteries: No matter how high μ is ranked
by the agent, for some mixture, μ0 is still strictly preferred to this mixture of μ and μ00 .
Similarly for μ00 : No matter how low it is ranked, μ0 is still strictly worse than some
mixture of μ and μ00 .
3
The Mixture Space Theorem states that a binary relation % on D satisfies these three
axioms if and only if there exists a real-valued function u on D representing % that
is linear in ∗ and unique up to positive affine transformations. If D = Δ0 ( X ), the von
Neumann and Morgenstern Theorem (von Neumann and Morgenstern, 1953) establishes
the existence of a real-valued function U on X that is also unique up to positive affine
R
transformations and such that u(μ) = Udμ. The result extends to the case D = Δ( X )
if there exists a metric d on X such that ( X, d) is separable, and if % is continuous in the
topology of weak convergence.
The function U in the von Neumann and Morgenstern Theorem represents %X , as
U ( x ) = u(δx ). Hence, the problem is to establish a link between properties of % and
supermodularity of U. The obvious link is given by the following axiom, (S):
Axiom (S). For all x, x ∈ X, ∗
1
2 , δx ∧ x 0 , δx ∨ x 0
%∗
1
2 , δx , δx 0
.
Axiom (S) states that, for any two outcomes, the 50-50 mixture between the “highest”
and the “lowest” of the two (under ≥ X ) is weakly preferred to the 50-50 mixture between
the outcomes. If we think of X as the product of two lattices, the axiom can be read
as saying that a 50-50 mixture between “all-high” or “all-low” coordinates is weakly
preferred to a 50-50 lottery between elements that feature both high and low coordinates.
Thus, it can be read as an axiom about “complementarity across dimensions.”
Theorem 1. Let ( X, ≥ X ) be a lattice and let % be a binary relation on Δ0 ( X ). Then, % satisfies
axioms (a), (b), (c), and (S), if and only if there exists a supermodular real-valued function
U : X → R such that u : Δ0 ( X ) → R given by u (μ) = ∑ x∈supp(μ) U ( x )μ({ x }) represents %.
Moreover, U is unique up to positive affine transformations.
The 50-50 mixture specified by axiom (S) is crucial in the proof of Theorem 1. For
other mixtures, quasi-supermodularity follows instead. Consider the following weaker
version of axiom (S), (qS):
Axiom (qS). For all x, x ∈ X, there exists some α ∈ (0, 1) such that ∗ (α, δx∧ x0 , δx∨ x0 ) %
∗ (α, δx , δx0 ).
Axiom (qS) states that, for any two outcomes, there exists a (strict) mixture between
the “highest” and the “lowest” of the two that is weakly preferred to the same mixture
4
between the outcomes themselves. However, this mixture may be different that 1/2, and
it may depend on the choice of x, x 0 ∈ X.1
Theorem 2. Let ( X, ≥ X ) be a lattice and let % be a binary relation on Δ0 ( X ). Then, % satisfies
axioms (a), (b), (c), and (qS), if and only if there exists a quasi-supermodular real-valued function
U : X → R such that u : Δ0 ( X ) → R given by u (μ) = ∑ x∈supp(μ) U ( x )μ({ x }) represents %.
Moreover, U is unique up to positive affine transformations.
3
Increasing Differences in Expected Utility
In this section, ( X, ≥ X ) is a poset (not necessarily a lattice), and RΘ := %θ : θ ∈ Θ
is an indexed family of complete preorders on D ⊆ Δ( X ). The index set Θ is also
endowed with a partial order, denoted by ≥Θ . Following Milgrom and Shannon (1994),
a function F : X × Θ → R satisfies the single-crossing property if, for each x, x 0 ∈ X
and each θ, θ 0 ∈ Θ such that x 0 > X x and θ 0 >Θ θ, F ( x 0 , θ ) ≥ F ( x, θ ) implies F ( x 0 , θ 0 ) ≥
F ( x, θ 0 ), and F ( x 0 , θ ) > F ( x, θ ) implies F ( x 0 , θ 0 ) > F ( x, θ 0 ); if we have F ( x 0 , θ 0 ) − F ( x, θ 0 ) ≥
F ( x 0 , θ ) − F ( x, θ ), then F has increasing differences.
Just as with supermodularity and quasi-supermodularity, the property of increasing
differences is a cardinal property, and it has the single-crossing property as an ordinal
implication. As in Section 2, the relevant link is the link between increasing differences of
representations of preferences over outcomes, %θX , and properties of the corresponding
preferences over lotteries, %θ , for each θ ∈ Θ. To simplify the analysis, I will maintain
the following assumption:
Condition 1. There exist some x0 , x1 ∈ X such that δx1 θ δx0 for all θ ∈ Θ.
Condition 1 means that there are two outcomes x0 , x1 ∈ X such that all preference
relations in RΘ agree that the lottery that pays x1 with certainty is strictly preferred to
the one that pays x0 with certainty. Under this condition, the desired link is given by the
following axiom, which will be called axiom (e):
1 If
the mixture in axiom (qS) is uniform across x, then representations will satisfy the following property, weaker than supermodularity but stronger than quasi-supermodularity. A function f : X → R
defined on a lattice ( X, ≥ X ) is α-supermodular if there exists some α ∈ [0, 1] such that, for all x, x 0 ∈ X,
α f ( x ∧ x 0 ) + (1 − α) f ( x ∨ x 0 ) ≥ max{α f ( x ) + (1 − α) f ( x 0 ), α f ( x 0 ) + (1 − α) f ( x )}.
5
0
Axiom (e).
that x 0 >
θ 0 >Θ θ, and for all
For all x, x ∈ X such
X x, for all θ, θ ∈ Θ such
that
0
e ≥ 0, ∗ 1+1 e , δx0 , δx0 %θ ∗ 1+1 e , δx , δx1 implies ∗ 1+1 e , δx0 , δx0 %θ ∗ 1+1 e , δx , δx1 .
Axiom (e) states that if any mixture of the higher of two elements with x0 is weakly
preferred under some preference relation to the corresponding mixture between the
lower of the two and x1 , then the same ranking applies to all preference relations identified by higher indices. Thus, it can be read as an axiom about “risk comparisons” across
preferences in the family.
Theorem 3. Let RΘ be an indexed family of binary relations on Δ0 ( X ). Then, the relations
in RΘ satisfy axioms (a), (b), (c), and (e) if and only if there exists a real-valued function
U : X × Θ → R with increasing differences such that, for every θ ∈ Θ, u(∙, θ ) : Δ0 ( X ) → R
given by u (μ, θ ) = ∑ x∈supp(μ) U ( x, θ )μ({ x }) represents %θ . Moreover, for each θ ∈ Θ, U (∙, θ )
is unique up to positive affine transformations.
If the implication in (e) can only be guaranteed for some rather than for all e ≥ 0, and
if the e’s on each side of the implication may be different, then we get the single-crossing
property instead. The weaker axiom that captures this ordinal property will be called
axiom (e0 ):
0
0
0
Axiom (e0 ). For all x, x ∈ X such
that x >X x and
for all θ, θ ∈ Θ such that θ >Θ θ, if there
exists some e ≥ 0 such that ∗ 1+1 e , δx0 , δx0 %θ ∗ 1+1 e , δx , δx1 , then there exists some e0 ≥ 0
0 such that ∗ 1+1e0 , δx0 , δx0 %θ ∗ 1+1e0 , δx , δx1 .
Axiom (e0 ) states that if some mixture of the higher of two elements with x0 is weakly
preferred under some preference relation to the corresponding mixture between the
lower of the two and x1 , then some other mixtures are similarly weakly preferred under
preferences with higher indices.
Theorem 4. Let RΘ be an indexed family of binary relations on Δ0 ( X ). Then, the relations
in RΘ satisfy axioms (a), (b), (c), and (e0 ) if and only if there exists a real-valued function U :
X × Θ → R with the single-crossing property such that, for every θ ∈ Θ, u(∙, θ ) : Δ0 ( X ) → R
given by u (μ, θ ) = ∑ x∈supp(μ) U ( x, θ )μ({ x }) represents %θ . Moreover, for each θ ∈ Θ, U (∙, θ )
is unique up to positive affine transformations.
6
A
Proofs
Proof of Theorem 1. Assume that % are represented by u(μ) = ∑ x∈supp(μ) U ( x )μ({ x })
for some real-valued supermodular function U. Then, u is linear in ∗. That % satisfies
axioms ( a), (b), and (c), follows from the Mixture Space Theorem. Take any x, x 0 ∈ X.
Using linearity in ∗ of u and supermodularity of U,
1
U ( x ∧ x0 ) + U ( x ∨ x0 )
U ( x) + U ( x0 )
1
.
u ∗
, δ 0, δ 0
, δ x , δx 0
=
≥
=u ∗
2 x∧x x∨x
2
2
2
Thus, Axiom (S) follows. Conversely, assume that preferences satisfy axioms ( a), (b), (c),
and (S). The von Neumann and Morgenstern Theorem produces a function U : X → R
such that u(μ) = ∑ x∈supp(μ) U ( x )μ({ x }). Clearly, U represents %X . Supermodularity of
U is a simple consequence of (S) and linearity of u:
1
1
≥ 2u ∗
= u(δx ) + u(δx0 ),
, δ 0, δ 0
, δx , δx0
u(δx∧ x0 ) + u(δx∨ x0 ) = 2u ∗
2 x∧x x∨x
2
and thus U ( x ∨ x 0 ) + U ( x ∧ x 0 ) = u(δx∨ x0 ) + u(δx∧ x0 ) ≥ u(δx ) + u(δx0 ) = U ( x ) + U ( x 0 )
for any two x, x 0 ∈ X, as desired. The statement about uniqueness up to positive affine
transformations follows from the von Neumann and Morgenstern Theorem.
Proof of Theorem 2. Assume that % are represented by u(μ) = ∑ x∈supp(μ) U ( x )μ({ x })
for some quasi-supermodular real-valued function U. As before, axioms ( a), (b), and
(c), are consequences of the Mixture Space Theorem. Take any x, x 0 ∈ X. Without loss of
generality, assume that U ( x ) ≥ U ( x 0 ). If U ( x ) > U ( x ∧ x 0 ), by quasi-supermodularity,
we have U ( x ∨ x 0 ) > U ( x 0 ). Define h : [0, 1] → R as h(α) = (1 − α)[U ( x ∨ x 0 ) − U ( x 0 )] +
α[U ( x ∧ x 0 ) − U ( x )]; this function is linear and satisfies h(0) > 0 and h(1) < 0. Thus,
there exists some α∗ ∈ (0, 1) close enough to 0 such that h(α∗ ) > 0, which implies
u (∗ (α∗ , δx∧ x0 , δx∨ x0 )) > u (∗ (α∗ , δx , δx0 )). If U ( x ) = U ( x ∧ x 0 ), quasi-supermodularity
implies U ( x ∨ x 0 ) ≥ U ( x 0 ); thus, U ( x ∨ x 0 ) + U ( x ∧ x 0 ) ≥ U ( x 0 ) + U ( x ∧ x 0 ) = U ( x 0 ) +
U ( x ). In this case, we can take α = 12 . Finally, consider the case U ( x ) < U ( x ∧ x 0 ). If
U ( x ∨ x 0 ) ≥ U ( x 0 ), for any α ∈ (0, 1), we have 1−α α (U ( x ∨ x 0 ) − U ( x 0 )) + U ( x ∧ x 0 ) >
U ( x ); rearranging terms yields the desired ranking. If U ( x ∨ x 0 ) < U ( x 0 ), we can still
e
find some e > 0 such that U ( x ) < U ( x ∧
x0 )
− e. Let α :=
|U ( x∨ x0 )−U ( x0 )|
;
e
1+
|U ( x∨ x0 )−U ( x0 )|
the same
inequality as before follows. Conversely, assume that preferences satisfy axioms ( a), (b),
(c), and (qS). Let U be as in the proof of Theorem 1. Take any two x, x 0 ∈ X. By (qS), there
exists some α ∈ (0, 1) such that αU ( x ∨ x 0 ) + (1 − α)U ( x ∧ x 0 ) ≥ αU ( x ) + (1 − α)U ( x 0 ),
7
or α[U ( x ∨ x 0 ) − U ( x )] ≥ (1 − α)[U ( x 0 ) − U ( x ∧ x 0 )], and so U ( x 0 ) > U ( x ∧ x 0 ) implies
U ( x ∨ x 0 ) > U ( x ). Again, the last statement in the proposition follows from the von
Neumann and Morgenstern Theorem.
Lemma 1. A function F : X × Θ → R satisfies increasing differences if and only if, for all
x, x 0 ∈ X such that x 0 > X x and θ, θ 0 ∈ Θ such that θ 0 >Θ θ: F ( x 0 , θ ) ≥ F ( x, θ ) + e implies
F ( x 0 , θ 0 ) ≥ F ( x, θ 0 ) + e for all e ≥ 0.
Proof. If F satisfies increasing differences, for any x, x 0 ∈ X such that x 0 > X x and any
θ, θ 0 ∈ Θ such that θ 0 >Θ θ, for any e ≥ 0, F ( x 0 , θ 0 ) − F ( x, θ 0 ) ≥ F ( x 0 , θ ) − F ( x, θ ) ≥ e.
Conversely, assume that there exists some x00 > X , x00 ∈ X and θ0 , θ00 ∈ Θ such that
F ( x00 , θ00 ) − F ( x0 , θ00 ) < F ( x00 , θ0 ) − F ( x0 , θ0 ). Then, there exists some e0 > 0 such that
F ( x00 , θ00 ) − F ( x0 , θ00 ) < e0 < F ( x00 , θ0 ) − F ( x0 , θ0 ).
Proof of Theorem 3. The only portion of the theorem that remains to be shown corresponds to axiom (e). Given some U : X × Θ → R, consider the family of preferences
with expected-utility representation induced by U (∙, θ ) : θ ∈ Θ. We can normalize these
representations so that U ( x0 , θ ) = 0 and U ( x1 , θ ) = 1 for each θ ∈ Θ.2 Take x, x 0 ∈ X
such that x 0 > X x, and θ, θ 0 ∈ Θ such that θ 0 >Θ θ. For e ≥ 0, assume that 1+1 e U ( x 0 , θ ) =
u ∗ 1+1 e , δx0 , δx0 , θ ≥ u ∗ 1+1 e , δx , δx1 , θ = 1+1 e U ( x, θ ) + 1+e e . If U has increasing differences, U ( x 0 , θ 0 ) − U ( x, θ 0 ) ≥ U ( x 0 , θ ) − U ( x, θ ) = e; the implication in axiom
(e) follows. Conversely, under axiom (e), we get 1+1 e U ( x 0 , θ 0 ) ≥ 1+1 e U ( x, θ 0 ) + 1+e e , or
U ( x 0 , θ 0 ) ≥ U ( x, θ 0 ) + e; the result follows by Lemma 1.
0
Proof of Theorem 4. Take x, x 0 ∈ X such that x 0 > X x, and θ, θ 0 ∈
Θ θ.
Θ such that θ >
1
1
0
Assume that there exists some e ≥ 0 such that 1+e U ( x , θ ) = u ∗ 1+e , δx0 , δx0 , θ ≥
u ∗ 1+1 e , δx , δx1 , θ = 1+1 e U ( x, θ ) + 1+e e . Then, U ( x 0 , θ ) − U ( x, θ ) ≥ e. If U has the
0
0 0
0
single-crossing
property,
then
e := U ( x , θ) − U ( x, θ ) ≥ 0, and rearranging terms yields
∗ 1+1e0 , δx0 , δx0 , θ 0 ≥ u ∗ 1+1e0 , δx , δx1 , θ 0 . Conversely, if e := U ( x 0 , θ ) − U ( x, θ ) ≥
0, we can find some e0 ≥ 0 such that U ( x 0 , θ 0 ) − U ( x, θ 0 ) ≥ e0 ≥ 0. Thus, U has the
single-crossing property.
References
Fishburn, P. (1982). The Foundations of Expected Utility. D. Reidel Publishing Company.
2 For
e (∙, θ ) :=
each θ ∈ Θ, take U
U (∙,θ )−U ( x0 ,θ )
.
U ( x1 ,θ )−U ( x0 ,θ )
8
Herstein, I.N. and Milnor, J. (1953). An axiomatic approach to measurable utility. Econometrica 21(2):291–297.
Kreps, D. (2013). Microeconomic Foundations I: Choice and Competitive Markets. Princeton
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Li Calzi, M. (1990). Generalized symmetric supermodular functions. Mimeo, Stanford
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Milgrom, P. and Shannon, C. (1994).
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9