RATIONALIZATION AND IDENTIFICATION OF
DISCRETE GAMES WITH CORRELATED TYPES ∗
NIANQING LIU? , QUANG VUONG† , AND HAIQING XU‡
A BSTRACT. This paper studies the rationalization and identification of discrete games where
players have correlated private information/types. Allowing for correlation across types
is crucial, for instance, in models with social interactions as it represents homophily. Our
approach is fully nonparametric in the joint distribution of types and the strategic effects
in the payoffs. First, under monotone pure Bayesian Nash Equilibrium strategy, we characterize all the restrictions if any on the distribution of players’ choices imposed by the
game-theoretic model as well as restrictions associated with three assumptions frequently
used in the empirical analysis of discrete games. Namely, we consider additive separability
of private information, exogeneity of payoff shifters relative to private information, and
mutual independence of private information given payoff shifters. Second, we study the
nonparametric identification of the payoff functions and types distribution. In particular, we
show that the model with exogenous payoff shifters and separable private information is
fully identified up to a single location–scale normalization under some exclusion restrictions
and rank conditions. Third, we discuss partial identification, multi–dimensional types and
multiple equilibria. Lastly, we briefly point out the implications of our results for model
testing and estimation.
Keywords: Rationalization, Identification, Discrete Game, Social Interactions
JEL: C14, C18, C31, C35, C52, and Z13
Date: April, 2014.
∗ We thank Laurent Mathevet, Bernard Salanie, Steven Stern, Elie Tamer, Xun Tang, Neil Wallace and Nese
Yildiz for useful comments. We also thank seminar participants at Pennsylvania State University, Texas
Metrics Camp 2013, Texas A&M University, University of Chicago, Brown University, University of Texas at
Austin, Columbia University, Rice University, New York University, the North American Summer Meeting at
University of Southern California, and the 2013 Asian Meeting of the Econometric Society at National University
of Singapore. The second and third authors respectively acknowledge financial supports from the National
Science Foundation through grant SES 1148149 and the Summer Research Fellowship of University of Texas.
? School of Economics, Shanghai University of Finance and Economics, Shanghai, China, [email protected].
† (corresponding author) Department of Economics, New York University, 19 W. 4th Street, 6FL, New York, NY,
10012, [email protected].
‡ Department of Economics, University of Texas at Austin, [email protected].
1
1. I NTRODUCTION
Over the last decades, games with incomplete information have been much successful
to understand the strategic interactions among agents in the analysis of various economic
and social situations. A leading example is auctions with e.g. Vickrey (1961), Riley and
Samuelson (1981), Milgrom and Weber (1982) for the theoretical side and Porter (1995),
Guerre, Perrigne, and Vuong (2000) and Athey and Haile (2002) for the empirical component.
In this paper, we study the identification of static binary games of incomplete information
where players have correlated types.1 We also characterize all the restrictions if any imposed
by such games on the observables, which are the players’ choice probabilities. Following the
work by Laffont and Vuong (1996) and Athey and Haile (2007) for auctions, our approach is
fully nonparametric.
The empirical analysis of static discrete games is almost thirty years old. In particular, the
range of applications includes labor force participation (e.g. Bjorn and Vuong, 1984, 1985;
Kooreman, 1994; Soetevent and Kooreman, 2007), firms’ entry decisions (e.g. Bresnahan
and Reiss, 1990, 1991; Berry, 1992; Tamer, 2003; Berry and Tamer, 2006; Jia, 2008; Ciliberto
and Tamer, 2009), and social interactions (e.g. Kline, 2012). These papers deal with discrete
games under complete information. More recently, discrete games under incomplete information have been used to analyze social interactions by Brock and Durlauf (2001, 2007); Xu
(2011) among others, firm entry and location choices by Seim (2006), timing choices of radio
stations commercials by Sweeting (2009), stock market analysts’ recommendations by Bajari,
Hong, Krainer, and Nekipelov (2010), capital investment strategies by Aradillas-Lopez
(2010) and local grocery markets by Grieco (2013). This list is far from being exhaustive and
does not even mention the growing literature on estimating dynamic games.
Our paper contributes to this literature in several aspects. First, we focus on monotone
pure strategy Bayesian Nash equilibria (BNE) throughout to bridge discrete game modeling
with empirical analysis. Monotonicity is a desirable property in many applications for both
theoretical and empirical reasons. For instance, White, Xu, and Chalak (2011) show that
monotone strategies are never worse off than non–monotone strategies in a private value
1
To simplify, we focus on binary games in this paper. We left the extension of our approach to general discrete
games for future research.
2
auction model. Moreover, when a structural model has nonseparable errors, the recent
literature heavily relies on the monotonicity of the structural functions in latent variables
for identification analysis (see, e.g., Matzkin, 2003; Chesher, 2003, 2005; Chernozhukov
and Hansen, 2005). On theoretical grounds, Athey (2001) provides seminal results on the
existence of a monotone pure strategy BNE whenever a Bayesian game obeys the Spence–
Mirlees single–crossing restriction. Relying on the powerful notion of contractibility, Reny
(2011) extends Athey’s results and related results by McAdams (2003) to give weaker
conditions ensuring the existence of a monotone pure strategy BNE. Using Reny’s results,
we establish the existence of a monotone pure strategy BNE under a weak monotonicity
condition on the expected payoff in our setting. This condition is satisfied in most models
used in the recent literature. For instance, in empirical IO, it is satisfied when the types
are conditionally independent given payoff shifters. In social interaction games, it is also
satisfied with strategic complement payoffs and positively regression dependent types.
Exceptions include Aradillas-Lopez and Tamer (2008); Xu (2014).
Second, we allow players’ private information/types to be correlated. Allowing correlated private information is motivated primarily by empirical concerns. In oligopoly entry,
for example, correlation among types allows us to know “whether entry occurs because of
unobserved profitability that is independent of the competition effect” (Berry and Tamer,
2006). Moreover, correlation among players’ types is crucial in sociology as it represents the
“homophily” effects. Homophily is the principle that people involved in an interaction tend
to be similar (see McPherson, Smith-Lovin, and Cook, 2001). Since people tend to be similar
to be friends, the positive correlation among friends’ behaviors might be simply caused by
the unobserved similarity in preferences. Clearly, homophily is different from peer effects,
which are essentially (strategic) interactions among friends. Both effects can be identified
and distinguished from each other in our approach.
In contrast, mutual independence of private information is widely assumed in the
empirical game literature. See, e.g., Brock and Durlauf (2001); Pesendorfer and SchmidtDengler (2003); Seim (2006); Aguirregabiria and Mira (2007); Sweeting (2009); Bajari, Hong,
Krainer, and Nekipelov (2010); Tang (2010); De Paula and Tang (2012); Lewbel and Tang
(2012). To our knowledge, the only exceptions are Aradillas-Lopez (2010), Wan and Xu
3
(2010), Xu (2014) and Liu and Xu (2012). Such an independence of types is a convenient
assumption, but imposes strong restrictions such as the mutual independence of players’
choices given covariates, a property which can be invalidated by the data.2 On the other
hand, when private information is correlated, the BNE solution concept requires that each
player’s beliefs on rivals’ choices depend on her private information, which invalidates
the usual two–step identification/estimation procedure, see, e.g., Bajari, Hong, Krainer,
and Nekipelov (2010). With such type–dependent beliefs, Wan and Xu (2010) establish
some upper/lower bounds for the beliefs in a semiparametric setting with linear–index
payoffs. Alternatively, Aradillas-Lopez (2010) adopts a different equilibrium concept related
to Aumann (1987), in which each player’s equilibrium beliefs do not rely on her private
information, but on her actual action.
Third, our analysis is fully nonparametric in the sense that players’ payoffs and the
joint distribution of players’ private information are subject to some mild smoothness
conditions only. As far as we know, with the exception of De Paula and Tang (2012) and
Lewbel and Tang (2012), every paper analyzing empirical discrete games has imposed
parametric restrictions on the payoffs and/or the distribution of private information. For
instance, Brock and Durlauf (2001); Seim (2006); Sweeting (2009) and Xu (2014) specify both
payoffs and the private information distribution parametrically. In a semiparametric context,
Aradillas-Lopez (2010); Tang (2010); Wan and Xu (2010) and Liu and Xu (2012) parameterize
players’ payoffs, while Bajari, Hong, Krainer, and Nekipelov (2010) parameterize the
distribution of private information. On the other hand, De Paula and Tang (2012) and
Lewbel and Tang (2012) do not introduce any parameter but impose some restrictions on
the payoffs’ functional form. In particular, they impose multiplicative separability in the
strategic effect and assume that it is a known function (e.g. sum) of the other players’ choices.
In addition to being fully nonparametric, we do not require either that players’ private
information enter additively in the payoffs. Consequently, our baseline discrete game is
the most general one and closest to that considered in game theory. We show that such a
model imposes essentially no restrictions on the distribution of players’ choices. In other
2
A model with unobserved heterogeneity and independent private information also generates dependence
among players’ choices conditional on covariates (see e.g. Aguirregabiria and Mira, 2007; Grieco, 2013).
4
words, monotone pure strategy BNEs can explain almost all observed choice probabilities
in discrete games.
In view of the preceding results, we consider three assumptions that are frequently used
in the empirical analysis of discrete games. First, we consider the assumption that private
information enters additively in the player’s payoff. To the best of our knowledge, such
an assumption has been made in every paper analyzing discrete games empirically. We
show again that the resulting model imposes essentially no restrictions on the distribution
of players’ choices. We also show that the players’ payoffs and the joint distribution
of the players’ private information are not identified nonparametrically whether private
information are additively separable or not.
A second assumption that is frequently imposed in empirical work is exogeneity of
some variables shifting players’ payoffs relative to players’ private information. Papers
using such an assumption are, e.g., Brock and Durlauf (2001); Seim (2006); Sweeting (2009);
Aradillas-Lopez (2010); Bajari, Hong, Krainer, and Nekipelov (2010); De Paula and Tang
(2012) and Lewbel and Tang (2012). We show that the resulting model restricts the distribution of players’ choices conditional upon payoff shifters and we characterize all those
restrictions. Specifically, the exogeneity assumption restricts the joint choice probability
to be a monotone function of the corresponding marginal choice probabilities. With the
exogeneity assumption, we show that one can identify the equilibrium belief of the player
at the margin under a mild support condition. We then characterize the partially identified
set of payoffs and the distribution of private information under the exogeneity assumption
and the support condition. In particular, the partially identified region is unbounded and
quite large unless one imposes additional restrictions on the payoffs’ functional form.
Further, we consider some identifying restrictions, namely some exclusion restrictions
and rank conditions, to achieve identification in both separable and non–separable setups.
We show that the copula function of the types’ distribution is identified on an appropriate
support in the nonseparable case. Then, the players’ payoffs are identified up to scale for
each fixed value of the exogenous state variables, as well as up to the marginal distributions
of players’ private information. Moreover, with additive separability in payoffs, we show
that both the players’ payoffs and distribution of types are fully identified up to a single
5
location–scale normalization. Our model can be viewed as an extension to game theoretic
setup from traditional threshold–crossing models considered by, e.g., Matzkin (1992). An
important difference is that the game setup allows us to exploit exclusion restrictions for
identification. Such restrictions are frequently used in the empirical analysis of discrete
games. See, e.g., Aradillas-Lopez (2010); Bajari, Hong, Krainer, and Nekipelov (2010); Wan
and Xu (2010); Lewbel and Tang (2012).
For completeness, we consider a third assumption, namely the mutual independence
of players’ private information given payoff shifters. Specifically, we characterize all
the restrictions imposed by exogeneity and mutual independence as considered almost
exclusively in the empirical game literature. We show that all the restrictions under this
assumption can be summarized by the conditional mutual independence of players’ choices
given the payoff shifters. In particular, we show that the restrictions imposed by mutual
independence are stronger than those imposed by exogeneity, separability and monotonicity
of the equilibrium. In other words, any of the latter becomes redundant in terms of
explaining players’ choices as soon as mutual independence and a single equilibrium are
imposed.
The paper is arranged as follows. We introduce our baseline model in Section 2. We
define and establish the existence of a monotone pure strategy BNE. We also characterize
such equilibrium strategies under additive separability of private information. In Section 3,
we study the restrictions imposed by the baseline model, whether private information is
additively separable or not. We also derive all the restrictions imposed by the exogeneity
and mutual independence assumption. In Section 4, we establish the nonparametric identification of the model primitives for the nonseparable and additively separable cases under
some exclusion restrictions and rank conditions. In Section 5, without exclusion restrictions,
we study the partial identification of the payoffs under both additive separability and non–
separability in types, respectively. We also discuss three related issues: multi–dimensional
types for each player, multiple equilibria in the DGP, and unobserved heterogeneity. Section
6 concludes with a brief discussion on estimation and testing of the model.
6
2. M ODEL AND M ONOTONE P URE S TRATEGY BNE
We consider a discrete game of incomplete information. There is a finite number of
players, indexed by i = 1, 2, · · · , I. Each player simultaneously chooses a binary action
Yi ∈ {0, 1}. Let Y = (Y1 , · · · , YI ) be an action profile and A = {0, 1} I be the space of action
profiles. Following the convention, let Y−i and A−i denote an action profile of all players
except i and the corresponding action profile space, respectively. Let X ∈ SX ⊂ Rd be a
vector of payoff relevant variables, which are publicly observed by all players and also by
the researcher.3 For instance, X can include individual characteristics of the players as well
as specific variables for the game environment. For each player i, we further assume that
the error term Ui ∈ R is her private information,4 i.e., Ui is observed only by player i, but
not by other players. To be consistent with the game theoretic literature, we also call Ui
as player i’s “type” (see, e.g., Fudenberg and Tirole, 1991). Let U = (U1 , · · · , U I ) and FU |X
be the conditional distribution function of U given X. The conditional distribution FU |X is
assumed to be common knowledge.5
The payoff of player i is described as follows:

 π (Y , X, U ),
i −i
i
Πi (Y, X, Ui ) =

0,
if Yi = 1,
if Yi = 0,
where πi is a structural function of interest. In particular, πi is not separable in the error
term Ui . The zero payoff for action Yi = 0 is a standard payoff normalization in binary
response models.6
3See Section 5.4 on unobserved heterogeneity when this is not the case. See also Grieco (2013) who analyzes a
discrete game that has some payoff relevant variables publicly observed by all players, but not by the researcher.
4See Section 5.2 on multi–dimensional signals when U is no longer a scalar.
i
5For the standard notion of common knowledge, see, e.g., Fudenberg and Tirole (1991), Chapter 14.
6Here we understand “normalization” from the view of observational equivalence: Suppose the payoffs take
the general form:
πi1 (Y−i , X, Ui ), if Yi = 1;
Πi (Y, X, Ui ) =
πi0 (Y−i , X, Ui ), if Yi = 0.
It can be shown that this model with associated assumptions later is observationally equivalent to the above
game with payoff πi (Y−i , X, Ui ) = πi1 (Y−i , X, Ui ) − πi0 (Y−i , X, Ui ).
7
Following the literature on Bayesian games, player i’s decision rule takes the form as a
function of her type:
Yi = δi ( X, Ui ),
where δi : Rd × R → {0, 1} maps all the information she knows to a binary decision. For
δ ( a | x, u ) be the conditional probability
any given strategy profile δ = (δ1 , · · · , δI ), let σ−
i
i −i
of other players choosing a−i ∈ A−i given X = x and Ui = ui , i.e.,
δ
σ−
i ( a−i | x, ui ) ≡ Pδ (Y−i = a−i | X = x, Ui = ui ) = P δj ( X, U j ) = a j , ∀ j 6 = i | X = x, Ui = ui ,
where Pδ represents the (conditional) probability measure under the strategy profile δ.
The equilibrium concept we adopt is the pure strategy Bayesian Nash equilibrium (BNE).
Mixed strategy equilibria are not considered in this paper, since a pure strategy BNE
generally exists under weak conditions in our model.
We now characterize the equilibrium solution in our discrete game. Fix X = x ∈ SX .
In equilibrium, player i with Ui = ui chooses action 1 if and only if her expected payoff is
greater than zero, i.e.,
"
δi∗ ( x, ui ) = 1
∑ πi (a−i , x, ui ) × σ−∗ i (a−i |x, ui ) ≥ 0
#
, ∀ i,
(1)
a −i
where δ∗ ≡ (δ1∗ , · · · , δ∗I ), as a profile of functions of u1 , · · · , u I respectively, denotes the
∗
∗ ( a | x, u ) is a shorthand notation for σ δ ( a | x, u ).
equilibrium strategy profile and σ−
i
i
i −i
−i −i
∗ depends on δ∗ . Hence, (1) for i = 1, · · · , I defines a simultaneous equation
Note that σ−
i
−i
system in δ∗ referred as “mutual consistency” of players’ optimal behaviors. A pure strategy
BNE is a fixed point δ∗ of such a system, which holds for all u = (u1 , . . . , u I ) in the support
SU |X = x . Ensuring equilibrium existence in Bayesian games is a complex and deep subject
in the literature. It is well known that a solution of such an equilibrium generally exists in a
broad class of Bayesian games (see, e.g., Vives, 1990).
The key to our approach is to employ a particular equilibrium solution concept of
BNE — monotone pure strategy BNEs, which exist under additional weak conditions.
Recently, much attention has focused on monotone pure strategy BNEs. The reason is that
monotonicity is a natural property and has proven to be powerful in many applications
8
such as auctions, entry, and global games. In our setting, a monotone pure strategy BNE is
defined as follows:
Definition 1. Fix x ∈ SX . A pure strategy profile (δ1∗ ( x, ·), · · · , δ∗I ( x, ·)) is a monotone pure
strategy BNE if (δ1∗ ( x, ·), · · · , δ∗I ( x, ·)) is a BNE and δi∗ ( x, ui ) is (weakly) monotone in ui for all i.
Monotone pure strategy BNEs are relatively easier to characterize than ordinary BNEs.
Fix X = x. In our setting, a monotone equilibrium strategy can be explicitly defined as a
threshold function (recall that δi∗ can take only binary values). Formally, in a monotone pure
strategy BNE, player i’s equilibrium strategy can be written as either δi∗ = 1 [ui ≤ ui∗ ( x )]
or δi∗ = 1 [ui > ui∗ ( x )],7 where ui∗ ( x ) is the cutoff value that might depend on x. Let
u∗ ( x ) ≡ (u1∗ ( x ), · · · , u∗I ( x )) ∈ R I be the profile of equilibrium strategy thresholds. Without
loss of generality, hereafter we restrict our attention to monotone decreasing pure strategy
(m.d.p.s.) BNEs, which essentially is a normalization of the game structure. To see this,
fix X = x. Suppose player i’s equilibrium strategy is monotone increasing in ui , i.e.
δi∗ = 1 [ui > ui∗ ( x )]. Then, we can construct an observationally equivalent structure [π̃; F̃U |X ]
by letting π̃i (·, x, ui ) = πi (·, x, −ui ) for all ui , π̃ j = π j for all j 6= i, and F̃U |X (·| x ) = FUe |X (·| x )
e differs from U only in its i–th argument: U
e i = −Ui . For the constructed structure,
where U
it can be shown that player i’s equilibrium strategy becomes monotone decreasing, i.e.
δ̃i∗ = 1 [ui ≤ −ui∗ ( x )], while other players’ equilibrium strategies stay the same. Further,
we can verify that the new equilibrium in the constructed structure leads to the same
distribution of observables.
In an m.d.p.s. BNE, the mutual consistency condition for a BNE solution defined by (1)
requires that for each player i,
ui ≤ ui∗ ( x ) ⇐⇒
∑ πi (a−i , x, ui ) × σ−∗ i (a−i |x, ui ) ≥ 0.
(2)
a −i
A simple but key observation is that under certain weak conditions introduced later, (2)
implies that player i with margin type ui∗ ( x ) should be indifferent between choosing 1 and
0, i.e.,
∑ πi (a−i , x, ui∗ (x)) × σ−∗ i (a−i |x, ui∗ (x)) = 0,
∀i.
a −i
7The left–continuity of strategies considered hereafter is not restrictive given our assumptions below.
9
Therefore, we can solve u∗ ( x ) from the above I–equation system.
The seminal work on the existence of a monotone pure strategy BNE in games of incomplete information was first provided by Athey (2001) in both supermodular and logsupermodular games, and later extended by McAdams (2003) and Reny (2011). Applying Reny (2011)
Theorem 4.1, we establish the existence of monotone pure strategy BNEs in our binary game
under some weak regularity assumptions.
Assumption R (Conditional Radon–Nikodym Density). For every x ∈ SX , the conditional
distribution of U given X = x is absolutely continuous w.r.t. Lebesgue measure and has a continuous
positive conditional Radon–Nikodym density f U |X (·| x ) a.e. over the nonempty interior of its
hypercube support SU |X = x .
Assumption R allows the support of U conditional on X = x to be bounded, namely of
the form ×i=1,...,I [ui ( x ), ui ( x )] for some finite endpoints ui ( x ) and ui ( x ) as frequently used
when Ui is i’s private information, or unbounded as when SU |X = x = R I used typically in
binary models. As a matter of fact, assumption R can be greatly weakened as shown by
Reny (2011) (see Appendix B.2 for more details).
For any strategy profile δ, let Eδ denote the (conditional) expectation under the strategy
profile δ. Without causing any confusion, we will suppress the subscript δ∗ in Eδ∗ (or Pδ∗ )
when the expectation (or probability) is measured under the equilibrium strategy profile.
Assumption M (Monotone Decreasing Expected Payoff). For any monotone decreasing pure
strategy profile δ and x ∈ SX , the (conditional) expected payoff Eδ πi (Y−i , X, Ui ) X = x, Ui = ui
is a monotone decreasing function in ui ∈ SUi |X = x .
Assumption M guarantees that each player’s best response is also monotone decreasing
in type given that all other players adopt monotone decreasing pure strategies. Note that
for the existence of m.d.p.s. BNE, assumption M is sufficient but may not be necessary in
most cases. It should also be noted that assumption M holds trivially if πi is monotone
decreasing in Ui , e.g. πi ( a−i , x, ui ) = β i ( a−i , x ) − ui for some β i , and all Ui s are conditionally
independent of each other given X.
Lemma 1. Suppose assumptions R and M hold. For any x ∈ SX , there exists an m.d.p.s. BNE.
10
Proof. See Appendix A.1
By Lemma 1, m.d.p.s. BNEs generally exist in a large class of binary games. Lemma 6
in Appendix A.2 replaces assumption M with more primitive sufficient conditions. In
particular, we assume positive regression dependence across Ui s given X, strategic complementarity of players’ actions and monotone payoffs in the Ui s. As far as we know, with the
only exception of Aradillas-Lopez and Tamer (2008) and Xu (2014), every paper analyzing
empirical discrete games of incomplete information so far has imposed certain restrictions
to guarantee the equilibrium strategies to be threshold–crossing.
Though there is a lack of attention on non–monotone strategy BNEs in the literature, it is
worth pointing out that this kind of equilibrium could exist and sometimes even stands as
the only type of equilibrium. It could happen when some player is quite sensitive to others’
choices and types are highly correlated. Here we provide a simple example to illustrate.8
Example 1. Let I = 2 and πi = Xi − αi Y−i − Ui , where (U1 , U2 ) conforms to a joint normal
distribution with mean zero, unit variances and correlation parameter ρ ∈ (−1, 1).9
Case 1: suppose ( X1 , X2 ) = (1, 0) and (α1 , α2 ) = (2, 0). Then, regardless of the value of ρ, there
is always a unique pure strategy BNE: Clearly, player 2 has a dominant strategy which is monotone
in u2 : choosing 1 if and only if u2 ≤ 0. Thus, player 1’s best response must be: choosing 1 if and
ρu
only if 1 − 2Φ − √ 1 2 − u1 ≥ 0. Further, it can be shown that player 1’s equilibrium strategy
1− ρ
q
π
is not monotone in u1 if and only if ρ ∈
2+ π , 1 ).
Case 2: suppose ( X1 , X2 ) = (1, 1) and (α1 , α2 ) = (2, 2). First, note that the monotone
q
strategy profile {1(u1 ≤ 0); 1(u2 ≤ 0)} is a BNE as long as ρ ∈ − 1, 2+ππ . Moreover, it
can be verified that this equilibrium is the unique BNE if and only if ρ ∈ − 1, 2π+−π2 . When
ρ ∈ 2π+−π2 , 1 , we can find two other equilibria of the game: {1(u1 ≤ u∗ ); 1(u2 ≤ −u∗ )} and
q
ρ
∗
∗
{1(u1 ≤ −u∗ ); 1(u2 ≤ u∗ )} where u∗ > 0 solves 1 − 2Φ( 11+
−ρ · u ) + u = 0.
In Example 1, the non–monotone strategy BNE occurs due to the large positive correlation
between U1 and U2 (relative to α1 ), which violates assumption M. On the other hand, for
8We thank Steven Stern and Elie Tamer for their comments and suggestions on the following examples.
9In a similar fully parametric setting, Xu (2014) proposes an inference approach based on first identifying a
subset of the covariate space where the game admits a unique monotone strategy BNE.
11
any given value of structural parameters, the existence of non–monotone strategy BNE
could also depend on the realization of ( X1 , X2 ).
Monotone pure strategy BNEs are convenient and powerful for empirical analysis. In
particular, we can represent each player’s monotone equilibrium strategy by a semi–linear–
index binary response model. To see this, we first make an assumption that is commonly
assumed in the literature.
Assumption S (Additive Separability). We have πi ( a−i , x, ui ) = β i ( a−i , x ) − ui for some
function β i and for each i, a−i , x and ui .
In assumption S, the negative slope of ui is only for notational simplicity. Assumption S
could be violated when there are interactions between a−i and ui in the payoffs for some
applications, e.g., auction participations (see, e.g., Li and Zheng, 2009; Marmer, Shneyerov,
and Xu, 2013).
Lemma 2. Suppose assumptions R, M and S hold and the equilibrium is an m.d.p.s. BNE. Then,
player i’s equilibrium strategy can be written as follows:
h
Yi = 1 Ui ≤
∑ βi (a−i , X ) × σ−∗ i (a−i |X, ui∗ (X ))
i
.
(3)
a −i
Proof. See Appendix A.3.
Here is an informal argument to provide some intuitions for Lemma 2: Under assumption
S, we have that for any x ∈ SX ,
E πi (Y−i , X, Ui ) X = x, Ui = ui =
∑a− βi (a−i , x) × σ−∗ i (a−i |x, ui ) − ui ,
i
which is continuously decreasing in ui under assumptions R and M. Then, player i with the
margin type ui∗ ( x ) should be indifferent between choosing 1 and 0, i.e.10
∑a− βi (a−i , x) × σ−∗ i (a−i |x, ui∗ (x)) − ui∗ (x) = 0.
i
(4)
Thus, (3) follows directly from Yi = 1 Ui ≤ ui∗ ( X ) .
10In (4), it is understood that u∗ ( x ) = u ( x ) and u∗ ( x ) = u ( x ) if E ∗ β (Y , X ) X = x, U = u ( x ) < u ( x )
i
δ
i −i
i
i
i
i
i
i
and Eδ∗ β i (Y−i , X ) X = x, Ui = ui ( x ) > ui ( x ), respectively.
12
The semi–linear–index model in (3) also relates to single-agent binary threshold crossing
models studied e.g. by Matzkin (1992). We will show later in the identification section that
∗ in (3) can be nonparametrically identified under additional weak conditions.
the beliefs σ−
i
3. R ATIONALIZATION
In this section, we study the baseline model defined by assumptions R and M as well
as three other models obtained by imposing additional assumptions frequently made in
the empirical game literature such as assumption S. Specifically, we characterize all the
restrictions imposed on the distribution of observables (Y, X ) by each of these models. We
say that a conditional distribution FY |X is rationalized by a model if and only if it satisfies
all the restrictions of the model. Equivalently, FY |X is rationalized by the model if and
only if there is a structure (not necessarily unique) in the model that generates such a
distribution. In particular, rationalization logically precedes identification as the latter,
which is addressed in Section 4, makes sense only if the observed distribution can be
rationalized by the model under consideration.
Besides assumption S, we also consider the exogeneity of X relative to U, an assumption
that is frequently made in the literature, e.g., Bajari, Hong, Krainer, and Nekipelov (2010).
The exceptions, to our knowledge, are De Paula and Tang (2012) and Wan and Xu (2010).
Assumption E (Exogeneity). X and U are independent of each other.11
Another assumption called as mutual independence is also widely used in the literature.
For examples, see an extensive list of references in two recent surveys: Bajari, Hong, and
Nekipelov (2010) and de Paula (2012).
Assumption I (Mutual Independence). U1 , · · · , U I are mutually independent conditional on X.
11Our results can be easily extended to the weaker assumption that X and U are independent from each other
conditional on W, where W are other observed payoff relevant variables.
13
Let S ≡ [π; FU |X ], where π = (π1 , · · · , π I ). We now consider the following models:
M1 ≡ S : assumptions R and M hold and a single m.d.p.s. BNE is played ,
M2 ≡ {S ∈ M1 : assumption S holds} ,
M3 ≡ {S ∈ M2 : assumption E holds} ,
M4 ≡ {S ∈ M3 : assumption I holds} .
Clearly, M1 ) M2 ) M3 ) M4 .
The last requirement in M1 is not restrictive when the game has a unique equilibrium
which has to be an m.d.p.s. BNE. When this is not the case, we follow most of the literature
by assuming that the same equilibrium is played in the DGP for any given x. Such an
assumption is realistic if the equilibrium selection rule is actually governed by some game
invariant factors, like culture, social norm, etc. See, e.g., de Paula (2012) for a detailed
discussion. Relaxing such a requirement has been addressed in recent work and will be
discussed in Section 5.3.
We introduce some key notation for the following analysis. For any S ∈ M1 , let αi ( x ) ≡
FUi |X (ui∗ ( x )| x ). By monotonicity of the equilibrium strategy, we have that αi ( x ) = E(Yi | X =
x ). For each p = 2, · · · , I, and 1 ≤ i1 < · · · < i p ≤ I, let CUi
1
,··· ,Ui p | X
be the conditional
copula function of (Ui1 , · · · , Ui p ) given X, i.e., for any (αi1 , · · · , αi p ) ∈ [0, 1] p and x ∈ SX ,
CUi
,··· ,Ui p | X ( αi1 , · · ·
1
, αi p | x ) ≡ FUi
,··· ,Ui p | X
1
FU−i1|X (αi1 | x ), · · · , FU−i1|X (αi p | x ) x .
1
p
The next proposition characterizes the collection of distributions of Y given X that can be
rationalized by M1 .
Proposition 1. A conditional distribution FY |X is rationalized by M1 if and only if for all x ∈ SX
and a ∈ A, P(Y = a| X = x ) = 0 implies that P(Yi = ai | X = x ) = 0 for some i.
Proof. See Appendix B.1
By Proposition 1, M1 rationalizes all distributions for Y given X that belong to the interior
of the 2 I − 1 dimensional simplex, i.e. distributions with strictly positive choice probabilities.
The distributions that cannot be rationalized by M1 must have P(Y = a| X = x ) = 0 for
14
some a ∈ A, i.e distributions for which there are “structural zeros.” In other words, our
baseline model M1 imposes no essential restrictions on the distribution of observables.
All those distributions that cannot be rationalized by M1 arise because of assumption
R. As noted earlier, one can replace assumption R by Reny (2011)’s weaker conditions.
Lemma 7 (in Appendix B.2) then shows that any distribution for Y given X can be rationalized.
Next, we show that M2 and M1 are observationally equivalent, i.e., M2 does not impose
additional restrictions on FY |X .
Proposition 2. A conditional distribution FY |X is rationalized by M1 if and only if it is rationalized
by M2 .
Proof. See Appendix B.3
In particular, additive separability of private information in the payoffs (assumption S)
does not impose any additional restrictions relative to model M1 . Then, it follows from
Proposition 1 that M2 can also rationalize all the conditional distributions of observables
that belong to the interior of the 2 I − 1 dimensional simplex.
We now characterize all the restrictions imposed on FY |X by model M3 . These additional
restrictions come from assumption E.
Proposition 3. A conditional distribution FY |X rationalized by M2 is also rationalized by M3 if
and only if for each p = 2, · · · , I and 1 ≤ i1 < · · · < i p ≤ I,
p
p
R1: E ∏ j=1 Yi j X = E ∏ j=1 Yi j αi1 ( X ), · · · , αi p ( X ) .
p
R2: E ∏ j=1 Yi j |αi1 ( X ) = ·, · · · , αi p ( X ) = · is strictly increasing on Sαi
1
( X ),··· ,αi p ( X )
at values for which some coordinates are zero.
p
R3: E ∏ j=1 Yi j |αi1 ( X ) = ·, · · · , αi p ( X ) = · is continuously differentiable on Sαi
Proof. See Appendix B.4.
1
except
( X ),··· ,αi p ( X ) .
Proposition 3 shows that the joint choice probabilities rationalized by M3 are monotone strictly increasing and continuously differentiable functions of the marginal choice
probabilities. In Proposition 3, the most stringent restriction is R1, which requires that
15
the joint choice probability depend on X only through the corresponding marginal choice
probabilities. In other words, suppose x, x 0 ∈ SX and αi ( x ) = αi ( x 0 ) for i = 1, · · · , I, then
R1 implies that P Y = a| X = x = P Y = a| X = x 0 for all a ∈ A. Moreover, under
restriction R1 and R2, the condition α( x ) ≥ α( x 0 ) implies that P(Yi1 = 1, · · · , Yi p = 1| X =
x ) ≥ P(Yi1 = 1, · · · , Yi p = 1| X = x 0 ) for all tuples {i1 , · · · , i p }. Note that αi ( x ) is identified
by αi ( x ) = E(Yi | X = x ). Therefore, all the restrictions R1–R3 are testable in principle. This
is discussed further in the Conclusion.
It is worth pointing out that our proofs for Propositions 1 to 3 are constructive. Specifically,
we construct an I–single–agent decision structure that rationalizes the given distributions
satisfying the corresponding restrictions. Hence, it is evident that without additional model
restrictions beyond the assumptions of M1 , M2 or M3 , a discrete Bayesian game model
with strict interactions cannot be empirically distinguished from an alternative model of
I–single–agent decisions. In contrast, with exclusion restrictions, Section 4 shows that these
two models can be distinguished from each other.
For completeness, we also study the restrictions on observables imposed by M4 , which
makes the additional assumption I. It should be noted that assumption M is implied when
we assume assumptions S and I. In other words,
M4 = {S : assumptions R, S, E, I hold and a single m.d.p.s. BNE is played}.
In the literature, several special cases of M4 have been considered under some parametric
or functional form restrictions, see, e.g., Bajari, Hong, Krainer, and Nekipelov (2010) and
Lewbel and Tang (2012).
Proposition 4. A conditional distribution FY |X can be rationalized by M4 if and only if Y1 , · · · , YI
are conditionally independent given X, i.e. for each p = 2, · · · , I and 1 ≤ i1 < · · · < i p ≤ I,
p
p
E ∏ j=1 Yi j | X = ∏ j=1 αi j ( X ).
Proof. See Appendix B.5.
Note that the conditional independence restriction in Proposition 4 implies conditions
R1–R3 in Proposition 3, as well as the necessary and sufficient conditions in Proposition 1.
Therefore, the conditional independence restriction summarizes all the restrictions imposed
16
by M4 . Moreover, we can replace M4 in Proposition 4 with
M40 ≡ {S : assumption I holds and a single BNE is played},
since M40 also implies the conditional independence restriction. Note that the monotonicity
of the BNE is not required for M40 . Because M4 ⊂ M40 , we have the following corollary.
Corollary 1. Model M4 imposes the same restrictions on the distribution of observables as M40 ,
i.e., both models are observationally equivalent.
This is a surprising result: separability (assumption S) and more importantly exogeneity
of the payoff shifters (assumption E) become redundant in terms of restrictions on the
observables, as soon as mutual independence of types conditional on X (assumption I) and
a single BNE condition are imposed on the baseline model. Moreover, the monotonicity
of equilibrium is not restrictive either under these two conditions. Last, if we are willing
to maintain assumption I, then Proposition 4 and Corollary 1 will give us a test of a single
equilibrium being played, as rejecting the conditional independence of Yi s given X indicates
the presence of multiple equilibria. A related result in terms of correlation was obtained by
De Paula and Tang (2012) in a partial–linear setting.
4. N ONPARAMETRIC I DENTIFICATION
In this section we study the nonparametric identification of the baseline game-theoretic
model M1 , and its special cases M2 , M3 and M4 . As a preliminary to the identification of
M3 , we also consider the identification of
M30 ≡ {S ∈ M1 : assumption E holds},
which is the nonseparable extension of M3 , and which is of interest by itself. The recent
literature has focused on the parametric or semiparametric identification of structures in
M4 , see, e.g., Brock and Durlauf (2001); Seim (2006); Sweeting (2009); Bajari, Hong, Krainer,
and Nekipelov (2010), and Tang (2010). As far as we know, Lewbel and Tang (2012) is the
only paper that studies the nonparametric identification of a model in M4 under additional
restrictions on the functional form of payoffs.
17
In our context, identification of each model is equivalent to identification of the payoffs πi ,
the marginal distribution function FUi |X and the copula function CU |X of the joint distribution
of private information. Let QUi |X be the quantile function of FUi |X . Because the quantile
function is the inverse of the CDF, i.e. QUi |X = FU−1|X , throughout the identification of models
i
reduces to the identification of the triple π; { QUi |X }iI=1 ; CU |X .
In this section we provide new identification results for the payoffs and types’ distribution
under weak conditions. In model M30 , we establish the identification of CU |X and some
features of the structure under additional identifying restrictions, i.e. some exclusion
restrictions and rank conditions. In model M3 , we provide the full identification of the
triple π; { QUi |X }iI=1 ; CU |X up to a single location–and–scale normalization on the payoff
function. The identification of M4 requires slightly weaker support restrictions than those
for M3 , though the differences are not essential.
4.1. Nonidentification of M1 and M2 . We begin with the most general model M1 and
M2 , in which we show that the structure is typically unidentified.
Proposition 5. Neither M1 nor M2 is identified nonparametrically.
The non–identification of M2 follows directly from the observational equivalence between any structure S in M2 and a collection of I–single-agent binary responses models: Let
S̃ ≡ π̃; F̃U |X , in which π̃i ≡ ũi ( x ) − ui , where ũi ( x ) is arbitrarily chosen, and F̃U |X satisfies assumption R with F̃Ui
1
,··· ,Ui p | X ( ũi1 ( x ), · · ·
, ũi p ( x )| x ) = FUi
1
∗
,··· ,Ui p | X ( ui1 ( x ), · · ·
, ui∗p ( x )| x )
for all x ∈ SX and all tuples {i1 , · · · , i p }. Thus, S̃ and S are observationally equivalent
thereby establishing the non–identification of M2 . The non–identification of M1 follows
immediately from Proposition 2.12
Next, we turn to the identification of M30 and its sub–models M3 and M4 . First note
that we maintain assumption E in all these models, i.e. that X and U are independent
of each other; see footnote 11 for weaker assumptions. It follows that QUi |X = QUi and
CU |X = CU . Thus, the identification of these models reduces to the identification of the
triple π; { QUi }iI=1 ; CU .
12Note that even if we assume the exclusion restriction introduced later, these two models are still not identified.
18
4.2. Identification of M30 . The essential model restrictions of M30 are a single m.d.p.s.
BNE being played and the exogeneity condition (assumption E). In particular, it allows for
nonseparable private information in payoffs which is often the case in empirical IO. By
Propositions 2 and 3 and their proofs, it can be easily shown that a conditional distribution
FY |X rationalized by M1 is also rationalized by M30 if and only if conditions R1–R3 are
satisfied. In other words, if we maintain M1 , the only restrictions imposed by the exogeneity
assumption are conditions R1–R3.
Let α( x ) ≡ (α1 ( x ), · · · , α I ( x )) be a profile of the marginal choice probabilities. Note that
αi ( x ) is identified by E(Yi | X = x ). Under assumption E, the copula function CU is nonparametrically identified on an appropriate domain, namely, the extended support of α( X ) deS
fined as Sαe(X ) ≡ {α : α j = 0 for some j}
α : (αi1 , · · · , αi p ) ∈ Sαi (X ),··· ,αi p (X ) ; other αi j =
1
1 . We have CU (α) = 0 if α j = 0 for some j; otherwise,13
CU (α) = P U1 ≤ QU1 (α1 ), · · · , U I ≤ QUI (α I )
(
)
n
o
I
= P Uj ≤ QUj (α j ), ∀ j ∈ {i : αi 6= 1} = E ∏ Yi α j ( X ) = α j , ∀ j ∈ {i : αi 6= 1} . (5)
i =1
Key among those conditions for the nonparametric identification of CU is the assumption
that a single m.d.p.s. BNE is played for the DGP. Such a restriction implies that conditional
on α j ( X ) = α j , Uj ≤ QUj (α j ) is equivalent to Yj = 1. In particular, the single equilibrium restriction combined with assumption I implies that Yj s are (conditionally) mutually
independent given X. Then CU is identified to be the independent copula.
∗ (·| x, u∗ ( x )) can also be nonparametrically
As mentioned above, the equilibrium belief σ−
i
i
identified, for which we need a support condition on α( X ).
Assumption SC (Support Condition). The support Sα(X ) is a convex subset of [0, 1] I with full
dimension, i.e. dim Sα(X ) = I.
Note that assumption SC implies that the relative interior and the interior Sα◦(X ) of Sα(X )
are equal. Hence, the dimension of the interior of Sα(X ) is I. See e.g. Rockafellar (1997).
13In (5), it is understood that C (α) = 1 if α = (1, · · · , 1).
U
19
Therefore, we can take derivatives in all directions of an arbitrary smooth function defined
on Sα◦(X ) .14
The main restriction in assumption SC is the full dimensionality of the support Sα(X ) ,
which requires the payoff shifters X to have sufficient variations. It is worth noting that our
support condition is weaker than the conditions required in the identification–at–infinity
approach (see, e.g. Wan and Xu, 2010). Moreover, given identification of α( x ), assumption
SC is verifiable.
Lemma 3. Let S ∈ M30 . Suppose assumption SC holds. Fix x ∈ SX◦ . Then the equilibrium beliefs
∗ (·| x, u∗ ( x )) is identified, namely, for all a
σ−
−i ∈ A −i ,
i
i
∂P
Y
=
1;
Y
=
a
|
α
(
X
)
=
α
(
)
i
−i
−i
∗
∗
σ−
(
a
|
x,
u
(
x
))
=
−
i
i
i
∂αi
.
(6)
α=α( x )
Proof. See Appendix C.1.
a
Under assumption I, the probability P (Yi = 1; Y−i = a−i |α( X )) = αi ( X ) · ∏ j6=i α j j ( X )[1 −
∗ ( a | x, u∗ ( x )) =
α j ( X )]1−a j becomes a (known) linear function in αi ( X ). Thus, we have σ−
i −i
i
a
j
∏ j6=i α j ( X )[1 − α j ( X )]1−a j , so we don’t need assumption SC to identify the equilibrium
beliefs, see, e.g., Bajari, Hong, Krainer, and Nekipelov (2010).
∗ is acquired from the local–instrumental–variables
The intuition for our identification of σ−
i
method developed in Heckman and Vytlacil (1999, 2005): the local variation in player i’s
choice probability αi ( X ) (after controlling for α−i ( X )) provides the identification power for
the conditional choice probability of the other players given that the latent variable is at the
margin. To see this, we use a two–player game to illustrate.
Example 2. Fix S ∈ M30 and let I = 2. Note that for any α ∈ [0, 1]2 , we have
P(Y1 = 1, Y2 = 1|α( X ) = α) = P [U1 ≤ u1∗ ( X ), U2 ≤ u2∗ ( X )|α( X ) = α]
= P [U1 ≤ QU1 (α1 ), U2 ≤ QU2 (α2 )] = CU (α1 , α2 ),
14As a matter of fact, for the boundary points, we can take directional derivatives as well.
20
where the second equality follows from αi ( X ) = αi being equivalent to ui∗ ( X ) = QUi (αi ) and from
the independence of U and X. Further, we have
∂CU (α1 , α2 )
= P U−i ≤ QU−i (α−i )|Ui = QUi (αi ) ,
∂αi
(7)
see, e.g., Darsow, Nguyen, and Olsen (1992). Because QUi (αi ( x )) = ui∗ ( x ), it follows that
∂P(Y1 = 1, Y2 = 1|α( X ) = α) = P U−i ≤ u∗−i ( x )|Ui = ui∗ ( x )
∂αi
α=α( x )
= P (Y−i = 1| X = x, Ui = ui∗ ( x )) = σ−∗ i (1| x, ui∗ ( x )).
Similarly, we have
∂P(Yi = 1, Y−i = 0|α( X ) = α) = σ−∗ i (0| x, ui∗ ( x )).
∂αi
α=α( x )
Equation (7) is related to the treatment effect literature, e.g., Heckman and Vytlacil
(1999, 2005), Carneiro and Lee (2009) and Jun, Pinkse, and Xu (2011). Taking derivative
with respect to the propensity score identifies the conditional quantile/expectation of the
treatment effect at the margin. Lemma 3 extends this result to the multivariate case by using
the law of iterated expectation.
Now, we are ready to discuss the identification of features of M30 , for which we begin
with a weak assumption.
Assumption C. The payoff functions πi ( a−i , x, ui ) are continuous in ui for all a−i and x in the
support.
In particular, under assumption S, i.e. πi ( a−i , x, ui ) = β i ( a−i , x ) − ui , assumption C holds
trivially.
Fix X = x such that αi ( x ) ∈ (0, 1). By assumptions R, M and C, the conditional expected
∗ ( a | x, u ) is continuously decreasing in u . Then, we
payoffs ∑ a−i ∈A−i πi ( a−i , x, ui ) × σ−
i
i
i −i
can represent the equilibrium condition (2) by
0=
∑
∗
∗
πi ( a−i , x, ui∗ ( x )) × σ−
i ( a−i | x, ui ( x )),
a−i ∈A−i
21
(8)
∗ is known by Lemma 3. Next, we will exploit (8) for the identification of the
in which σ−
i
∗ (·| x, u∗ ( x )) while keeping π (·, x, u∗ ( x )) fixed, for which
payoffs πi . The idea is to vary σ−
i
i
i
i
we need the following exclusion restriction.
Assumption ER (Exclusion Restriction). Let X = ( X1 , · · · , X I ). For all i, a−i , x and ui , we
have πi ( a−i , x, ui ) = πi ( a−i , xi , ui ).15
In the context of discrete games, the identification power of exclusion restrictions was first
demonstrated in Pesendorfer and Schmidt-Dengler (2003), Tamer (2003), and was used by
Bajari, Hong, Krainer, and Nekipelov (2010) in a semiparametric setting.
∗ (·| X, u∗ ( X )) as Σ∗ ( X ), a
For notational simplicity, we denote the random vector σ−
i
i
−i
∗ ( a | x, u∗ ( x )) = 1, then Σ∗ ( X )
column vector of dimension 2 I −1 . By definition, ∑ a−i ∈A−i σ−
−i
i
i
−i
is distributed on a hyperplane in R2
I −1
. Given Lemma 3, we treat Σ∗−i ( X ) as observables
hereafter. Moreover, for each ( xi , αi ) ∈ SXi ,αi (X ) , let
h
i
Ri ( xi , αi ) ≡ E Σ∗−i ( X )Σ∗−i ( X )> Xi = xi , αi ( X ) = αi .
Conditional on Xi = xi and αi ( X ) = αi , suppose there is “no linear correlation” among
the variables in Σ∗−i ( X ). Then the matrix Ri ( xi , αi ) will have full rank 2 I −1 . By (8), full rank
of Ri ( xi , αi ( x )) implies that πi ( a−i , xi , ui∗ ( x )) = 0 for all a−i ∈ A−i , i.e., there is no strategic
interaction on player i in the equilibrium. In the next proposition, we give more detailed
identification results for M30 .
Proposition 6. Let S ∈ M30 . Suppose that assumptions SC, C and ER hold. Fix xi ∈ SXi
and αi ∈ Sαi (X )|Xi = xi
(0, 1). If Ri ( xi , αi ) has full rank 2 I −1 , then Sαi (X )|Xi =xi must be the
singleton {αi } and πi ·, xi , QUi (αi ) = 0. If Ri ( xi , αi ) has rank 2 I −1 − 1, then πi ·, xi , QUi (αi )
T
is identified up to scale. In addition, if there exists α0 ∈ Sα(X )|Xi = xi such that αi0 ∈ (0, 1), αi0 6= αi
and (αi , α0−i ) ∈ Sα(X ) ,16 then for each a−i ∈ A−i , the sign of πi a−i , xi , QUi (αi ) is also identified.
Proof. See Appendix C.2.
15As a matter of fact, X s can have some common variables. To simplify, we assume that X s partition X.
i
i
16The existence of such an α0 can be guaranteed if for some α , (α , α ) is an interior point in S
−i
i −i
α( X )| Xi = xi .
22
Proposition 6 shows that fixing xi , the nonseparable payoffs are identified as zero, or
identified up to scale and up to the marginal distributions of players’ types on an appropriate domain, which is essentially the support of the FUi -quantile associated with E(Yi | X )
controlling for Xi = xi . If Ri ( xi , αi ) has full rank 2 I −1 for some αi ∈ (0, 1), then the domain
has to be a singleton. On the other hand, if Ri ( xi , αi ) has rank 2 I −1 − 1 for all αi ∈ (0, 1),
more variations in E(Yi | X ) as X−i varies, the larger will be this domain.17
It is of particular interest to consider and identify the case where there are no strategic
interactions. Fix Xi = xi . Suppose player i with some type ui∗ is indifferent to other players’
moves, as well as her own action, i.e., πi ( a−i , xi , ui∗ ) = πi ( a0−i , xi , ui∗ ) = 0 for all a−i 6= a0−i .
Then, assumption M implies that player i’s best response has to be δi∗ ( x, ui ) = 1{ui ≤ ui∗ }
no matter what other players move. Given Xi = xi , player i’s equilibrium strategy must
always be 1{ui ≤ ui∗ }, as if she is in a single agent environment. On the other hand, the lack
of strategic effects can be detected by the full rank of Ri ( xi , αi ) for some αi ∈ (0, 1). Further,
from the discussion above, the latter implies that Sαi (X )|Xi = xi = {αi }, which is a restriction
on the model specification.
When there exist strategic effects on player i with the threshold type ui∗ ( x ) ∈ SU◦ |X = x ,
i
which implies that
πi ( a−i , xi , ui∗ ( x ))
6=
πi ( a0−i , xi , ui∗ ( x ))
for some a−i 6=
a0−i ,
then
2 I −1
−1
is the largest rank that the matrix Ri ( xi , αi ( x )) can possibly have. This is because of (8) —
after controlling for Xi = xi and αi ( X ) = FUi (ui∗ ( x )), the existence of a non–zero solution of
πi (·, xi , ui∗ ( x )) requires that Ri ( xi , αi ( x )) should not have full rank.
In the non–degenerate case, Proposition 6 shows the up–to–scale and sign identification
of πi (·, xi , QUi (αi )) for each αi ∈ Sαi (X )|Xi = xi
T
(0, 1) under the rank and additional support
conditions. It follows that that the signs of the strategic effects, i.e. πi ( a−i , xi , QUi (αi )) −
πi a0−i , xi , QUi (αi ) for all a−i 6= a0−i , are also identified. In a different setting, De Paula and
Tang (2012) develop a special approach for nonparametrically identifying the signs of the
strategic effects by exploiting the identification power of multiple equilibria. Our approach
does not require assumption I, but relies on assumptions E, ER while being applicable when
there is only one (m.d.p.s.) equilibrium.
17This domain excludes the boundaries Q (0) and Q (1) because (8) holds only for those x’s with α ( x ) ∈
Ui
Ui
i
(0, 1).
23
It should also be noted that we cannot compare the payoffs across different xi s and ui s.
This arises because the scale depends on xi in Proposition 6. In other words, the payoffs
as a function of xi are not identified. In addition, Proposition 6 is silent about the quantile
function QUi , which is not identified either. In contrast to the nonseparable single agent
binary response model discussed in Matzkin (2003), here the non–identification of πi is
essential, even if we specified some distribution function for FUi .
4.3. Identification of M3 . By definition, M3 is a sub–model of M30 . Under the additional
separability restriction (i.e., assumption S), we can achieve the full identification of the structural parameters under a single location–scale normalization on the payoffs. Assumption S
has been widely imposed in the literature, see Bajari, Hong, and Nekipelov (2010); de Paula
(2012). Combined with the exclusion restriction, the equilibrium condition (8) becomes
∑
∗
∗
β i ( a−i , xi ) × σ−
i ( a−i | x, ui ( x )) = QUi ( αi ( x )).
(9)
a−i ∈A−i
In (9), we can treat the unobserved thresholds ui∗ ( x ) as fixed effects and control them using
the marginal choice probability αi ( x ).
Similar to Matzkin (2003), our normalization is imposed on the payoffs function at
∗
some xi∗ ∈ SXi . To begin with, let Σ−i ( X ) ≡ Σ∗−i ( X ) − E Σ∗−i ( X )| Xi , αi ( X ) and Ri ( xi ) =
i
h
I −1
∗
∗
E Σ−i ( X )Σ−i ( X )> Xi = xi . Note that ι0 Σ∗−i ( X ) = 1 a.s., where ι ≡ (1, · · · , 1)0 ∈ R2 . It
∗
∗
follows that ι0 Σ−i ( X ) = 0. Thus, Σ−i ( X ) consists of a vector of linearly dependent variables.
Indeed, the largest possible rank of the matrix Ri ( xi ) is 2 I −1 − 1.
Lemma 8 in Appendix C.3 is similar to Proposition 6 imposing separability to obtain a
weaker rank condition in terms of Ri ( xi ).18 This is because (9) implies that
∑
n
∗
o
∗
∗
∗
β i ( a−i , xi ) · σ−
(
a
|
x,
u
(
x
))
−
E
σ
a
|
X,
u
(
X
))
|
X
=
x
,
α
(
X
)
=
α
(
x
)
= 0.
−
i
−
i
i
i
i
i
i
i
−i
i
a−i ∈A−i
In Lemma 8, we show that the lack of strategic effects case occurs, i.e., β i ( a−i , xi ) =
β i ( a0−i , xi ) for all a−i 6= a0−i when Ri ( xi ) has the largest possible rank 2 I −1 − 1; moreover, β i (·, xi ) is identified up to location and scale depending on xi when the rank of
T
18It can be shown that if the rank of R ( x , α ) equals k (1 ≤ k ≤ 2 I −1 ) for some α ∈ S
i i i
i
αi ( X )| Xi = xi (0, 1), then
the rank of Ri ( xi ) is no less than k − 1.
24
Ri ( xi ) is 2 I −1 − 2. Further, if Sα(X )|Xi =xi∗ contains two different elements α and α0 such
that αi , αi0 ∈ (0, 1), αi 6= αi0 and (αi , α0−i ) ∈ Sα(X ) , we identify the sign of strategic effects
β i ( a−i , xi ) − β i ( a0−i , xi ).
Assumption N (Payoff Normalization). For some xi∗ ∈ SXi satisfying (i) the rank of Ri ( xi∗ ) is
2 I −1 − 2; (ii) Sα(X )|Xi = xi∗ contains two different elements α and α0 such that αi , αi0 ∈ (0, 1), αi 6= αi0
and (αi , α0−i ) ∈ Sα(X ) , we set β i a0−i , xi∗ = 0 and k β i (·, xi∗ )k = 1.19
Let S ∈ M3 . Suppose that assumptions SC, ER and N hold. By Lemma 8, the payoffs
β i (·, xi∗ ) is point identified. By (9), QUi is identified on the support Sαi (X )|Xi = xi∗
QUi (αi ) =
∑
T
(0, 1), since
∗
∗
∗
β i ( a−i , xi∗ ) × E σ−
i ( a−i | X, ui ( X )) Xi = xi , αi ( X ) = αi .
(10)
a−i ∈A−i
Further, for each xi ∈ SXi , suppose that the rank of Ri ( xi ) equals 2 I −1 − 2, and that
Sαi (X )|Xi = xi
T
Sαi (X )|Xi = xi∗
T
(0, 1) contains two elements αi , αi0 ∈ (0, 1). By Lemma 8, β i (·, xi )
is identified up to location and scale. Note that the quantiles QUi (αi ) and QUi (αi0 ) are known
since αi , αi0 ∈ Sαi (X )|Xi = xi∗
T
(0, 1). Therefore, we can figured out the location and scale of
β i (·, xi ) from the following two equations
∑
∗
∗
β i ( a−i , xi ) × E σ−
i ( a−i | X, ui ( X ))| Xi = xi , αi ( X ) = αi = QUi ( αi );
∑
∗
∗
0
0
β i ( a−i , xi ) × E σ−
i ( a−i | X, ui ( X ))| Xi = xi , αi ( X ) = αi = QUi ( αi ).
a−i ∈A−i
a−i ∈A−i
Moreover, we can identify QUi on the support Sαi (X )|Xi ∈{ xi∗ ,xi }
T
(0, 1). Repeating such
an argument, we can show that β i (·, xi ) can be point identified for all xi s in a collection,
denoted as Ci∞ , and QUi is identified on the support Sαi (X )|Xi ∈Ci∞
T
(0, 1).
Definition 2. Let the subset Ci∞ in SXi be defined by the following iterative scheme. Let C0i = { xi∗ }.
Then, for all t ≥ 0, Cit+1 consists of all elements xi ∈ SXi such that at least one of the following
conditions is satisfied: (i) xi ∈ Cit ; (ii) Ri ( xi ) has rank 2 I −1 − 2 and there exists an xi0 ∈ Cit such
that Sαi (X )|Xi = xi
T
Sαi (X )|Xi = xi0
T
(0, 1) contains at least two different elements; and (iii) Ri ( xi )
has rank 2 I −1 − 1 and there exists an xi0 ∈ Cit such that Sαi (X )|Xi = xi ⊆ Sαi (X )|Xi = xi0
T
(0, 1).
19W.l.o.g., we set a0 = (0, · · · , 0) ∈ A . Note that k β (·, x ∗ ) − β ( a0 , x ∗ )k 6= 0 because of the non–degeneracy
−i
i
i −i i
i
−i
of the support Sαi ( X )| Xi = x∗ .
i
25
In view of Lemma 8 in the appendix, condition (ii) in Definition 2 corresponds to the case
where there are strategic effects. This case is the key to effectively expand the collection of
xi s in an iterative manner by enlarging Sαi (X )|Xi ∈Ct to Sαi (X )|Xi ∈Ct+1 .
i
i
Proposition 7. Let S ∈ M3 . Suppose assumptions SC, ER and N hold. Then β i and QUi are point
identified on the support A−i × Ci∞ and Sαi (X )|Xi ∈Ci∞
T
(0, 1), respectively.
Proof. See Appendix C.4
It is interesting to note that our identification argument does not apply to a nonparametric
single–agent binary response model, see e.g. Matzkin (1992).20 This is because the support
Sαi (X )|Xi = xi∗ is a singleton in a single–agent binary response model, i.e., we are always in the
case of condition (iii) in Definition 2. With interactions and exclusion restrictions, in contrast,
we can exploit variations of X−i while controlling for Xi to identify a set of quantiles of FUi .
Note that {Cit : t ≥ 1} is an expending sequence on the support of SXi , which ensures
that the limit Ci∞ is well defined (and may not be bounded if SXi is unbounded). The
domain and size of Ci∞ depend on the choice of xi∗ as well as the variation of Sαi (X )|Xi = xi
across different xi s. Regarding the choice of the starting point xi∗ , intuitively we should
choose it in a way such that Ci∞ is the largest. However, it can be shown that for any xi0
satisfying assumption N and xi0 ∈ Ci∞ , we will end up with the same Ci∞ .
The iterative mechanism might not be necessary when xi∗ is chosen in a way that has the
largest variations in player i’s marginal choice probability conditional on Xi .
Assumption N. (iii) Sαi (X )|Xi = xi ⊆ Sαi (X )|Xi = xi∗ for all xi ∈ SXi .
Corollary 2. Let S ∈ M3 . Suppose assumptions SC, ER and N (i) to (iii) hold. Then the results in
Proposition 7 hold, where Ci∞ = { xi ∈ SXi : Rank of Ri ( xi ) ≥ 2 I −1 − 2}.
Last, an alternative normalization to assumption N is to specify two quantiles of the
marginal distributions. Specifically, let τi1 , τi2 ∈ Sαi (X )|Xi = xi∗ (0, 1). Instead of choosing
β i a0−i , xi∗ = 0 and k β i (·, xi∗ )k = 1 in assumption N, we can set the quantiles QUi (τi1 )
T
and QUi (τi2 ) at some values, as long as (strict) monotonicity is satisfied. Proposition 7 still
20In single–agent binary response models, Matzkin (1992) establishes nonparametric identification results
under the separability condition and additional model restrictions (see e.g. assumptions W.2, W.4 and G.2).
26
holds. We may ask whether we can use the usual mean/variance normalization in latent
variable models. This is possible under a full support condition. Namely, suppose Ri ( xi∗ )
has rank 2 I −1 − 2 and (0, 1) ⊆ Sαi (X )|Xi = xi∗ . Then we can choose E(Ui ) = 0 and Var(Ui ) = 1.
Specifically, by Lemma 8, β i is identified up to location and scale. Hence, all the quantiles
QUi are identified up to location and scale by (10). The latter are determined by the mean
and variance normalization.
4.4. Identification of M4 . Model M4 is nonparametrically identified. The argument does
not essentially differ from that of M3 : assumption I only relaxes the support condition for
∗ in Lemma 3. We illustrate this in the next lemma.
identification of σ−
i
∗ (·| x, u∗ ( x )) is identified by
Lemma 4. Let S ∈ M4 . Fix x ∈ SX . Then σ−
i
i
∗
∗
σ−
i ( a−i | x, ui ( x )) = P (Y−i = a−i | X = x ) .
The proof is straightforward, hence omitted. By the proofs for Proposition 3, we can also
∗ ( a | x, u∗ ( x )) = P (Y = a | α ( X ) = α ( x )). Similar results can be found in
show that σ−
−i
−i
i −i
i
Aguirregabiria and Mira (2007); Bajari, Hong, Krainer, and Nekipelov (2010), among others.
Further, the identification of β i and QUi in M4 follows Lemma 8 and Proposition 7.
5. D ISCUSSION
In this section, we consider four issues related to our identification analysis. First, without
assumption ER, we examine the partial identification of structures in M30 and M3 . Second,
we study the case in which each player’s type is a multiple dimensional random vector,
which is of particular interest in some IO applications. Third, we relax the single equilibrium
assumption, i.e., we allow multiple m.d.p.s. BNEs in the DGP, under which the observed
data is a mixture of distributions from all these equilibria. Fourth, we discuss the issue of
unobserved heterogeneity.
5.1. Partial Identification. It is worth emphasizing that the lack of point identification of a
structure here is not due to multiple equilibria, but to the lack of identifying restrictions, i.e.,
the exclusion restrictions and the rank conditions. This is similar to, e.g., Shaikh and Vytlacil
(2011) who study partial identification of the average structural function in a triangular
27
model without imposing a restrictive support condition. When there are multiple monotone
pure strategy BNEs, we still maintain the assumption of a single equilibrium being played
for generating the distribution of observables.
By the same argument as for the identification of M30 , the copula function CU is point–
identified on the extended support Sαe(X ) . Let C be the set of strictly increasing (on (0, 1] I )
and continuously differentiable copula functions mapping [0, 1] I to [0, 1]. Then, the identification region of CU can be characterized by
n
o
eU ∈ C : C
eU (α) = CU (α), ∀α ∈ S e
CI = C
α( X ) .
eU ∈ C I , suppose we set FeU to be the uniform distribution on [0, 1] and π
ei (·, x, ui ) =
For each C
i
e FeU is observationally equivalent to the
αi ( x ) − ui . Clearly, the constructed structure π;
underlying structure. Thus, C I is the sharp identification region for CU .
Next, we turn to the set identification of the quantile function QUi . By assumption R, QUi
belongs to the set of strictly increasing and continuously differentiable functions mapping
[0, 1] to R, denoted as Q. The next lemma shows that M3 (or M30 ) imposes no restrictions
on QUi and its identification region is Q.
eU , · · · , Q
eU ) ∈ Q I , there exists an observationally
Lemma 5. Let S ∈ M3 (or M30 ). For any ( Q
I
1
eU , · · · , Q
e U ).
equivalent structure Se ∈ M3 (or M30 ) with the marginal quantile function profile ( Q
I
1
Proof. See Appendix D.1
Now we discuss the sharp identification region for πi . Let G be the set of functions
mapping A−i × SX to R.
Proposition 8. Let S ∈ M3 . Suppose assumption SC holds. Then the sharp identification region
n
o
e {Q
eU } I ; C
eU : ( Q
eU , C
eU ) ∈ (Q, C I ), βe ∈ Θ I { Q
eU } I , C
eU , where
is given by β;
i i =1
i
i i =1
eU } I , C
eU ) ≡
Θ I ({ Q
i i =1
eU (αi ( x )) =
Q
i
n
βe ∈ G I : ( a) for all x ∈ SX and i,
∑
∗
∗
βei ( a−i , x ) × σ−
i ( a−i | x, ui ( x )); ( b ) for any m.d.p.s. profile δ :
a−i ∈A−i
h
i
o
e U ( αi ) − Q
eU (αi ) is decreasing in αi ∈ (0, 1) .
Eδ βei (Y−i , X ) | X = x, Ui = Q
i
i
28
Proof. See Appendix D.2.
In the definition of Θ I , condition (a) requires that β i (·, x ) should belong to a hyperplane,
for which the slopes are given by the identified beliefs Σ∗−i ( x ); condition (b) is weak as it
does not impose much restriction on the structural parameters. Clearly, Θ I is nonempty
and convex.21
The identification region is unbounded and quite large. To see this, fix an arbitrary non–
negative function κi ( x ) ≥ 0. Let ψi : R → R satisfy: (i) ψi is a continuously differentiable
and monotone increasing function; and (ii) for all x, κi ( x ) QUi (αi ) − ψi ( QUi (αi )) is monotone decreasing in αi ∈ (0, 1). Note that condition (ii) is equivalent to: infui ∈SUi ψi0 (ui ) ≥
eU =
supx∈SX κi ( x ). Clearly, there are plenty of choices for such a function ψi . Let further Q
i
ψi ( QUi ) and β̃ i ( a−i , x ) = ξ i ( x ) + κi ( x ) × β i ( a−i , x ), in which ξ i ( x ) = ψi ( QUi (αi ( x ))) −
e {Q
eU } I ; C
eU
κi ( x ) × QU (αi ( x )). Then, it can be verified that the constructed structure [ β;
i
i
i =1
belongs to the identified set. To narrow down the identification region, additional restrictions need to be introduced. Instead of imposing assumption ER, an alternative approach
is to make assumptions on the payoffs’ functional form. For instance, De Paula and Tang
(2012) set β i ( a−i , x ) = β∗i ( x ) + gi ( a−i ) × hi∗ ( x ), where gi is a function known to all players
as well as to the econometrician, and ( β∗i , hi∗ ) are structural parameters in their model.
When Sα(X ) = (0, 1) I , C I degenerates to the singleton {CU }. In this case, the sharp identification region for ( β, { QUi }iI=1 ) can be characterized in a more straightforward manner:
n
e {Q
eU } I ) ∈ G I × Q I : ( a0 ) for all x ∈ SX and i,
Θ∗I = ( β,
i i =1
eU (αi ( x )) =
Q
i
∑
∗
∗
0
I −1
βei ( a−i , x ) × σ−
,
i ( a−i | x, ui ( x )); ( b ) and for all α−i ∈ [0, 1]
a−i ∈A−i
∑
o
α
e
βei ( a−i , x ) × σ−
i ( a−i , α−i , αi ) − QUi ( αi ) is decreasing in αi ∈ (0, 1) ,
a−i ∈A−i
α ( a , α , α ) = P (C ≤ α ∀ a = 1; C > α ∀ a = 0|C = α ).
where σ−
CU
j
j
j
j
j
j
i
i
i −i −i i
For M30 , we can similarly characterize the sharp identification region for the structural
parameters. Let G 0 be the set of functions mapping A−i × SX × R to R.
21 To see the nonemptiness, we can simply take β
ei (·, x ) = Q
eU (αi ( x )).
i
29
Proposition 9. Let S ∈ M30 . Suppose assumption SC holds. Then the sharp identification region
nh
i
o
0 ({ Q
eU } I , C
eU : ( Q
eU , C
eU ) ∈ (Q, C I ), π
eU } I , C
eU ) , in which
e {Q
e
is given by π,
∈
Θ
i i =1
i
i i =1
I
n
I
eU } I , C̃U ) = π
e ∈ G 0 : for all x ∈ SX and i,
Θ0I ({ Q
i i =1
eU (αi ( x )) × σ∗ ( a−i | x, u∗ ( x )); and for any m.d.p.s. profile δ
ei a−i , x, Q
0= ∑ π
−i
i
i
a−i ∈A−i
h i
o
eU (αi ) | X = x, Ui = Q
eU (αi ) is decreasing in αi ∈ (0, 1) .
ei Y−i , X, Q
Eδ π
i
i
The proof is similar to that of Proposition 8, and therefore omitted.
5.2. Multi–dimensional Signals. In some applications, players may have multi-dimensional
signals. For example, a potential entrant to some local market could individually observe
idiosyncratic demand shocks, as well as her private cost information.
Without loss of generality, let Ui ∈ Rd , where d ≥ 2. Let T be the class of functions
mapping Rd to R and
n
o
ei (·, ·, Ti (ui )) for some π
ei .
TS = ( T1 , · · · , TI ) ∈ T I : πi (·, ·, ui ) = π
Let Vi ≡ Ti (Ui ). The idea is to apply m.d.p.s. BNEs on the transformed scalar type Vi .
Under additional conditions below, we show that there exists a BNE in which each player’s
strategy δi∗ is a monotone decreasing function of Vi .
It is worth pointing out that the collection TS is always non–empty. By general topology,
there exists a collection of functions that are canonical bijections from Rd to R, i.e., one–
to–one and onto mappings from Rd to R. This is true because the space Rd and R have
the same cardinality — a genuine measure for the “number of elements of a set”.22 For
ei (·, ·, vi ) = πi (·, ·, Ti−1 (vi )) for the condition in the
any bijection Ti , we can simply choose π
definition of TS . Note, however, that we are not requiring Ti to be a bijection in TS .
Assumption T. There exists some (possibly unknown) function profile ( T1∗ , · · · , TI∗ ) ∈ TS satisfying
(1) Ti∗ is measurable and FV |XUi = FV |XVi , where Vi = Ti∗ (Ui ) and V = (V1 , · · · , VI ).
22Following the Cantor–Schroeder–Bernstein Theorem, we could explicitly construct one example of such a
mapping. It is worth pointing out that the constructed mapping is not homeomorphic due to discontinuity.
30
(2) For every x ∈ SX , the conditional distribution of V given X = x is absolutely continuous
w.r.t. Lebesgue measure and has a continuous positive conditional Radon–Nikodym density
f V |X (·| x ) a.e. over the nonempty interior of its hypercube support SV |X = x .
(3) Fix any x ∈ SX . For any pure strategy profile δ in which δj (j = 1, · · · , I) is monotone
ei (Y−i , X, Vi ) X = x, Vi = vi is also monotone decreasing
decreasing in v j ≡ Tj∗ (u j ), Eδ π
in vi .
In assumption T, conditions (2) and (3) are analogous to assumption R and M, respectively.
Condition (1) requires that the dependence of types U be fully captured by the dependence
of the transformed types V, for which the measurability of Ti∗ is needed.23
Under assumption T, we can shift our focus from the original structure [π, { QUi }iI=1 , CU ]
eV } I , C
eV ], which admits an m.d.p.s. (monotone decreasing in the
e {Q
in M3 (or M30 ) to [π,
i i =1
transformed type Vi ) BNE by Lemma 1. If Ti∗ is a bijection for all i. Then, the structures
eV } I , C
eV ] are also in a bijective relationship to each other. Let
e {Q
[π, { QUi }iI=1 , CU ] and [π,
i i =1
n
o
f3 ≡ Se : assumptions T, S and E hold and a single m.d.p.s. (in Vi ) BNE is played .
M
f3 is identified under additional
Similar to Lemma 8 and proposition 7, we can show that M
identifying conditions.
In some applications, Ti∗ might be a known function, or known up to some structural
parameters. For example, suppose Ui = (UiD , UiC ) ∈ R2 and πi ( a−i , x, ui ) = β i ( a−i , x ) +
γi uiD − uiC . In addition, suppose one assumes Fγ N U D −U C |X,U D ,U C = Fγ N U D −U C |X,γi U D −U C .
i
i
i
i
Then it follows naturally that Ti∗ (ui ) = γi uiD − uiC , a transformation parameterized by γi .
Though we don’t know γi , it’s still possible to identify the function β i in player i’s payoffs.
5.3. Multiple Equilibria in DGP. Our (point and partial) identification analysis relies on
the maintained assumption that data come from only a single m.d.p.s. BNE for each x, as
it is common in the empirical game literature to rule out mixtures of distributions from
multiple equilibria, e.g., Aguirregabiria and Mira (2007) and Bajari, Hong, Krainer, and
Nekipelov (2010).
23We thank Nese Yildiz for pointing this out.
31
To relax this assumption, we follow Henry, Kitamura, and Salanié (2010) by introducing
an instrumental variable Z, which does not affect players’ payoffs, the distribution of types,
or the set of equilibria in the game, but can effectively change the equilibrium selection. For
each x ∈ SX , let E ( x ) be the set of equilibria in the game with X = x. Note that E ( x ) could
be an infinite collection and the number of equilibria depend on the value of x. To simplify,
we assume that players only focus on a subset Γ( x ) of E ( x ) for the DGP, i.e., the set of
equilibria that will be played in the data. We further assume that the number of elements in
Γ( x ) is finite and bounded above by a constant J (J ≥ 2) for all x.
Let λ be a probability distribution { p1λ , · · · , pλJ } on the support {1, · · · , J } such that
the j–th equilibrium occurs with probability pλj . λ can be degenerate, which means that
the number of equilibria in Γ( x ) is strictly less than J. Essentially, λ summarizes the
mixture of equilibrium distributions due to the equilibrium selection mechanism. Following
Henry, Kitamura, and Salanié (2010), we assume that λ = λ( X, Z ), where Z is a vector of
instrumental variables that does not affect either E ( X ) or Γ( X ), but has influence on the
equilibrium selection and λ.
In Henry, Kitamura, and Salanié (2010), it is shown that the set of component distributions
are partially identified in the space of probability distributions. For example, for J = 2, the
observed distribution FY |X = x is a convex combination of the two component distributions
generated from the two equilibria in Γ( x ). Then, variations of the instrumental variable
Z cause the mixture distribution to move along a straight line in the function space of
probability distributions.
Further, we can point identify the set of distributions corresponding to Γ( x ) if Z has
sufficient variations. To see this, w.l.o.g. let J = 2. For each x ∈ SX , suppose λ( x, z) = (0, 1)
and λ( x, z0 ) = (1, 0) for some z, z0 ∈ SZ|X = x . In the space of probability distributions,
therefore, the two equilibrium distributions can be identified as the two extreme points
of the convex hull (which is a straight line) of the collection of distributions FY |X = x,Z=z
for all z ∈ SZ|X = x . Either one of them represents a probability distribution from a single
m.d.p.s. BNE for given x, which thereafter provides the identification of the underlying
game structure as we discussed in the identification section.
32
5.4. Unobserved heterogeneity. Within a paradigm where private signals are independent
unconditionally or conditionally given X, a well–known approach for obtaining correlation
among actions given X is to introduce unobserved heterogeneity. In a fully parametric
setting, Aguirregabiria and Mira (2007) and Grieco (2013) introduce unobserved heterogeneity through some payoff relevant variables ζ publicly observed by all players, but not
by the researcher. An important question is whether one can distinguish this model from
our model with correlated types. Because M1 and M2 can rationalize any distributions
generated by a model with unobserved heterogeneity and independent types, we consider
below M3 or M30 .
Consider the following payoffs with unobserved heterogeneity:

 π (Y , X, ζ, U ), if Y = 1,
i −i
i
i
Πi (Y, X, ζ, Ui ) =

0,
if Y = 0,
i
where ζ is a discrete variable that is unobserved to the researcher. Let the support of ζ
be {z1 , · · · , z J } and p j ( x ) = P(ζ = z j | X = x ). To simplify, we assume there are only two
players with U1 ⊥U2 and (U1 , U2 )⊥( X, ζ ). We assume further that there is only a single
equilibrium in the DGP for every ( x, z j ) ∈ SXζ , which is also assumed in our model M3
(or M30 ). Then, the joint choice probability for the model with unobserved heterogeneity
and independent types is
J
E(Y1 Y2 | X = x ) =
∑ E(Y1Y2 |X = x, ζ = z j ) · p j (x) =
j =1
J
∑ α1 (x, z j ) · α2 (x, z j ) · p j (x)
j =1
where αi ( x, z j ) = E(Yi | X = x, ζ = z j ). Moreover, the marginal choice probability is
J
αi ( x ) =
∑ αi (x, z j ) · p j (x),
for i = 1, 2,
j =1
where αi ( x ) = E(Yi | X = x ).
We now argue that these joint and marginal choice probabilities can violate restriction R1
in Proposition 3. Specifically, R1 requires that E(Y1 Y2 | X = x ) be a function of (α1 ( x ), α2 ( x ))
only. The model with unobserved heterogeneity and independent types does not exclude the
possibility that there exist x, x 0 ∈ SX such that αi ( x ) = αi ( x 0 ) for i = 1, 2, but E(Y1 Y2 | X =
33
x ) 6= E(Y1 Y2 | X = x 0 ). For instance, suppose J = 2, p j ( x ) = p j ( x 0 ) = 0.5 for j = 1, 2,
α1 ( x, z1 ) = α2 ( x, z1 ) = 0.4, α1 ( x, z2 ) = α2 ( x, z2 ) = 0.6, α1 ( x 0 , z1 ) = 0.3, α2 ( x 0 , z1 ) = 0.7,
α1 ( x 0 , z2 ) = 0.7, α2 ( x 0 , z2 ) = 0.3. Therefore, α1 ( x ) = α1 ( x 0 ) = 0.5 and α2 ( x ) = α2 ( x 0 ) = 0.5,
but E(Y1 Y2 | X = x ) = 0.26 6= E(Y1 Y2 | X = x 0 ) = 0.21. Consequently, the model with
unobserved heterogeneity and independent types can be distinguished from model M3 (or
M30 ). We have focused on R1 above, but monotonicity in αi ( X ) (see R2) could be violated
as well for similar reasons.
6. C ONCLUSION
This paper studies the rationalization and identification of discrete games with correlated
types within a fully nonparametric framework. Allowing for correlation across types
is important structurally as it represents homophily in models with social interactions.
Regarding rationalization, we show that our baseline game–theoretical model M1 with a
single m.d.p.s. BNE in the DGP does not impose any essential restrictions on observables,
and hence is not testable in view of players’ choice probabilities only. We also show that
separability is not testable in M2 or M3 , i.e., in the models without or with exogeneity.
On the other hand, we can test exogeneity with or without separability, because R1–R3
in Proposition 3 characterize all the restrictions imposed by exogeneity. For instance, we
can view R1 as a regression of the joint choices on covariates that depends on the latter
only through the marginal choice probabilities. Thus, to test R1 we can extend Fan and Li
(1996) and Lavergne and Vuong (2000) significance tests by allowing for estimation of the
marginal choice probabilities.
Model M4 is mostly adopted in empirical work within a parametric or semiparametric
setting. We show that all its restrictions reduce to the mutual independence of choices
conditional on covariates. This can be tested by using conditional independence tests, see
e.g. Su and White (2007, 2008). Moreover, Proposition 4 and Corollary 1 show that the same
restriction characterizes model M40 which only assumes mutual independence of types
conditional on covariates and a single (not necessary monotone) BNE in the DGP. These two
assumptions seem to be unrelated, but actually are two sides of the same coin. Maintaining
a single equilibrium in the DGP, we can use the mutual independence of choices given
34
covariates to test mutual independence of types which is widely assumed in the literature.
On the other hand, maintaining mutual independence of types, we can use the same mutual
independence of choices to test for a single equilibrium versus multiple equilibria. See, e.g.,
De Paula and Tang (2012).
It is worth noting that the above tests do not rely on identification and consequently
on the assumptions used to identify the primitives of the various models. In particular,
we show that model M3 is identified up to a single location–scale normalization under
exclusion restrictions, rank conditions and a non–degenerate support condition. The
exclusion restrictions take the form of excluding part of a player’s payoff shifters from all
other players’ payoffs as frequently assumed in the literature. Specifically, the dependence
of players’ joint choices on the marginal choice probabilities identifies the dependence
across types, while the dependence of a player’s marginal choice probabilities on her
equilibrium beliefs identifies her payoffs. Without exclusion restrictions, we show that the
sharp identification region of players’ payoffs is unbounded. If one is interested in the signs
of strategic effects only, then we show that they are identified under slightly stronger rank
conditions in an nonseparable extension of M3 , i.e. model M30 .
Our identification results are useful for estimation of social interaction models. In a
semiparametric setup, Liu and Xu (2012) propose an estimation procedure for our model
M3 with linear payoffs, and establish the root–n consistency of the linear payoff coefficients.
A fully nonparametric estimation, however, deserves to be developed. A strategy could
rely on the identification results and propose sample–analog estimators for the players’
payoffs and the joint distribution of private information. The equilibrium condition (9) is
the key estimating equation. It has the nice feature to be partially linear, namely, linear
in the payoffs and nonparametric in the quantile. A difficulty is to take into account the
estimation of the beliefs of the player at the margin and the marginal choice probabilities.
Part of the problem could be addressed by using the recent literature on nonparametric
regression with generated covariates, see e.g. Mammen, Rothe, and Schienle (2012). An
important question is to determine the optimal (best) rate at which the primitives of M3
can be estimated from players’ choices. Proposition 3 which characterizes all the restrictions
imposed by M3 can be useful, see e.g. Guerre, Perrigne, and Vuong (2000).
35
R EFERENCES
A GUIRREGABIRIA , V.,
AND
P. M IRA (2007): “Sequential estimation of dynamic discrete
games,” Econometrica, 75(1), 1–53.
A RADILLAS -L OPEZ , A. (2010): “Semiparametric estimation of a simultaneous game with
incomplete information,” Journal of Econometrics, 157(2), 409–431.
A RADILLAS -L OPEZ , A., AND E. TAMER (2008): “The identification power of equilibrium in
simple games,” Journal of Business and Economic Statistics, 26(3), 261–283.
ATHEY, S. (2001): “Single crossing properties and the existence of pure strategy equilibria
in games of incomplete information,” Econometrica, 69(4), 861–889.
ATHEY, S., AND P. A. H AILE (2002): “Identification of standard auction models,” Econometrica, 70(6), 2107–2140.
(2007): “Nonparametric approaches to auctions,” Handbook of Econometrics, 6, 3847–
3965.
A UMANN , R. (1987): “Correlated equilibrium as an expression of Bayesian rationality,”
Econometrica: Journal of the Econometric Society, pp. 1–18.
B AJARI , P., H. H ONG , J. K RAINER , AND D. N EKIPELOV (2010): “Estimating static models of
strategic interactions,” Journal of Business and Economic Statistics, 28(4), 469–482.
B AJARI , P., H. H ONG ,
AND
D. N EKIPELOV (2010): “Game theory and econometrics: a
survey of some recent research,” in Econometric Society World Congress in Shanghai.
B ERRY, S. (1992): “Estimation of a model of entry in the airline industry,” Econometrica:
Journal of the Econometric Society, pp. 889–917.
B ERRY, S., AND E. TAMER (2006): “Identification in models of oligopoly entry,” Econometric
Society Monographs, 42, 46.
B JORN , P., AND Q. V UONG (1984): “Simultaneous equations models for dummy endogenous
variables: a game theoretic formulation with an application to labor force participation,”
working Papers.
B JORN , P. A.,
AND
Q. H. V UONG (1985): “Econometric modeling of a Stackelberg game
with an application to labor force participation,” California Institute of Technology, Division
of the Humanities and Social Sciences Working Papers.
36
B RESNAHAN , T., AND P. R EISS (1990): “Entry in monopoly market,” The Review of Economic
Studies, 57(4), 531.
B RESNAHAN , T. F., AND P. C. R EISS (1991): “Empirical models of discrete games,” Journal of
Econometrics, 48(1-2), 57–81.
B ROCK , W., AND S. D URLAUF (2001): “Discrete choice with social interactions,” The Review
of Economic Studies, 68(2), 235–260.
(2007): “Identification of binary choice models with social interactions,” Journal of
Econometrics, 140(1), 52–75.
C ARNEIRO , P.,
AND
S. L EE (2009): “Estimating distributions of potential outcomes using
local instrumental variables with an application to changes in college enrollment and
wage inequality,” Journal of Econometrics, 149(2), 191–208.
C HERNOZHUKOV, V., AND C. H ANSEN (2005): “An IV model of quantile treatment effects,”
Econometrica, 73(1), 245–261.
C HESHER , A. (2003): “Identification in nonseparable models,” Econometrica, 71(5), 1405–
1441.
(2005): “Nonparametric identification under discrete variation,” Econometrica, 73(5),
1525–1550.
C ILIBERTO , F., AND E. TAMER (2009): “Market structure and multiple equilibria in airline
markets,” Econometrica, 77(6), 1791–1828.
D ARSOW, W., B. N GUYEN , AND E. O LSEN (1992): “Copulas and Markov processes,” Illinois
Journal of Mathematics, 36(4), 600–642.
DE
PAULA , Á. (2012): “Econometric analysis of games with multiple equilibria,” Annual
Review of Economics, 5(1).
D E PAULA , A., AND X. TANG (2012): “Inference of signs of interaction effects in simultaneous games with incomplete information,” Econometrica, 80(1), 143–172.
FAN , Y.,
AND
Q. L I (1996): “Consistent model specification tests: omitted variables and
semiparametric functional forms,” Econometrica: Journal of the Econometric Society, pp.
865–890.
F UDENBERG , D., AND J. T IROLE (1991): Game Theory. MIT Press, Cambridge, MA.
37
G RIECO , P. (2013): “Discrete games with flexible information structures: an application to
local grocery markets,” RAND Journal of Economics, Forthcoming.
G UERRE , E., I. P ERRIGNE , AND Q. V UONG (2000): “Optimal nonparametric estimation of
first–price auctions,” Econometrica, 68(3), 525–574.
H ECKMAN , J.,
AND
E. V YTLACIL (2005): “Structural equations, treatment effects, and
econometric policy evaluation,” Econometrica, 73(3), 669–738.
H ECKMAN , J. J.,
AND
E. J. V YTLACIL (1999): “Local instrumental variables and latent
variable models for identifying and bounding treatment effects,” Proceedings of the National
Academy of Sciences, 96(8), 4730–4734.
H ENRY, M., Y. K ITAMURA , AND B. S ALANIÉ (2010): “Identifying finite mixtures in econometric models,” working paper.
J IA , P. (2008): “What happens when Wal-Mart comes to town: an empirical analysis of the
discount retailing industry,” Econometrica, 76(6), 1263–1316.
J UN , S. J., J. P INKSE , AND H. X U (2011): “Tighter bounds in triangular systems,” Journal of
Econometrics, 161(2), 122–128.
K LINE , B. (2012): “Identification of complete information games,” working paper.
K OOREMAN , P. (1994): “Estimation of econometric models of some discrete games,” Journal
of Applied Econometrics, pp. 255–268.
L AFFONT, J.,
AND
Q. V UONG (1996): “Structural analysis of auction data,” The American
Economic Review, 86(2), 414–420.
L AVERGNE , P., AND Q. V UONG (2000): “Nonparametric significance testing,” Econometric
Theory, 16(4), 576–601.
L EHMANN , E. (1955): “Ordered families of distributions,” The Annals of Mathematical
Statistics, 26(3), 399–419.
L EWBEL , A., AND X. TANG (2012): “Identification and estimation of discrete Bayesian games
with multiple equilibria using excluded regressors,” Discussion paper, working paper,
Boston College.
L I , T., AND X. Z HENG (2009): “Entry and competition effects in first-price auctions: theory
and evidence from procurement auctions,” The Review of Economic Studies, 76(4), 1397–
1429.
38
L IU , P. N.,
AND
H. X U (2012): “Semiparametric analysis of binary games of incomplete
information,” working Paper.
M AMMEN , E., C. R OTHE ,
AND
M. S CHIENLE (2012): “Nonparametric regression with
nonparametrically generated covariates,” The Annals of Statistics, 40(2), 1132–1170.
M ARMER , V., A. S HNEYEROV,
AND
P. X U (2013): “What model for entry in first-price
auctions? A nonparametric approach,” Journal of Econometrics.
M ATZKIN , R. (1992): “Nonparametric and distribution-free estimation of the binary threshold crossing and the binary choice models,” Econometrica: Journal of the Econometric Society,
pp. 239–270.
(2003): “Nonparametric estimation of nonadditive random functions,” Econometrica,
71(5), 1339–1375.
M C A DAMS , D. (2003): “Isotone equilibrium in games of incomplete information,” Econometrica, 71(4), 1191–1214.
M C P HERSON , M., L. S MITH -L OVIN , AND J. M. C OOK (2001): “Birds of a feather: Homophily
in social networks,” Annual review of sociology, pp. 415–444.
M ILGROM , P. R., AND R. J. W EBER (1982): “A theory of auctions and competitive bidding,”
Econometrica: Journal of the Econometric Society, pp. 1089–1122.
P ESENDORFER , M., AND P. S CHMIDT-D ENGLER (2003): “Identification and estimation of
dynamic games,” Discussion paper, National Bureau of Economic Research.
P ORTER , R. H. (1995): “The role of information in US offshore oil and gas lease auction,”
Econometrica, 63(1), 1–27.
P OWELL , J., J. S TOCK , AND T. S TOKER (1989): “Semiparametric estimation of index coefficients,” Econometrica: Journal of the Econometric Society, pp. 1403–1430.
R ENY, P. (2011): “On the existence of monotone pure–strategy equilibria in Bayesian games,”
Econometrica, 79(2), 499–553.
R ILEY, J. G.,
AND
W. F. S AMUELSON (1981): “Optimal auctions,” The American Economic
Review, 71(3), 381–392.
R OCKAFELLAR , R. T. (1997): Convex analysis, vol. 28. Princeton university press.
S EIM , K. (2006): “An empirical model of firm entry with endogenous product-type choices,”
The RAND Journal of Economics, 37(3), 619–640.
39
S HAIKH , A.,
AND
E. V YTLACIL (2011): “Partial identification in triangular systems of
equations with binary dependent variables,” Econometrica, 79(3), 949–955.
S OETEVENT, A., AND P. K OOREMAN (2007): “A discrete-choice model with social interactions: with an application to high school teen behavior,” Journal of Applied Econometrics,
22(3), 599–624.
S U , L., AND H. W HITE (2007): “A consistent characteristic function-based test for conditional
independence,” Journal of Econometrics, 141(2), 807–834.
(2008): “A nonparametric Hellinger metric test for conditional independence,”
Econometric Theory, 24(4), 829.
S WEETING , A. (2009): “The strategic timing incentives of commercial radio stations: An
empirical analysis using multiple equilibria,” The RAND Journal of Economics, 40(4),
710–742.
TAMER , E. (2003): “Incomplete simultaneous discrete response model with multiple equilibria,” The Review of Economic Studies, 70(1), 147–165.
TANG , X. (2010): “Estimating simultaneous games with incomplete information under
median restrictions,” Economics Letters, 108(3), 273–276.
V ICKREY, W. (1961): “Counterspeculation, auctions, and competitive sealed tenders,” The
Journal of finance, 16(1), 8–37.
V IVES , X. (1990): “Nash equilibrium with strategic complementarities,” Journal of Mathematical Economics, 19(3), 305–321.
WAN , Y.,
AND
H. X U (2010): “Semiparametric estimation of binary decision games of
incomplete information with correlated private signals,” working paper.
W HITE , H., H. X U ,
AND
K. C HALAK (2011): “Causal discourse in a game of incomplete
information,” working paper.
X U , H. (2011): “Social interactions: a game theoretic approach,” working paper.
X U , H. (2014): “Estimation of discrete games with correlated types,” The Econometrics
Journal.
40
A PPENDIX A. E XISTENCE OF M . D . P. S . BNE S
A.1. Proof of Lemma 1. First, assumptions G1–G6 of Reny (2011) are satisfied in our discrete game
under assumption R. Moreover, by assumption M, when other players employ monotone decreasing
pure strategies, player i’s best response is also a joint–closed set of monotone decreasing pure
strategies. By Reny (2011, Theorem 4.1), the conclusion follows.
A.2. Existence of monotone pure strategy BNEs under primitive conditions.
Definition 3. a set A ⊆ Rd is upper if and only if its indicator function is non–decreasing, i.e., for any
x, y ∈ Rd , x ∈ A and x ≤ y imply y ∈ A, where x ≤ y means xi ≤ yi for i = 1, · · · , d.
Assumption PR (Positive Regression Dependence). For any x ∈ SX and any upper set A ⊆ R I −1 , the
conditional probability P (U−i ∈ A| X = x, Ui = ui ) is non–decreasing in ui ∈ SUi | X = x .
Assumption SCP (Strategic Complement Payoffs). For any x ∈ SX and ui ∈ SUi | X = x , suppose
a−i ≤ a0−i , then πi ( a−i , x, ui ) ≤ πi ( a0−i , x, ui ).
Assumption DP (Non-increasing Payoffs). ∀ i = 1, · · · , I and ∀ ( a−i , x ) ∈ A−i × SX , πi ( a−i , x, ·) is
non-increasing in ui ∈ SUi | X = x .
Lemma 6. Suppose assumptions R, PR, SCP, and DP hold. For any x ∈ SX , there exists an m.d.p.s. BNE.
Proof. By Lemma 1, it suffices to show that assumption M holds. Fix x ∈ SX . Given an arbitrary
m.d.p.s. profile: for i = 1, · · · , I, δi ( x, ui ) = 1[ui ≤ ui ( x )], where ui (·) is arbitrarily given. By
assumptions PR and SCP, and Lehmann (1955), for any ui < ui0 in the support, we have
Eδ πi (Y−i , X, ui )| X = x, Ui = ui0 ≤ Eδ [πi (Y−i , X, ui )| X = x, Ui = ui ] .
Further, by assumption DP,
Eδ πi (Y−i , X, ui0 )| X = x, Ui = ui0 ≤ Eδ πi (Y−i , X, ui )| X = x, Ui = ui0 .
Thus, Eδ [πi (Y−i , X, Ui )| X = x, Ui = ui ] is a non–increasing function of ui .
A.3. Proof of Lemma 2. Fix X = x. By assumption S, there is Eδ∗ πi (Y−i , X, Ui ) X = x, Ui = ui =
∗ ( a | x, u ) − u . Because σ ∗ ( a | x, u ) are continuous in u under assumption R,
∑ a−i β i ( a−i , x )σ−
i
i
i
i
i −i
−i −i
∗ ( a | x, u ) − u is a continuously decreasing function in u .
then ∑ a−i β i ( a−i , x )σ−
i
i
i
i −i
Suppose ui ( x ) < ui∗ ( x ) < ui ( x ). It follows that
∑a−i βi (a−i , x)σ−∗ i (a−i |x, ui∗ (x)) − ui∗ (x) = 0.
41
Hence, conditional on ui ( X ) < ui∗ ( X ) < ui ( X ), we have
"
Yi = 1 [Ui ≤
ui∗ ( X )]
= 1 Ui ≤ ∑
a −i
#
∗
β i ( a−i , X )σ−
i
a−i | X, ui∗ ( X )
.
∗ ( a | x, u ( x )) − u ( x ) ≥ 0, which implies that
Suppose ui∗ ( x ) = ui ( x ). Then ∑ a−i β i ( a−i , x )σ−
i
i
i −i
conditional on ui∗ ( X ) = ui ( X ), there is
"
Yi = 1 [Ui ≤ ui ( X )] ≤ 1 Ui ≤
∑
a −i
#
∗
β i ( a−i , X )σ−
i
a−i | X, ui ( X )
.
Because 1 [Ui ≤ ui ( X )] = 1 a.s., thus
"
Yi = 1 [Ui ≤ ui ( X )] = 1 Ui ≤
∑
a −i
#
∗
β i ( a−i , X )σ−
i
a−i | X, ui ( X )
a.s.
Similar arguments hold for the case ui∗ ( X ) = ui ( X ).
A PPENDIX B. R ATIONALIZATION
B.1. Proof of Proposition 1. Prove only if part first: Proofs by contradiction. Let FY | X be rationalized
by M1 , i.e., some S ∈ M1 can generate FY | X . Fix X = x and let equilibrium be characterized by
(u1∗ ( x ), · · · , u∗I ( x )). For some a ∈ A, w.l.o.g., a = (1, · · · , 1), suppose P(Y = a| X = x ) = 0 and
P(Yi = ai | X = x ) > 0 for all i. It follows that P(U1 ≤ u1∗ ( x ), · · · , U I ≤ u∗I ( x )| X = x ) = 0 and
P(Ui ≤ ui∗ ( x )| X = x ) > 0 for all i, which violates assumption R. Then S 6∈ M1 . Contradiction.
Proofs for the if part: Fix an arbitrary x ∈ SX . First, we assume P(Y = a| X = x ) > 0 for all
a ∈ A, which will be relaxed later. Now we construct a structure in M1 that will lead to FY | X (·| x ).
Let πi ( a−i , x, ui ) = E(Yi | X = x ) − ui for i = 1, · · · , I. Note that there is no strategic effect by
construction and assumption M is satisfied. Now we construct FU | X (·| x ). Let FUi | X (·| x ) be uniformly
distributed on [0, 1]. So it suffices to construct the copula function CU | X (·| x ) on [0, 1] I . We first
construct CU | X (·| x ) on a finite sub–support: {E(Y1 | X = x ), 1} × · · · × {E(YI | X = x ), 1}. Then
we extend it to a proper copula function with the full support [0, 1] I . Let CU | X (α1 , · · · , α I | x ) =
p
E(∏ j=1 Yi j | X = x ) where i1 , · · · , i p are all the indexes such that αi j = E(Yi j | X = x ); while other
indexes have αk = 1. Because P(Y = a| X = x ) > 0 for all a ∈ A, CU | X (·| x ) is monotone increasing
in each index on the finite sub–support. Thus it is straightforward that we can extend CU | X (·| x )
to the whole support [0, 1] I as a monotone increasing (on the support (0, 1] I ) and smooth copula
function. By construction, it is straightforward that the constructed structure can generate FY | X (·| x ).
42
When P(Y = a| X = x ) = 0 for some a’s in A. By the condition in Proposition 1, the conditional
distribution of Y given X = x is degenerated in some indexes. W.l.o.g., let {1, · · · , k } be set of
indexes such that P(Yi = 1| X = x ) = 0 or 1, and let {k + 1, · · · , I } satisfy 0 < P(Yi = 1| X = x ) < 1.
Then let πi ( a−i , x, ui ) = E(Yi | X = x ) − ui for i = 1, · · · , I. For player i = k + 1, · · · , I, we can
construct a sub–copula function CUk+1 ,··· ,UI | X (·| x ) as described above such that CUk+1 ,··· ,UI | X (·| x )
is monotone increasing and smooth. Further, we can extend CUk+1 ,··· ,UI | X (·| x ) to a proper copula
function having the full support [0, 1] I . Similarly, the constructed structure generates FY | X (·| x ).
B.2. Rationalizing All Probability Distributions. Suppose we replace assumption R with the
following conditions in Reny (2011): For every x ∈ SX ,
G.2. The distribution FUi | X (·| x ) on SUi | X = x is atomless.
G.3. There is a countable subset SU0 | X = x of SUi | X = x such that every set in SUi | X = x assigned
i
positive probability by FUi | X (·| x ) contains two points between which lies a point in SU0 | X = x .
i
Note that it is straightforward that assumptions G.1 and G.4 through G.6 in Reny (2011) are all
satisfied in our discrete game because the action space A is finite and the conditional distribution
of U given X = x has a hypercube support in R I . Thus, the conclusion in Lemma 1 still holds (i.e.,
existence of a monotone pure strategy BNE) under assumptions G.2, G.3 and M. Moreover, let M10 ≡
{S : G.2, G.3 and M hold and a single m.d.p.s. BNE is played}. Then, we generalize Proposition 1.
Lemma 7. Any conditional distribution FY | X can be rationalized M10 .
Proof. We prove by construction. Fix x. Let πi ( a−i , x, ui ) = αi ( x ) − ui for all i. Note that there is no
strategic effect by construction and assumption M is satisfied. Now we construct FU | X (·| x ). Let [0, 1] I
be the support of the distribution and partition it into 2 I disjoint events:
NI
i =1 {[0, αi ( x )), [ αi ( x ), 1]}
24
.
Further, we define a conditional distribution FU | X = x,U ∈ Bj as a uniform distribution on Bj , where Bj is
the j–th event in the partition of the support. Moreover, let P(U ∈ Bj | X = x ) = P(Y = a( j)| X = x )
where a( j) ∈ A and satisfies ai ( j) = 0 if the i–th argument of event Bj is [αi ( x ), 1], and ai ( j) = 1 if
the i–th argument is [0, αi ( x )). With such construction, the marginal distribution of Ui given X = x
is a uniform distribution on [0, 1] which satisfies assumptions G.2 and G.3. It can be verified that the
constructed structure leads to FY | X (·| x ).
B.3. Proof of Proposition 2.
24To have meaningful partition, it is understood that
0.
43
{[0, αi ( x )), [αi ( x ), 1]} becomes {{0}, (0, 1]} when αi ( x ) =
Proof. The if part is trivial. It suffices to show the only if part.
We now show that for any given structure S ≡ [π; FU | X ] ∈ M1 , there always exists an observationally equivalent structure S̃ ≡ [π̃; F̃U | X ] ∈ M2 . Fix x ∈ SX . Let F̃U | X (·| x ) = FU | X (·| x ) and
π̃i ( a−i , x, ui ) = ui∗ ( x ) − ui where u1∗ ( x ), . . . , u∗I ( x ) is the profile of equilibrium threshold points
under S. It is straightforward that S̃ belongs to M2 and is observationally equivalent to S.
B.4. Proof of Proposition 3.
Proof. We first show the "only if" part. Suppose that the distribution FY | X (·|·) rationalized by M2 is
derived from Se = [π̃; F̃U | X ] ∈ M3 . Then
p
E
∏ Yij |X
!
= P Yi1 = 1, · · · , Yi p = 1| X
j =1
= P Ui1 ≤ ũi∗1 ( X ), · · · , Ui p ≤ ũi∗p ( X )| X = CeUi
1
,··· ,Ui p
α i1 ( X ), · · · , α i p ( X ) .
Similarly,
p
∏ Yij |αi1 (X ), · · · , αi p (X )
E
!
= CeUi
j =1
,··· ,Ui p
1
α i1 ( X ), · · · , α i p ( X ) .
Thus, we have condition R1. Further, R2 and R3 obtain by the properties of the copula function
eU
C
i
1
,··· ,Ui p .
Proofs for the if part. For any x ∈ SX , let π̃i ( a−i , x, ui ) = αi ( x ) − ui . Let further F̃Ui be uniformly
distributed on [0, 1]. For all 1 ≤ i1 < · · · < i p ≤ I, (αi1 , · · · , αi p ) ∈ Sαi
define FeUi
1
,··· ,Ui p (·, · · ·
, ·) as follows: for each αi1 , · · · , αi p ∈ Sαi
"
FeUi
1
1
1
( X ),··· ,αi p ( X )
and x ∈ SX ,
( X ),··· ,αi p ( X ) ,
#
,··· ,Ui p ( αi1 , · · · , αi p ) = E ∏ Yi j αi1 ( X ) = αi1 , · · · , αi p ( X ) = αi p .
p
j =1
Thus, we define FeU on the support {α : (αi1 , · · · , αi p ) ∈ Sαi
By Proposition 1, we have that FeUi
1
,··· ,Ui p ,Uk ( αi1 , · · ·
1
( X ),··· ,αi p ( X ) ; other
, αi p , αk ) < FeUi
1
α i j = 1}.
,··· ,Ui p ( αi1 , · · ·
, αi p ) for any
k 6= i j , j = 1, · · · , p, αi j > 0 and αk < 1. Further, under conditions R2, R3, FeU is strictly increasing
and continuously differentiable on {α : (αi1 , · · · , αi p ) ∈ Sαi
can extend it to the whole support
[0, 1] I
1
( X ),··· ,αi p ( X ) ; other
αi j = 1}. Hence, we
as a proper distribution function such that it is strictly
monotone increasing and continuously differentiable on [0, 1] I . The extended FeU (·) will yield a
positive and continuous conditional Radon–Nikodym density on [0, 1] I .
44
By construction, [π̃; FeU ] ∈ M3 . Fix X = x. The constructed structure [π̃; F̃U (·)] will generate the
same marginal distribution αi ( x ) for all i. Moreover, for any tuple {i1 , · · · , i p } from {1, · · · , I },
e (Yi = 1, · · · , Yi = 1| X = x ) = FeU
P
p
i
1
=E
h
,··· ,Ui p
1
α i1 ( x ), · · · , α i p ( x )
p
∏ Yij αi1 (X ) = αi1 (x), · · · , αi p (X ) = αi p (x)
i
=E
j =1
h
p
∏ Yij X = x
i
.
j =1
Because the tuple {i1 , · · · , i p } is arbitrary, then [π̃, FeU ] generates the distribution FY | X (·| x ).
B.5. Proof of Proposition 4.
Proof. The only if part follows directly from assumption I and the single equilibrium condition. It
suffices to show the if part.
Fix a distribution FY | X that satisfies the condition. Let F̃Ui | X = F̃Ui be uniformly distribution on
[0, 1] and F̃U |X = ∏iI=1 F̃Ui . Moreover, let π̃i ( a−i , x, ui ) = αi ( x ) − ui for any x ∈ SX . By construction,
[π̃; F̃U |X ] satisfies assumptions R, M, S, E, and I. Hence, [π̃; F̃U |X ] ∈ M4 .
It suffices to show that the constructed structure [π̃; F̃U | X ] can generate FY | X . Fix x. By construction,
e (Yi = 1| X = x ) = αi ( x ). Moreover, for any tuple {i1 , · · · , i p } from {1, · · · , I },
we have that P
e (Yi = 1, · · · , Yi = 1| X = x ) = FeU
P
p
i
1
,··· ,Ui p
1
α i1 ( x ), · · · , α i p ( x )
p
= ∏ αi j ( x ) = P(Yi1 = 1, · · · , Yi p = 1| X = x ).
j =1
Because the tuple {i1 , · · · , i p } is arbitrary, then [π̃, FeU ] generates the distribution FY | X (·| x ).
A PPENDIX C. I DENTIFICATION
C.1. Proof of Lemma 3.
Proof. Our proof is an extension of the copula argument in Darsow, Nguyen, and Olsen (1992). Fix
X = x. By law of iterated expectation,
P (Yi = 1; Y−i = a−i |α( X ) = α)
= EUi [P (Yi = 1; Y−i = a−i |α( X ) = α, Ui )]
=
Z QU ( α )
i i
QUi (0)
=
Z α
i
0
[P (Y−i = a−i |α( X ) = α, Ui = ui )] dFUi (ui )
P Y−i = a−i |α−i ( X ) = α−i , Ui = QUi (vi ) dvi
45
where the second equality comes from assumption E and the fact that P[Yi = 1|α( X ) = α, Ui ≤
QUi (αi )] = 1 and P[Yi = 1|α( X ) = α, Ui > QUi (αi )] = 0, and the last from a change–in–variable
(vi = FUi (ui )) in the integration.
Therefore, we have
∂P (Yi = 1; Y−i = a−i |α( X ) = α)
= P Y−i = a−i |α( X ) = α, Ui = QUi (αi ) .
∂αi
Note that QUi (αi ( x )) = ui∗ ( x ). Then, it follows that
∗
∗
∗
σ−
i ( a−i | x, ui ( x )) ≡ P [Y−i = a−i | X = x, Ui = ui ( x )]
= P [Y−i = a−i |α( X ) = α( x ), Ui = ui∗ ( x )] =
∂P (Yi = 1; Y−i = a−i |α( X ) = α) . ∂αi
α=α( x )
C.2. Proof of Proposition 6.
Proof. When αi ∈ Sαi (X )| Xi = xi
T
(0, 1), there exists some x ∈ S X such αi ( x ) = αi . By (8), we have
∑ πi (a−i , xi , ui∗ (x)) × σ−∗ i (a−i |x, ui∗ (x)) = 0.
a −i
Since ui∗ ( x ) = QUi (αi ), then we have
∑
a−i ∈A−i
∗
∗
πi a−i , xi , QUi (αi ) × σ−
i ( a−i | x, ui ( x )) = 0.
(11)
Thus, if Ri ( xi , αi ) has full rank 2 I −1 , (11) only admits a unique solution: πi (·, xi , QUi (αi )) = 0.
Moreover, by assumption M, for any ui ∈ SUi and m.d.p.s. profile δ we have: if ui > QUi (αi ),
Eδ πi (Y−i , Xi , Ui ) X = x, Ui = ui < Eδ πi (Y−i , Xi , Ui ) X = x, Ui = QUi (αi ) = 0
and if ui < QUi (αi ),
Eδ πi (Y−i , Xi , Ui ) X = x, Ui = ui > Eδ πi (Y−i , Xi , Ui ) X = x, Ui = QUi (αi ) = 0.
Therefore, regardless other players’ strategies δ−i , player i’s best response is always 1 ui ≤ QUi (αi )
as her dominant strategy. Hence, Sαi (X )| Xi = xi must be a singleton {αi }.
Further, if Ri ( xi , αi ) has rank 2 I −1 − 1, then conditioning on Xi = xi and αi ( X ) = αi , the random
vector Σ∗−i ( X ) consists of a set of random variables which can be linearly independent if we exclude
∗ a0 | X, u∗ ( X ) from Σ∗ ( X ), the random variables left in
some variable. By excluding, w.l.o.g., σ−
i
i
−i
−i
46
Σ∗−i are conditionally linearly independent given Xi = xi and αi ( X ) = αi . Because
∑
∗
∗
∗
∗
0
0
πi a−i , xi , QUi (αi ) × σ−
i ( a−i | x, ui ( x )) = − σ−i ( a−i | x, ui ( x )) πi a−i , xi , QUi ( αi ) ,
a−i ∈A−i \{ a0−i }
the rank condition implies that the πi ( a−i , xi , QUi (αi )) are identified for all a−i ∈ A−i \{ a0−i } up to
the scale πi a0−i , xi , QUi (αi ) .
Last, we establish the identification of the sign of πi ·, xi , QUi (αi ) . W.L.O.G., let αi0 < αi . Let
further x 0 ∈ SX | Xi = xi such that α( x 0 ) = α0 . Then by assumption M, we have
∑
a−i ∈A−i
∗
0
πi a−i , xi , QUi (αi ) × σ−
i ( a−i | x , QUi ( αi ))
<
∑
a−i ∈A−i
∗
0
0
πi a−i , xi , QUi (αi0 ) × σ−
i ( a−i | x , QUi ( αi )) = 0
∗ ( a | x 0 , Q ( α )) is also identified. This is because
Note that σ−
Ui i
i −i
∗
0
σ−
i ( a−i | x , QUi ( αi ))
= P(Y−i = a−i | X = x 0 , Ui = QUi (αi ))
= P(δj∗ ( x 0 , Uj ) = a j , ∀ j 6= i | X = x 0 , Ui = QUi (αi ))
= P(δj∗ ( x 0 , Uj ) = a j , ∀ j 6= i |Ui = QUi (αi ))
= P(Y−i = a−i |α−i ( X ) = α−i ( x 0 ), Ui = QUi (αi )),
where the RHS of the last equality is identified by Lemma 3.
Moreover, since πi ·, xi , QUi (αi ) are identified up to the unknown scale πi a0−i , xi , QUi (αi ) , let
πi ·, xi , QUi (αi ) = pi ·, xi , QUi (αi ) × πi a0−i , xi , QUi (αi )
where pi ∈ R2
(
I −1
∑
is known. By definition, pi a0−i , xi , QUi (αi ) = 1. Thus
)
∗
0
0
pi a−i , xi , QUi (αi ) × σ−
i ( a−i | x , QUi ( αi )) × πi a−i , xi , QUi ( αi ) < 0,
a−i ∈A−i
from which we identify the sign of πi a0−i , xi , QUi (αi ) and hence the sign of πi a−i , xi , QUi (αi ) .
C.3.
To prove Proposition 7, we present a result that parallels Proposition 6.
Lemma 8. Let S ∈ M3 , assumptions SC and ER hold. Fix xi ∈ SXi such that Sαi (X )| Xi = xi
T
(0, 1) 6= ∅.
If the rank of Ri ( xi ) is 2 I −1 − 1, then Sαi (X )| Xi = xi must be a singleton {αi } and β i (·, xi ) is identified up to
the αi –quantile of FUi , i.e. β i (·, xi ) = QUi (αi ). If the rank of Ri ( xi ) is 2 I −1 − 2, then β i (·, xi ) is identified
47
up to location and scale that depend on xi , or equivalently, β i (·, xi ) − β i ( a0−i , xi ) is identified up to scale
for any a0−i ∈ A−i . In addition, if there exist α, α0 ∈ Sα(X )| Xi = xi such that αi0 ∈ (0, 1), αi0 6= αi and
(αi , α0−i ) ∈ Sα(X ) , then the sign of β i ( a−i , xi ) − β i ( a0−i , xi ) is also identified for each a−i ∈ A−i .
Proof. With additive separability in the payoffs as well as assumption ER, we can obtain the following
equilibrium condition by Lemma 2: for all x ∈ SX such that αi ( x ) ∈ (0, 1),
∑
∗
∗
β i ( a−i , xi )σ−
i a−i | x, ui ( x ) = QUi ( αi ( x )).
(12)
∗
∗
β i ( a−i , xi )E σ−
i a−i | X, ui ( X )) | Xi = xi , αi ( X ) = αi ( x ) = QUi ( αi ( x ))
(13)
a−i ∈A−i
It follows that
∑
a−i ∈A−i
The difference between (12) and (13) yields
∑
β i ( a−i , xi ) × σ∗−i a−i , x = 0
(14)
a−i ∈A−i
∗ a | x, u∗ ( x ) − E σ ∗ a | X, u∗ ( X ) | X = x , α ( X ) = α ( x ) .
where σ∗−i a−i , x ≡ σ−
i
i i
i
i
i
−i −i
i −i
When xi is fixed, we can identify β i (·, xi ) as coefficients by varying σ∗−i ( a−i , x ) through x−i .
Suppose Ri ( xi ) has rank 2 I −1 − 1. By a similar argument as that for Proposition 6, β i (·, xi ) are
identified up to scale. Note that ∑ a−i ∈A−i σ∗−i a−i , x = 0. Hence, β i ( a−i , xi ) equals the same constant
for all a−i ∈ A−i . By (12), we have β i (·, xi ) = QUi (αi ( x )). Moreover, similarly to Proposition 6,
Sαi (X )| Xi = xi has to be a singleton set, i.e. {αi ( x )} for some x ∈ SX | Xi = xi .
Next, suppose Ri ( xi ) has rank 2 I −1 − 2. Then we can pick a vector β0i (·, xi ) ∈ R2
I −1
such that
β0i ( a−i , xi ) 6= β0i ( a0−i , xi ) for some a−i , a0−i ∈ A−i , and β0i (·, xi ) satisfy
∑
a−i ∈A−i
β0i ( a−i , xi ) × σ∗−i a−i , x = 0.
Note that we also have ∑ a−i ∈A−i 1 × σ∗−i a−i , x = 0. By linear algebra, β i can be written as
β i (·, xi ) = ci ( xi ) + pi ( xi ) × β0i (·, xi )
where ci , pi : SXi → R. Hence, β i are identified up to location (ci ) and scale (pi ).
The identification of the sign of β i ( ai , xi ) − β i ( a0i , xi ), is similar to the proof for Proposition 6: let
x, x 0 ∈ SX | Xi = xi , α( x ) = α, α( x 0 ) = α0 , and w.l.o.g., αi0 < αi . Then
∑
a−i ∈A−i
∗
0
β i ( a−i , xi )σ−
i a−i | x , QUi ( αi ) < QUi ( αi ( x )) =
∑
a−i ∈A−i
48
∗
β i ( a−i , xi )σ−
i a−i | x, QUi ( αi ) ,
from which we have
pi ( xi ) ×
∑
a−i ∈A−i
∗
0
∗
< 0.
β0i ( a−i , xi ) × σ−
i a−i | x , QUi ( αi ) − σ−i a−i | x, QUi ( αi )
Thus we identify the sign of pi ( xi ). It follows that the sign of β i ( a−i , xi ) − β i ( a0−i , xi ) = pi ( xi ) ×
0
β i ( a−i , xi ) − β0i ( a0−i , xi ) is also identified.
The interpretation of the rank conditions in Lemma 8 is quite similar to that for Proposition 6.
When Sαi (X )| Xi = xi is not degenerate, we know that player i’s payoffs are strategically affected by
others. Then, β i (·, xi ) is identified up to location and scale under the “full” rank condition. To see
this, consider the following equation system in terms of c,
∗
c0 Σ−i ( x ) = 0, ∀ x ∈ SX | Xi = xi
It has two different solutions, (1, · · · , 1)0 ∈ R2
Appendix, which is derived from (9). Hence,
I −1
2 I −1
and β i (·, xi ); the latter is due to (14) in the
− 2 is the largest possible rank of the matrix
Ri ( xi ) when there are strategic interactions.
It is worth pointing out that we can identify the payoffs in model M3 using the single–index
structure suggested in Lemma 2. To see this, let βi ( a−i , xi ) = β i ( a−i , xi ) − β i ( a0−i , xi ). Then
E Yi | Xi = xi , Σ∗−i ( X ) = Σ∗−i ( x )
= FUi β i
a0−i , xi
+
∑
!
∗
βi ( a−i , xi ) × σ−
i
a−i | x, ui∗ ( x )
.
(15)
a−i ∈A/{ a0−i }
Similarly to Powell, Stock, and Stoker (1989), we can identify βi (·, xi ) up to scale by differentiating
(15) with respect to Σ∗−i ( x ). Thus, the normalized payoffs functions βi (·, xi ) are identified up to
scale. This strategy, however, involves an additional support condition on SΣ∗
−i ( X )| Xi = xi
for taking
the derivative.
C.4. Proof of Proposition 7.
Proof. By (10), it suffices to show that the identification of β i ( a−i , ·) on Cit implies its identification
on Cit+1 . Because of Case (i) in Definition 2, it suffices to consider xi ∈ Cit+1 /Cit .
Suppose that Case (ii) occurs, i.e. Ri ( xi ) has rank 2 I −1 − 2 and there exists xi0 ∈ Cit such that
Sαi (X )| Xi = xi
T
Sαi (X )| Xi = x0
i
T
(0, 1) contains at least two different elements 0 < αi0 < αi < 1. Let
x, x 0 ∈ SX | Xi = xi , αi ( x ) = αi and αi ( x 0 ) = αi0 . Because xi0 ∈ Cit , then by assumption β i (·, xi0 ) are
identified. Then both QUi (αi ) and QUi (αi0 ) are identified by (10). Further, because Ri ( xi ) has rank
49
2 I −1 − 2, then by Lemma 8, β i (·, xi ) is identified up to location and scale, i.e. ∃ ci ( xi ), pi ( xi ) ∈ R and
a known vector β0i (·, xi ) ∈ R2
αi , αi0 ∈ Sαi (X )| Xi = xi
T
I −1
, such that β i (·, xi ) = ci ( x ) + pi ( x ) × β0i (·, xi ). Moreover, because
(0, 1), then we have
∑
a−i ∈A−i
∑
a−i ∈A−i
∗
∗
β i ( a−i , xi ) × σ−
i ( a−i | x, ui ( x )) = QUi ( αi ),
∗
0 ∗ 0
0
β i ( a−i , xi ) × σ−
i ( a−i | x , ui ( x )) = QUi ( αi ).
It follows that
ci ( xi ) + pi ( xi ) ×
∑
a−i ∈A−i
∑
ci ( xi ) + pi ( xi ) ×
a−i ∈A−i
∗
∗
β0i ( a−i , xi ) × σ−
i ( a−i | x, ui ( x )) = QUi ( αi ),
∗
0 ∗ 0
0
β0i ( a−i , xi ) × σ−
i ( a−i | x , ui ( x )) = QUi ( αi ).
∗ ( a | x, u∗ ( x )) and σ ∗ ( a | x 0 , u∗ ( x 0 )) are all known terms.
Note that QUi (αi ), QUi (αi0 ), β0i (·, xi ), σ−
i −i
i
−i −i
i
Because QUi (αi0 ) < QUi (αi ), the determinant of the equation system cannot be zero. Then we can
identify ci ( xi ) and pi ( xi ) from the above two equations. Therefore, β i (·, xi ) are identified.
Suppose that Case (iii) occurs, i.e. Ri ( xi ) has rank 2 I −1 − 1 and there exists xi0 ∈ Cit such that
Sαi (X )| Xi = xi ⊆ Sαi (X )| Xi = x0
i
T
(0, 1). By Lemma 8, β i (·, xi ) is identified by QUi (αi ), which is known
since Sαi (X )| Xi = xi ⊆ Sαi (X )| Xi = x0
i
T
(0, 1).
A PPENDIX D. E XTENSIONS
D.1. Proof of Lemma 5.
Proof. First, we construct a structure Se ∈ M3 (which implies that Se ∈ M30 ) such that (1) Se has
eU (·) = CU (·) on [0, 1] I ; (3) for any x ∈ SX ,
eU , · · · , Q
eU ; (2) C
the marginal quantile functions Q
I
1
eU (E(Yi | X = x )). By construction, it is straightforward that
i, and a−i ∈ A−i , let βei ( a−i , x ) = Q
i
assumptions R, M, S and E are satisfied.
Now it suffices to verify the observational equivalence between Se and S. Fix x ∈ SX . Note that in
n
o
eU (E(Yi | X = x )) for
the structure Se there is no strategic effects, then the equilibrium is: 1 ui ≤ Q
i
i = 1, · · · , I. Here we only verify the observational equivalence for action profile (1, · · · , 1) and the
proofs for other action profiles follow similarly:
eU (E(Y | X = x ))
e (Y1 = 1; · · · ; YI = 1| X = x ) = C
P
= CU (E(Y | X = x )) = P(Y1 = 1; · · · ; YI = 1| X = x ). 50
D.2. Proof of Proposition 8.
Proof. It is straightforward that β ∈ Θ I ({ QUi }iI=1 , CU ). For sharpness, it suffices to show that for
any β̃ ∈ Θ I ({ Q̃Ui }iI=1 , C̃U ), then S̃ ≡ ( β̃, { Q̃Ui }iI=1 , C̃U ), which belongs to M3 by the definition of
Θ I ({ Q̃Ui }iI=1 , C̃U ), is observationally equivalent to the underlying structure S ≡ ( β, { QUi }iI=1 , CU ).
Fix X = x. It suffices to verify that δ∗ = 1 u1 ≤ Q̃U1 (α1 ( x )) , · · · , 1 u I ≤ Q̃UI (α I ( x ))
is a
BNE solution for the constructed structure. Because C̃U ∈ C I and by the proof for Lemma 3,
∗
∗
P̃δ∗ Y−i = a−i | X = x, Ui = Q̃Ui (αi ( x )) = σ−
i ( a−i | x, ui ( x )).
Then, by the conditions in the definition of Θ I ({ Q̃Ui }iI=1 , C̃U ), 1 ui ≤ Q̃Ui (αi ( x )) is the best re∗ . Thus δ∗ is a BNE.
sponse to δ−
i
51