RATIONALIZATION AND IDENTIFICATION OFDISCRETE GAMES WITH CORRELATED TYPES∗

NIANQING LIU?, QUANG VUONG†, AND HAIQING XU‡

ABSTRACT. This paper studies the rationalization and identification of discrete games where

players have correlated private information (i.e. types). Our approach is fully nonparametric.

First, under monotone pure strategy Bayesian Nash Equilibrium, 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 that have been frequently

used in the empirical analysis of discrete games. Namely, we consider additive separability

of the private information in the payoffs, exogeneity of the payoff shifters relative to the

private information, and mutual independence of the private information conditional on the

payoff shifters. Second, we study the nonparametric identification of the payoff functions and

types distribution under exclusion restrictions and rank conditions. In particular, we show

that our structural model is identified up to a location-scale normalization in the separable

case. Third, without imposing exclusion restrictions, we characterize the sharp identification

region for the payoff functions and types’ distribution.

Keywords: Rationalization, Identification, Discrete Game, Bayesian Nash Equilibrium

JEL: C14, C35, C62 and C72

Date: August, 2013.∗We thank Laurent Mathevet, Eugenio Miravete, Bernard Salanie, Steven Stern, Elie Tamer and Nese Yildiz foruseful comments. We also thanks seminar participants at the Texas Metrics Camp 2013, Texas A&M University,University of Chicago, Brown University, University of Texas at Austin, Columbia University, the NorthAmerican Summer Meeting at University of South California, and the 2013 Asian Meeting of the EconometricSociety at National University of Singapore.?Department of Economics, The Pennsylvania State University, 408 Kern Graduate Building, University Park,PA, 16802, nliu@psu.edu.†Department of Economics, New York University, 19 W. 4th Street, 6FL, New York, NY, 10012, qvuong@nyu.edu.‡Department of Economics, University of Texas at Austin, BRB 3.160, Mailcode C3100, Austin, TX, 78712,h.xu@austin.utexas.edu.

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1. INTRODUCTION

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 models on the observables, which are the players’ choices. Following the work by

Laffont and Vuong (1996) and Athey and Haile (2007) for auctions, our approach is fully

nonparametric.

The empirical analysis of discrete games is almost thirty years old. In particular, the range

of applications includes, among others, 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; Berry and Tamer, 2006; Ciliberto and Tamer,

2009; Jia, 2008). 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); Kline (2012) 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 (2011).

Our paper contributes to this literature in several aspects. First, we focus on monotone

pure strategy Bayesian Nash equilibria (BNE) throughout. 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 auction model. On the other hand, when a structural

model has nonseparable errors, the recent literature heavily relies on the monotonicity of

1To simplify, we focus on binary games in this paper. We left the extension of our approach to general discretegames for future research.

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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) provided seminal results on the existence of a monotone pure strategy BNE

whenever a Bayesian game obeys a Spence–Mirlees single–crossing restriction. Relying on

the powerful notion of contractibility, Reny (2011) has extended 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 show the existence of a monotone pure strategy

BNE under a high–level assumption in our setting. In particular, such a high–level condition

is satisfied if the game is of strategic complement and private information are positively

regression dependent.

Second, we allow players’ private information (i.e. types) to be correlated. Allowing

correlated private information is motivated primarily by empirical concerns. In oligopoly

entry, for example, the correlation among types allows us to see “whether entry occurs

because of unobserved profitability that is independent of the competition effect” (Berry

and Tamer, 2006). In contrast, mutual independence of private information has been widely

assumed in the empirical literature. See, e.g., Brock and Durlauf (2001); Pesendorfer and

Schmidt-Dengler (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); exceptions include Aradillas-Lopez (2010) and Wan and Xu (2010). Such

an independence of types is a convenient assumption for identification, but imposes strong

restrictions such as the mutual independence of players’ choices, a property which can be in-

validated by the data.2 On the other hand, when private information are 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. They nonparametrically estimate these bounds in the first

step and then apply a modified maximum score estimator approach (see Manski and Tamer,

2A model with unobserved heterogeneity and independent private information also generates dependenceamong players’ choices conditional on observed regressors (see Grieco, 2011).

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2003) to construct a set estimator for the payoff coefficients. Alternatively, Aradillas-Lopez

(2010) adopts a different equilibrium concept suggested by 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 the 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 in-

stance, Brock and Durlauf (2001); Seim (2006); Sweeting (2009) and Xu (2010) have specified

both payoffs and the private information distribution parametrically. In a semiparametric

context, Aradillas-Lopez (2010); Tang (2010) and Wan and Xu (2010), among others, param-

eterize players’ payoffs, while Bajari, Hong, Krainer, and Nekipelov (2010) parameterize

the private information distribution. 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 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

words, monotone pure strategy BNE can explain almost all observed choice probabilities in

discrete games.

In view of the preceding results, we consider three assumptions that have been 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 the private

information are additively separable or not.

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A second assumption that has been frequently imposed in empirical work is the exogene-

ity of some variables shifting the 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 the payoff shifters and we characterize all

those restrictions. Specifically, the exogeneity assumption restricts the joint choice probabil-

ity 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 we impose 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. With additive separability in payoffs, however, we show

that both the players’ payoffs and distribution of types are identified up to one scale-

location normalization on the payoffs function. Our identification results can be viewed

as an extension to game theoretic models of the nonparametric analysis in traditional

threshold–crossing models considered by, e.g., Matzkin (1992). An important difference is

that the discrete game setup allows us to exploit the exclusion restrictions for identification.

Such restrictions have been used frequently 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); exceptions include De Paula and Tang (2012).

For completeness, we consider a third assumption, namely the mutual independence

of players’ private information. Specifically, we characterize all the restrictions imposed

by exogeneity and mutual independence as considered by Brock and Durlauf (2001); Seim

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(2006); Sweeting (2009); Bajari, Hong, Krainer, and Nekipelov (2010); Lewbel and Tang

(2012), among others. 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. In other words, exogeneity is redundant in

terms of explaining players’ choices as soon as mutual independence is 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 the private information. In

Section 3, we study the restrictions imposed by the baseline model, whether the private

information are additively separable or not. We also derive all the restrictions imposed

by the exogeneity and mutual independence assumptions. In Section 4, we establish the

nonparametric identification of the model primitives for the additively nonseparable and

separable cases under some exclusion restriction 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 two possible

extensions: one considers the multiple dimensional types for each player and the other

allows multiple equilibria the DGP. Section 6 concludes with a brief discussion on estimation

and testing of the model.

2. MODEL AND MONOTONE PURE STRATEGY 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

3Grieco (2011) analizes a discrete game that has some payoff relevant variables publicly observed by all players,but not by the researcher.

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the error term Ui ∈ R is her private information, i.e., Ui is observed by only player i, but

not by other players. Consistent with the theoretic literature, we also call Ui as player i’s

“type” (see, e.g., Fudenberg and Tirole, 1991). Let U = (U1, · · · , UI) and FU|X(·|·) be the

conditional distribution function of U given X. The conditional distribution FU|X(·|·) is

assumed to be common knowledge.4

The payoff of player i is described as follows:

Πi(Y, X, Ui) =

πi(Y−i, X, Ui), if Yi = 1,

0, 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.

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

any given strategy profile δ = (δ1, · · · , δI), let σδ−i(a−i|x, ui) be the conditional probability

of other players choosing a−i ∈ A−i given X = x and Ui = ui, i.e.,

σδ−i(a−i|x, ui) ≡ Pδ (Y−i = a−i|X = x, Ui = ui) = P

[δj(X, Uj) = aj, ∀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.,

δ∗i (x, ui) = 1

[∑a−i

πi(a−i, x, ui)σ∗−i(a−i|x, ui) ≥ 0

], (1)

4For the standard notion of common knowledge, see, e.g., Fudenberg and Tirole (1991), Chapter 14.

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where δ∗ ≡ (δ∗1 , · · · , δ∗I ), as functions of (u1, · · · , uI) respectively, denotes the equilibrium

strategy profile and σ∗−i(a−i|x, ui) is a shorthand notation for σδ∗−i(a−i|x, ui). Note that σ∗−i

depends on δ∗−i. Hence, eq. (1) for i = 1, · · · , I defines a simultaneous equation system of δ∗

referred to 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, . . . , uI) 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, however, 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

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 (δ∗1 (x, ·), · · · , δ∗I (x, ·)) is a monotone pure

strategy BNE if (δ∗1 (x, ·), · · · , δ∗I (x, ·)) is a BNE and δ∗i (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 monotonic equilibrium strategy can be explicitly defined as a

threshold function (recall that δ∗i can take only binary values). Formally, in a monotone pure

strategy BNE, player i’s equilibrium strategy can be written as either δ∗i = 1 [ui ≤ u∗i (x)]

or δ∗i = 1 [ui > u∗i (x)],5 where u∗i (x) is the cutoff value that might depend on x. Let

u∗(x) ≡ (u∗1(x), · · · , u∗I (x)) ∈ RI 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.

δ∗i = 1 [ui > u∗i (x)]. Then, we can construct an observationally equivalent structure [π; FU|X]

by letting πi(·, x, ui) = πi(·, x,−ui) for all ui, πj = πj (j 6= i), and FU|X(·|x) = FU|X(·|x)

where U differs from U only in its i–th argument: Ui = −Ui. For the constructed structure,

5The left–continuity of strategies considered hereafter is not restrictive given our assumptions below.

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it can be shown that player i’s equilibrium strategy becomes monotone decreasing, i.e.

δ∗i = 1 [ui ≤ −u∗i (x)], while other players’ equilibrium strategies keep the same. Further,

we can verify that the new equilibrium in the constructed structure will lead to the same

distribution of observables.

In an m.d.p.s. BNE, the mutual consistency condition for a BNE solution defined by

eq. (1) requires that for each player i,

ui ≤ u∗i (x)⇐⇒∑a−i

πi(a−i, x, ui)σ∗−i(a−i|x, ui) ≥ 0. (2)

A simple, but key, observation is that under certain weak conditions introduced later, eq. (2)

implies that player i with the margin type u∗i (x) should be indifferent between choosing 1

and 0, i.e.,

∑a−i

πi(a−i, x, u∗i (x))σ∗−i(a−i|x, u∗i (x)) = 0, ∀i.

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 incom-

plete information was first provided by Athey (2001) in both supermodular and logsupermod-

ular 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 fU|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 = RI 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).

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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 payoffs 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 the type given 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 and all Uis 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.

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 extends Lemma 1 by replacing assumption M with more primitive conditions.

In particular, we assume positive regression dependence across Uis given X, strategic

complementarity of players’ actions and monotone payoffs in the Uis. As far as we know,

with the only exception of Aradillas-Lopez and Tamer (2008) and Xu (2010), 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 equilibria could exist and sometimes even stand 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.6

Example 1. (Xu, 2010) Let I = 2 and πi = Xi− αiY−i−Ui, where (X1, X2) = (1, 0), (α1, α2) =

(2, 0), and (U1, U2) conforms to a joint normal distribution with mean zero, unit variances and

correlation parameter ρ ∈ (−1, 1).

6We thank Steven Stern and Elie Tamer for their commons and suggestions on the following examples.

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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 only if 1− 2Φ(− ρu1√

1−ρ2

)− u1 ≥ 0. Further, it can be shown

that player 1’s equilibrium strategy is monotone in u1 if and only if ρ ∈(− 1,

√π

2+π

).

In Example 1, the non–monotone strategy BNE occurs due to the large positive correlation

between U1 and U2 (relative to α1). On the other hand, for any given value of (α1, α2, ρ), the

existence of non–monotone strategy BNE could also depend on the realization of (X1, X2).

Xu (2010) characterize a subset of the covariate space where the game only admits monotone

pure strategy BNEs.

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) = βi(a−i, x) − ui for some

function βi 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., the auction participations (see, e.g., Li and Zheng, 2009; Marmer,

Shneyerov, and Xu, 2013).

Lemma 2. Suppose that 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:

Yi = 1[Ui ≤∑

a−i

βi(a−i, X)σ∗−i(a−i|X, u∗i (X))], (3)

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−i

βi(a−i, x)σ∗−i(a−i|x, ui)− ui,

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which is continuously decreasing in ui under assumptions R and M. Then, player i with the

margin type u∗i (x) should be indifferent between choosing 1 and 0, i.e.7

∑a−iβi(a−i, x)σ∗−i(a−i|x, u∗i (x))− u∗i (x) = 0. (4)

Thus, eq. (3) follows directly from Yi = 1[Ui ≤ u∗i (X)

].

The semi–linear–index model in eq. (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 the beliefs σ∗−i in eq. (3) can be nonparametrically identified under additional

weak conditions.

3. RATIONALIZATION

In this section, we will 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 will characterize all the

restrictions imposed on the distribution of observables (Y, X) by each of these models. We

will say that one 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 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 Sections 5, 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 has been frequently made in the literature, e.g., Bajari, Hong, Krainer, and Nekipelov

(2010) and De Paula and Tang (2012). The only exception, to our knowledge, is Wan and Xu

(2010).

Assumption E (Exogeneity). X and U are independent of each other.8

7In eq. (4), it is understood that u∗i (x) = ui(x) and u∗i (x) = ui(x) if Eδ∗[βi(Y−i, X)

∣∣X = x, Ui = ui(x)]< ui(x)

and Eδ∗[βi(Y−i, X)

∣∣X = x, Ui = ui(x)]> ui(x), respectively.

8Our results can be easily extended to the weaker assumption that X and U are independent from each otherconditional on W, where W are observed random variables.

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Another assumption called as mutual independence has also been 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, · · · , UI are mutually independent conditional on X.

Let S ≡ [π; FU|X], where π = (π1, · · · , πI). We now consider the following models

(classes of structures):

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 inM1 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 a 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. Further, relaxing such a requirement has been addressed in recent work and

will be discussed in the extension section.

We introduce some key notation for the following analysis. For any S ∈ M1, let αi(x) ≡

FUi |X(u∗i (x)|x). By the equilibrium strategy, we have that αi(x) = E(Yi|X = x). For each

p = 2, · · · , I, and 1 ≤ i1 < · · · < ip ≤ I, let CUi1 ,··· ,Uip |X(·, . . . , ·|·) be the conditional copula

function of (Ui1 , · · · , Uip) given X, i.e., for any (αi1 , · · · , αip) ∈ [0, 1]p and x ∈ SX,

CUi1 ,··· ,Uip |X(αi1 , · · · , αip |x) ≡ FUi1 ,··· ,Uip |X

(F−1

Ui1 |X(αi1 |x), · · · , F−1

Uip |X(αip |x)

∣∣∣x) .

The next proposition characterizes the collection of distributions of Y given X that can be

rationalized byM1.

Proposition 1. A conditional distribution FY|X is rationalized byM1 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.

13

Proof. See Appendix B.1 �

By Proposition 1,M1 rationalizes all distributions for Y given X that belong to the interior

of the simplex in R2I−k(0 ≤ k ≤ I). In particular,M1 rationalizes all the distributions with

strictly positive choice probabilities. Proposition 1 also indicates that the only possible

distributions that cannot be rationalized byM1 must have P(Y = a|X = x) = 0 for some

a ∈ A, i.e distributions for which there are “structural zeros.” In other words, our baseline

modelM1 imposes no essential restrictions on the distribution of observables.

All those distributions that cannot be rationalized byM1 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 rational-

ized.

Next, we show thatM2 andM1 are observationally equivalent, i.e.,M2 does not impose

additional restrictions on FY|X.

Proposition 2. A conditional distribution FY|X is rationalized byM1 if and only if it is rationalized

byM2.

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 modelM1. Then, it follows from

Proposition 1 thatM2 can also rationalize all the conditional distributions of observables

that belong to the interior of the simplex in R2I−kwith 0 ≤ k ≤ I.

We now characterize all the restrictions imposed on FY|X by modelM3. These additional

restrictions come from assumption E.

Proposition 3. A conditional distribution FY|X rationalized byM2 is also rationalized byM3 if

and only if for each p = 2, · · · , I and 1 ≤ i1 < · · · < ip ≤ I,

R1: we have that E(

∏pj=1 Yij

∣∣∣X) = E(

∏pj=1 Yij

∣∣∣αi1(X), · · · , αip(X))

.

R2: E(

∏pj=1 Yij |αi1(X) = ·, · · · , αip(X) = ·

)is monotone strictly increasing on Sαi1 (X),··· ,αip (X)

except at values for which some coordinates are zero.

14

R3: E(

∏pj=1 Yij |αi1(X) = ·, · · · , αip(X) = ·

)is continuously differentiable on Sαi1 (X),··· ,αip (X).

Proof. See Appendix B.4. �

Proposition 3 shows that the joint choice probabilities rationalized byM3 are monotone

strictly increasing and continuously differentiable functions of the marginal choice proba-

bilities. The intuition of Proposition 3 comes from ?? in Appendix B, which characterizes

the sufficient and necessary conditions for two structures in M2 being observationally

equivalent.

In Proposition 3, the most essential restrictions is R1, which requires the joint choice

probability depends on X only through the corresponding marginal choice probabilities.

Moreover, note that αi(x) is identified by the fact that αi(x) = E(Yi|X = x). Therefore, all

the restrictions R1–R3 are testable in principle. This is discussed further in the Conclusion.

For completeness, we also study the restrictions on observables imposed byM4, which

make the additional assumption I. It should be noted that assumption M is implied when

we assume assumptions S and I. In other words, assumption M is redundant inM4.

In the literature, several special cases ofM4 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 always be rationalized byM4 if and only if

Y1, · · · , YI are conditionally independent given X.

Proof. See Appendix B.5. �

Note that the conditional independence assumption in Proposition 4 implies the con-

ditions R1–R3 in Proposition 3. Therefore, we can replace M4 in Proposition 4 with

M′4 ≡ {S ∈ M1 : assumption I holds}, sinceM′

4 also implies the conditional indepen-

dence condition. Above discussion is summarized in the following corollary.

Corollary 1. ModelM4 imposes the same restrictions on the distribution of observables asM′4,

i.e., both models are observationally equivalent.

15

This is a surprising result: exogenous payoff shifters (assumption E) becomes redundant

in terms of restrictions on the observables, as soon as mutual independence of private

information conditional on X (assumption I) has been imposed on the baseline model.

4. NONPARAMETRIC IDENTIFICATION

In this section we study the nonparametric identification of the baseline game-theoretic

modelM1, and its special casesM2,M3 andM4. As a preliminary to the identification of

M3, we also consider the identification ofM′3 ≡ {S ∈ M1 : assumption E holds} which

is the nonseparable extension ofM3, and which is of interest by itself. The recent literature

has focused on the parametric or semiparametric identification of structures inM4, 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 inM4 under additional

restrictions on the functional form of payoffs.

In our context, the identification of each model is equivalent to the 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 = F−1Ui |X

, throughout the identification

of models refers to the identification of the triple[π; {QUi |X}

Ii=1; CU|X

].

A main contribution of this paper is to give new identification results for the payoffs and

types’ distribution under weak conditions. In modelM′3, we show the identification of

CU|X and some features of the structure under additional identifying restrictions, i.e. the

exclusion restrictions and rank condition. In modelM3, we provide the full identification

of the triple[π; {QUi |X}

Ii=1; CU|X

]up to a location–and–scale normalization on the payoff

function. Further, the identification ofM4 requires slightly weaker support restrictions

than those forM3, though the differences are not essential.

4.1. Nonidentification ofM1 andM2. We begin with the most general modelM1 and

M2, in which we show that the structure is typically unidentified.

Proposition 5. NeitherM1 norM2 is identified nonparametrically.

16

The non–identification ofM2 follows directly from the observational equivalence be-

tween any structure S inM2 and a collection of I–single-agent binary responses models: Let

S ≡(

π; FU|X

)in which πi(·, x) ≡ πi(x) is arbitrarily chosen, and a distribution function

FU|X satisfies assumption R with FU|X(π(x)|x) = FU|X(u∗(x)|x) for all x ∈ SX. Thus, S and

S are observationally equivalent thereby establishing the non–identification ofM2. The

non–identification ofM1 follows immediately from Proposition 2.

Next, we turn to the identification of the identification ofM′3 and its sub–modelsM3

and M4. First note that we maintain assumption E in all these models, i.e. that X and

U are independent of each other. It follows that QUi |X = QUi and CU|X = CU . Thus, the

identification of these models becomes the identification of the triple[π; {QUi}I

i=1; CU].

4.2. Identification ofM′3. Let α(x) ≡ (α1(x), · · · , αI(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 can be nonparametrically identified inM′3 on an appropriate domain, which

is essentially the support of the FU–quantile associated with α(X): for all α ∈ Sα1(X) ∪

{0, 1} × · · · ×SαI(X) ∪ {0, 1}, we have that CU(α) = 0 if αj = 0 for some j; otherwise,9

CU(α) = P{

Uj ≤ F−1j (αj)

}= E

{I

∏i=1

Yi∣∣αj(X) = αj, ∀j ∈ {i : αi 6= 1}

}. (5)

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.

As mentioned above, the equilibrium belief σ∗−i(·|x, u∗i (x)) can also be nonparametrically

identified, for which we need a rank condition on the support of α(X).

Assumption RC (Rank Condition). The support Sα(X) is a convex and compact subset of [0, 1]I

with dim(Sα(X)

)= I.

The convexity and compactness in assumption RC is not essential and can be relaxed

at the cost of notational complexity. The main restriction in assumption RC is the full

dimensionality of the support Sα(X), which requires the payoff shifters X to have sufficient

variations. Given the identification of α(x), assumption RC is verifiable.

9In eq. (5), it is understood that CU(α) = 1 if α = (1, · · · , 1).

17

Lemma 3. Let S ∈ M′3. Suppose that assumptions E and RC hold. Fix x ∈ SX. Then the

equilibrium beliefs σ∗−i(·|x, u∗i (x)) is identified, i.e. for all a−i ∈ A−i,

σ∗−i(a−i|x, u∗i (x)) =∂P (Yi = 1; Y−i = a−i|α(X) = α)

∂αi

∣∣∣∣∣α=α(x)

. (6)

Proof. See Appendix C.1. �

In Lemma 3, the rank condition RC is needed for taking the derivative in eq. (6). In

addition, suppose assumption I holds. Then, the probability P (Yi = 1; Y−i = a−i|α(X) = α)

becomes a (known) linear function in αi and we don’t need assumption RC to identify the

equilibrium beliefs, see, e.g., Bajari, Hong, Krainer, and Nekipelov (2010).

The intuition for our identification of σ∗−i is acquired from the local–instrumental–variables

method developed in Heckman and Vytlacil (2005): the local variation of player i’s choice

probability αi(X) (after controlling for α−i(X)) provides the identification power for the con-

ditional probability given that the latent variable is at the margin. Here we use a two–player

game to illustrate.

Example 2. Fix S ∈ M′3 and let I = 2. Note that for any α ∈ [0, 1]2, we have that

P(Y1 = 1, Y2 = 1|α(X) = α) = P [U1 ≤ u∗i (X), U2 ≤ u∗2(X)|α(X) = α]

= P [U1 ≤ QU1(α1), U2 ≤ QU2(α2)] = CU(α1, α2).

Further, we have that ∂CU(α1, α2)/∂αi = P(U−i ≤ QU−i(α−i)|Ui = QUi(αi)

), see, e.g., Darsow,

Nguyen, and Olsen (1992). Note that QUi(αi(x)) = u∗i (x). It follows that

∂P(Y1 = 1, Y2 = 1|α(X) = α)

∂αi

∣∣∣α=α(x)

= P(U−i ≤ u∗−i(x)|Ui = u∗i (x)

)= P (Y−i = 1|X = x, Ui = u∗i (x)) = σ∗−i(1|x, u∗i (x)).

Similarly, we have that ∂P(Yi = 1, Y−i = 0|α(X) = α(x))/

∂αi = σ∗−i(0|x, u∗i (x)).

Now, we are ready to discuss the the identification of features ofM′3, for which we begin

with a weak assumption.

18

Assumption C. The payoff functions πi(a−i, x, ui) is continuous in ui for all a−i and x in the

support.

Note that under assumption S, i.e., πi(a−i, x, ui) = βi(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

payoffs ∑a−i∈A−iπi(a−i, x, ui)× σ∗−i(a−i|x, ui) is continuously decreasing in ui. Then, we

can represent the equilibrium condition, the eq. (2), by

0 = ∑a−i∈A−i

πi(a−i, x, u∗i (x))× σ∗−i(a−i|x, u∗i (x)), (7)

in which σ∗−i is already known by Lemma 3. Next, we will exploit eq. (7) for the identification

of payoffs πi. The idea is to vary σ∗−i(·|x, u∗i (x)) while we keep πi(·, x, u∗i (x)) fixed, for

which we need the following exclusion restriction.

Assumption ER (Exclusion Restriction). Let X = (X1, · · · , XI). For all i, a−i, x and ui, we

have πi(a−i, x, ui) = πi(a−i, xi, ui). 10

In the context of discrete games, the identification power of exclusion restrictions was

first demonstrated in Pesendorfer and Schmidt-Dengler (2003), and later was discussed by

Bajari, Hong, Krainer, and Nekipelov (2010) in a semiparametric setting.

For notational simplicity, we denote the random vector σ∗−i (·|X, u∗i (X)) as Σ∗−i(X), a

column vector of dimension 2I−1. By definition, ∑a−i∈A−iσ∗−i (a−i|x, u∗i (x)) = 1, then Σ∗−i(X)

is distributed on a hyperplane in R2I−1. Given Lemma 3, we treat Σ∗−i(X) as observables

hereafter. Moreover, for each (xi, αi) ∈ SXi ,αi(X), let

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 multicollinearity” among

the variables in Σ∗−i(X). Then the matrixRi(xi, αi) will have the full rank 2I−1. By eq. (7),

the full rank ofRi(xi, αi(x)) implies that πi (a−i, xi, u∗i (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 inM′3.

10As a matter of fact, Xis can have some common variables. To simplify, we assume that Xis partition X.

19

Proposition 6. Let S ∈ M′3. Suppose that assumptions RC, C and ER hold. Fix xi ∈ SXi

and αi ∈ Sαi(X)|Xi=xi∩ (0, 1). If Ri(xi, αi) has the full rank 2I−1, then Sαi(X)|Xi=xi

must be a

singleton {αi} and πi(·, xi, QUi(αi)

)= 0. IfRi(xi, αi) has rank 2I−1 − 1, then πi

(·, xi, QUi(αi)

)is identified up to scale. In addition, if there exists α′ ∈ Sα(X)|Xi=xi

∩ (0, 1) such that αi 6= α′i and

(αi, α′−i) ∈ Sα(X),11 then for each a−i ∈ A−i, the sign of πi

(a−i, xi, QUi(αi)

)is also identified.

Proof. See Appendix C.2. �

Proposition 6 shows that fixing xi, the nonseparable payoffs are identified as zero, or

identified up to scale, as well as 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. IfRi(xi, αi) has the full rank 2I−1 for some αi ∈ (0, 1), then

the domain has to be a singleton. IfRi(xi, αi) has rank 2I−1 − 1 for all αi ∈ (0, 1), however,

the more variations in E(Yi|X) as X−i varies, the larger will be this domain.12

It is of particular interest to consider and identify the case that the strategic interactions

do not exist. Fix Xi = xi. Suppose player i with some type u∗i is indifferent to other players’

moves, as well as her own action, i.e., πi(a−i, xi, u∗i ) = πi(a′−i, xi, u∗i ) = 0 for all a−i 6= a′−i.

Then, assumption M implies that player i’s best response has to be δ∗i (x, ui) = 1{ui ≤ u∗i }

no matter what other players move. Given Xi = xi, player i’s equilibrium strategy must

always be 1{ui ≤ u∗i }, 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 ofRi(xi, αi), where αi is essentially

the choice probability FUi(u∗i ). Further, from the discussion above, the full rank ofRi(xi, αi)

implies that Sαi(X)|Xi=xi= {αi}, which is essentially a testable restriction on the model

specification.

When there exist strategic effects on player i with the threshold type u∗i (x) ∈ (u∗i (x), u∗i (x)),

i.e., πi(a−i, xi, u∗i (x)) 6= πi(a′−i, xi, u∗i (x)) for some a−i 6= a′−i, then 2I−1 − 1 is the largest

rank that the matrix Ri(xi, αi(x)) could possibly have. This is because of eq. (7) — after

controlling for Xi = xi, αi(X) = αi(x), the existence of a non–zero solution of πi(·, xi, u∗i (x))

requires thatRi(xi, αi) should not have full rank.

11The existence of such an α′ can be guaranteed if (αi, α−i) is an interior point in Sα(X).12This domain excludes the boundaries QUi (0) and QUi (1) for technical reasons.

20

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∩ (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(a′−i, xi, QUi(αi)

)for all a−i 6= a′−i, are also identified. In a different setting, De Paula and

Tang (2012) also develop a novel approach for nonparametrically identifying the signs of the

strategic effects by exploiting the identification power of multiple equilibria. Our approach

needs not assume assumption I, but relies on assumptions E, ER and the assumption of a

single m.d.p.s. BNE for DGP.

It should also be noted that we can not compare the payoffs across different xis and uis. In

other words, the payoffs as a function of xi or ui is not identified. In addition, Proposition 6

is also silent about the quantile function QUi , which is indeed not identified either. In

contrast to the nonseparable 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 ofM3. By definition,M3 is a sub–model ofM′3. Under the additional

model restriction (i.e., assumption S) and one location–scale normalization on the payoffs,

we can achieve the full identification of the structural parameters.

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,

i.e., eq. (7), becomes

∑a−i∈A−i

βi(a−i, xi)× σ∗−i(a−i|x, u∗i (x)) = u∗i (x). (8)

In eq. (8), we can treat the unobserved thresholds u∗i (x) as fixed effects and control it using

the marginal choice probability αi(x). To illustrate the identification power of assumption

S, we first present a result that is parallel to Proposition 6. After that, we give the full

identification ofM3.

Let Σ∗−i(X) ≡ Σ∗−i(X)−E[Σ∗−i(X)|Xi, αi(X)

]andRi(xi) = E

[Σ∗−i(X)Σ∗−i(X)>

∣∣Xi = xi

].

Note that ι′Σ∗−i(X) = 1 a.s., where ι ≡ (1, · · · , 1)′ ∈ R2I−1. It follows that ι′Σ∗−i(X) = 0.

Thus, Σ∗−i(X) consists of a set of linearly dependent variables. Indeed, the largest possible

21

rank of the matrix Ri(xi) is 2I−1 − 1, which occurs only if βi(a−i, xi) = βi(a′−i, xi) for all

a−i 6= a′−i.

Proposition 7. Let S ∈ M3. Suppose that assumptions RC and ER hold. Fix xi ∈ SXi such

that Sαi(X)|Xi=xi∩ (0, 1) 6= ∅. If the rank of Ri(xi) is 2I−1 − 1, then Sαi(X)|Xi=xi

must be a

singleton {αi} and βi(·, xi) is identified up to the αi–quantile of FUi , i.e. βi(·, xi) = QUi(αi). If

the rank ofRi(xi) is 2I−1 − 2, then βi(·, xi) is identified up to location and scale, or equivalently,

βi(·, xi)− βi(a0−i, xi) is identified up to scale, where a0

−i ≡ (0, · · · , 0). In addition, if there exist

α, α′ ∈ Sα(X)|Xi=xisuch that αi 6= α′i and (αi, α′−i) ∈ Sα(X), then the sign of βi(a−i, xi) −

βi(a0−i, xi) is also identified for each a−i ∈ A−i.

Proof. See Appendix C.3. �

The interpretation of the rank conditions in Proposition 7 is quite similar to that for

Proposition 6. When Sαi(X)|Xi=xiis not degenerate, we know that player i’s payoffs are

strategically affected by others. Then, βi(·, 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,

c′Σ∗−i(x) = 0, ∀x ∈ SX|Xi=xi

It has two different solutions, (1, · · · , 1)′ ∈ R2I−1and βi(·, xi); the latter is due to eq. (8).

Hence, 2I−1 − 2 is the largest possible rank of the matrix Ri(xi) when there are strategic

interactions.

It should be noted that the rank conditions discussed in Proposition 7 are weaker than

those in Proposition 6. It could be shown that if the rank ofRi(xi, αi) equals k (1 ≤ k ≤ 2I−1)

for some αi ∈ Sαi(X)|Xi=xi∩ (0, 1), then the rank ofRi(xi) is no less than k− 1.

It is also worth pointing out that we could identify the payoffs in modelM3 using the

single–index structure suggested in Lemma 2. To see this, let βi(a−i, xi) = βi(a−i, xi)−

βi(a0−i, xi). Then

E(Yi|Xi = xi, Σ∗−i(X) = Σ∗−i(x)

)= FUi

(βi(a0−i, xi

)+ ∑

a−i∈A/{a0−i}

βi(a−i, xi)× σ∗−i(a−i|x, u∗i (x)

)). (9)

22

Similarly to Powell, Stock, and Stoker (1989), we could identify βi(·, xi) up to scale by

differentiating eq. (9) 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=xifor taking the derivative.

Next, we discuss the full identification ofM3 under a scale–location normalization for

each player’s payoffs. Similar to Matzkin (2003), our normalization is imposed on the

payoffs function at some x∗i ∈ SXi .

Assumption N (Payoff Normalization). Let some x∗i ∈ SXi satisfy: (i) the rank of Ri(x∗i ) is

2I−1 − 2; (ii) Sα(X)|Xi=xi

⋂(0, 1) contains two different elements α and α′ such that αi 6= α′i and

(αi, α′−i) ∈ Sα(X). We set βi(a0−i, x∗i

)= 0 and ‖βi(·, x∗i )‖ = 1.13

Let S ∈ M3. Suppose that assumptions RC, ER and N hold. By Proposition 7, the payoffs

βi(·, x∗i ) is point identified. It follows that QUi is identified on the support Sαi(X)|Xi=x∗i

⋂(0, 1)

by

QUi(αi) = ∑a−i∈A−i

βi(a−i, x∗i )×E[σ∗−i(a−i|X, u∗i (X))

∣∣Xi = x∗i , αi(X) = αi]

. (10)

Further, for each xi ∈ SXi , suppose that the rank of Ri(xi) equals to 2I−1 − 2, and that

Sαi(X)|Xi=xi

⋂Sαi(X)|Xi=x∗i

⋂(0, 1) contains at least two elements. Then, βi(·, xi) is also iden-

tified by Proposition 7, which thereafter implies the identification of QUi on the support

Sαi(X)|Xi∈{x∗i ,xi} ∩ (0, 1). Repeating such an argument, we can show that βi(·, xi) can be

point identified for all xis in a collection, denoted as C∗i , and QUi is identified on the support

Sαi(X)|Xi∈C∗i

⋂(0, 1)

Definition 2. C∗i is the subset C∞i in SXi defined by the following iterative scheme. Let C0

i = {x∗i }.

Then, for all t ≥ 0, Ct+1i consists of all elements xi ∈ SXi such that at least one of the following

conditions is satisfied: (i) xi ∈ Cti ; (ii) Ri(xi) has rank 2I−1 − 2 and there exists an x′i ∈ Ct

i such

that Sαi(X)|Xi=xi

⋂Sαi(X)|Xi=x′i

⋂(0, 1) contains at least two different elements; and (iii) Ri(xi)

has rank 2I−1 − 1 and there exists an x′i ∈ Cti such that Sαi(X)|Xi=xi

⊆ Sαi(X)|Xi=x′i

⋂(0, 1).

In Definition 2, condition (ii) is the key to effectively expand the collection of xis in an

iterative manner by enlarging Sαi(X)|Xi∈Cti

to Sαi(X)|Xi∈Ct+1i

.

13Note that ‖βi(·, x∗i )− πi(a0−i, x∗i )‖ 6= 0 because of the non–degeneracy of Sαi(X)|Xi=x∗i

.

23

Proposition 8. Let S ∈ M3. Suppose assumptions RC, ER and N hold. Then βi and QUi are point

identified on the support A−i ×C∗i and Sαi(X)|Xi∈C∗i

⋂(0, 1), respectively.

Proof. See Appendix C.4 �

In Proposition 8, the domain C∗i essentially depends on x∗i and the variation of Sαi(X)|Xi=xi

across different xis. Regarding the choice of the starting point x∗i , intuitively, we should

choose it in a way such that C∗i is the largest. However, it can be shown that for any x′isatisfying conditions in assumption N, suppose x′i ∈ C∗i . Then we will end up with the

same C∗i if we normalize payoffs function βi at x′i instead of x∗i .

It is interesting to notice that our normalization does not apply to a nonparametric

single–agent binary response model (see, e.g., Matzkin, 1992). This is because the support

Sαi(X)|Xi=x∗iis a singleton in a single–agent binary response model. With interactions,

in contrast, we can exploit variations of X−i while controlling for Xi to identify a set of

quantiles of FUi .

4.4. Identification ofM4. The identification ofM4 does not essentially differ from that of

M3: assumption I only relaxes the rank condition for identification of σ∗−i in Lemma 3. We

illustrate this in the next lemma.

Lemma 4. Let S ∈ M4. Fix x ∈ SX. Then σ∗−i(·|x, u∗i (x)) is identified by

σ∗−i(a−i|x, u∗i (x)) = P (Y−i = a−i|X = x) . (11)

The proof is straightforward, hence omitted. By the proofs for Proposition 3, we can also

show that σ∗−i(a−i|x, u∗i (x)) = P (Y−i = a−i|α(X) = α(x)). Similar results can be found in

Aguirregabiria and Mira (2007); Bajari, Hong, Krainer, and Nekipelov (2010), among others.

Further, the identification of βi and QUi inM4 follows Propositions 7 and 8.

5. EXTENSIONS

In this section, we consider three main extensions of our identification analysis. First,

without assuming assumption ER, we examine the partial identification of structures inM′3

andM3. Second, we study the case in which each player’s type is a multiple dimensional

random vector, which is of particular interest to many IO applications. Last, we relax the

24

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 the distributions from all these equilibria.

5.1. Partial Identification. Partial identification is a notion closely related to rationalization

and identification studied in Sections 3 and 4. Given a probability distribution of observ-

ables, i.e. FY|X, we are interested in which structure or which set of structures in modelM

can generate the observed distribution.

For each S ∈ M, let φ : M → F be a correspondence mapping a structure to a set of

conditional distributions of observables that could be generated from the given structure,

and let φ−1 be the inverse correspondence defined as

φ−1(FY|X) = {S ∈ M : FY|X ∈ φ(S)}.

Regarding the set φ−1(FY|X), there could be two distinct possibilities

Case I: if φ−1 (FY|X)

is an empty set, then we say that FY|X is unrationalizable byM.

Case II: if φ−1 (FY|X)

is a non–empty set, then we say FY|X is rationalized byM. Further,

if φ−1(FY|X) is a singleton, we sayM is point identified, or identified given FY|X; otherwise

we sayM is partially identified given FY|X.

Our rationalization results in Section 3 characterize whether the set φ−1 (FY|X)

is empty

or not. For instance, given the characterization of the set of rationalizable distributions

in Section 3, it follows that those probability distributions which violate the conditions

in Propositions 1 and 3 have an empty identified set. In other words, those distributions

are not generated by any structure inM3. On the other hand, our Section 4 studies point

identification under some identifying assumptions such as assumption ER.

In this section, we relax the identifying assumption (assumption ER) and characterize

the set of structures inM3 orM′3 that could generate the observed distribution FY|X.

Before proceeding, it is worth emphasizing that the lack of point identification of a

structure is not due to multiple equilibria, but to the lack of identifying restrictions, namely,

the exclusion restriction. This is similar to, e.g., Shaikh and Vytlacil (2011) who study partial

identification of the average structural function in a triangular model without imposing

a restrictive support condition. When there are multiple monotone pure strategy BNEs,

25

we still maintain the assumption of a single equilibrium being played for generating the

distribution of observables.

By the same argument in the section of identification inM′3, the copula function CU is

point–identified on the support Sα1(X) ∪ {0, 1} × · · · ×SαI(X) ∪ {0, 1}. 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

CI ={

C ∈ C : CU(α) = CU(α), ∀α ∈ Sα1(X) ∪ {0, 1} × · · · ×SαI(X) ∪ {0, 1}}

.

For each CU ∈ CI , suppose we set FUi to be the uniform distribution on [0, 1] and

πi(·, x, ui) = αi(x)− ui. Clearly, the constructed structure[π; FU

]is observationally equiva-

lent to the underlying structure. Thus, CI 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 from [0, 1] to R, denoted as Q. The next lemma shows thatM3 (orM′3) imposes

no restrictions on QUi and its identification region is Q.

Lemma 5. Let S ∈ M3 (orM′3). For any (QU1 , · · · , QUI ) ∈ Q I , then there exists an observa-

tionally equivalent structure S ∈ M3 (therefore, S ∈ M′3) with the marginal quantile function

proflie (QU1 , · · · , QUI ).

Proof. See Appendix D.1 �

Now we discuss the sharp identification region for πi. Let G be the set of real functions

mapping A−i ×SX to R.

Proposition 9. Let S ∈ M3. Suppose assumption RC holds. Then the sharp identification region of

the structural parameters is given by{[

β; {QUi}Ii=1; CU

]: QUi ∈ Q, (β, CU) ∈ ΘI({QUi}I

i=1)}

,

26

in which

ΘI({QUi}Ii=1) ≡

{(β, CU) ∈ G I × CI : (a) for all x ∈ SX and i,

QUi(αi(x)) = ∑a−i∈A−i

βi(a−i, x)× σ∗−i(a−i|x, u∗i (x)); (b) for any m.d.p.s. profile δ;

Eδ

[βi (Y−i, X) |X = x, Ui = QUi(αi)

]− QUi(αi) is decreasing in αi ∈ (0, 1)

}.

Proof. See Appendix D.2. �

In the definition of ΘI({QUi}Ii=1), condition (a) requires that βi(·, 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,

ΘI({QUi}Ii=1) is nonempty and convex.14

Indeed, 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 continu-

ously 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∈SUiψ′i(ui) ≥ supx∈SX

κi(x). Clearly, there are plenty of choices for such a func-

tion ψi. Let further QUi = ψi(QUi) and βi(a−i, x) = ξi(x) + κi(x) × βi(a−i, x), in which

ξi(x) = ψi(QUi(αi(x)))− κi(x)×QUi(αi(x)). Then, it can be verified that the constructed

structure [β; {QUi}Ii=1; CU

]belongs to the identified set. To narrow down the identification

region, additional restrictions are necessary to be introduced. Instead of imposing assump-

tion ER, an alternative approach is to make assumptions on the payoffs’ functional form.

For instance, De Paula and Tang (2012) set βi(a−i, x) = β∗i (x) + gi(a−i)× h∗i (x), where gi is

a function known to all players as well as to the econometrician, and (β∗i , h∗i ) are structural

parameters in their model.

14 To see the nonemptiness, we can simply take βi(·, x) = QUi (αi(x)) and CU = CU .

27

When Sα(X) = [0, 1]I , CI degenerates to the singleton {CU}. In this case, the sharp identi-

fication region for (β, {QUi}Ii=1) can be characterized in a more straightforward manner:

Θ∗I ={(β, QU) ∈ G I ×Q I : (a′) for all x ∈ SX and i,

QUi(αi(x)) = ∑a−i∈A−i

βi(a−i, x)× σ∗−i(a−i|x, u∗i (x)); (b′) and for all α−i ∈ [0, 1]I−1,

∑a−i∈A−i

βi(a−i, x)× σ∗−i (a−i|x, QUi(αi))− QUi(αi) is decreasing in αi ∈ (0, 1)}

.

ForM′3, we can similarly characterize the sharp identification region for the structural

parameters. Let G ′ be the set of functions mapping A−i ×SX ×R to R.

Proposition 10. Let S ∈ M′3. Suppose that assumption RC holds. Then the sharp identification re-

gion for [π; {QUi}Ii=1; CU ] is given by

{[π, {QUi}I

i=1, CU

]: QUi ∈ Q, (π, CU) ∈ Θ′I({QUi}I

i=1)}

,

in which

Θ′I(QU) ={(π, CU) ∈ G ′

I × CI : for all x ∈ SX and i,

0 = ∑a−i∈A−i

πi

(a−i, x, QUi(αi(x))

)× σ∗−i(a−i|x, u∗i (x)); and for any m.d.p.s. profile δ

Eδ

[πi

(Y−i, X, QUi(αi)

)|X = x, Ui = QUi(αi)

]is decreasing in αi ∈ (0, 1)

}.

The proofs are similar to those for Proposition 9, and therefore omitted.

5.2. Multiple Dimensional Signals. In some applications, players might observe multiple

dimensional signals. For example, a potential entrant to some local market could individu-

ally observe the idiosyncratic demand shocks, as well as her private cost information.

Without loss of generality, let Ui ∈ Rd, where d ≥ 2. Let further T be the class of

functions mapping from Rd to R and

TS ={(T1, · · · , TI) ∈ T I : πi(·, ·, ui) = πi (·, ·, Ti(ui)) for some πi

}.

By such a transformation, i.e., let Vi ≡ Ti(Ui), the idea is to apply m.d.p.s. BNE concept on

the transformed scalar type Vi. Therefore, we could show that there exist BNEs in which

each player’s strategy δ∗i is a monotone decreasing function of Vi.

28

It is worth pointing out that the collection TWS 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”.15 For

any bijection g, we can simply choose πi(·, ·, vi) = πi(·, ·, g−1(vi)) for the condition in the

definition of TS.

Assumption T. There exists some (maybe unknown) function profile (T∗1 , · · · , T∗I ) ∈ TS satisfying

(1) T∗i is measurable and FV|X,Ui= FV|X,Vi

, where Vi = T∗i (Ui) and V = (V1, · · · , VI).

(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

fV|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

decreasing in vj ≡ T∗j (uj), Eδ

[πi(Y−i, X, Vi)

∣∣X = x, Vi = vi]

is also monotone decreasing

in vi.

In assumption T, the conditions (2) and (3) are the analogous of assumption R and M,

respectively. Condition (1) requires that the dependence of types (U) is fully captured by

the dependence of the transformed types (V), for which the measurability of T∗i is needed.16

Under assumption T, we can shift our focus on the original structure S = [π, {QUi}Ii=1, CU ]

inM3 (orM′3) to S = [π, {QVi}I

i=1, CV ], which admits an m.d.p.s. (monotone decreasing in

the transformed type Vi) BNE by Lemma 1. If T∗i is a bijection for all i. Then, the structures

S and S also one to one and onto map to each other. Let further

M3 ≡{

S : assumptions T, S and E hold and a single m.d.p.s. BNE is played}

.

Similar to Propositions 7 and 8, we can show that M3 can be identified under additional

identifying conditions.

In some applications, T∗i might be a known function, or known up to some structural pa-

rameter. For example, suppose Ui = (UDi , UC

i ) ∈ R2 and πi(a−i, x, ui) = βi(a−i, x) + γuDi −

15Following the Cantor–Schroeder–Bernstein Theorem, we could explicitly construct one example of such amapping.16We thank Nese Yildiz for pointing this out.

29

uCi . In addition, suppose one would like to assume FγUD

−i−UC−i |X,UD

i ,UCi= FγUD

−i−UC−i |X,γUD

i −UCi

.

Then it follows naturally that T∗i (ui) = γuDi − uc

i , a transformation parametrized by γ.

Though we don’t know γ, it’s still possible to identify function βi in the payoffs.

5.3. Multiple Equilibria in DGP. Our (point and partial) identification analysis relies on

the maintained assumption that data comes from only a single m.d.p.s. BNE for each

x, though it is quite common in the empirical game literature to rule out the mixture of

distributions from multiple equilibria, e.g., Aguirregabiria and Mira (2007) and Bajari, Hong,

Krainer, and Nekipelov (2010).

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 changes the equilibrium selection

among multiple solutions. 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. For the purpose of simplification, we assume that players only

focus on a subset Γ(x) of E (x) for DGP, i.e., the set of equilibria that will be played in data.

We further assume the number of elements in Γ(x) is bounded above by a constant J (J ≥ 2)

for all x. Without loss of generality, we simply take J = 2 in the following discussion.

Let λ be a probability distribution {pλ1 , · · · , pλ

J } on the support {1, · · · , J} such that the j–

th equilibrium will get to play with the probability pλj . λ could 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 J = 2, the observed

distribution FY|X=x is essentially a convex combination of the two component distributions

generated from the two equilibria in Γ(x). Then, the variations of the instrumental variable

Z cause the mixture distribution to move along a straight line in the functional space of

probability distributions.

30

Further, we can point identify the set of distributions corresponding to Γ(x) if Z has

sufficient variations. For each x ∈ SX, suppose λ(x, z) = (0, 1) and λ(x, z′) = (1, 0) for

some z, z′ ∈ 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.

6. CONCLUSION

This paper addresses the rationalization and identification of discrete games with corre-

lated types in a fully nonparametric way. We show that our baseline game-theoretical model

does not impose any essential restriction on observables. This implies that binary Bayesian

games are not testable in view of players’ choices only. We also characterize all the restric-

tions on players’ choices imposed by three assumptions frequently made in the empirical

analysis of discrete games. We then exploit exclusion restrictions to identify our structural

model nonparametrically in both nonseparable and separable settings. These restrictions

take the form of excluding part of a player’s payoff shifters from all other players’ payoffs

as frequently assumed in the empirical discrete game literature. We also characterize the

sharp identification region of the structural parameters without the exclusion restrictions.

Further, we generalize our nonparametric identification analysis in two model aspects.

First, we allow multiple dimensional type for each player, which is of particular interest to

some IO applications. Second, we relax the assumption of the same m.d.p.s. BNE in the

DGP. By introducing an instrumental variable that only enters the equilibrium selection

mechanism and has sufficient variations, each component of the mixture distribution can

be point identified. Alternatively, one could also follow the set–identification approach

initiated by Tamer (2003) in the presence of multiple equilibria.

A second line of research, which needs to be developed, concerns model testing. Our

Propositions 3 and 4 become especially useful as they characterize all the restrictions in

terms of observables imposed byM3 andM4, respectively. Thus such restrictions are in

31

principle testable for the purpose of model specifications. In particular, one could extend

Fan and Li (1996) and Lavergne and Vuong (2000) by allowing generated covariates to test

the restriction R1 in Proposition 3. Further, the restriction given in Proposition 4 can be

tested by using conditional independence tests developed in statistics and econometrics

(see, e.g. Su and White, 2007, 2008). It is also worthnoting that such tests do not rely

on identification of the model and consequently on the assumptions used to identify the

primitives.

Lastly, a third line of research deals with the nonparametric estimation of the various

models. In a semiparametric setup, Liu and Xu (2012) propose an estimation procedure for

our modelM3 with linear payoff, and establish the root-N consistency of the linear payoff

coefficients. A fully nonparametric estimation, however, deserves future investigation. A

strategy could rely on the identification results and propose a sample-analog type of estima-

tors for the players’ payoffs and the joint distribution of private information. Establishing

the asymptotic properties of such an estimation procedure is left for future research. The

main difficulty relies on the generated covariates, namely the belief of the player at the

margin which appears in the expected payoff. Such a problem could be addressed by using

the most recent literature on nonparametric regression with nonparametrically generated

covariates (see, e.g. Mammen, Rothe, and Schienle, 2012).

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XU, H. (2010): “Estimation of Discrete Games with Correlated Types,” working paper.

(2011): “Social interactions: a game theoretic approach,” working paper.

37

APPENDIX A. EXISTENCE OF M.D.P.S. BNES

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 ⊆ RI−1, the

conditional probability P (U−i ∈ A|X = x, Ui = ui) is non–decreasing in ui ∈ SUi |X=x.

Assumption SC (Strategic Complementarity). For any x ∈ SX and ui ∈ SUi |X=x, suppose a−i ≤ a′−i,

then πi(a−i, x, ui) ≤ πi(a′−i, x, ui).

Assumption DP (Non-increasing Payoffs). ∀ i = 1, · · · , I and ∀ (a−i, x) ∈ A−i ×SX, πi(a−i, x, ·)

are non-increasing functions in ui ∈ SUi |X=x.

Lemma 6. Suppose assumptions R, PR, SC, 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: δi(x, ui) = 1[ui ≤ ui(x)] for i = 1, · · · , I, where ui(x) is an arbitrary function. By

assumptions PR and SC, and Lehmann (1955), for any ui < u′i in the support, we have

Eδ

[πi(Y−i, X, ui)|X = x, Ui = u′i

]≤ Eδ [πi(Y−i, X, ui)|X = x, Ui = ui] .

Further, by assumption DP,

Eδ

[πi(Y−i, X, u′i)|X = x, Ui = u′i

]≤ Eδ

[πi(Y−i, X, ui)|X = x, Ui = u′i

].

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−iβi(a−i, x)σ∗−i(a−i|x, ui)− ui. Because σ∗−i(a−i|x, ui) are continuous in ui under assumption R,

then ∑a−iβi(a−i, x)σ∗−i(a−i|x, ui)− ui is a continuously decreasing function in ui.

Suppose ui(x) < u∗i (x) < ui(x). It follows that

∑a−iβi(a−i, x)σ∗−i(a−i|x, u∗i (x))− u∗i (x) = 0.

38

Hence, conditional on ui(X) < u∗i (X) < ui(X), we have that

Yi = 1 [Ui ≤ u∗i (X)] = 1

[Ui ≤∑

a−i

βi(a−i, X)σ∗−i(a−i|X, u∗i (X)

)].

Suppose u∗i (x) = ui(x). Then ∑a−iβi(a−i, x)σ∗−i(a−i|x, ui(x)) − ui(x) ≥ 0, which implies that

conditional on u∗i (X) = ui(X), there is

Yi = 1 [Ui ≤ ui(X)] ≤ 1

[Ui ≤∑

a−i

βi(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

βi(a−i, X)σ∗−i(a−i|X, ui(X)

)]a.s.

Similar arguments hold for the case u∗i (X) = ui(X). �

APPENDIX B. RATIONALIZATIONS

B.1. Proof of Proposition 1. Prove only if part first: Proofs by contradiction. Let FY|X be rationalized

byM1, i.e., some S ∈ M1 can generate FY|X. Fix X = x and let equilibrium be characterized by

(u∗1(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 ≤ u∗1(x), · · · , UI ≤ u∗I (x)|X = x) = 0 and

P(Ui ≤ u∗i (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 inM1 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, · · · , αI |x) =

E(∏pj=1 Yij |X = x) where i1, · · · , ip are all the indexes such that αij = E(Yij |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).

39

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 {k + 1, · · · , I} satisfying 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 S 0Ui |X=x of SUi |X=x such that every set in SUi |X=x assigned

positive probability by FUi |X(·|x) contains two points between which lies a point in S 0Ui |X=x.

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 RI . 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, letM′1 ≡

{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. For any x ∈ SX , FY|X(·|x) can be rationalized by a structure S ≡ [π1, · · · , πI ; FU|X ] ∈ M′1.

Proof. Fix x ∈ SX. Now we construct a structure inM′1 to give us FY|X(·|x). Let πi(a−i, x, ui) =

αi(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 [0, 1]I be the support of the distribution and

partition it into 2I disjoint events:⊗I

i=1{[0, αi(x)), [αi(x), 1]} 17. Further, we define a conditional

distribution FU|X=x,U∈Bjas 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

S ≡ [π1, · · · , πI ; FU|X ] ∈ M′1 leads to FY|X(·|x). �

17To have meaningful partition, it is understood that {[0, αi(x)), [αi(x), 1]} becomes {{0}, (0, 1]} when αi(x) =0.

40

B.3. Proof of Proposition 2.

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 obser-

vationally equivalent structure S ≡ [π; FU|X ] ∈ M2. Fix x ∈ SX. Let FU|X(·|x) = FU|X(·|x) and

πi(a−i, x, ui) = u∗i (x)− ui where(u∗1(x), . . . , u∗I (x)

)is the profile of equilibrium threshold points

under S. It is straightforward that S belongs toM2 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

S = [π; FU|X ] ∈ M2 is also rationalized by S = [π; FU|X ] ∈ M3 . Then

E

(p

∏j=1

Yij |X)

= P(

Yi1 = 1, · · · , Yip = 1|X)

= P(

Ui1 ≤ u∗i1(X), · · · , Uip ≤ u∗ip(X)|X

)= CUi1

,··· ,Uip

(αi1(X), · · · , αip(X)

).

Similarly,

E

(p

∏j=1

Yij |αi1(X), · · · , αip(X)

)= CUi1

,··· ,Uip

(αi1(X), · · · , αip(X)

).

Thus, we have condition R1. Further, R2 and R3 obtain by the properties of the copula function

CUi1,··· ,Uip

.

Proofs for the if part. For any x ∈ SX , let πi(a−i, x, ui) = αi(x)− ui. Let further FUi be uniformly

distributed on [0, 1]. For all 1 ≤ i1 < · · · < ip ≤ I, (αi1 , · · · , αip) ∈ Sαi1(X),··· ,αip (X) and x ∈ SX,

define FUi1,··· ,Uip

(·, · · · , ·) as follows: for each αi1 , · · · , αip ∈ Sαi1(X),··· ,αip (X),

FUi1,··· ,Uip

(αi1 , · · · , αip) = E

[p

∏j=1

Yij

∣∣αi1(X) = αi1 , · · · , αip(X) = αip

].

Thus, we define FU on the support Sα1(X) ∪ {1} × · · · ×SαI(X) ∪ {1}.

By Proposition 1, we have that FUi1,··· ,Uip ,Uk (αi1 , · · · , αip , αk) < FUi1

,··· ,Uip(αi1 , · · · , αip) for any

k 6= ij, j = 1, · · · , p, and αk < 1. Further, under conditions R2, R3, FU is strictly increasing and

continuously differentiable on Sα1(X) ∪ {1} × · · · ×SαI(X) ∪ {1}. Hence, we can extend it to the

whole support [0, 1]I as a proper distribution function such that it is strictly monotone increasing

and continuously differentiable on (0, 1]I . The extended FU(·) will yield a positive and continuous

conditional Radon–Nikodym density on [0, 1]I .

41

By construction, [π; FU ] ∈ M3. Fix X = x. The constructed structure [π; FU(·)] will generate the

same marginal distribution αi(x) for i = 1, · · · , I. Moreover, for any subsequence i1, · · · , ip from

{1, · · · , I}, where 1 ≤ p ≤ I,

P(Yi1 = 1, · · · , Yip = 1|X = x) = Fi1,··· ,ip

(αi1(x), · · · , αip(x)

)= E

[p

∏j=1

Yij

∣∣αi1(X) = αi1 , · · · , αip(X) = αip

]= E

[p

∏j=1

Yij

∣∣X = x

].

Because i1, · · · , ip is arbitrary, then [π, FU ] generates the distribution FY|X(·|x). �

B.5. Proof of Proposition 4.

Proof. The only if part follows directly from assumption I. It suffices to show the if part.

Fix a distribution FY|X that satisfies the condition. Let FUi |X(·|x) = FUi be uniformly distribution

on [0, 1] and FU|X = ∏Ii=1 FUi . Moreover, let πi(a−i, x, ui) = αi(x) − ui for any x ∈ SX. By

construction, [π; FU|X ] satisfies assumptions R, M, S, E, and I. Hence, [π; FU|X ] ∈ M4.

It suffices to show that the constructed structure [π; FU|X ] can generate FY|X . Fix any x ∈ SX . By

construction, we have that P(Yi = 1|X = x) = αi(x). Moreover, for any subsequence i1, · · · , ip from

{1, · · · , I}, where 1 ≤ p ≤ I,

P(Yi1 = 1, · · · , Yip = 1|X = x) = Fi1,··· ,ip

(αi1(x), · · · , αip(x)

)=

p

∏j=1

αip(x) = P(Yi1 = 1, · · · , Yip = 1|X = x).

Because i1, · · · , ip is arbitrary, then [π, FU ] generates the distribution FY|X(·|x). �

APPENDIX C. IDENTIFICATIONS

C.1. Proof of Lemma 3.

Proof. Our following proofs are essentially similar to 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)]

=∫ QUi

(αi)

QUi(0)

[P (Y−i = a−i|α(X) = α, Ui = ui)] dFUi (ui)

=∫ αi

0P[Y−i = a−i|α(X) = α, Ui = QUi (vi)

]dvi

42

where the second equality is due to the fact P[Yi = 1|α(X) = α, Ui ≤ QUi (αi)] = 1 and P[Yi =

1|α(X) = α, Ui > QUi (αi)] = 0, and the last one changes the variable (vi = FUi (ui)) in the integration.

Therefore, we have that

∂P (Yi = 1; Y−i = a−i|α(X) = α)

∂αi= P

[Y−i = a−i|α(X) = α, Ui = QUi (αi)

].

Note that QUi (αi(x)) = u∗i (x). Then, it follows that

σ∗−i(a−i|x, u∗i (x)) ≡ P [Y−i = a−i|X = x, Ui = u∗i (x)]

= P [Y−i = a−i|α(X) = α(x), Ui = u∗i (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∩ (0, 1), there exists some x ∈ SX such αi(x) = αi. By eq. (7), we have

that

∑a−i

πi(a−i, xi, u∗i (x))× σ∗−i(a−i|x, u∗i (x)) = 0.

Since u∗i (x) = QUi (αi), then we have

∑a−i∈A−i

πi(a−i, xi, QUi (αi)

)× σ∗−i(a−i|x, u∗i (x)) = 0. (12)

Thus, ifRi(xi, αi) has full rank 2I−1, eq. (12) 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, Sαi(X)|Xi=ximust be a singleton {αi}.

Further, ifRi(xi, αi) has rank 2I−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

some variable. By excluding, w.l.o.g., σ∗−i(a0−i|X, u∗i (X)

)from Σ∗−i(X), the random variables left in

Σ∗−i are conditionally linearly independent given Xi = xi and αi(X) = αi. Because

∑a−i∈A−i\{a0

−i}πi(a−i, xi, QUi (αi)

)× σ∗i (a−i|x, u∗i (x)) = −σ∗i (a0

−i|x, u∗i (x))πi(a0−i, xi, QUi (αi)

),

43

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)

).

Now we show the identification of the sign of πi(·, xi, QUi (αi)

). W.L.O.G., let α′i < αi. Let further

x′ ∈ SX such that α(x′) = α′. Then by assumption M, there is

∑a−i∈A−i

πi(a−i, xi, QUi (αi)

)× σ∗i (a−i|x′, QUi (αi))

< ∑a−i∈A−i

πi(a−i, xi, QUi (α

′i))× σ∗i (a−i|x′, QUi (α

′i)) = 0

Note that σ∗i (a−i|x′, QUi (αi)) is also 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)

)= ki

(·, xi, QUi (αi)

)× πi

(a0−i, xi, QUi (αi)

)where ki ∈ R2I−1

is known. By definition, ki(a0−i, xi, QUi (αi)

)= 1. Thus{

∑a−i∈A−i

ki(a−i, xi, QUi (αi)

)× σ∗i (a−i|x′, QUi (αi))

}× πi

(a0−i, xi, QUi (αi)

)< 0,

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. Proof of Proposition 7. 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),

∑a−i∈A−i

βi(a−i, xi)σ∗−i(a−i|x, u∗i (x))

)= QUi (αi(x)). (13)

It follows that

∑a−i∈A−i

βi(a−i, xi)E[σ∗−i(a−i|X, u∗i (X))

)|Xi = xi, αi(X) = αi(x)

]= QUi (αi(x)) (14)

The difference between eq. (13) and (14) yields

∑a−i∈A−i

βi(a−i, xi)× σ∗−i(a−i, x

)= 0 (15)

where σ∗−i(a−i, x

)≡ σ∗−i

(a−i|x, u∗i (x)

)−E

[σ∗−i(a−i|X, u∗i (X)

)|Xi = xi, αi(X) = αi(x)

].

When xi is fixed, we can identify βi(·, xi) as coefficients by varying σ∗−i(a−i, x) through x−i.

Suppose Ri(xi) has rank 2I−1 − 1. By a similar argument as that for Proposition 6, βi(·, xi) are

identified up to scale. Note that ∑a−i∈A−iσ∗−i(a−i, x

)= 0. Hence, βi(a−i, xi) equals to the same

44

constant for all a−i ∈ A−i. By eq. (13), we have that βi(·, xi) = QUi (αi(x)). Moreover, similarly to

Proposition 6, Sαi(X)|Xi=xihas to be a singleton set, i.e. {αi(x)} for some x ∈ SX|Xi=xi

.

Further, supposeRi(xi) has rank 2I−1 − 2. Then we can pick a vector β0i (·, xi) ∈ R2I−1

such that

β0i (a−i, xi) 6= β0

i (a′−i, xi) for some a−i, a′−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 that ∑a−i∈A−i1× σ∗−i

(a−i, x

)= 0. By linear algebra, βi can be written as

βi(·, xi) = ci(xi) + ki(xi)× β0i (·, xi)

where ci(xi), ki(xi) : SXi → R. Hence, βi are identified up to location (ci) and scale (ki).

The identification of the sign of βi(ai, xi)− βi(a0i , xi), it is similar to the proofs for Proposition 6:

let x, x′ ∈ SX|Xi=xi, α(x) = α, α(x′) = α′, and w.l.o.g., α′i < αi. Then

∑a−i∈A−i

βi(a−i, xi)σ∗−i(a−i|x′, QUi (αi)

)< QUi (αi(x)) = ∑

a−i∈A−i

βi(a−i, xi)σ∗−i(a−i|x, QUi (αi)

),

from which we have

ki(xi)× ∑a−i∈A−i

β0i (a−i, xi)×

[σ∗−i(a−i|x′, QUi (αi)

)− σ∗−i

(a−i|x, QUi (αi)

)]< 0.

Thus we identify the sign of ki(xi). It follows that the sign of βi(a−i, xi)− βi(a′−i, xi) = ki(xi)×[β0

i (a−i, xi)− β0i (a′−i, xi)

]is also identified. �

C.4. Proof of Proposition 8.

Proof. It suffices to show that the identification of βi(·, xi) for all xi ∈ Cti implies that βi(·, xi) is

point identified for all xi ∈ Ct+1i . Let xi ∈ Ct+1

i . The claim is straightforward if xi ∈ Cti . Hence, let

x ∈ Ct+1i /Ct

i . Then it could be Case (ii) or (iii).

Suppose that Case (ii) occurs, i.e. Ri(xi) has rank 2I−1 − 2 and there exists x′i ∈ Cti such that

Sαi(X)|Xi=xi

⋂Sαi(X)|Xi=x′i

⋂(0, 1) contains at least two different elements: 0 < α′i < αi < 1. Because

x′i ∈ Cti , then by assumption πi(·, x′i) are identified. Then both QUi (αi) and QUi (α

′i) are identified by

eq. (10).

Further, because Ri(xi) has a rank 2I−1 − 2, then by Proposition 7, βi(·, xi) is identified up

to location and scale, i.e. ∃ ci(xi), ki(xi) ∈ R and a known vector β0i (·, xi) ∈ R2I−1

, such that

45

βi(·, xi) = ci(x) + ki(x)× β0i (·, xi). Moreover, because αi, α′i ∈ Sαi(X)|Xi=xi

⋂(0, 1), then we have

∑a−i∈A−i

βi(·, xi)× σ∗−i(a−i|x, u∗i (x)) = QUi (αi),

∑a−i∈A−i

βi(·, xi)× σ∗−i(a−i|x′, u∗i (x′)) = QUi (α′i).

It follows that

ci(xi) + ki(xi)× ∑a−i∈A−i

β0i (·, xi)× σ∗−i(a−i|x, u∗i (x)) = QUi (αi),

ci(xi) + ki(xi)× ∑a−i∈A−i

β0i (·, xi)× σ∗−i(a−i|x′, u∗i (x′)) = QUi (α

′i).

Note that QUi (αi), QUi (α′i), β0

i (·, xi), σ∗−i(a−i|x, u∗i (x)) and σ∗−i(a−i|x′, u∗i (x′)) are all known terms

and QUi (α′i) < QUi (αi). Then we can identify ci(xi) and ki(xi) from above two equations. Therefore,

βi(·, xi) are identified.

Suppose that Case (iii) occurs, i.e. Ri(x′i) has a rank 2I−1 − 1 and there exists x′i ∈ Cti such that

Sαi(X)|Xi=xi⊆ Sαi(X)|Xi=x′i

⋂(0, 1). Because of the rank condition, βi(·, xi) is identified by QUi (αi),

which is known by the fact Sαi(X)|Xi=xi⊆ Sαi(X)|Xi=x′i

⋂(0, 1). �

APPENDIX D. EXTENSIONS

D.1. Proof of Lemma 5.

Proof. First, we construct a structure S ∈ M3 (which implies that S ∈ M′3) such that (1) S has

the marginal quantile functions(

QU1 , · · · , QUI

); (2) CU(·) = CU(·) on [0, 1]I ; (3) for any x ∈ SX,

i, and a−i ∈ A−i, let βi(a−i, x) = QUi (E(Yi|X = x)). By construction, it is straightforward that

assumptions R, M, S and E are satisfied.

Now it suffices to verify the observational equivalence between S and S. Fix x ∈ SX . Note that

in the structure S there is no strategic effects, then the equilibrium is: 1{

ui ≤ QUi (E(Yi|X = x))}

for i = 1, · · · , I. Then similar to the proof for Lemma 3, here we only verify the observational

equivalence for action profile (1, · · · , 1) and the proofs for other action profiles follow similarly:

P(Y1 = 1; · · · ; YI = 1|X = x) = CU (E(Y|X = x))

= CU (E(Y|X = x)) = P(Y1 = 1; · · · ; YI = 1|X = x). �

D.2. Proof of Proposition 9.

46

Proof. It is straightforward that (β, CU) ∈ ΘI(QU). For sharpness, it suffices to show that for any

(β′, C′U) ∈ ΘI({Q′Ui}I

i=1), then S′ ≡ (β′, {Q′Ui}I

i=1, C′U), which belongs toM3 by the definition of

ΘI({Q′Ui}I

i=1), is observationally equivalent to the underlying structure S ≡ (β, {QUi}Ii=1, CU).

Fix X = x. It suffices to verify that δ∗ =(

1{

u1 ≤ QU1(α1(x))}

, · · · , 1{

uI ≤ QUI (αI(x))} )

is a

BNE solution for the constructed structure. Because C′U ∈ CI and by the proof for Lemma 3,

Pδ∗{

Y−i = a−i|X = x, Ui = QUi (αi(x))}= σ∗−i(a−i|x, u∗i (x)).

Then, by the conditions in the definition of ΘI({Q′Ui}I

i=1), 1{

ui ≤ QUi (αi(x))}

is the best response

to δ∗−i. Thus δ∗ is a BNE. �

47