Simultaneous Choice Models:
The Sandwich Approach to Nonparametric Analysis
Natalia Lazzatiy
November 09, 2013
Abstract
We study collective choice models from a revealed preference approach given limited data.
By limited data we mean that we observe a single equilibrium from the equilibrium set of
a …nite collection of related models. The objective of our analysis is twofold: We …rst use
the data to provide out-of-sample predictions of equilibrium points via a novel use of the
monotone comparative statics method. We then propose conditions on the data so that
the empirical evidence can be rationalized as the Nash equilibria of a supermodular game.
Our approach is nonparametric and does not rely on any equilibrium selection mechanism.
It only depends on monotone and exclusion restrictions.
JEL codes: C31, D01
Keywords: Simultaneous choice models, supermodular games, partial identi…cation,
out-of-sample predictions, monotone comparative statics.
Natalia Lazzati, Department of Economics, University of Michigan, Ann Arbor (e-mail: nlazzati@umich.edu).
y
I thank Dan Ackerberg, Amilcar Menichini, and John Quah for very useful comments. I also thank
participants at the economics seminars of Cornell U, U of Pittsburgh and CMU, and Binghamton U.
1
1
Introduction
Much of the empirical content of economics lies in the comparative statics
predictions it generates [Kreps (2013)].
The celebrated monotone comparative statics (MCS) for games is a powerful yet simple
method to contrast equilibrium points for an indexed family of models as we vary the parameter of interest.1 From a theoretical perspective, MCS has proved very useful. There are
at least two plausible explanations for its success. First, it works under conditions that are
often easy to justify on economic grounds. Second, it allows us to compare equilibrium points
in games with multiple solutions. From an empirical perspective, on the other hand, it has not
received much attention. One of the reasons is that, with partial observability of equilibrium
points, the standard predictions of MCS are neither testable nor useful for counterfactual
analysis unless we impose speci…c equilibrium selection rules on the data-generating process.
One may conclude too readily that, in contrast to our initial statement, MCS is of little use
in the empirical analysis of games.
We give a new look at the importance of MCS for counterfactual predictions in the context
of simultaneous choice models. The idea is simple: There is a set of agents that receive
treatments and make interdependent decisions. We think of a treatment in a very general way.
For instance, if individual behavior derives from an underlying game, then the treatment may
a¤ect the payo¤ function of the player and/or the collection of feasible strategies. Observable
choices are assumed to be consistent with equilibrium behavior. We have limited data on
past choices for various realized treatments. The data is limited in that, for each realized
treatment, we observe only one equilibrium from the equilibrium set. The objective of our
analysis is twofold: We …rst want to predict the actions that could be taken by the group
members if they were to receive a treatment that di¤ers from the ones we observe in the data.
We refer to this set of choices as either the counterfactual or the out-of-sample predictions.
Our second objective is to address whether our limited data can be rationalized as the Nash
equilibria of an underlying supermodular game among the agents.
The approach we propose to achieve our goals is nonparametric and does not impose any
equilibrium selection mechanism on the data-generating process. It alternatively relies on
1
See, e.g., Milgrom and Roberts (1990), Milgrom and Shannon (1994), and Vives (1990).
2
exclusion restrictions at the individual level and monotone shape conditions on the primitives
of the model (i.e., the choice functions). An advantage of our assumptions is that they are
indeed testable. That is, they can be proved false given the data. By exclusion restrictions
we mean that individual treatments may a¤ect the choice of one agent without having a
direct impact on the choice functions of the other agents in the group. For instance, in a
model of demand with network e¤ects, income levels are natural candidates for the exclusion
restrictions. The reason is that the income level of a given person only conditions the budget
set of that particular individual. In the same model, prices may or may not play the role of
exclusion restriction depending on whether we allow for price discrimination. This restriction
allows us to track the choice function of each agent for any action pro…le of the others. We
elaborate next on the monotone assumptions we sustain.
The primitives of our model are a system of interdependent choice functions that relate
the choice of each individual in a group to the treatment and the decisions of the other group
members. The two monotone conditions we impose are as follows: the choice function of
each individual increases with the action pro…le of the others; and it varies monotonically
with the treatment to be received by the agent. These conditions imply clear restrictions
on the equilibrium set: The system of structural equations leads to an increasing function
that maps feasible choices into itself, so that the set of solutions of the model coincides
with the set of …xed points of this arti…cial function. By Tarski’s …xed point theorem, the
…rst condition guarantees the system has a minimal and a maximal solution, i.e., the model
is coherent. The second restriction shifts the function up or down, inducing the extremal
solutions to vary monotonically with the treatments. This is the celebrated MCS of equilibrium points for games. Since this comparative statics result only refers to the extremal
equilibria, its implications regarding equilibrium behavior are neither testable nor useful for
making counterfactual predictions unless we impose speci…c equilibrium selection rules on the
data-generating process, e.g., coordination on extremal equilibria. As we described above,
we develop this model with two objectives in mind. We …rst show that while the standard
implications of the MCS method are uninformative regarding counterfactual predictions, the
two monotone conditions that are behind this result are yet extremely powerful. Speci…cally,
when combined with the exclusion restrictions we mentioned above, they allow us to construct
sharp bounds for individual choice functions given the empirical evidence. By sharp we mean
3
that the true set of choice functions must lie between the bounds we propose, and that we
cannot reject the hypothesis that the true set actually coincides with one or the other. We
then use these extremal choice functions to bound equilibrium behavior by an approach that
relies on a novel use of the MCS method. This result addresses our initial objective of analysis,
namely, deriving out-of-sample predictions of equilibrium points. We refer to the latter
as the sandwich approach to the out-of-sample predictions.
The second objective of our analysis is to show that, if the data satis…es certain monotone
conditions, then the empirical evidence can always be rationalized as the Nash equilibria of
a supermodular game. (Thus, the monotone restrictions we derive capture all the empirical
content of the supermodular games.) To achieve our goal we show that any set of monotone
choice functions can be rationalized as a set of best-reply functions of a supermodular game
among the agents. The proof follows by construction. That is, we construct a set of payo¤
functions, one for each agent, that have the choice functions as the corresponding maximizers.
Because the consistent set of choice functions is multivalued and has extremal elements, we
construct extremal supermodular games that are consistent with the data. These games di¤er
regarding the strength of the interaction e¤ects: The lowest set of consistent choice functions
is associated with the supermodular game with the weakest interaction e¤ects. The opposite
is true regarding the highest set. We refer to this result as the sandwich approach to
the supermodular rationalization.
We …nally extend the idea to allow for multivalued choice functions (or choice correspondences) and discuss some other relevant features of our model.
Literature Review
This study is intimately related to the theoretical literature of supermodular games [see, e.g.,
Milgrom and Roberts (1990), Milgrom and Shannon (1994), Vives (1990), and Topkis (1979)].
We contribute to this work by elaborating on the empirical content of monotone comparative
statics for equilibrium points [see also Aradillas-López (2011), De Paula (2009), Jia (2008),
and Molinari and Rosen (2008)]. Our work also relates to some recent studies on the testable
implications of collective choice models [see, e.g., Browning and Chiappori (1998), Cherchye
et al. (2013), Carbajal et al. (2013-a), Chambers et al. (2010), Deb et al. (2012), and
4
Sprumont (2000)]. We emphasize on the advantages of both exclusion and monotone shape
restrictions to recover the objects of interest. Varian (1982) and De Clippel and Rozen (2013)
provide a nonparametric analysis of individualistic models of rational behavior and bounded
rationality, respectively. We extend their results to models of interdependent decisions. As
we mentioned above, we construct a family of supermodular games that rationalize the set of
choice functions that are consistent with both our assumptions and the data. The family we
construct is bounded from below and above by two games that di¤er regarding the strength of
the interaction e¤ects. This result relates to the approach proposed by Amir (2008) to obtain
comparative statics analysis of equilibrium points in games that may not be supermodular.
Echenique and Komunjer (2009) provide a test for MCS. They show that, under a general
stochastic equilibrium selection rule that places positive probability on the extremal equilibria,
MCS has observable implications on the extremal quantiles of the distribution of equilibrium
points. Their result applies to models of individual behavior with multiple maximizers and
certain games with multiple equilibria. Along the study, we show that, while the implications of MCS in terms of equilibrium behavior for games are non-testable unless we impose
some structure on the equilibrium selection rule, its underlying assumptions can be proved
false given the data without such a restriction. This result opens new avenues for testing
complementarities under Nash-behavior or for testing Nash-behavior in games of strategic
complements (see also Carbajal et al. (2013-b)).
The econometric literature has recently emphasized on the relevance of monotone restrictions to provide identi…cation results. In this line of research, Kline and Tamer (2012),
Lazzati (2013), and Manski (1997, 2011) take advantage of monotone conditions to obtain
non-parametric partial identi…cation results of di¤erent counterfactuals. Our closest precedent is Lazzati (2013), who studies identi…cation in a model of treatment response with social
interactions. A key di¤erence between the present study and the former is that, in order to
bound equilibrium behavior, we …rst recover individual choice functions. This intermediate
step allows us to dispense with any assumption about equilibrium selection but it requires
exclusion restrictions at the level of each agent. Thus, the two studies complement each other.
Uetake and Watanabe (2013) also exploit MCS to provide estimation results for supermodular
games. Their approach is parametric and the bounds they obtain are (generally) not sharp.
The rest of the paper is organized as follows. Section 2 presents the model and the
5
objective of our analysis. Section 3 shows why the standard implications of the MCS method
are not useful for counterfactual predictions and proposes an alternative approach to tackle
this di¢ culty. Section 4 uses the previous results to achieve our twofold objective, namely, outof-sample prediction of equilibrium points and rationalization of the data by a supermodular
game. Section 5 includes some extensions and remarks, and Section 6 concludes. All proofs
are collected in Section 7.
2
The Model
This section presents the model and elaborates on the objective of our analysis.
We consider a …nite set of agents N = f1; 2; :::; ng that make interdependent decisions.
Agent i 2 N selects an element ai from her set of available options Ai ( i )
Ai : We describe
her behavior by the choice function
Ci ( i ; a i ) : T i
where a
i
A
i
! Ai
with
Ci ( i ; a i ) 2 Ai ( i )
= (aj : j 2 N; j 6= i) is the vector of other agents’choices and A
We refer to
i
i
=
j2N;j6=i Aj .
2 Ti as the treatment received by the agent. Thus, the choice function of agent
i indicates the action that she selects for each possible treatment and pro…le of actions of the
other agents in N. This last e¤ect is the source of interdependence in our model.
If the choice functions derive from an underlying game, then
Ci ( i ; a i ) = arg maxai fUi (ai ; a i ;
i)
: ai 2 Ai ( i )g
(1)
where Ui is the payo¤ function of agent i and Ci can be interpreted as her best-reply function.
In (1), the treatment a¤ects the choice of the agent by modifying both her payo¤ function
and her set of available options.
Let V = (Ci : i 2 N). We refer to V as the microeconomic speci…cation of the model. Thus,
our model can be described by the tuple
= (N; V; ) where
=(
i
: i 2 N) 2 T =
i2N Ti
is a treatment vector. We next introduce our solution concept.
De…nition (Solution Concept): We say
rium behavior generated by
i(
( )=(
i(
) : i 2 N) is consistent with equilib-
if, for all i 2 N,
) = Ci ( i ; a i )
with a
6
i
=
j
( ) : j 2 N; j 6= i :
We indicate by
:T
That is, we say
i2N Ai ,
with
i2N Ai ( i ),
the equilibrium correspondence:
( ) is consistent with equilibrium behavior if each agent selects an option
according to both her choice function and the equilibrium behavior of the other agents. In
the context of a game, our solution concept is a Nash equilibrium in pure strategies.
We are now ready to describe the objective of our analysis.
Objective of Analysis: We have limited data on realized choices for a set of treatments
am = (am
i : i 2 N) and
m
=(
m
i
: i 2 N) for m = 1; :::; M:
We assume these observations are consistent with equilibrium behavior. Our data is limited
because for each realized treatment we only observe one element of the equilibrium set, i.e.,
the equilibrium selected by the group. Our …rst goal is to learn about the actions that
could be taken by this set of agents if they were to receive a treatment . In other words, we
observe an equilibrium outcome
(
m)
2
m
for a set of realized treatments (
and want to use this information to learn about the equilibrium set
m
:m
M)
for an alternative .
Our second goal is to provide conditions on the data so that the empirical evidence can be
rationalized as the Nash equilibria of an underlying game. This is the same as to ask whether
we can …nd a set of payo¤ functions P = (Ui : i 2 N), with each Ui given as in (1), that can
generate the data as the Nash equilibria of the induced game.
The approach we propose to achieve our goals relies on monotone conditions on the microeconomic speci…cation of our model (V) and exclusion restrictions at the individual level.
The next example illustrates the concepts and ideas that we just described and motivates our
main assumptions.
Example 1: Let us consider a demand model with network e¤ects. The choice function
of each agent indicates the amount of the network good, X, that the person acquires. The
treatment is the price the individual has to pay per unit of the good and her income level.
Thus,
i
=
p; mi and Ai ( i ) = [0; mi =p] : (We write
p instead of p to ful…ll our monotone
restrictions.) We have that
Ci (p; mi ; x i ) : R++
R+
n
R+
1
! R+ with Ci (p; mi ; x i ) 2 [0; mi =p]
7
where x
i
= (xj : j 2 N; j 6= i) is the amount of the network good acquired by the others. In
this application, Ci represents the demand function of consumer i. An equilibrium
(p; m),
with m = (mi : i 2 N), is a vector in Rn+ that indicates the amount of the network good that
has been acquired by each person in the economy.
We have limited data on past consumption decisions for a set of di¤erent prices and income
levels. We want to use the empirical evidence to predict market demand for a vector of prices
and income levels that di¤er from the ones we observe in the data. The method we propose
relies on monotone restrictions on V. When applied to this model, it requires the decision
of each consumer to be increasing in the choices of the others and her income level, and
decreasing with respect to the price of the good. Our approach also depends on exclusion
restrictions. Since the income level of consumer i a¤ects her demand function without having
a direct impact on the choices of the others, income levels are natural exclusion restrictions in
this application. These exclusion restrictions allow us to track the demand function of each
consumer as a function of the choices of other consumers in the economy.
The method we propose allows for price discrimination. That is, we could let prices
depend on the names of the agents by substituting p by pi : In this context, the counterfactual
predictions we derive could be used to evaluate target marketing strategies that aim to identify
the in‡uential consumers, that is, the set of consumers that have a large impact on the choices
of other consumers in the economy (see, e.g., Candogan et al. (2012)).
The next section introduces our required conditions and presents a few propositions that
will allow us to make out-of-sample predictions. It also connects our assumptions with the
so-called supermodular games.
3
Another Look at Monotone Comparative Statics
Simultaneous choice models may have no equilibrium. Lack of equilibrium existence means
that we cannot use our solution concept to predict consistent behavior. From an econometrics
perspective this possibility introduces additional di¢ culties into the analysis. For instance,
data on past choices may not be informative about the beliefs of each agent with respect
to the behavior of the others and it is therefore hard to recover their choice functions. To
address this di¢ culty, we …rst impose simple conditions on V that guarantee the model has
8
at least one equilibrium.2 We then provide comparative statics results and elaborate on our
strategy for making out-of-sample predictions given the limited data. We …nally connect all
our assumptions with the class of supermodular games.
With a slight abuse of notation, we use
to compare elements of both the action and the
treatment spaces. When applied to either Rm or Zm ,
is simply the standard coordinatewise
order on the reals. Along the analysis, Ti is a partially ordered set. When applied to the
treatments, the meaning of
depends on the speci…c application.
Our …rst condition imposes some structure on the choice sets.
(A1) For all i 2 N, Ai ( i ) and Ai are non-empty compact intervals of either Rm or Zm and
ai ( i ) = inf (Ai ( i )) and ai ( i ) = sup (Ai ( i )) increase in
i:
The monotonicity of the extremal elements of the choice sets is a natural restriction given
the nature of the approach we propose for the nonparametric analysis. In Example 1
Ai ( i ) = [0; mi =p]
is a non-empty compact interval in R and ai ( i ) = 0 and ai ( i ) = mi =p increase in
i
=
p; mi : Thus, this model satis…es A1.
The next condition requires the choice function of each agent to be increasing in the action
pro…le of the other agents.
(A2) For all i 2 N, Ci ( i ; a i ) increases in a i :
Under condition A2, our model displays positive interaction e¤ects. In Example 1, cellphones and software are goods for which researchers and practitioners often believe this restriction is naturally satis…ed. The next result states that, if A1 and A2 hold, then the solution
set is non-empty and has extremal elements for each treatment level.
Proposition 1 Under A1 and A2,
extremal equilibria,
2
is a non-empty complete lattice. Thus, there exist
and , such that any
of
satis…es
( )
( ):
Chesher and Rosen (2012) provide identi…cation results for models that have no solution for some covariate
values.
9
We next provide a sketch of the proof for two agents. The proof of Proposition 1 relies on
a mapping M : A1
A2 ! A1
A2 , de…ned as follows
M (a1 ; a2 ) = (C1 ( 1 ; a2 ) ; C2 ( 2 ; a1 )) :
(2)
Function M maps pairs of actions into itself. Given our solution concept, any …xed point of
M corresponds to a pro…le of actions
consistent with equilibrium behavior generated by
: Thus, the proof of Proposition 1 consists in showing that the set of …xed points of M is
a non-empty complete lattice. Under A2, the result follows by Tarski’s …xed point theorem.
Though important, the existence result we just described does not impose enough structure
on the model in order to make the empirical evidence informative. To do so, we need to connect
the alternative models in such a way that we can use the equilibrium behavior revealed by the
data for a set of treatments to learn about the equilibrium set corresponding to a di¤erent
treatment level. We next add a second restriction on V that establishes this connection.
(A3) For all i 2 N, Ci ( i ; a i ) increases in
i:
When applied to Example 1, this condition requires the demand function of consumer i
to be increasing in the income level and decreasing in the market price. Note that we can
always reverse the order of some of the elements in the treatment to satisfy this condition.
We just need to be consistent with A1.
The next proposition states that, under (A1, A2, and A3), we can compare the extremal
elements of the equilibrium sets for every pair of ordered treatment vectors.
Proposition 2 (MCS) Under A1, A2, and A3,
( ) and
( ) increase in :
Assumption A3 shifts map M up (in equation (2)), inducing an increase in the extremal
solutions. Proposition 2 is the standard MCS result of equilibrium points.
The next example serves various purposes: First, it helps us to illustrate our claim in the
introduction. That is, we show that the MCS result, in and of itself, is not useful for counterfactual analysis with limited data. We then show that its underlying assumptions— namely,
A2 and A3— are nevertheless extremely powerful for deriving out-of-sample predictions. We
…nally use the example to highlight the role of the exclusion restrictions in our identi…cation
analysis.
10
Example 2: Let N = f1; 2g, T1 = T2 = f ; g (with
), and Ai = fai ; ai g (with ai
ai )
for i = 1; 2: The data we have and the problem we face are described by the next tables
Data
Out-of-Sample Prediction
a2
1
= ;
2
=
a2
a2
a1
1
= ;
2
=
a1
1
= ;
2
=
a2
a1
a1
a2
a2
a2
a2
a1
a1
1
= ;
2
=
a1
a1
That is, we know the options selected by these two agents under three di¤erent treatments: The
selected pro…les are indicated by dots in the tables on the left. We also know that these data
were generated by equilibrium behavior. Our goal is to describe all possible equilibrium pro…les
for the low treatment levels that are consistent with the empirical evidence and satisfy our
initial assumptions. We will write X for consistent pro…les, and reserve x for the inconsistent
ones.
It is readily veri…ed that the MCS result, in and of itself, does not allow us to rule out
any action pro…le. That is, by only relying on this prediction, every single pair of choices is
consistent with equilibrium behavior given the available data.
Out-of-Sample Prediction with MCS
1
= ;
2
=
a2
a2
a1
X
X
a1
X
X
Nevertheless, if we know that A2 and A3 are satis…ed, then we can make a unique prediction.
The table below captures this observation.
11
Out-of-Sample Prediction with A2 and A3
1
We now elaborate on how we found
= ;
2
=
a2
a2
a1
X
x
a1
x
x
( ; ) under A2 and A3. We know the data corresponds
to equilibrium behavior. Thus, each data point allows us to recover an element of the choice
functions, e.g., C1 ( ; a2 ) = a1 . Once we recover all these elements we can invoke A2 and
A3 to infer the other ones. For instance, since C1 ( ; a2 ) = a1 , then, by A2, it must be that
C1 ( ; a2 ) = a1 . Proceeding in this way, there is only one element for which we do not have
any information, C2 ( ; a1 ) : For this element, all we can say is that C2 ( ; a1 ) 2 fa2 ; a2 g.
Thus, we …nd two sets of choice functions that are consistent with the data and satisfy A2
and A3, namely, U and W below.
U
W
=
=
0
@
0
@
B1 ( 1 ; a2 ) = a1 1 ( 1 ; a2 < ; a2 ) + a1 [1
1 ( 1 ; a2 < ; a2 )]
B2 ( 2 ; a1 ) = a2 1 ( 2 ; a1
1 ( 2 ; a1
; a1 ) + a2 [1
; a1 )]
D1 ( 1 ; a2 ) = a1 1 ( 1 ; a2 < ; a2 ) + a1 [1
1 ( 1 ; a2 < ; a2 )]
D2 ( 2 ; a1 ) = a2 1 ( 2 ; a1
1 ( 2 ; a1
; a1 ) + a2 [1
; a1 )]
1
A
1
A
We …nally compute all possible equilibria for U and W. These action pro…les constitute our
out-of-sample prediction. After doing so, we …nd only one equilibrium, namely, (a1 ; a2 ). In
summary, while the MCS result is neither testable nor informative for out-of-sample analysis,
the opposite is true regarding assumptions A2 and A3. This is the leading force of our
nonparametric analysis in the following section.
In this example, by exclusion restrictions we mean that the treatment level can take on
di¤erent values for the two individuals. This allowed us to track the choice function of each
agent for a given treatment and alternative choices of the other one. While the method we
propose for out-of-sample predictions can still be applied without these exclusion restrictions,
if they do not hold, then our bounds for equilibrium behavior would almost always be just the
trivial ones, i.e., the extremal elements of the set of available options. To illustrate this point,
12
note that, in absence of the exclusion restrictions, the available data and the corresponding
predictions would be as follows.
Data
Out-of-Sample Prediction with A2 and A3
a2
1
= ;
2
=
a2
a1
1
= ;
2
=
a1
a2
a2
a1
X
X
a1
X
X
Thus, even assuming that A2 and A3 hold, we cannot rule out any action pro…le.
Example 2 shows the power of (A1, A2, and A3) and the exclusion restrictions for making
counterfactual predictions. The problem with the approach we just described is that it is
non-tractable when we add more feasible options. Moreover, it does not work if the choice set
of each agent is of in…nite dimension, e.g., a compact interval on the real line as in Example
1. The key idea we want to convey is that, in our model, in order to provide sharp bounds
for equilibrium points we only need to recover the extremal elements of the consistent set
of choice functions, e.g., U and W above. We formalize this idea in the following section by
invoking a theoretical result that we present next. We will refer to this result as the sandwich
approach to the nonparametric analysis.
Let Bi and Di be two possible speci…cations for the choice function of agent i. Our last
proposition relies on the next condition.
(A4) For all i 2 N, Di ( i ; a i )
Ci ( i ; a i )
Bi ( i ; a i ) for all
i
and a i .
Let us indicate by
U = (Bi : i 2 N)
and
W = (Di : i 2 N)
two microeconomic speci…cations for the model. Let ;
and ;
be the extremal equilibria
corresponding to U and W, respectively, when such equilibria exist. The next proposition
states that if A1 holds and U, V, and W satisfy A2 and A4, then the extremal equilibria
associated to U and W bound every element of the equilibrium set corresponding to V.
Proposition 3 Assume A1 holds and U, V, and W satisfy A2 and A4. Then, for all
( )
( )
( )
and
13
( )
( )
( ):
2 T,
From a theoretical perspective, Proposition 3 is just a variant of Proposition 2.3 It says
that if we …nd two sets of monotone choice functions that sandwich the one we are interested
in, then the extremal equilibria of these two sets of functions bound any equilibrium behavior
that is consistent with the true one. We next o¤er a sketch of proof for the case of two agents.
In addition to M , we de…ne two other mappings
L (a1 ; a2 ) = (B1 ( 1 ; a2 ) ; B2 ( 2 ; a1 ))
and N (a1 ; a2 ) = (D1 ( 1 ; a2 ) ; D2 ( 2 ; a1 )) .
Under A4, we have that, for all (a1 ; a2 ) 2 A1
N (a1 ; a2 )
A2 ,
M (a1 ; a2 )
L (a1 ; a2 ) :
Functions L , M , and N map pairs of actions into themselves. Under A2, the three mappings
are increasing. Thus, the three of them have extremal …xed points and these extremal …xed
points respect the order of the mappings.
The next section invokes this result to provide out-of-sample predictions for equilibrium
points. To this end, we will show that, if the empirical evidence satis…es certain monotone
conditions, then we can always use the data to construct extremal sets of monotone choice
functions such as U and W in Example 2. The result will then follow as a simple corollary of
Proposition 3.
We next connect our modeling restrictions with the theoretical literature on supermodular
games.
Supermodular Games with Monotone Best-Reply Functions: Let P = (Ui : i 2 N)
be the set of payo¤ functions of all players in N. We assume that condition A1 holds.
When the choice functions result from an underlying game, then
Ci ( i ; a i ) = arg maxai fUi (ai ; a i ;
i)
: ai 2 Ai ( i )g
where Ui is the payo¤ function of agent i and Ci is her best-reply function. If Ui is strictly
concave in ai (on all Rm ); then Ci ( i ; a i ) is indeed non-empty and single-valued.4
3
4
Amir (2008) shows that a similar result can be obtained even if the intermediate mapping is not increasing.
If Ai Zm ; then the standard de…nition of strict concavity needs to be modi…ed as follows. We say Ui is
strictly concave in a if there exists an extention of Ui on the convex hull of Ai that is strictly concave.
14
We say Ui is supermodular in ai if, for all ai ; a0i 2 Ai ,
Ui ai ^ a0i ; a i ;
i
+ Ui ai _ a0i ; a i ;
i
Ui (ai ; a i ;
i)
In addition, Ui has increasing di¤erences in (ai ; a i ) if, for all ai
Ui (ai ; a i ;
Ui a0i ; a i ;
i)
Ui ai ; a0 i ;
i
i
+ Ui a0i ; a i ;
a0i and a
Ui a0i ; a0 i ;
:
a0 i ,
i
i
i
:
That is, if the extra payo¤ of increasing own action increases with the choices of the others.
When Ui is supermodular in ai and has increasing di¤erences in (ai ; a i ), then
Ci ( i ; a i ) increases in a
i
and A2 holds. These two conditions are the distinctive feature
of any supermodular game.
We say Ui has increasing di¤erences in (a;
Ui (ai ; a i ;
i)
Ui a0i ; a i ;
i
i)
a0i and
if, for all ai
Ui ai ; a i ;
0
i
0,
i
i
Ui a0i ; a i ;
0
i
:
That is, if the extra payo¤ of increasing own action increases with the treatment. If Ui
is supermodular in ai and has increasing di¤erences in (ai ;
increases in
i
i ),
then Ci ( i ; a i )
and A3 holds.
Thus, our initial assumptions are satis…ed by certain types of supermodular games with
single valued best-reply functions and our results can be thereby applied to them. We will
show later that the reverse is also true. That is, if the available data satisfy certain monotone
restrictions, then we can always construct a set of payo¤ functions that rationalize the empirical evidence as the Nash equilibria of a supermodular game with the above properties. We
will also show that our results apply to certain games with multi-valued best-replies.
We next illustrate our last concepts by using an extension of Example 1.
Example 3: Let X be the network good and let Y be a numerarie. Suppose that consumer
i selects X and Y by solving the standard optimization problem
maxx;y Ui (x; y; x i ) : x; y 2 R2+ and xp + y
mi :
Let Ui be strictly concave and strictly increasing in y. It follows that
Ci ( i ; x i ) = arg maxx fUi (x; mi
15
xp; x i ) : x 2 [0; mi =p]g :
Thus, we can think of Ci as the best-reply function of person i induced by a game among
potential consumers. In this context, the equilibrium demand is just a Nash equilibrium.
In this example, Ui has increasing di¤erences in (x; x i ) if, for all x
Ui (x; mi
xp; x i )
Ui x0 ; mi
xp; x
Ui x; mi
i
xp; x0
i
x0 and x
Ui x0 ; mi
i
xp; x0
x0 i ,
i
:
When Ui is continuously di¤erentiable, this condition holds if
MRSx;y = (@Ui (x; y; x i ) =@x) = (@Ui (x; y; x i ) =@y) increases in x i :
That is, if the marginal rate of substitutions between the network good and the numerarie
increases with the amount of the network good acquired by the other consumers. When this
condition is satis…ed, then A2 holds and the game among potential consumers is said to be
supermodular.
In addition, Ui has increasing di¤erences in (x; mi ; p) if, for all x
x0 and mi ; p
m0i ; p0 ,
Ui (x; mi
xp; x i )
Ui x0 ; mi
xp; x
Ui x; m0i
i
xp0 ; x
i
Ui x0 ; m0i
xp0 ; x
i
:
When Ui is continuously di¤erentiable, this condition holds if
@Ui (x; y; x i ) =@x increases in y:
That is, if the marginal utility with respect to the network good increases with the numerarie.
Finally, the constraint set is ascending in
of the interval [0; mi =p] increase in
i
=
i
=
p; mi in the sense that the extremal elements
p; mi . Thus, A1 holds. It follows that A3 is also
satis…ed.
The next section elaborates on the nonparametric analysis of our model: We …rst use our
previous results to provide out-of sample predictions of equilibrium points given the limited
data. We then address rationalization of the data by a supermodular game.
4
4.1
The Sandwich Approach to Nonparametric Analysis
Out-of-Sample Predictions
This section addresses our …rst goal. That is, we provide out-of-sample predictions of
equilibrium behavior for alternative treatments given our limited data. We do so without im16
posing functional form restrictions on the choice functions. Taking advantage of our previous
results, we follow what we call the sandwich approach to nonparametric analysis. Kline and
Tamer (2012), Lazzati (2013), and Manski (1997, 2011) exploit similar monotone conditions
to obtain nonparametric partial identi…cation results of counterfactuals in di¤erent models.
In what follows, we maintain the next restriction.
Identi…cation Assumption (IA): (A1, A2, and A3) hold and Ai ( i ) is known for all i 2 N.
Recall that we have limited data on realized choices for a set of treatments
am = (am
i : i 2 N) and
m
=(
By Proposition 1, IA guarantees that, for each
m
i
: i 2 N) for m = 1; :::; M:
2 T , there is at least one equilibrium. Since
the empirical evidence is consistent with equilibrium behavior, the data identi…es
for all m
(
m)
= am
M: Given our solution concept, for each i 2 N,
Ci
m m
i ;a i
= am
i for all m
Thus, Ci ( i ; a i ) is identi…ed by the data for all
i; a i
M:
m ; am
i
i
2
:m
M . This also
means that, if IA holds, then the data must satisfy the next monotone restriction.
(M) For all i 2 N and m; t
am
i
ati if
M,
m
; ami
t
t
; at i and am
i = ai if
m
; ami =
t
; at i :
We next elaborate on the other direction. That is, we show that if the data satis…es condition
M, then we can always …nd a set of monotone choice functions V = (Ci : i 2 N) such that
each Ci extends [Ci
m ; am
i
i
= am
i for all m
M ] to all other arguments. This is precisely
what we did in Example 2 to provide bounds for equilibrium behavior.
Functions Bi and Di below constitute the smallest and the largest speci…cations of Ci that,
being consistent with the data, satisfy conditions A2 and A3. In the following expressions,
inf and sup are taken with respect to Ai .
Bi ( i ; a i ) = inf ai : ai 2 Ai ( i ) ; ai
sup am
i :
m ; am
i
i
i; a i; m
M
Di ( i ; a i ) = sup ai : ai 2 Ai ( i ) ; ai
inf am
i :
m ; am
i
i
i; a i; m
M
The next proposition formalizes our claim.
17
Proposition 4 Given the data and IA, for all i 2 N,
Di ( i ; a i )
Ci ( i ; a i )
Bi ( i ; a i ) for all
i
and a i :
Moreover, the bounds are sharp.
A key step in the proof of Proposition 4 consists of showing that Bi and Di are indeed
monotone. We now use the previous result to provide bounds for
U = (Bi : i 2 N)
and let
and
and
. Let us de…ne
W = (Di : i 2 N)
indicate the smallest and the largest equilibrium corresponding to U and W,
respectively. Given Proposition 4, the next result follows as a corollary of Proposition 3.
Proposition 5 Given the data and IA; for all
( )
2T :
( ) for all
2
:
Moreover, the bounds are sharp.
We next summarize our main …ndings.
The Sandwich Approach to Out-of-Sample Predictions: If the empirical evidence
satis…es condition M, then, for any treatment , we can always construct two extremal sets of
choice functions such that the true one must lie between them and can actually coincide with
one or the other. The extremal equilibria of these two sets of functions bound any element of
the equilibrium set corresponding to the treatment of interest .
In the following section we show that if the data satis…es condition M, then the empirical
evidence can be always rationalized as the Nash equilibria of a supermodular game.
4.2
Rationalization by a Supermodular Game
This section addresses our second goal. That is, we show that if the data satisfy
condition M in the previous section, then we can always construct a supermodular game with
monotone best-replies that rationalizes the empirical evidence as its Nash equilibria.
18
In the last section, we showed that if M holds, then we can always construct extremal sets
of monotone choice functions U = (Bi : i 2 N) and W = (Di : i 2 N) such that the true set
of choice functions V = (Ci : i 2 N) lies between them and can actually coincide with one or
the other. To achieve our second goal we now show that for any given set of monotone choice
functions V there always exists a set of payo¤ functions, one for each agent, that has V as the
set of maximizers. Moreover, by construction, these payo¤ functions satisfy the conditions of
the supermodular games with monotone best-reply functions that we discussed at the end of
Section 3.
Before presenting the main proposition, we de…ne the concept of supermodular rationalization.
De…nition (Supermodular Rationalization): We say a set of payo¤ functions P = (Ui :
i 2 N ), with Ui : Ai
A
i
T ! Ai , rationalize V if, for all i 2 N,
Ci ( i ; a i ) = arg maxai fUi (ai ; a i ;
i)
: ai 2 Ai (ai )g :
If in addition, for all i 2 N, Ui is supermodular in ai and has increasing di¤erences in both
(ai ; a i ) and (ai ;
i ),
then we say P is a supermodular rationalization of V.
The next theorem formalizes our previous claim.
Proposition 6 If V satis…es A2 and A3, then there exists a supermodular rationalization P
of V.
The proof of this proposition follows by construction. That is, we construct a set of payo¤
functions P = (Ui : i 2 N ) that satisfy the properties of a supermodular game with monotone
best-replies and rationalize V. We elaborate next on this idea.
Let us de…ne, for all i 2 N,
Ui (ai ; a i ;
with
ij
:A
i
i)
=
Xm
j=1
(1=2) (aij )2 +
T ! R and aij 2 R for all j
Xm
j=1
ij
(a i ;
Zm
(3)
m (so that ai 2 Rm ): We next show that if Ci
is monotone, then we can always …nd a set of increasing functions
such that (3) rationalizes Ci : (If Ci : Ti
i ) aij
(n 1)
i
=(
:i
n; j
m)
! Zm , then we rede…ne Ci as the smallest
or the largest monotone extension of the initial choice function to all Rm
19
ij
(n 1) .)
The …rst-order condition for an interior maximizer is
@Ui (ai ; a i ;
i ) =@aij
=
aij +
ij
(a i ;
i)
= 0 for all j
Since Ui is strictly concave, this condition is also su¢ cient.
m:
If this agent were utility-
maximizer, then
Cij (a i ;
i)
+
ij
(a i ;
i)
= 0 for all j
where Cij is the jth coordinate of Ci : Thus, by letting
i (a i ; i )
m
= Ci (a i ;
i ),
we get that
P = (Ui : i 2 N ) rationalizes V. It is readily veri…ed that Ui is supermodular in ai . Since Ci
is increasing, then
(ai ; a i ) and (ai ;
The functions
i
i ).
is an increasing function. Thus, Ui has increasing di¤erences in both
It follows that P = (Ui : i 2 N ) is a supermodular rationalization of V.
’s in our construction capture the strength of the interaction e¤ects.
Moreover, we have that
@Ui (ai ; a i ;
i ) =@aij
=
aij +
Thus, Ui has increasing di¤erences in (ai ;
imizer of Ui increases in
i:
ij
(a i ;
i ).
i)
increases in
ij
for all j
m:
It follows, by Topkis’theorem, that the max-
Therefore, in our construction, the game that rationalizes U
= (Bi : i 2 N) has lower interaction e¤ects than the one that rationalizes W = (Di : i 2 N) :
This observation leads to our last statement.
The Sandwich Approach to the Supermodular Rationalization: If the empirical
evidence satis…es condition M, then we can always construct two extremal supermodular
games that rationalize the data and di¤er regarding the strength of the interaction e¤ects.
For any other consistent set of choice functions, the corresponding supermodular game will
be sandwiched by the last two games.
5
5.1
Extensions of the Model and Remarks
Choice Correspondences
Along the study we described the behavior of each agent by a choice function. We next propose
an alternative set of conditions to show that we can still make out-of-sample predictions if we
substitute choice functions by choice correspondences.
20
Let us suppose the behavior of agent i is described by the choice correspondence
Ci ( i ; a i ) : T i
where a
i
A
i
= (aj : j 2 N; j 6= i) and A
i
Ai
=
with
Ci ( i ; a i )
j2N;j6=i Aj .
Ai ( i )
In this context, we need to adapt our
solution concept as follows.
De…nition (Solution Concept): We say
rium behavior generated by
i(
( )=(
i(
) : i 2 N) is consistent with equilib-
if, for all i 2 N,
) 2 Ci ( i ; a i )
with a
i
=
j
( ) : j 2 N; j 6= i :
The extra di¢ culty we face when we allow for correspondences is that observed choices
are not enough to describe Ci even for those arguments that satisfy
m
i; a i
m ; am
i
i
=
for some
M: We next show that, with additional restrictions, this di¢ culty can be overcome.
To compare correspondences we will use the set order
B be two sets. We write A
S
B if a
S
that we de…ne next. Let A and
b for any a 2 A and b 2 B: That is, if each element in
A is larger than every element in B: The next two restrictions are the analogue to conditions
A2 and A3.
(A20 ) For all i 2 N, Ci ( i ; a i )
S
Ci
0
i; a i
(A30 ) For all i 2 N, Ci ( i ; a i )
S
Ci ( 0i ; a i ) for all
for all a
i
i
> a0 i :
0:
i
>
These two conditions guarantee that any selection from Ci is increasing in both
i
and
a i : When the choice correspondences derive from an underlying game, A20 holds if Ui is
supermodular in ai and satis…es the next condition. For all ai > a0i and a
Ui (ai ; a i ;
i)
Ui a0i ; a i ;
i
> Ui ai ; a0 i ;
i
i
Ui a0i ; a0 i ;
> a0 i ,
i
:
That is, if it has strict increasing di¤erences in (ai ; a i ). In addition, A30 is satis…ed if, for all
ai > a0i and
i
>
0,
i
Ui (ai ; a i ;
i)
Ui a0i ; a i ;
i
> Ui ai ; a i ;
and assumption A1 holds.
21
0
i
Ui a0i ; a i ;
0
i
:
Given A1, A20 and A30 , any ai 2 Ci ( i ; a i ) satis…es
ati
ai
am
i whenever
t t
i; a i
>
i; a i
>
m m
i ;a i
for some m; t 2 M:
That is, we can still use the data to bound the choice function. The di¤erence with respect
to our previous analysis is that in order to provide bounds for the choice correspondence
evaluated at
i; a i
we need information regarding two data points associated with pairs of
treatments and choices of other agents that are strictly smaller and strictly larger than
the argument we are interested in.
It can be easily shown that Propositions 4 and 5 remain valid if we slightly modify the
sharp bounds for Ci , Bi , and Di , as we show next.
B0i ( i ; a i ) = inf ai : ai 2 Ai ( i ) ; ai
sup am
i :
m ; am
i
i
<
i; a i; m
M
D0i ( i ; a i ) = sup ai : ai 2 Ai ( i ) ; ai
inf am
i :
m ; am
i
i
>
i; a i; m
M
That is, if we change two of the weak inequalities by strict ones.
5.2
Identi…cation of Choice Functions vs Equilibrium Points
Manski (2010) clearly explains that identi…cation of the choice functions by shape restrictions
is quite di¤erent from identi…cation of the equilibrium points. By using examples of models
where the solution set is a singleton, he shows that the assumptions we need to impose on the
primitives of the models to identify one or the other object can be quite di¤erent. We next
explain why the issue is more delicate when the solution set has multiple elements.
Our approach to identify solution sets relies on previous identi…cation of the choice functions. Thus, in terms of Manski’s discussion, the question is whether we can provide bounds
for
by imposing monotone conditions on V without (partially) identifying …rst V =
fCi : i 2 Ng : The answer to this question would be positive if, for instance, we were able
to provide conditions on V so that
S
where
2
S
0
for all
0
is de…ned as in the previous subsection. If we were able to do so, then, for any
,
22
inf fam :
sup fa : a 2 A ( ) ; a
sup fam :
inf fa : a 2 A ( ) ; a
m
m
;m
M gg
;m
M gg :
The problem with this approach is that, so far, there is no …xed point theorem that, allowing
for multiple solutions, provides conditions on the primitives so that
0
S
. In view of the
result we provided in the last subsection, this negative statement highlights a key di¤erence
between the MCS method for maximizers as compared to …xed points.
5.3
Equilibrium Selection and Out-of-Sample Predictions
Along the study, we obtained out-of-sample predictions without assuming any equilibrium
selection rule. It is nevertheless interesting to emphasize that the equilibrium selection rule
in the data-generating process will a¤ect the informativeness of our predictions. We illustrate
this idea via a simple example.
Example 4: Let N = f1; 2g, T1 = T2 = f ; g (with
), and Ai = fai ; ai g (with ai
ai )
for i = 1; 2: The next table describes the equilibrium set for the high treatment
Equilibrium Set
a2
1
= ;
2
=
a2
a1
a1
Let us suppose the data was generated by equilibrium behavior and A2 and A3 hold. Moreover,
let us assume …rst that these agents always coordinate in the largest equilibrium (though the
researcher does not know it). The problem we face is captured by the next tables
Data
Out-of-Sample Predictions
a2
1
= ;
2
=
a2
a1
1
a1
23
= ;
2
=
a2
a2
a1
X
X
a1
X
X
That is, A1 and A2 are not informative in this case.
Let us next suppose that these agents always coordinate in the smallest equilibrium
(though the researcher does not know it). Now the situation is
Data
Out-of-Sample Predictions
a2
1
= ;
2
=
a2
a1
1
a1
= ;
2
=
a2
a2
a1
X
x
a1
x
X
Under this alternative equilibrium selection rule our prediction is more informative.
We hypothesize that, with more data points, a stochastic equilibrium selection rule would
lead, on average, to smaller bounds as compared to extremal selections. The reason is that
a stochastic equilibrium selection rule would generate more variation in the selected action
pro…les and this would allow us to track individual choice functions more precisely.
6
Conclusion
Monotone comparative statics (MCS) has proved extremely useful in economic theory due
to the wide variety of models where this method allows us to compare equilibrium points.
This paper shows that this method for comparative statics can be equally valuable for the
nonparametric analysis of simultaneous choice models, e.g., supermodular games.
The primitives of our model are a system of interdependent choice functions. When they
arise from an underlying game, they constitute the best-reply functions of the various players.
We show that, while the standard implications of the MCS method for games may be neither
testable nor useful for counterfactual predictions with partial observability of equilibrium
points, its underlying assumptions are yet quite informative. Speci…cally, they allow us to
partially identify the set of choice functions from observed behavior. We then propose an
alternative use of the MCS method that translates the latter into bounds for counterfactual
prediction of equilibrium behavior. We refer to this method as the sandwich approach to
the out-of-sample predictions. We …nally show that if the data satis…es certain monotone
restrictions, then the empirical evidence can be always rationalized as the Nash equilibria of
24
a supermodular game. We illustrate all our ideas with an IO model of demand with network
e¤ects.
25
7
Proofs
Proof of Proposition 1: Let us consider a mapping M :
i2N Ai
!
i2N Ai ,
de…ned as
follows
M (a1 ; a2 ; :::; an ) = (C1 ( 1 ; a
1 ) ; C2 ( 2 ; a 2 ) ; :::; Cn ( n ; a n )) .
Function M maps n-dimensional actions into itself. Given our solution concept, the set of
…xed point of M coincides with
: Thus, the proof of Proposition 1 reduces to show that
the set of …xed points of M is a non-empty complete lattice. Any compact interval of Rm
or Zm is a complete lattice. It is well-known that if a product lattice is a product of all
complete lattices, then it is itself complete. Thus,
i2N Ai
is a complete lattice. Under A2,
M is increasing. The result follows by Tarskis0 …xed point theorem (see Tarski (1955)).
Proof of Proposition 2: Under A3, we have that, for all
M (a1 ; a2 ; :::; an )
where M and M
0
0,
M 0 (a1 ; a2 ; :::; an )
are de…ned as in the proof of Proposition 1. Under A1, they map complete
lattices into themselves. Under A2, M and M
0
are increasing. Thus, the result follows by
Topkis (1998, Corollary 2.5.2).
Proof of Proposition 3: Let us consider two mapping L and N from
i2N Ai
into itself,
de…ned as follows
L (a1 ; a2 ; :::; an ) = (Bi ( i ; a i ) : i 2 N)
and N (a1 ; a2 ; :::; an ) = (Di ( i ; a i ) : i 2 N) .
Under A4, we have that
N (a1 ; a2 ; :::; an )
M (a1 ; a2 ; :::; an )
L (a1 ; a2 ; :::; an )
where M is de…ned as in the proof of Proposition 1. By our argument in the proof of Proposition 2, they map complete lattices into themselves. Under A2, the three mappings are increasing. Thus, the result follows by Topkis (1998, Corollary 2.5.2).
Proof of Proposition 4: Under IA, we get that:
Bi
m m
i ;a i
= Di
m m
i ;a i
26
= am for all m
M:
Thus, U and W can generate equilibrium behavior consistent with the data. The intersection
of two complete sublattices is itself a complete sublattice. Thus, under A1, Di ( i ; a i ) and
Bi ( i ; a i ) are non-empty. By construction, Di ( i ; a i ) ;Bi ( i ; a i ) 2 Ai ( i ) for all
i
2 Ti :
We next show that Bi ( i ; a i ) and Di ( i ; a i ) satisfy A2 and A3.
Recall that
Bi ( i ; a i ) = inf ai : ai 2 Ai ( i ) ; ai
To show that A2 and A3 hold, let
am
i :
m m
i ;a i
0 0
i; a i,
0 ; a0 :
i
i
i; a i
m
sup am
i :
i; a i; m
M
:
It is readily veri…ed that
am
i :
M
m m
i ;a i
m m
i ;a i
i; a i,
m
M :
0 ; a0
i
i
Thus, whenever Bi ( i ; a i ) reaches the least possible value, the same happens for Bi
By A1, ai ( i )
:
ai ( 0i ). In addition,
sup am
i :
m m
i ;a i
It follows that Bi ( i ; a i )
i; a i,
m
0 ; a0
i
i
Bi
sup am
i :
M
whenever
m m
i ;a i
0 0
i; a i,
0 ; a0
i
i
i; a i
m
M :
and A2 and A3 hold.
Recall that
inf am
i :
Di ( i ; a i ) = sup ai : ai 2 Ai ( i ) ; ai
To show that A2 and A3 hold, let
am
i :
m m
i ;a i
Thus, whenever Di
By A1, ai ( i )
inf am
i :
0 ; a0
i
i
i; a i,
0 ; a0 :
i
i
i; a i
m
i; a i; m
M
:
It follows immediately that
am
i :
M
m m
i ;a i
m m
i ;a i
0 0
i; a i,
m
M :
reaches the highest possible value, the same happens for Di ( i ; a i ) :
ai ( 0i ) : In addition,
m m
i ;a i
It follows that Di ( i ; a i )
i; a i,
Di
m
M
0 ; a0
i
i
inf am
i :
whenever
m m
i ;a i
i; a i
0 0
i; a i,
0 ; a0
i
i
m
M :
and A2 and A3 are
satis…ed. Thus, Bi ( i ; a i ) and Di ( i ; a i ) satisfy A2 and A3.
Moreover, by construction, Bi ( i ; a i ) and Di ( i ; a i ) are the smallest and the largest
extensions of
Bi
m m
i ;a i
= Di
m m
i ;a i
that satisfy A2 and A3.
27
= am for all m
M
Proof of Proposition 5: By Proposition 4, U and W are the minimal and maximal microeconomic speci…cations for our model that are consistent with the data and satisfy IA. Thus,
the result follows by Proposition 3.
Proof of Proposition 6: Let gij (aij ) =
Ui (ai ; a i ;
i)
=
Xm
j=1
(1=2) (aij )2 : Then,
(1=2) (aij )2 +
Xm
j=1
ij
(a i ;
i ) aij .
Di¤erentiating this expression with respect to ai we get
@Ui (ai ; a i ;
@aij
i)
=
Cij (a i ;
aij = C i (a
i; i)
(a i ;
= Cij (a i ;
i)
+
ij
(a i ;
i)
= 0 for all j
We then de…ne
ij
By conditions A2 and A3,
ij
i)
i)
for all i
n and j
m:
is increasing. By construction, U rationalizes V.
28
n:
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