Supermodular Comparative Statics∗

Pawe l Dziewulski† Lukasz Woźny‡

March 2019. Preliminary draft. Please do not circulate.

Abstract

An important set of questions in economics concern how changes in the model’s

exogenous parameters (income, wealth, productivity, distortions, information, etc.)

impact individual choices and market outcomes. In this paper, we develop a theory

of supermodular comparative statics that addresses this set of issues. Specifically,

we show ordinal and cardinal conditions one should impose on the optimization

problem so that its solution is a supermodular function or a supermodular cor-

respondence. We illustrate application to industrial organization, supermodular

stochastic orders and extensive form games with strategic complements.

Keywords: comparative statics, supermodularity, supermodular correspondences,

strategic complements, arg max, policy functions, increasing differences

JEL classification: C61, D90, E21.

1 Introduction

Comparative statics results based on the implicit function theorem or the monotonic-

ity theorems of Topkis (1978), Milgrom and Shannon (1994) or Quah (2007), provide

∗We want to acknowledge talks with Rabah Amir, Bernard Cornet, Federico Echenique, Martin

Jensen, Marco Li Calzi, Kevin Reffett, Tarun Sabarwal as well as Ed Shlee concerning this topic. We

also thank participants of our presentations on Time, Uncertainties and Strategies III Conference in Paris

2017; ASU Economic Seminar (2018), SAET Conferences in Taipei 2018 and Ischia 2019, and Workshop

at the University in Kansas, May 2019. They have vastly shaped our thinking about this project and

significantly improved this paper.†Department of Economics, University of Sussex, UK.‡Department of Quantitative Economics, Warsaw School of Economics, Warsaw, Poland. Address:

al. Niepodleg lości 162, 02-554 Warszawa, Poland. E-mail: lukasz.wozny@sgh.waw.pl.

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conditions under which an optimization problem’s endogenous variables increase or de-

crease when exogenous variables change. Such results are of course immensely valuable

for economic analysis. Recently, however, apart from monotonicity a number of ques-

tions emerge when comparing solutions to optimization problem over multidimensional

parameters. Specifically, economists are interested in conditions guaranteeing that pa-

rameterized solution to an optimization problem has complementarities in parameters.

This can be captured by a function solving an optimization problem being supermodular.

In this paper we develop conditions that assure that function solving a parameterized

optimization problem is supermodular in optimization problem parameters. Our con-

ditions are ordinal, yet yield cardinal conclusions. We then characterize our condition

via their cardinal version and finish with simple conditions related to partial derivatives.

The main characteristic of our lattice type method is that it relies essentially on critical

assumptions for the desired supermodularity conclusions and dispenses with superfluous

assumptions that are often imposed only by the use of the classical method based on the

implicit function theorem, including smoothness, inferiority and concavity. The main in-

sight from our conditions is indeed quite simple. For a single peaked function, it controls

how the differences with respect to optimization argument change with respect to param-

eters along with ”supermodularity direction”. For a single peaked function our sufficient

conditions are also necessary in a given sense. Our method and results recall some recent

results of Jensen (2018) on (quasi)concavity of the argmax function. We then extend this

basic result to cover multivalued and multidimensional argmaxes. When doing so, we

generalize a notion of supermodularity to Rn valued functions and develop a notion of

supermodular correspondences.

We discuss three applications. First, we consider models under which parameter is a

random variable. So suppose a decision maker can condition his action on the realization

of a random variable. Specifically, he receives information through a signal x drawn from

a distribution µ, and chooses an action y ∈ R so as to maximize his payoff functionf(y, x). We then consider a question, when the expected decision (with respect to distri-

bution µ) depends on shifts in distribution µ with respect to supermodular order. This

is an important question, as the stochastic supermodular order captures the complemen-

tarities between various dimensions of the random variable. Recall that precisely such

order it often used to measure multidimensional inequalities (see Meyer and Strulovici

(2015)) e.g. between countries. Our second application concern industrial organization,

specifically two period two players Stackelberg games. In this one, we seek conditions for

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the strategy of the second player in the subgame perfect Nash equilibrium to be given by

a supermodular function of the first players actions. Again, this has implications on the

properties of the first player payoff function and hence structure of the subgame perfect

Nash equilibrium set. Finally, we consider a multiplayer dynamic games, where each

period a number of players takes actions. We then provide conditions under which a sub-

game perfect Nash equilibrium is given by supermodular functions of games parameters,

states in stochastic games or players own signals. This has important consequences for

the preservations of supermodularities between periods in perfect equilibria in dynamic

games and hence yields conditions for a dynamic game to be supermodular in its extensive

form (see also Echenique (2004)). To see the last point, consider a two-stage game, when

two players invest in some variable at the first stage, say promotion effort that fosters

brand loyalty, and then compete in the second stage, say in prices. Suppose, further-

more, that strategies at the second stage are strategic complements. The questions we

are trying to answer is when are promotion strategies strategic complements, when price

competition with the same property is anticipated? In this case, when rivals increase

promotion expenditure, we also want to. That is, under what conditions are the strate-

gies in the reduced-form first-stage game, obtained by folding back second-stage payoffs

at a subgame-perfect equilibrium, also strategic complements? When does an increase in

promotion expenditure by any firm at the first stage induce higher prices at the second?

We provide sufficient conditions on the primitives of the game to answer these questions.

The rest of the paper is structured as follows. First, in section 1.1 we state the main

terminology from the lattices programming that we use in the paper. Next, in section 1.2,

we start with an attempt to answer our question (conditions for a supermodular argmax)

by the first order approach an use of the implicit function theorem. It is done to grasp

intuition on sufficient conditions and can be served as a benchmark for our main re-

sult. The main result is contained in section 2, its cardinal version 3 and differentiable

characterization in section 4. Section 7 contains our applications.

1.1 Preliminaries on lattices and supermodularity

Here we state main definitions concerning lattice programming that we use throughout

the paper. For a reference we refer a reader to books of Birkhoff (1967), Topkis (1998)

or notes of Veinott (1992).

Posets and lattices: A partially ordered set (or poset) is a set P ordered with a

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reflexive, transitive, and antisymmetric relation. If any two elements of C ⊂ P arecomparable, C is referred to as a linearly ordered set, or chain. A lattice is a set L

ordered with a reflexive, transitive, and antisymmetric relation ≥ such that any twoelements x and x′ in L have a least upper bound in L, denoted x ∧ x′, and a greatestlower bound in L, denoted x ∨ x′. L1 ⊂ L is a sublattice of L if it contains the sup andthe inf (with respect to L) of any pair of points in L1. A lattice is complete if any subset

L1 of L has a least upper bound and a greatest lower bound in L. L1 is subcomplete if

it is complete and a sublattice. In a poset P , if every subchain in C ⊂ P is complete,then C is referred to as a chain complete poset (or CPO). If every countably subchain in

C is complete, then C is referred to as a countably chain complete poset (or CCPO). Let

[a) = {x|x ∈ P, x ≥ a} be the upperset of a, and (b] = {x|x ∈ P, x ≤ b} the lowerset ofb. We say P is an ordered topological space if [a) and (b] are closed in the topology on P.

An order interval is defined to be [a, b] = [a) ∩ (b], a ≤ b.Isotone (or order preserving) mappings on a poset: Let (X,≥X) and (Y,≥Y )

be posets. A mapping f : X → Y is increasing (or isotone) on X if f(x′) ≥Y f(x), whenx′ ≥X x, for x, x′ ∈ X. If f(x′) >Y f(x) when x′ >X x, we say f is strictly increasing.1

The mapping f : X → Y is join preserving (resp, meet preserving) if we have for anycountable chain C, f(∨C) = ∨f(C) (resp, f(∧C) = ∧f(C)). A mapping that is bothjoin and meet preserving is order continuous.

A correspondence (or multifunction) F : X → 2Y is ascending in a binary set relationB on 2Y if F (x′) B F (x), when x′ ≥X x. Let X be a poset, Y a lattice, and define therelation B =≥v on the range L(Y) of all nonempty sublattices of Y, where for L1, L2 ∈L(X) we say L1 ≥v L2 in Veinott’s Strong Set order if for all x2 ∈ L2, x1 ∈ L1, x1∨x2 ∈ L1,x1 ∧ x2 ∈ L2.

Supermodular functions on lattices Let (X,≥X) be a lattice and (Y,≥Y ) be apartially ordered set. Function f : X → R is quasisupermodular over X whenever, forany x, x′ in X, f(x) ≥ f(x ∧ x′) implies f(x ∨ x′) ≥ f(x′), and f(x) > f(x ∧ x′) impliesf(x∨x′) > f(x′). Next f : X → R is supermodular over lattice X if for any x, x′ in X wehave f(x∧x′)+f(x∨x′) ≥ f(x)+f(x′). Observe that it is a cardinal property, hence neednot be preserved by arbitrary increasing transformation, say g(f(x)), unless g is increasing

and convex and f apart from being supermodular is also increasing (the property we will

1To avoid using references to “isotone mapping”, we will often use the more traditional terminology

in economics “increasing”. In the literature on partially ordered sets, an “increasing map” often denotes

something slightly different (e.g., f(x′) ≥Y f(x) when x′ >X x for x, x′ ∈ X).

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use extensively in applications part). A C2 function on R` is supermodular iff ∂2f∂xi∂xj

(·) ≥ 0for any i and j 6= i.

Function g : X × Y → R has single crossing differences in (x, y) on X × Y whenever,for any x′ ≥X x in X and y′ ≥Y y in Y , g(x′, y) ≥ g(x, y) implies g(x′, y′) ≥ g(x, y′),and g(x′, y) > g(x, y) implies g(x′, y′) > g(x, y′). Next g : X × Y → R has increasingdifferences, if for any x, x′ in X, with x′ > x the function t→ g(x′, t)−g(x, y) is increasingwith t.

1.2 First order approach

In this subsection we first consider conditions that yield supermodularity of the argmax

functions using the first order approach and implicit function theorem. This would serve

as a benchmark for ordinal conditions obtained in our main theorem.

Consider a function f : R×X → R, and problem

maxy∈Y

f(y, x),

where X ⊂ R` is a compact lattice, Y ⊂ R is convex and compact, while f(·, x) continu-ous. Denote:

φ(x) = arg maxy∈Y

f(y, x).

For a differentiable f accept the following notation: f ′′′yij(·) =∂3f

∂y∂xi∂xj(·), and similarly for

other partial derivatives. We show the following:

Proposition 1. Suppose f ∈ C3 and that for any x ∈ X there exists the unique interiorsolution φ(x) in C2, then if:

• f ′′yy(·) < 0,

• f ′′yi(·) ≥ 0 for any i,

• f ′′′yyi(·) ≥ 0 for any i,

• f ′′′yij(·) ≥ 0 for any i, j,

• f ′′′yyy(·) ≥ 0,

then φ : X → R is increasing and supermodular.

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Proof. For interior solution we must have f ′y(φ(x), x) = 0, hence by implicit function

theorem:∂φ

∂xi(x) =

f ′′yi(φ(x), x)

−f ′′yy(φ(x), x).

If φ is C2 then:

∂2φ

∂xi∂xj(·) =

[−f ′′′yyj(·)φ′j(·)− f ′′′yij(·)]f ′′yy(·) + f ′′yi(·)[f ′′′yyy(·)φ′j(·) + f ′′′yyj(·)][f ′′yy(·)]2

≥ 0

The result follows from our assumptions on partial derivatives.

Proposition contains the first set of assumptions sufficient to obtain our supermodu-

larity result. In fact, as we will show in the next section some of them are not necessary

(like differentability, interiority or concavity). Here, we only stress that the above propo-

sition contains a hypothesis that is difficult to verify at the level of primitives, namely

twice differentiability of the argmax function.

2 Ordinal conditions and the main theorem

Consider a function f : R × X → R, defined over a lattice (X,≥X). Function f issingle-peaked over R for some x ∈ X, whenever there is some y∗ ∈ R such that f(·, x)is strictly increasing on (−∞, y∗] and strictly decreasing on [y∗,∞). Clearly, this impliesthat correspondence φ : X → R, given by φ(x) := argmaxy∈R f(y, x) is a well-definedfunction.

Function f satisfies property QUASISUPERMODULAR DIFFERENCES CONDI-

TION if, for any numbers y ∈ R and δ ∈ R+, as well as x, x′ ∈ X we have

(i) if f(·, x ∧ x′) decreases, while f(·, x) increases on [y, y + δ], then there is a y′ ∈ Rsuch that

f(·, x′) decreases, while f(·, x ∨ x′) increases on [y′, y′ + δ];

(ii) if f(·, x ∨ x′) decreases, while f(·, x′) increases on [y, y + δ], then there is a y′ ∈ Rsuch that

f(·, x) decreases, while f(·, x ∧ x′) increases on [y′, y′ + δ];

Clearly, the above property is ordinal. That is, it is preserved by an arbitrary strictly

monotone transformation of f .

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Theorem 1. Suppose that function f : R × X → R is single-peaked on R. Function fobeys (∗) if and only if function φ : X → R, φ(x) := argmaxy∈R f(y, x) is supermodular.

Proof. Since function f is single-peaked, correspondence φ is uniquely well-defined. In

order to prove (⇒), take any x, x′ in X and consider three cases.Case 1. min

{φ(x ∧ x′), φ(x ∨ x′)

}≥ max

{φ(x), φ(x′)

}. Result holds trivially.

Case 2. Suppose that φ(x) ≥ φ(x∧x′). Define δ := φ(x)−φ(x∧x′) and y := φ(x∧x′).Then f(·, x ∧ x′) is decreasing on [y, y + δ]. As y + δ = φ(x), then f(·, x) increases over[y, y + δ].

Property QSD(i) implies that there is some y′ ∈ R such that function f(·, x′) de-creases, while f(·, x∨x′) increases on [y′, y′+ δ]. Hence φ(x′) ≤ y′ and φ(x∨x′) ≥ y′+ δ.In particular, we have φ(x ∨ x′)− φ(x′) ≥ δ. Therefore,

φ(x)− φ(x ∧ x′) = δ ≤ φ(x ∨ x′)− φ(x′).

Case 3. Suppose that φ(x′) ≥ φ(x ∨ x′) and φ(x ∧ x′) > max{φ(x), φ(x′)

}. Define

δ := φ(x′)− φ(x ∨ x′) and denote y := φ(x ∨ x′). Notice that f(·, x ∨ x′) decreases, whilef(·, x′) increases on [y, y + δ]. By property QSD(ii), this implies that there is some y′

such that f(·, x ∧ x′) increases and f(·, x) decreases on [y′, y′ + δ]. Given that function fis single-peaked, we have φ(x) ≤ y′ and φ(x ∧ x′) ≥ y′ + δ. This implies that

φ(x)− φ(x ∧ x′) ≤ −δ = φ(x ∨ x′)− φ(x′).

This concludes the first part of the proof.

We prove (⇐) by contradiction. First, suppose that function φ is supermodular, butproperty (∗)(i) fails to hold. Assume that there is some y ∈ R and δ ∈ R+, as well asx ∈ X and vectors v, v′ ∈ R`+ such that f(·, x) decreases, while f(·, x + v) increases on[y, y + δ]. Since function f(·, x) is single-peaked, for all x ∈ X, this condition impliesthat φ(x + v) − φ(x) ≥ δ. Whenever property (∗)(i) is violated, there is no y′ ∈ R suchthat f(·, x+ v′) decreases, while f(·, x+ v+ v′) increases on [y′, y′+ δ]. In particular, theconditions fails for y′ := φ(x+ v′). Clearly, function f(·, x+ v′) decreases over [y′, y′+ δ].Therefore, there must be some z ∈ [y′, y′+ δ] such that function f(·, x+ v+ v′) decreaseson [z′, y′ + δ]. Hence, it must be that φ(x+ v + v′) < y′ + δ.

Denote z := x + v and z′ := x + v′. Since (v ∧ v′) = 0, we have (z ∧ z′) = x and(z ∨ z′) = x+ v + v′. Given this, the above observation implies that

φ(z ∨ z′)− φ(z′) < δ ≤ φ(z)− φ(z ∧ z′),

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Figure 1: Quasisupermodular differences condition.

which contradicts that function φ is supermodular. Using an analogous argument, we

show that supermodularity of φ is violated once property (∗)(ii) fails to hold.

Observe that ordinal condition QSD on the primitives yield cardinal results. More-

over, it is straightforward to see that we can generalize the result to discrete Y ⊂ R orgeneralize the result to Y = [y, y] ⊂ R but QSD condition must be satisfied on [y, y].

Example 1. f : R× R2+ with

f(y, x1, x2) =

ey if y ≤ x1x2,−ey if y > x1x2.Observe that f is single-peaked but is non continuous. Take x′1 > x1 and x

′2 > x2. Then

f(·, x1x2) is decreasing while f(·, x′1x2) is increasing on [x1x2, x1x2 + (x′1 − x1)x2]. QSDrequires existence of some [y′, y′ + (x′1 − x1)x2] such that f(·, x1x′2) is decreasing whilef(·, x′1x′2) is increasing. Indeed, f(·, x1x′2) is decreasing while f(·, x′1x′2) is increasing on[x1x

′2, x1x

′2+(x

′1−x1)x′2], where (x′1−x1)x′2 > (x′1−x1)x2. Clearly argmax φ(x1, x2) = x1x2

is SPM.

Function f : R×X → R obeys property (?) if for any x, x′ ∈ X the following conditionholds: whenever for y′ > y

f(y′, x ∧ x′) > f(y, x ∧ x′)⇒ f(ỹ′, x) > f(ỹ, x) for ỹ′ > ỹ

with ỹ′ ≥ y′ then

f(z′, x′) > f(z, x′) with z′ > z ⇒ f(z̃′, x ∨ x′) > f(z̃, x ∨ x′) for z̃′ > z̃

with z̃′ > (resp. ≥) z′.

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Proposition 2 (Quasisupermodular Comparative Statics). Suppose that function f sat-

isfies property (?) and f(·, x) is single-peaked for all x ∈ X. Then function φ(x) :=argmaxy∈Y f(y, x) is (resp. weakly) quasi-supermodular.

Proof. Take any x, x′ ∈ X and assume that φ(x) > φ(x ∧ x′). We need to show thatφ(x ∨ x′) > φ(x′). Denote y′ := φ(x ∧ x′). Given that f(·, x) is single-peaked, for allx ∈ X, while φ(x) > φ(x ∧ x′), it must be that

f(y′, x ∧ x′) > f(y, x ∧ x′) implies f(ỹ′, x) > f(ỹ, x), for all ỹ′ > ỹ,

with ỹ′ ≥ y′ Let z′ := φ(x′). Condition (?) guarantees that

f(z′, x′) > f(z, x′) implies f(z′, x ∨ x′) > f(z, x ∨ x′), for all z̃′ ≤ z̃,

with z̃′ > z′. This suffices for φ(x ∨ x′) > z′ = φ(x′).

We believe this result is important also in a context provided by Chambers and

Echenique (2009) that every weakly increasing, quasisupermodular function on finite

domain can be transformed into weakly increasing and supermodular function.

3 Cardinal conditions and characterization

In this section we provide a set of cardinal conditions that imply our QSD property.

Define ∆1f(y, x) := f(y + δ′, x)− f(y, x).

Theorem 2. Suppose a single-peaked function f satisfies for any y, x, x′, as well as

y2 ≥ y1 and x2 ≥ x1:

1. ∆1f(y, x)−∆1f(y, x ∧ x′) ≤ ∆1f(y, x ∨ x′)−∆1f(y, x′),

2. ∆1f(y1, x2)−∆1f(y1, x1) ≤ ∆1f(y2, x2)−∆1f(y2, x1),

3. ∆1f(y, x1) ≤ ∆1f(y, x2)

where ∆1f(y, x) := f(y+δ′, x)−f(y, x). Then φ is increasing and supermodular function.

One can obtain that φ is decreasing and supermodular function, if inequalities in (2)

and (3) are reversed.

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Proof. Denoting ∆1f(y, x) := f(y + δ, x)− f(y, x) the first property says:

∆1f(y, x)−∆1f(y, x ∧ x′) ≤ ∆1f(y, x ∨ x′)−∆1f(y, x′) (1)

Now suppose we have f satisfying 1 and 2 and 3 that is also single-peaked. Suppose the

antecedents from the QSD-property are satisfied, i.e. ∆1f(y, x) > 0 and ∆1f(y, x∧x′) <0. It implies by 1 that ∆1f(y, x∨ x′)−∆1f(y, x′) > 0. For this to be true, we must haveone of the three cases:

1. ∆1f(y, x ∨ x′) > 0 and ∆1f(y, x′) < 0; as required by the consequent of the QSD-property (for y′ = y)

2. ∆1f(y, x∨x′) < 0 and ∆1f(y, x′) < 0; but that violates condition 3 as 0 > ∆1f(y, x∨x′) ≥ ∆1f(y, x) > 0

3. ∆1f(y, x ∨ x′) > 0 and ∆1f(y, x′) > 0.

So consider the third case. By property 2 we have for any y′ > y

∆1f(y, x ∨ x′)−∆1f(y, x′) ≤ ∆1f(y′, x ∨ x′)−∆1f(y′, x′) (2)

Hence ∆1f(y′, x∨ x′)−∆1f(y′, x′) > 0 for any y′ > y. Clearly by single-peaked property

of f there must be y′ > y such that ∆1f(y′, x ∨ x′) > 0 but ∆1f(y′, x′) < 0 as required

by the consequent of the ∗-property. The (ii) part of the QSD-property is satisfied byanalogous argument.

4 Differentiable characterization

In this section we provide a set of easily checked conditions on partial derivatives that

imply cardinal assumptions of theorem 2 and hence our (∗) property.

Theorem 3. Suppose that X = RL, a single peaked function f ∈ C3 and the followingholds (for any i,j) f

′′′yij ≥ 0, f

′′′yyi ≥ 0 and f

′′yi ≥ 0, then φ is increasing and supermodular.

Proof. We show that conditions of theorem 2 are satisfied. We conduct the proof for

f′′′yij ≥ 0l other conditions follow from similar arguments. So suppose f

′′′yij ≥ 0. This

means that f′′ij(y

′, x) ≥ f ′′ij(y, x) for any y′ ≥ y. And hence∫ x′jxjf

′′ij(y

′, xj, x−j)dxj ≥∫ x′jxjf

′ij(y, xj, x−j)dxj. Hence

f′

i (y′, x′j, x−j)− f

′

i (y′, xj, x−j) ≥ f

′

i (y, x′j, x−j)− f

′

i (y, xj, x−j).

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Hence∫ x′ixi

(f′

i (y′, x′j, x−j)− f

′

i (y′, xj, x−j))dxi ≥

∫ x′ixi

(f′

i (y, x′j, x−j)− f

′

i (y, xj, x−j)dxi),

and this implies that

f(y′, x′j, x′i, x−i−j)− f(y′, x′j, xi, x−i−j)− (f(y′, xj, x′i, x−i−j)− f(y′, xj, xi, x−i−j)) ≥

f(y, x′j, x′i, x−i−j)− f(y, x′j, xi, x−i−j)− (f(y, xj, x′i, x−i−j)− f(y, xj, xi, x−i−j)),

which implies desired property.

Theorem 4. Suppose that X = RL, a single peaked function f ∈ C3 and the followingholds (for any i,j) f

′′′yij ≥ 0, f

′′′yyi ≤ 0 and f

′′yi ≤ 0, then φ is decreasing and supermodular.

5 Constrained case

Now consider a problem:

φ(x) := argmaxy∈Γ(x)

f(y),

where

Γ(x) = [Γ(x),Γ(x)] ⊂ R,

and f : R→ R is single-peaked on R. When φ is supermodular?

Theorem 5 (Constrained SCS). Suppose that f : R×X → R is single-peaked on R. If

• f is increasing on [y, y+δ1] ⊂ [Γ(x ∧ x′),Γ(x)] ANDf is decreasing on [y, y+δ2] ⊂ [Γ(x ∧ x′),Γ(x)] THEN2 ∃[y′, y′+δ3] ⊂ [Γ(x′),Γ(x ∨x′)] on which f is increasing AND ∃[y′, y′+δ4] ⊂ [Γ(x′),Γ(x ∨ x′)] on which f isdecreasing such that

δ3 + δ4 ≥ δ1 + δ2.

• f is increasing on [y, y+δ1] ⊂ [Γ(x ∨ x′),Γ(x′)] ANDf is decreasing on [y, y+δ2] ⊂ [Γ(x∨x′),Γ(x′)] THEN ∃[y′, y′+δ3] ⊂ [Γ(x),Γ(x∧x′)]on which f is increasing AND ∃[y′, y′+δ4] ⊂ [Γ(x),Γ(x∧x′)] on which f is decreasingsuch that δ3 + δ4 ≥ δ1 + δ2.

2If some of these intervals are empty take δi = 0.

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Figure 2: Quasisupermodular differences condition.

Then φ is supermodular.

Proof. Case 1. min{φ(x ∧ x′), φ(x ∨ x′)

}≥ max

{φ(x), φ(x′)

}. Result holds.

Case 2. Suppose that φ(x) ≥ φ(x ∧ x′). Define δ := φ(x) − φ(x ∧ x′). Since f is single peaked

there are 3 possibilities:

(a) Γ(x ∧ x′) ≤ Γ(x) and there exists a subset of [y, y + δ] ⊂ [Γ(x ∧ x′),Γ(x)], where f is

increasing OR

(b) Γ(x ∧ x′) ≤ Γ(x) and there exists a subset of [y, y + δ] ⊂ [Γ(x ∧ x′),Γ(x)], where f is

decreasing OR BOTH,

(c) Γ(x∧ x′) ≤ Γ(x) and φ(x)−φ(x∧ x′) = δ′+ δ′′, where f is increasing on [φ(x∧ x′), φ(x∧

x′) + δ′] and f is decreasing on [φ(x)− δ′′, φ(x)].

By theorem point 1, cases (a) or (b) imply that ∃[y′, y′+δ] ⊂ [Γ(x′),Γ(x ∨ x′)] on which f is

increasing or ∃[y′, y′+δ] ⊂ [Γ(x′),Γ(x ∨ x′)] on which f is decreasing. Any of those imply that

φ(x′) ≤ y′ and φ(x ∨ x′) ≥ y′ + δ. This implies that

φ(x ∨ x′)− φ(x′) ≥ y′ + δ − y′ = δ = φ(x)− φ(x ∧ x′).

By theorem point 1, case (c) implies that ∃[y′, y′+δ′] ⊂ [Γ(x′),Γ(x ∨ x′)] on which f is

increasing AND ∃[y′′, y′′+δ′′] ⊂ [Γ(x′),Γ(x ∨ x′)] on which f is decreasing. This can be only

possible if y′′ ≥ y′ + δ′. This implies that φ(x′) ≤ y′ and φ(x ∨ x′) ≥ y′′ + δ′′. As a result:

φ(x ∨ x′)− φ(x′) ≥ y′′ + δ′′ − y′ ≥ δ′ + δ′′ = φ(x)− φ(x ∧ x′).

hence the result follows.

Case 3. φ(x′) ≥ φ(x∨x′) holds similarly (with three possibilities) by point 2 of the theorem.

Obviously supermodular correspondence must have a spm selection. To verify some

other ppropoerties consuider the following example:

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Example 2. Consider f(y) = −(y−2)2 but assumptions of the theorem are not satisfiedfor: Γ(x ∧ x′) = [0, 1], Γ(x′) = [0, 2], Γ(x) = [0, 3], Γ(x ∨ x) = [0, 4], φ∗(x ∧ x′) = 1,φ∗(x) = φ∗(x′) = φ∗(x ∨ x′) = 2 even though constraints boundaries are supermodular.To fix that one needs to have Γ(x ∧ x′) = [0, 2] or Γ(x′) = [0, 1]. In fact: constraintsboundaries need not be supermodular for spm of the argmax. To see that consider:

Γ(x ∧ x′) = [0, 1], Γ(x′) = [1.5, 2], Γ(x) = [2, 3], Γ(x ∨ x′) = [3, 3.5], φ∗(x ∧ x′) = 1,φ∗(x) = φ∗(x′) = 2 and φ∗(x ∨ x′) = 3.

6 Some extensions

6.1 Multidimensional choice variable

We first consider extensions to multidimensional choice variables. For this reason consider:

Definition 1 (SPM Rn valued function). f : X → Rn is a supermodular function iff forany x, x′ ∈ X we have for i = 1, . . . , n:

fi(x ∧ x′) + fi(x ∨ x′) ≥ fi(x) + fi(x′).

Let X be a lattice. Consider f : Rn ×X → R, Function f(·, x) is single-peaked overRn, for any x ∈ X if there is some y∗ ∈ Rn such that:

f(y1, x) < f(y2, x), whenever

y1 ≤ y2 ≤ y∗ with y1 6= y2, or

y∗ ≤ y2 ≤ y1 with y1 6= y2.

Definition 2 (Multidimensional QSD). Function f : Rn ×X → R satisfies QSD iff fori = 1, . . . , n function fi : R×X → R is satisfies QSD.

Theorem 6 (Multidimensional SCS). Suppose that function f : Rn ×X → R is single-peaked on Rn. Function f obeys QSD if and only if function φ : X → Rn, φ(x) :=argmaxy∈Rn f(y, x) is supermodular.

6.2 Weakly single peaked functions

Consider a class of weak single peaked functions f : R × X → R, such that for eachx ∈ X there exists some y∗ ∈ R we have: f(·, x) is weakly increasing on (−∞, y∗] and

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weakly decreasing on [y∗,∞). Observe y∗(x) need not be unique. This implies thatcorrespondence Φ : X ⇒ R, given by

Φ(x) := arg maxy∈R

f(y, x)

is non-empty valued. By Φ and Φ denote the greatest and least elements of the closure

of Φ. We now define a notion of quasisupermodular differences for weakly single-peaked

functions (WQSD).

Definition 3. Function f satisfies WQSD if, for any numbers y ∈ R and δ ∈ R+, aswell as x, x′ ∈ X we have

(i) if f(·, x∧x′) weakly decreases, while f(·, x) weakly increases on [y, y+δ], then thereis a y′ ∈ R such thatf(·, x′) weakly decreases, while f(·, x ∨ x′) weakly increases on [y′, y′ + δ];

(ii) if f(·, x∨x′) weakly decreases, while f(·, x′) weakly increases on [y, y+δ], then thereis a y′ ∈ R such thatf(·, x) weakly decreases, while f(·, x ∧ x′) weakly increases on [y′, y′ + δ];

Theorem 7 (Supermodular Comparative Statics). Suppose that function f : R×X → Ris weak single-peaked on R. Function f obeys WQSD if and only if Φ(x) := arg maxy∈R f(y, x)

satisfies:

max{Φ(x ∨ x′) + Φ(x ∧ x′); Φ(x ∧ x′) + Φ(x ∨ x′)} ≥

min{Φ(x) + Φ(x′); Φ(x′) + Φ(x)}.

Note: Φ has a spm selection (we can construct it). And if Φ(x) is single valued then

Φ is spm.

Proof. Case 1. Suppose that Φ(x) < Φ(x∧x′) and Φ(x′) < Φ(x∨x′) then the result holdstrivially. Case 2. Suppose that Φ(x) ≥ Φ(x ∧ x′). Define δ := Φ(x)−Φ(x ∧ x′) and y :=Φ(x∧x′). Then f(·, x∧x′) is weakly decreasing on [y, y+δ]. As y+δ = Φ(x), then f(·, x)weakly increases over [y, y+δ]. Property WQSD(i) implies that there is some y′ ∈ R suchthat function f(·, x′) weakly decreases, while f(·, x ∨ x′) weakly increases on [y′, y′ + δ].Hence Φ(x′) ≤ y′ and Φ(x ∨ x′) ≥ y′ + δ. In particular, we have Φ(x ∨ x′) − φ(x′) ≥ δ.Therefore,

Φ(x)− Φ(x ∧ x′) = δ ≤ Φ(x ∨ x′)− Φ(x′).

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Case 3. Similarly consider Φ(x′) ≥ Φ(x ∨ x′) to obtain

Φ(x′)− Φ(x ∨ x′) ≤ Φ(x ∧ x′)− Φ(x).

Hence at least one of the inequalities must be true:

Φ(x ∨ x′) + Φ(x ∧ x′) ≥ min{Φ(x) + Φ(x′); Φ(x′) + Φ(x)}.

and

Φ(x ∨ x′) + Φ(x ∧ x′) ≥ min{Φ(x) + Φ(x′); Φ(x′) + Φ(x)}.

This implies the result.

6.3 Beyond a class of single peaked functions

Now consider a class of functions f : R×X → R, such that for each x ∈ X there existsγ(x) ≤ γ(x), such that f(·, x) is increasing on (−∞, γ(x)] and decreasing on [γ(x),∞).Take Γ(x),Γ(x) such that these intervals are the largest (in the set inclusion order).

Assume f(·, x) has a maximizer on R. Clearly Γ(x) ≤ φ(x) ≤ Γ(x) for any argmaxφ(x) ∈ Φ(x) := arg maxy∈R f(y, x). By [Φ(x),Φ(x)] ⊂ R denote the smallest (in the setinclusion order) interval containing Φ(x).

Theorem 8 (General case). For a class of functions defined above suppose the following

condition holds

(∗) Γ(x ∨ x′) + Γ(x ∧ x′) ≥ Γ(x) + Γ(x′) then

(∗∗) min{Φ(x ∨ x′) + Φ(x ∧ x′); Φ(x ∧ x′) + Φ(x ∨ x′)} ≥

max{Φ(x) + Φ(x′); Φ(x′) + Φ(x)}.

Few important observations follow:

• Condition (∗) implies that extremal selections Γ and Γ are spm

• Condition (∗∗) implies that Φ has a spm selection (we can construct it).

• If Φ(x) is singled valued then Φ is spm.

Proof.

Φ(x ∧ x′)− Φ(x) ≥ Γ(x ∧ x′)− Γ(x) ≥ Γ(x′)− Γ(x ∨ x′) ≥ Φ(x′)− Φ(x ∨ x′).

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Similarly we can derive that

Φ(x ∨ x′)− Φ(x′) ≥ Γ(x ∨ x′)− Γ(x′) ≥ Γ(x)− Γ(x ∧ x′) ≥ Φ(x)− Φ(x ∧ x′).

This implies the result.

6.4 Sufficient conditions for constrained optimization SCS

Consider a class of functions f : R → R, such that for each x ∈ X there exists amaximizer on the interval [Γ(x),Γ(x)]. clearly Γ(x) ≤ φ(x) ≤ Γ(x) for any argmaxφ(x) ∈ Φ(x) := arg maxy∈[Γ(x),Γ(x)] f(y). By [Φ(x),Φ(x)] ⊂ R denote the smallest (in theset inclusion order) interval containing Φ(x).

Theorem 9. Sufficient conditions For a class of functions defined above suppose the

following (∗) condition holds

Γ(x ∨ x′) + Γ(x ∧ x′) ≥ Γ(x) + Γ(x′) then

min{Φ(x ∨ x′) + Φ(x ∧ x′); Φ(x ∧ x′) + Φ(x ∨ x′)} ≥

max{Φ(x) + Φ(x′); Φ(x′) + Φ(x)}.

Again: this implies that: Γ and Γ are supermodular and in case Φ(x) is single valued

then Φ is spm. Compare notion of supermodular correspondence to the one introduced

by Dziewulski and Quah (2016).

7 Applications

Here we discuss three applications.

Stackelberg games Consider a two-period two-player Stackelberg games. In this one,

we seek for conditions for the strategy of the second player in the subgame perfect Nash

equilibrium be given by a supermodular function of the first players actions. Again, this

has implications on the properties of the first player payoff function and hence structure

of the subgame perfect Nash equilibrium set.

Specifically, consider a two stage game, where in the first period player 1 chooses

x ∈ X, while in the second stage player 2 chooses y ∈ Y . Suppose payoffs are givenby Π1(x, y) and Π2(x, y). We look for a subgame perfect Nash equilibrium. Suppose

Π2 : X × Y → R satisfies assumptions of proposition 2, then the best response mapping

16

y∗ : X → R is increasing and supermodular. Next consider x∗ = arg maxx∈X Π1(x, y∗(x)).We seek conditions for x→ H(x) := Π1(x, y∗(x)) to be a supermodular function on latticeX. This is true, if Π′y ≥ 0, Π′′ij ≥ 0, Π′′iy ≥ 0, Π′′yy ≥ 0.

For a specific application consider a two stage game, where in the first there are two

market liders simultaneously choosing prices of two goods pL1, pL2, while in the second

period a follower chooses price of its good p. We seek conditions on demands of both

leaders DL1, DL2 and the second period good Df so that goods L1 and L2 are strategic

complements in the first period on the subgame perfect Nash equilibrium path. Suppose

profit of the follower is given by Πf (p, pL1, pL2) = (p− cf )Df (p, pL1, pL2) and similarly forΠLi(p, pL1, pL2) = (pLi − cLi)DLi(p, pL1, pL2). Taking log (as strictly monotone transfor-mation preserves our ∗−property) the sufficient assumptions on Df to have p∗ increasingand supermodular are (logDf )

′′1i ≥ 0, (logDf )

′′′11i ≥ 0, (logDf )

′′′123 ≥ 0. Next to obtain

strategic complements in the first period game with payoffs ΠLi(p∗(pL1, pL2), pL1, pL2) we

have: (logDLi)′1 ≥ 0, (logDLi)

′′11 ≥ 0, (logDLi)

′′ij ≥ 0 for i 6= j (again strictly monotone

transformation preserves single-crossing differences).

Measuring interdependence First, consider a decision maker can condition his action

on the realization of a random variable. Specifically, he receives information through a

signal x drawn from a distribution µ, and chooses an action y ∈ R so as to maximizehis payoff function f(y, x). We then consider a question, when the expected decision

(with respect to distribution µ) depends on shifts in distribution µ with respect to su-

permodular order. This is an important question, as the stochastic supermodular order

captures the complementarities between various dimensions of the random variable. Re-

call that precisely such order it often used to measure multidimensional interdependencies

(see Meyer and Strulovici (2015)) e.g. for measuring systemic risk or income inequality

between countries.

Specifically, we say that that two measures µ2 and µ1 (i.e. multivariate distributions

on X with the same number of variables and finite support) are ordered with respect

to supermodular order (µ2 � µ1), whenever for any supermodular function on a latticef : X → R we have

∫Xf(x)µ2(dx) ≥

∫Xf(x)µ1(dx). Hence, a supermodular order

can be use to measure interdependencies that go beyond correlation. When measuring

interdependence using supermodular objective functions, we treat their variables as com-

plements, and their expectation increases as the realizations of the variables become more

aligned. Our theorem gives conditions that allow to obtain supermodular f that is used

in the integration as an argmax of some optimization problems.

17

Applications to dynamic games. Finally, we consider a multiplayer dynamic games,

where each period a number of players takes actions. We then provide conditions under

which a subgame perfect Nash equilibrium is given by supermodular functions of games

parameters, states in stochastic games or players own signals. This has important conse-

quences for the preservations of supermodularities between period in perfect equilibria in

dynamic games and hence yields conditions for a dynamic game to he supermodular in

its extensive form (see also Echenique (2004)). See also examples stated by Vives (2009).

To illustrate this example consider a two-stage two player game. Payoffs of player i

in period t are given by πti(x, y), where x is a state vector and y = (y1, y2) is an action

vector. We first seek conditions for the extremal (greatest and least) Nash equilibria in

the second stage supermodular game to be given by y∗1(x), y∗2(x) and y

∗1(x), y∗

2(x) i.e. by

a pair of supermodular functions. Suppose yi ∈ [y, y] ⊂ R and x ∈ Rl. Now considera function gi(yi; t) = π

ti(x, y) where t = (y−i, x). Under assumptions of our theorem 2

imposed on function gi, where yi is a choice variable, t is a parameter, we obtain that

(y−i, x) → BRi(y−i, x) is increasing and supermodular function. If πti is also continuouswith y, then (by Berge maximum theorem) BRi is also continuous. Clearly, in such

case extremal Nash equilibria of the supermodular game are limits of iterations on the

BR operator starting from the greatest and least elements of the strategy space (y, y)

and (y, y). Next to preserve supermodularity in the iterations (composition) of the BRn

one needs to assure that y−i → BRi(y−i, x) is also convex. For this reason we usecorollary 1 from Jensen (2018) applied to a function (yi, y−i) → πti(x, (yi, y−i)). Theseconditions together guarantee that BR maps increasing and supermodular functions to

increasing and supermodular functions. As the pointwise limit preserves monotonicity and

supermodularity one assures that extremal Nash equilibria y∗1, y∗2 and y

∗1, y∗

2are increasing

and supermodular. Next, we can define values V ti (x) = maxyi∈[y,y] πti(x, yi, y

∗−i(x)). Again,

whenever πti is supermodular, and is also increasing and convex with y−i, then Vti is

supermodular. This shows conditions under which supermodularity can be preserved via

the value function to the earlier periods. See Vives (2009) for the detailed examples.

References

Birkhoff, G. (1967): Lattice theory, vol. 25. American Mathematical Society Colo-

quium Publications, Providence, Rhode Island, 3 edn.

18

Chambers, C. P., and F. Echenique (2009): “Supermodularity and preferences,”

Journal of Economic Theory, 144(3), 1004–1014.

Dziewulski, P., and J. Quah (2016): “Supermodular correspondences,” Economics

Series Working Papers 795, University of Oxford, Department of Economics.

Echenique, F. (2004): “Extensive-form games and strategic complementarities,” Games

and Economic Behaviour, 46(348-354).

Jensen, M. K. (2018): “Distributional comparative statics,” The Review of Economic

Studies, 85(1), 581–610.

Meyer, M., and B. Strulovici (2015): “Beyond correlation: measuring interdepen-

dence through complementarities,” CSIO Working Paper 0134.

Milgrom, P., and C. Shannon (1994): “Monotone comparative statics,” Economet-

rica, 62(1), 157–180.

Quah, J. K.-H. (2007): “The comparative statics of constrained optimization problems,”

Econometrica, 75(2), 401–431.

Topkis, D. M. (1978): “Minimazing a submodular function on a lattice,” Operations

Research, 26(2), 305–321.

(1998): Supermodularity and complementarity, Frontiers of economic research.

Princeton University Press.

Veinott (1992): Lattice programming: qualitative optimization and equilibria. MS

Standford.

Vives, X. (2009): “Strategic complementarity in multi-stage games,” Economic Theory,

40(1), 151–171.

19

IntroductionPreliminaries on lattices and supermodularityFirst order approach

Ordinal conditions and the main theoremCardinal conditions and characterizationDifferentiable characterizationConstrained caseSome extensionsMultidimensional choice variableWeakly single peaked functionsBeyond a class of single peaked functionsSufficient conditions for constrained optimization SCS

ApplicationsReferences