ECONOMETRIC INFERENCE ON LARGE BAYESIAN GAMES WITHHETEROGENEOUS BELIEFS

KYUNGCHUL SONG

Abstract. Most existing econometric models on games assume observation of many repli-cations of a single small-player game. Such a framework is not adequate for a situation whereone observes multiple heterogeneous many-player games, as in models of social interactionsor heterogeneous complete information games. This paper focuses on a situation where theeconometrician observes multiple heterogeneous many-player games. The model does notrequire a common prior assumption, and allows for the players to form beliefs or higher orderbeliefs about the other players�types and beliefs di¤erently from each other. The main chal-lenge for the econometrician is to recover the beliefs an individual player forms about otherplayers�actions. By drawing on the main intuition of Kalai (2004), this paper introducesthe notion of a hindsight regret which measures each player�s ex post value of other players�type information, and establishes its belief-free bound. From this bound, this paper derivesa set of moment inequalities associated with the incentive constraints of a Bayesian Nashequilibrium, and develops an asymptotic inference procedure for the structural parameters.

Key words. Large Game; Incomplete Information; Heterogenous Beliefs; Bayesian NashEquilibria; Ex Post Stability; Hindsight Regrets; Cross-Sectional Dependence; Partial Iden-ti�cation; Moment Inequalities.

JEL Subject Classification. C13, C31.

1. Introduction

Many economic variables arise as a consequence of economic agents�rational decisions in

various strategic environments. Education decisions by people, entries into a market by �rms,

or behavioral choices by students such as smoking, are made directly under the in�uence of

others�choices. For an empirical researcher studying the way economic agents make decisions

in such environments, the main challenge lies in the dilemma that the statistical inference

based on the law of large numbers and the central limit theorem requires independence

Date: May 21, 2013.I thank Andres Aradillas-Lopez, Aureo de Paula, Hiro Kasahara, Jinwoo Kim, Sokbae Lee, Wei Li, VadimMarmer, and Mike Peters for useful conversations and comments, and seminar participants at Seoul NationalUniversity, UBC, University of Texas at Dallas, and University of Wisconsin-Madison for their comments.I also thank Yoram Halevy, Wei Li, Qingmin Liu, and Mike Peters for their kind answers to my numerouselementary questions on Bayesian games. Last but not least, I thank Bruce Hansen and Joon Park for theirencouragements at the early stage of this research. All errors are mine. I acknowledge that this researchwas supported by Social Sciences and Humanities Research Council of Canada. Corresponding address:Kyungchul Song, Department of Economics, University of British Columbia, Vancouver, BC, Canada. Emailaddress: kysong@mail.ubc.ca.

1

2 SONG

among observed cross-sectional outcomes, whereas the focus of interest is on the nature of

their dependence which potentially arises from their strategic interactions.

A strand of empirical models have been actively developed in response to this challenge

by separating statistical independence and strategic interdependence through modeling ex-

plicitly the strategic environments. This strand of models have a common feature that the

econometrician observes many replications of a single small representative game with i.i.d.

draws of certain payo¤ or information components, so that statistical independence is im-

posed across the replications, whereas strategic interdependence is maintained within each

replication. This separation enables one to deal with both asymmetric players within each

game and heterogeneity across the games. (See Bresnahan and Reiss (1991), Tamer (2003),

Ciliberto and Tamer (2009), Aradillas-Lopez (2010), Beresteanu, Molchanov, and Molinari

(2011), Aradillas-Lopez and Tamer (2008), and de Paula and Tang (2011), among many

others, for methodological contributions.1)

However, there are many strategic interactions where such a nice separation is far from

plausible. For example, consider studying peer e¤ects among students. One may view strate-

gic interactions among the students in each school as one game. Each school has a di¤erent

number of students. In order to view the games as arising from a single representative game

and to deal with heterogeneity in the number of the students, one needs to either ascribe it

to various observable school characteristics, or assume simply that there exists a potential

set of students from which students are randomly assigned to each school, or explicitly model

endogeneity in the assignments of the students. Then one needs to specify how the equilib-

rium is selected across schools or whether the same equilibrium is selected across schools or

not. In other words, one is forced to introduce various additional structures to transform the

original strategic environment into one that �ts the representative game framework. This

observation applies to many other examples, such as studies on neighborhood e¤ects on the

choice of housing location or on the e¤ect of friendship networks on the students�smoking

behavior. Many such examples are found in the literature of social interactions. (See Brock

and Durlauf (1995, 2001 and 2007) for their pioneering works on the structural modeling of

social interactions. See also surveys by Brock and Durlauf (2001) and a recent monograph

by Ioannides (2012) for this literature.)

This paper proposes an alternative modeling view in which heterogeneity across the games

and heterogeneity across the players are given characteristics of a single large game. For ex-

ample, in the study of the peer e¤ects mentioned before, this framework views interactions

1Also see Chernozhukov, Hong and Tamer (2007), Rosen (2008), Bugni (2010), Andrews and Soares (2010),Andrews and Shi (2011), Chernozhukov, Lee and Rosen (2013), among many others, for general theory ofinference for models under moment inequality restrictions. Note that the representative game models havebeen among the main motivation behind this literature.

ECONOMETRIC INFERENCE ON A LARGE GAME 3

within each school as subgames of a large game, where subgames are allowed to be het-

erogeneous in various aspects such as the number of the players, their payo¤ speci�cations,

and the way they form beliefs about other players�types or higher order beliefs about other

players�beliefs. Since we do not require that there be random components accounting for

game heterogeneity that are drawn from a common distribution, we do not need to model

the nature of the randomness of those components in relation to other components of the

game.

More speci�cally this paper introduces a static large Bayesian game which is not necessarily

reducible to many replications of a small representative game. To accommodate various

informational assumptions in games into a unifying framework, this paper de�nes two layers

of group structures on the large number of the players. The �rst group structure is an

information group structure where each agent�s payo¤ di¤erential at the change of his action

is never a¤ected by the actions of the players outside his group, and players in each group

observe commonly a group-speci�c signal (called a public signal here). The second group

structure is an acquaintance group structure such that each agent has a small group of

players whose types he observes. This paper calls this group the player�s acquaintance

group.2 Throughout this paper, it is assumed that the acquaintance groups form a partition

of the players that is �ner than that of information groups.

In this game, the beliefs that the players form about the types of other players may be

di¤erent across individual players, except that the type vectors across disjoint acquaintance

groups are conditionally independent given their public signal. Hence we do not make a

common prior assumption that ensures a direct link (through Bayes� rule) between the

objective probability that the Nature adopts for drawing types at the initial stage of the

game and the subjective probability that each player adopts for predicting the other players�

types.

The econometrician observes outcomes from a pure strategy Bayesian Nash equilibrium

(possibly among multiple equilibria), and attempts to make inference about the structural

parameters of the game. While the equilibrium is driven by the subjective beliefs of the

players, the validity of the econometrician�s inference is measured in terms of the Nature�s

objective probability.3

Examples in this paper�s framework are many. First, many small games of complete

information similar to those in Tamer (2003) and Ciliberto and Tamer (2009) belongs to this

paper�s framework as a special case. Each complete information entry game can be viewed as

2Kim and Che (1994) analyzed auction models with a group structure similar to the acquaintance groupstructure. See also Andreoni, Che and Kim (2007).3This paper con�nes attention to simultaneous-move games with an unordered �nite action space. Henceauction models with continuous bids are excluded. Global network games with endogenous network formationor matching games are also excluded because the action space increases as the number of players increases.

4 SONG

an acquaintance group in this paper�s game. The main di¤erence is that the games need not

have the same number of players or payo¤ structures. Second, models of social interactions

developed by Brock and Durlauf (2001b, 2007) can be accommodated in this framework.

One may model each game of social interactions as one information group with agents with

private information.

The main challenge for the econometrician in this data environment is that it is hard for the

econometrician to recover the heterogeneous beliefs of each player about other players�types

from observing only several large games. To address this challenge, this paper develops

what this paper calls a hindsight regret approach. Using the assumption of conditional

independence, this paper draws on the insights of Kalai (2004) and Deb and Kalai (2010),

and introduces the notion of hindsight regret for each player. Hindsight regret measures the

payo¤ loss due to the players�not being able to observe the other players�actions. This

paper establishes a belief-free bound for the regret, using a concentration inequality called

McDiarmid�s inequality. The bound for the hindsight regret obtained here does not depend

on the way each player forms beliefs about other players�types.

Using the belief-free hindsight regret, this paper derives moment inequalities in a spirit

similar to Ciliberto and Tamer (2009), and applies their partial identi�cation approach to

develop asymptotically valid con�dence sets for the structural parameters. When one ob-

serves large games, one might think that checking equilibrium constraints jointly for many

players in each game raises the problem of dimensionality in practice. However, this paper

formulates moment inequalities for each acquaintance group, not for the whole game, and

hence such a problem does not arise, as long as acquaintance groups are small.

The paper proposes two kinds of wild bootstrap methods and establishes their asymptotic

validity uniformly over the probabilities that the Nature adopts for drawing the players�

types. The paper also presents results from a Monte Carlo simulation study based on a

social interactions model. First, the results show that the larger the hindsight regrets, the

more conservative the inference becomes. This re�ects the di¢ cult nature of the problem

in which one cannot recover the beliefs while the beliefs about other players� types play

an important role in players�strategic decisions. Second, it is also found that even when

the coverage probabilities are reported to be 1, the power properties for the test of the

parameter values can be reasonably good. Third, the results also show that ignoring the

hindsight regrets and proceeding as if the Bayesian Nash equilibrium was from an ex post

Nash equilibrium may lead to invalid inference for a reasonable range of sample sizes. Hence

the hindsight regrets cannot be ignored in general.

Given the general set-up of the games in this paper, one might ask in what kinds of game

set-ups the asymptotic approach of the proposed method becomes reliable and produces

potentially a nontrivial result. The major condition is that many players in the game should

ECONOMETRIC INFERENCE ON A LARGE GAME 5

have small hindsight regrets. While this condition is mostly satis�ed by social interactions

models and some network models, the condition excludes the case where one observes many

small private information games as in Aradillas-Lopez (2010) and de Paula and Tang (2011).

This paper�s framework is relevant to many models of social interactions and a certain class

of social network models. See Brock and Durlauf (2001a) and Ioannides (2013) for a review

of the literature of social interactions. This paper�s approach is related to the structural

approach of Brock and Durlauf (2001b, 2007) who developed discrete choice-based social

interactions models. See also Krauth (2006) for a simulation-based method of inference

for such models, Li and Lee (2009) for investigating the role of subjective expectations in

social interactions models based on Brock and Durlauf (2001b). Blume, Brock, Durlauf and

Jayraman (2011) explored the issue of identi�cation in linear social network models. While

the perspective of a large game in this literature has already appeared in various forms

in Brock and Durlauf (2001, 2007), recent researches by Xu (2012), Bisin, Moro, and Topa

(2011), andMenzel (2012) are more explicit about the inferential issues in a large game model.

The inferential procedure of Xu (2012) employs conditions that yield uniqueness of the

equilibrium. On the other hand, Bisin, Moro, and Topa (2011) admit multiple equilibria, but

their method is built on the characterization of an equilibrium through aggregate quantities

of outcomes, where the asymptotic stability of the aggregate quantities (as the number of

the players increases) is part of their equilibrium concept. In contrast, this paper de�nes

the set of equilibria within a �nite player game, without prescribing a particular asymptotic

behavior of the equilibria as the number of the players grows to in�nity. Menzel (2012)

recently developed asymptotic theory for inference based on large games where type-action

pro�les are exchangeable sequences.

This research was originally motivated as an exploration of a way to elucidate the structural

source of cross-sectional dependence among the outcomes observed by the econometrician.

The literature of cross-sectional dependence assumes a cross-sectional dependence structure

directly on the observed outcomes either through spatial version of weak dependence or

using factor models. For example, see Conley (1999) for an introduction of cross-sectionally

stationary series, and see Jia (2008) for its application in the model of entry decisions by Wal

Mart. Also see Bai and Ng (2002) and Phillips and Sul (2003) for models of strong cross-

sectional dependence through factors, among many others. The structure of cross-sectional

dependence arises from the informational assumptions about the players in the game, and

are mainly due to two sources: the public signal commonly observed among the players in

each information group, and the types of the players shared among each other within each

acquaintance group.

This paper�s approach for inference is inspired by the work of Andrews (2005) who explored

how the presence of unobserved public signals a¤ect the asymptotic properties of estimators

6 SONG

from linear factor models with short panel data. Kuersteiner and Prutcha (2012) provides

asymptotic theory for a wide class of dynamic panel models with regressors that are not

strictly exogenous and may even exhibit strong cross-sectional dependence of unknown form.

As in Andrews (2005) and Kuersteiner and Prutcha (2012), the test statistic we are using

has a functional of a mixture normal distribution as its limiting distribution, but we cannot

use the random norming to pivotize the test statistic, because the restrictions here are

inequalities rather than equalities. Instead, this paper develops a modi�ed form of a wild

bootstrap procedure that is asymptotically valid uniformly over a large class of objective

probabilities that the Nature adopts.

This paper is organized as follows. The �rst section formally introduces a large Bayesian

game. The section also de�nes the notion of hindsight regret, and o¤ers its bound. Section

3 develops a general inference method and establishes its asymptotic validity. The proofs of

the results in this paper appear in the Appendix.

2. A Large Bayesian Game with Acquaintance Groups

2.1. Acquaintance Groups, Information Groups and Bayesian Nash Equilibria.In this section, we introduce a general form of a Bayesian game that essentially de�nes the

scope of the paper. The game is played by N players, where each player i 2 N � f1; 2; ���; Ngchooses an action from a common �nite action set A � f�a1; � � �; �aKg, after observing his ownpayo¤ types and some other signals.

More speci�cally, the nature draws an outcome ! from a sample space which realizes

the (payo¤) type pro�le

T (!) = (T1(!); � � �; TN(!)) 2 TN ;for the N players, where Ti(!) 2 T � Rt represents a type vector for player i. Let the

distribution of T adopted by the Nature be denoted by P .

To specify the cross-sectional correlation structure of the types, we suppose that there

exists a partition of f1; � � �; Ng = [Ss=1Ns, where for each s = 1; � � �; S, every player i 2 Nsobserves a common signal vector Cs, but does not observe Ct with t 6= s: For each s 2 S �f1; � � �; Sg, we call an information group s the set of players who commonly observe the

signal vector Cs. Hence there are S number of information groups, and each information

group s contains Ns � jNsj number of players.Each player i observes the payo¤ types of some other players in a set denoted by I(i).

We call I(i) player i�s acquaintance group, and each member of I(i) player i�s acquaintance.

Hence the type vector that player i observes is TI(i) � (Tj)j2I(i). We assume for the sake

of analytical simplicity that we have j 2 I(i) if and only if i 2 I(j), and the acquaintance

groups form a �ner partition of the players than that of information groups so that for all

ECONOMETRIC INFERENCE ON A LARGE GAME 7

i 2 N, I(i) � Ns for some s 2 S. Acquaintance groups are useful for representing situationswith complete information games and some network games, as we shall show later.

Once the Nature draws a type vector T that realizes to be t = (tj)j2N 2 TN , each playeri observes tI(i) � (tj)j2I(i) and forms a belief on T . The belief is denoted by Qi(�jtI(i)), aprobability measure on TN for each tI(i). The beliefs Qi are allowed to be heterogeneous

across players.

The probability P is the objective probability that the econometrician uses to express the

validity of his inference method. On the other hand, the probability Qi for each player i is a

subjective probability formed according to player i�s prior and possibly through higher order

beliefs about other players�beliefs. When the Nature�s P belongs to common knowledge,

we have Qi(�jtI(i)) = P (�jtI(i)) for all i 2 N and tI(i) 2 TjI(i)j; so that the distinction betweenthe objective probability that the econometrician�s inference centers on and the subjective

probabilities that the players take as their beliefs is not necessary. Here it is, as we are not

making such an assumption.

We introduce a conditional independence assumption for P and Qi�s. For this, we de�ne

the following notion: for each s 2 S, let Ms � Ns be such that for i; j 2 Ms; i 6= j if and

only if I(i)\ I(j) = ?. Hence players fromMs have no common acquaintance. We call such

a set Ms a disjoint selection from Ns.

Assumption 1: For each s 2 S and any disjoint selection Ms from Ns, fTI(i) : i 2 Msg isthe set of random vectors that are conditionally independent given Cs both under P and

under Qi(�jtI(i)) for all i 2 N and all tI(i) 2 TjI(i)j.

Assumption 1 assumes that the types of players from disjoint selections are conditionally

independent among each other given the information group�s public signal Cs according to

the Nature�s probability, and all the players know this fact. However, we do not require that

this conditional independence be common knowledge among the players.

Assumption 2: There exists a small � > 0 such that for any event B � TN such that

Qi(BjtI(i)) � 1� � for all i 2 N and all tI(i) 2 TjI(i)j, it is satis�ed that PfBjTI(i) = tI(i)g �1� � for all i 2 N and all tI(i) 2 TjI(i)j.

Assumption 2 says that any event that all the players in the game believe strongly to oc-

cur does occur with high probability according to the Nature�s conditional probability. This

assumption imposes a limited (one-sided) version of rational expectations on the players�

beliefs on events that are commonly believed to be highly likely among the players. The

version is one-sided in the sense that a high probability event (according to the Nature�s

8 SONG

experiment) does not need to be viewed as a high probability event by each player. Assump-

tion 2 is used later to translate the results about belief free bounds for hindsight regrets into

moment inequalities that the econometrician can use.

Without loss of generality, we assume that for each s 2 S; and each i 2 Ns; the publicsignal Cs is a deterministic function of Ti so that Cs is measurable with respect to the �-�eld

of Ti for all i 2 Ns.Once the Nature draws T = t with distribution P , each player i, facing the other players

choosing a�i 2 AN�1; receives payo¤ ui(ai; a�i; tI(i)) from choosing ai 2 A; so that the payo¤of player i is not directly a¤ected by the types of other players outside his acquaintance

group, although it is a¤ected by their actions.

It is assumed that the information group structure and the payo¤ functions are common

knowledge. A pure strategy yi for player i is an A-valued map on T, and a pure strategypro�le is a pro�le y = (y1; � � �; yN) of such maps yi. Given a strategy pro�le y, the expectedpayo¤ for player i is given by

Ui(yjtI(i)) =ZTN

ZANui(y(t); tI(i))Qi(dtjtI(i));

where y(t) = (y1(tI(1)); y2(tI(2)); � � �; yN(tI(N))): Then we say that a strategy pro�le y is a purestrategy Bayesian Nash equilibrium, if for each i 2 N; tI(i) 2 TjI(i)j; and any pure strategy y0ifor i;

(2.1) Ui(yjtI(i)) � Ui(y0i; y�ijtI(i)):

This paper does not place restrictions on Qi other than that Assumptions 1 and 2 hold.4

Instead of directly observing a strategy pro�le, the econometrician typically observes its

realized action pro�le. Given a pure strategy equilibrium y = (y1; � � �; yN), de�ne

(2.2) Yi � yi(TI(i)), i 2 N;

and let Y � (Y1; � � �; YN) 2 AN . The econometrician observes Yi�s and part of Ti. (We willspecify the econometrician�s observations in detail later.) The equation (2.2) is a reduced

form for Yi. When the game has multiple equilibria, this reduced form is not uniquely

determined by the game.

Given a pure strategy equilibrium y, let P y be the joint distribution of (y(T ); T ), with

y(T ) = (y1(TI(1)); � � �; yN(TI(N))), when its marginal distribution of T is equal to P . Also

4Existence of a Nash equilibrium in this general set-up is not always ensured. See Stinchcombe (2010) foran example of a Bayesian game that does not have an equilibrium, failing the absolute continuity conditionof Milgrom and Weber (1985). However, existence of an equilibrium can be established by invoking morespecial structure of the game in application. For example, see Milgrom and Weber (1985), Balder (1988),Athey (2001), McAdams (2003) and Reny (2011) and references therein for general results.

ECONOMETRIC INFERENCE ON A LARGE GAME 9

given y, let Qy = (Qy1; � � �; QyN), where Q

yi is the joint distribution of (y(T ); T ) representing

the player i�s beliefs about the actions and the types (in equilibrium y).

2.2. Belief-Free Hindsight Regrets. The equilibrium constraints in (2.1) can be rewrittenas follows: for all i 2 N and all �a 2 A;

(2.3) Eyi�ui(Y ;TI(i))� ui(�a; Y�i;TI(i))jTI(i) = tI(i)

�� 0,

where Eyi [�jTI(i) = tI(i)] denotes conditional expectation (under Qyi ) given TI(i) = tI(i): The

constraints are generally useful for the econometrician to generate moment inequalities for

observations. However, they cannot be directly used in our context, because there is no

general way for the econometrician to recover the subjective beliefs.

The hindsight regret approach in this paper replaces the inequality in (2.3) by the following

ex post version: for all i 2 N and all �a 2 A;

(2.4) ui(Y ;TI(i))� ui(�a; Y�i;TI(i)) � ���a;

with large probability according to his belief Qyi , where ��a � 0 is a compensation for playeri needed to prevent her from switching from her action Yi in equilibrium to action �a (with

large probability) after the types of all the players are revealed to her.

We seek to formulate the inequality (2.4) in a way that have three main features. First,

we would like the inequalities to hold jointly for all the players whose equilibrium actions

can be determined by player i, i.e., the players in player i�s acquaintance group. Behind the

inequality (2.4) is the assumption that other players also maintain their equilibrium strategy

after the types are revealed. But from the perspective of player i, it is hard to determine

the minimal compensation level for those players whose equilibrium action player i cannot

determine from his type information tI(i). Hence we envisage a scheme in which the com-

pensation level for such players is chosen so that the violation of the inequality in (2.4) due

to the deviation by such players�actions becomes highly unlikely from player i�s perspec-

tive. Second, for simplicity, we do not allow the compensation scheme to rely on any signal

about the beliefs or higher order beliefs of the players. Otherwise there may arise a complex

mechanism design issue regarding the compensation. We call this belief-free compensation

a belief-free hindsight regret, because the compensation represents a potential payo¤ loss to

the players in hindsight due to not being able to observe the types of other players out-

side his acquaintance group. Third, we seek to formulate a minimal amount of belief-free

hindsight regret. This is because the quality of prediction from the incentive constraints in

(2.4) becomes better with tighter compensation scheme. As we shall see later, the econo-

metrician makes inference about the payo¤ parameter by checking whether the equilibrium

constraints in (2.4) are plausible with each parameter value given the observed equilibrium

10 SONG

outcomes. Hence using a tighter hindsight regret tends to yield a sharper empirical result

for the econometrician.

To de�ne a belief-free hindsight regret formally, we introduce some notation. Given i 2 Nand any �i = (�i;�a)�a2A, �i;�a � 0; let

bi(�i) ��a 2 A : ui(a; Y�i;TI(i))� ui(�a; Y�i;TI(i)) � ��i;�a; 8�a 2 Anfag

:

When Yi 2 bi(�), it means that Yi remains a best response for player i to (Y�i; TI(i)) even

after all the types of the players are revealed, as long as player i is compensated by the

compensation scheme �i for adopting the response Yi.

Given small � > 0, we say that the vector �i = (�ij)j2I(i) (with �ij = (�ij;�a)�a2A and

�ij;�a � 0) is a �-hindsight regret for player i; if

Qyi�Yj 2 bj(�ij); 8j 2 I(i)jTI(i) = tI(i)

� 1� �:

A hindsight regret �i for player i lists the amount of compensations for players in I(i) to

induce them to maintain their strategies in equilibrium y with high probability.

Hindsight regrets are closely related to strategic interdependence among the players: the

regrets tend to be high for a player who does not observe the types of those players whose

action can have a large impact on his payo¤. To formally introduce this interdependence,

we de�ne a maximal variation of a real function. Suppose that f(x1; � � �; xN) is a real-valuedfunction on a set XN � RN . Then, we write

Vj(f) = max jf(x)� f(xj(x))j ;

where the maximum is over all x�s in XN and over all xj(x)�s in XN such that xj(x) is x

except for its j-th entry. We call Vj(f) a maximal variation of f at the j-th coordinate.

For i 2 N; and �a 2 A, let

u�i;m�a�i; tI(i); �a

�� ui(�am; a�i; tI(i))� ui(�a; a�i; tI(i));

and let for i 2 N; j 2 N, and tI(i) 2 TjI(i)j,

(2.5) �ij(tI(i); �a) � maxm=1;���;K

Vj(u�i;m

��; tI(i); �a

�):

The quantity �ij(tI(i); �a) measures the worst payo¤ loss for player i from his benchmark

payo¤ action �a that can be in�icted by player j�s choice of an action. Hence �ij(tI(i); �a)

measures strategic relevance of player j to player i. When �ij(tI(i); �a) = 0 for all tI(i) and

�a 2 A, there is no way player j can a¤ect the optimal choice of player i. We say that players

ECONOMETRIC INFERENCE ON A LARGE GAME 11

i and j are payo¤ disjoint, if �ij(tI(i); �a) = �ji(tI(j); �a) = 0 for all tI(i) 2 TjI(i)j; tI(j) 2 TjI(j)j,and all �a 2 A. 5

Assumption 3: Any two players from di¤erent information groups Ns and Nt with s 6= t

are payo¤ disjoint.

Assumption 3 simpli�es the framework by allowing that a player in one information group

Ns does not need to form beliefs about the types of the players from a di¤erent information

group Nt. In many empirical examples, the di¤erent information groups Ns can be thoughtof as constituting separate games observed by the econometrician.

To characterize the belief-free hindsight regrets, for each i 2 Ns, j 2 I(i); �a 2 A, and� 2 (0; 1); we let

(2.6) Bij;�(tI(j); �a) �

s�12 ij(tI(j); �a) � log

��

(K � 1)jI(i)j

�,

where

(2.7) ij(tI(j); �a) �X

l2NsnI(i)

0@Xk2I(l)

�jk(tI(j); �a)

1A2

:

The following theorem shows that Bij;� is a belief-free �-hindsight regret for player j (from

the perspective of player i.)

Theorem 1: Suppose that Assumptions 1-3 hold. Then for each pure strategy equilibrium

y, each � 2 (0; 1), for all i 2 N and all tI(i) 2 TI(i);

Qyi�Yj 2 bj(Bij;�(TI(i))); 8j 2 I(i)jTI(i) = tI(i)

� 1� �;

where Bij;�(TI(i)) = (Bij;�(TI(i); �a))�a2A:

The hindsight regret Bij;�(�) is fully determined only by the payo¤ functions and theacquaintance group structure I of the game, and does not depend on Qi�s. Hence it is belief

free. The bound is a consequence of McDiarmid�s inequality (McDiarmid (1989)) and the

Nash equilibrium constraints in (2.3).

Recall that Bij;� is the compensation for player j from player i�s perspective. Player

i knows all the types of the acquaintances of the players in I(i), and hence he can fully

determine their equilibrium actions, and the compensation level for them. The uncertainty

5When two players are payo¤ disjoint, this does not necessarily mean that one player�s change of action hasno way of having an impact on the other player�s payo¤. It only means that one player�s change of actionhas no way of having an impact on an optimal decision made by the other player.

12 SONG

faced by player i is due the actions of those players outside I(i): To maintain the ex post

incentive constraint with high probability, the compensation level has to take these players

into account, i.e. the impact the change of their types may have on player j�s payo¤.

Perturbing the types of those players outside I(i) can a¤ect player j�s payo¤ by a¤ecting the

actions of those players who observe the perturbed types. Thus the component ij(tI(j); �a)

in the de�nition of Bij;�(tI(j); �a) now measures the overall strategic relevance (to player j) of

those players who observe the types of the players outside I(i).

In the case of private information game, we have I(i) = fig. Hence in this case, Theorem 1is reduced to the following simple form: for each pure strategy equilibrium y, each � 2 (0; 1),for all i 2 N and all t 2 T;

Qyi fYi 2 bi(Bi;�(Ti))jTi = tg � 1� �;

where Bi;�(Ti) = (Bi;�(Ti; �a))�a2A,

Bi;�(ti; �a) �r�12 i(ti; �a) � log �,

and i(ti; �a) �P

l2Nsnfig�2il(ti; �a). The component i now summarizes the strategic rele-

vance to player i of all the other players in player i�s information group. The larger i is the

higher the hindsight regret Bi;�.

2.3. Examples.

2.3.1. Many Small Entry Games with Complete Information. We consider a large game con-

stituted byM markets where each marketm forms an acquaintance group. The acquaintance

group I(i) of i represents the group of players in the same market that player i belongs to.

The action space is A = f0; 1g, with 1 representing entry into the market, and 0 non-entry.For each player i in market s, the payo¤ di¤erential is speci�ed as

ui(1; a�i;Ti)� ui(0; a�i;Ti) = �>Ti;1 + �X

j2I(i)nfig

aj + Ti;2;

where Ti = (Ti;1; Ti;2). The payo¤ di¤erential shows that the players from di¤erent acquain-

tance groups are payo¤ disjoint. Each market constitutes a complete information game.

This type of payo¤ speci�cation was used by many researchers (Bresnahan and Reiss

(1991), Tamer (2003), Ciliberto and Tamer (2008).) If we focus only on pure strategies, it

turns out that ij(TI(j); �a) = 0 and hence Bij;�(TI(j); �a) = 0. This is not surprising; there

is no hindsight regret, because each player observes the types of all the players that are

strategically relevant to her (that is, her opponents in her market).

2.3.2. Large Games with Social Interactions. Suppose that we have S groups of players where

each group s has Ns number of players. Each group constitutes a large game with private

ECONOMETRIC INFERENCE ON A LARGE GAME 13

information. For player i in group s, we consider either of the following two speci�cations of

payo¤ functions:

ui(ai; a�i; ti) = v(ai; ti) +� � aiNs � 1

NsXj 6=i

aj;

or

ui(ai; a�i; ti) = v(ai; ti)��

2� ai �

1

Ns � 1

NsXj 6=i

aj

!2,

where a�i = (aj)Nsj=1;j 6=i and v(ai; ti) is a component depending only on (ai; ti). The parameter

� captures strategic interaction of player i with the other players within his group s: The �rst

speci�cation expresses interaction between player i�s action (ai) and the average actions of

the other players. The second speci�cation captures preference for conformity to the average

actions of the other players. (See Bernheim (1994).) If we replace the other players�actions

aj by EYj, i.e., the expected value of the actions, both speci�cations coincide with those

considered in the payo¤ speci�cation of Brock and Durlauf (2001).

In both cases,

u�i (a�i; ti; �a) = minm=1;���;K

(vm(ti; �a) +

� � (�am � �a)Ns � 1

NsXj 6=i

aj

),

where vm(ti; �a) = v(�am; ti) � v(�a; ti) in the �rst speci�cation, and vm(ti; �a) = v(�am; ti) �v(�a; ti)� � � (�a2m � �a2)=2 in the second speci�cation.Also, in both cases, ij(ti; �a) =

Pj2N�i �ij(ti; �a)

2 with�ij(ti; �a) = j�j�maxm(�am��a)=(Ns�1) and the belief-free hindsight regret in Theorem 1 becomes:

Bi;�(ti; �a) =j�j �maxm(�am � �a)p

Ns � 1�

s�12log

��

K � 1

�:

Hence the hindsight regret becomes asymptotically negligible, as long as the number of the

players in each information group Ns increases.

3. Econometric Inference

3.1. The Econometrician�s Observations and Parametrization. Suppose that theeconometrician observes (Y;X), where Y 2 AN is an N -dimensional vector of actions by

N players and X is an N�dX matrix whose i-th row is X>i ; with Xi being a subvector of Ti,

represents a covariate vector of player i. As for (Y;X), we make the following assumptions.

Assumption 4 (The Econometrician�s Observation): (i) The distribution of (Y; T )

is equal to P y associated with a pure strategy equilibrium y.

14 SONG

(ii) For each i 2 N, Ti = (�i; Xi), where Xi 2 RdX is observed but �i 2 H � Rd� is not

observed by the econometrician.

The distribution of (Y; T ) that the econometrician focuses on is originated from the Na-

ture�s objective probability P and a pure strategy equilibrium y. The econometrician does

not know which equilibrium the vector of observed outcomes Y is associated with. The play-

ers�subjective beliefs a¤ect the distribution of (Y; T ) through their impact on the associated

equilibrium y.

Assumption 4(ii) speci�es that Ti involves components �i and Xi which are unobserved

and observed by the econometrician respectively. Thus the econometrician may not observe

part of the type information each player has.

We �rst introduce parametrization of unobserved heterogeneity �i and payo¤s.

Assumption 5 (Parametrization of Unobserved Heterogeneity and Payoffs):

For all s 2 S; i 2 Ns, tI(i) 2 TI(i), and a 2 AjI(i)j,

P��I(i) � tI(i)jXI(i)

= Gs;�0

�tI(i)jXI(i)

�and ui(a; tI(i)) = ui;�0(a; tI(i));

where �0 2 � � Rd, Gs;�(�jXI(i)) and ui;�(�; tI(i)) are parametric functions parametrized by� 2 �.

Assumption 5 assumes that the conditional CDF of �I(i) givenXI(i) and the payo¤ function

are parametrized by a �nite dimensional vector � 2 �. Assumption 5 does not require thatthe payo¤ function be additive in unobserved heterogeneity �I(i). Therefore, the model

accommodates speci�cations where the observed covariates XI(i) are intertwined with �I(i)in a complicated manner as in random coe¢ cient speci�cations.

Assumption 6 (Conditional Independence): (i) For any s 2 S and disjoint selectionMs from N, the set fTI(i) : i 2 Msg is the set of conditionally independent random vectors

given C = (Cs)Ss=1 under P .

(ii) f�1; � � �; �N ; Cg is conditionally independent given X = (X1; � � �; XN) under P:

Assumption 6(i) requires that the type vectors across di¤erent acquaintance groups are

conditionally independent given C. This follows by Assumption 1 and the additional as-

sumption that (TI(i); Cs)i2Ns�s are independent across s = 1; � � �; S. Assumption 6(ii) requiresthat the unobserved components �1; � � �; �N and C are conditionally independent of X. Thiscondition is satis�ed, for example, if �1; � ��; �N are conditionally independent given X and for

each i 2 N, Xi = (Zs;Wi), where Wi is an idiosyncratic component and Zs is a component

common in information group s, and Cs is a nonstochastic function of Zs: Note that the

ECONOMETRIC INFERENCE ON A LARGE GAME 15

econometric may observe Xi, but may or may not observe Cs, although the players observe

Cs.

Assumption 6 is also concerned only with the primitives of the game. It does not impose

restrictions on the equilibrium y or the way beliefs of the agents are formed in equilibrium.

It is only concerned with the objective probability P that the Nature uses to draw types for

the players.

3.2. Moment Inequalities. In this subsection, we derive moment inequalities from the

Nash equilibrium constraints. For simplicity, we write ui(aI(i)) = ui;�(aI(i); Y�I(i);TI(i)),

suppressing Y�I(i) and TI(i) and � from the notation.

Let

�i�aI(i); TI(i)

�� 1

�Eyj�uj(aI(j))jTI(j)

�� max

c2AEyj�uj(c; aI(j)nfjg)jTI(j)

�; 8j 2 I(i)

�.

Recall that the conditional expectation Eyj��jTI(j)

�originates from player j�s subjective be-

liefs. Since Y = y(T ) for some pure strategy Nash equilibrium y, for any aI(i) 2 AjI(i)j andfor all values of TI(i) such that YI(i) = aI(i), we have �i

�aI(i); TI(i)

�= 1, i.e.,

(3.1) 1�YI(i) = aI(i)

� �i

�aI(i); TI(i)

�:

Similarly, for all values of TI(i) such that YI(i) 6= aI(i), we have YI(i) = cI(i) for some cI(i) 2AjI(i)jnfaI(i)g so that we have �i

�cI(i); TI(i)

�= 1 for some cI(i) 2 AjI(i)jnfaI(i)g. In other

words, we have

(3.2) 1�YI(i) 6= aI(i)

� 1

�9 cI(i) 2 AjI(i)jnfaI(i)g s.t. �i

�cI(i); TI(i)

�= 1:

We take conditional expectations (given XI(i)) of both sides in (3.1) and (3.2), and deduce

that for each i 2 N, and aI(i) 2 AjI(i)j;

(3.3) 1� ��i;L(aI(i)) � P�YI(i) = aI(i)jXI(i)

� ��i;U(aI(i)):

where

��i;U(aI(i)) � P� �i�aI(i); TI(i)

�= 1jXI(i)

and

��i;L(aI(i)) � P�9 cI(i) 2 AjI(i)jnfaI(i)g s.t. �i

�cI(i); TI(i)

�= 1jXI(i)

:

In fact the bounds in (3.3) can be viewed as a Bayesian-game version of what Ciliberto and

Tamer (2009) derived from a complete information game. As Tamer (2003) and Ciliberto and

Tamer (2009) illustrated well, when there are multiple equilibria and acquaintance groups

are not singletons, we may have

1 < ��i;U(aI(i)) + ��i;L(aI(i)):

16 SONG

This is because we may have both �i (aI(i); �I(i)) = 1 and �i (a0I(i); �I(i)) = 1 with di¤erent

aI(i) and a0I(i), when there are multiple equilibria. The inequalities in (3.3) become equalities

when there is a unique equilibria (with zero probability of a tie occurring) or acquaintance

groups are singletons as in the private information set-up. In the case of private information

(so that I(i) = I(i) = fig), we have for di¤erent actions ai and a0i, �i (ai; �i) = 1 if and

only if �i (a0i; �i) = 0 for almost all �i. Hence in this case, the inequalities in (3.3) become

equalities even when there are multiple equilibria

Unfortunately, the inequalities in (3.3) cannot be directly used in our set-up for inference

for two reasons. First, the sets H�i;L

�aI(i)

�and H�

i;U

�aI(i)

�involve heterogeneous subjective

beliefs which the econometrician has di¢ culty recovering from the observations. Second, the

probabilities in both bounds of (3.3) cannot be simulated, because the bounds depend on

the unknown distribution of Y�I(i).

We formulate the moment inequalities through probabilities that do not involve subjective

beliefs of players and can be simulated. First, we de�ne an ex post version of �i :

(3.4) i�aI(i); TI(i)

�� 1

�uj(aI(j)) � max

c2Auj(c; aI(j)nfjg)�Bij;�(TI(j)); 8j 2 I(i)

�.

Now we construct probabilities that can be simulated:

�i;U(aI(i)) � P� i�aI(i); �I(i)

�= 1jWi

and

�i;L(aI(i)) � P�9 cI(j) 2 AjI(j)jnfaI(j)g s.t. i

�cI(i); �I(i)

�= 1jWi

;

where Wi = (XI(i); Y�I(i); C): To see that �i;U(aI(i)) and �i;L(aI(i)) can be simulated, note

that the randomness of i;U�aI(i); �I(i)

�and i;L

�aI(i); �I(i)

�is due to (Y�I(i); �I(i); XI(i)), and

that �I(i) is conditionally independent of (Y�I(i); XI(i); Cs) given XI(i) by Assumption 5(ii).

Hence we can write

�i;U(aI(i)) =

ZHi;U (aI(i);XI(i))

dGs;�(�I(i)jXI(i)) and(3.5)

�i;L(aI(i)) =

ZHi;L(aI(i);XI(i))

dGs;�(�I(i)jXI(i));

where

Hi;U

�aI(i);XI(i)

��

��I(i) 2 Hd� jI(i)j : i

�aI(i); �I(i); XI(i)

�= 1and

Hi;L

�aI(i);XI(i)

��

��I(i) 2 Hd� jI(i)j : 9 cI(j) 2 AjI(i)jnfaI(i)g s.t. i

�cI(i); �I(i); XI(i)

�= 1:

Thus the quantities �i;U(aI(i)) and �i;L(aI(i)) can be simulated using the parametric speci�-

cation of the conditional distribution of �I(i) given XI(i) in Assumption 4.

To construct moment inequalities, we assume that jI(i)j � q for all i 2 N, for some integerq > 0. Given a = (a1; � � �; aJ) 2 Aq, we write ai = (a1; � � �; aI(i)) 2 AjI(i)j for each i 2 N.

ECONOMETRIC INFERENCE ON A LARGE GAME 17

Then, de�ne

ei;L(ai) � P�YI(i) = aijWi

��1� 1

1� �i� �i;L(ai)

�and

ei;U(ai) � P�YI(i) = aijWi

� 1

1� �i� �i;U(ai);

where �i � ��(c� i=(1+c� i)) with some constant c > 0, and � i = maxj2I(i)maxtI(j);�a ij(tI(j); �a).The choice of �i is made such that when there is no hindsight regret, i.e., � i = 0, �i is taken

to be zero. (In simulation studies, it works �ne to choose its extreme version with c = 1,i.e., �i = �1f� i > 0g:)We choose nonnegative functions gij on RjI(i)j to construct moment inequalities in a spirit

similar to Andrews and Shi (2012). For this, �rst, de�ne the event: for w = (wj;U(a); wj;L(a) :

a 2 Aq)Jj=1, wj;U(a) � 0; wj;L(a) � 0 and D � Aq;

M(w;D) =

(8j = 1; � � �; J; 8a 2 D :

1N

PNi=1 ei;L(ai)gij(XI(i)) + wj;L(a) � 0

1N

PNi=1 ei;U(ai)gij(XI(i))� wj;U(a) � 0

).

The event M(w;D) expresses the event that certain moment inequality restrictions hold.We aim to �nd a good bound w such that the probability ofM(w;D) becomes su¢ cientlylarge.

To de�ne the bound, we �rst note that �i;L(ai) and �i;U(ai) are nonstochastic functions

of (Y;X) from (3.5) and write

1

N

NXi=1

�1� 1

1� �i� �i;L(ai)

�gij(Xi) = fj;L(Y;X; a) and

1

N

NXi=1

�1

1� �i� �i;U(ai)

�gij(Xi) = fj;L(Y;X; a);

for some functions fj;L and fj;L: Then we let �ij;L(X; a) and �ij;U(X; a) be bounds for maximal

variations of fj;L(�; X; a) and fj;U(�; X; a) at the i-th coordinate, i.e.,

maxk2I(i)

Vk (fj;L(�; X; a)) � �ij;L(X; a) and maxk2I(i)

Vk (fj;U(�; X; a)) � �ij;U(X; a):

Since the payo¤s and the conditional distribution of �i given Xi are fully parametrized, we

can usually obtain reasonable bounds �ij;L(X; a) and �ij;U(X; a) as we illustrate later. Now

we are ready to state the main result of moment inequality restrictions.

Theorem 2: Suppose that Assumptions 1-6 hold. Then it is satis�ed that for any small

� 2 (0; 1), ��, and any D � Aq;

P [M(w(� ;X);D)jX;C] � 1� �

2

�1fj��U j > 0g+ 1fj��Lj > 0g

�; everywhere,

18 SONG

where w(� ;X) = (wj;U(� ; a; X); wj;L(� ; a; X) : a 2 Aq)Jj=1,

wj;L(� ; a; X) �

vuut�12

NXi=1

�2ij;L(X; a)

!log

��

2J jDj

�,

wj;U(� ; a; X) �

vuut�12

NXi=1

�2ij;U(X; a)

!log

��

2J jDj

�;

��U = maxa2Aq ;j=1;���;JPN

i=1 �2ij;U(X; a) and ��L = maxa2Aq ;j=1;���;J

PNi=1 �

2ij;L(X; a).

Theorem 2 o¤ers �nite sample moment inequalities in a general form. When the game

is private information and i = 0 for each i 2 N, so that there is no strategic interactionbetween players, all the players have zero hindsight regret and thus we have Bj;�(Ti) = 0,

��i = 0; �ij(a) = 0 and ei;L(ai) = ei;U(ai). We obtain the following result:

P

(1

N

NXi=1

ei;L(ai)g(Xi) = 0;8a 2 Aq)= 1:

One can use this restriction to construct the usual moment restrictions for discrete choice

models without strategic interactions.

On the other hand, when i is large so that the strategic relevance of the players among

each other is strong, we have larger Bi;�(Ti) and �2ij(a), so that the testable implications in

Theorem 2 become weaker. This can be viewed as a cost to the econometrician for not being

able to recover fully the beliefs of individual players despite their strategic interactions.

To appreciate the components of the bounds in Theorem 2 more concretely, let us consider

a private information large game where each player chooses an action from f0; 1g so that weonly consider the action 1. Furthermore, we specify the payo¤ function as follows:

ui(1; a�i;Ti)� ui(0; a�i;Ti) = �>1Xi +�2

Ns � 1

NsXj=1;j 6=i

aj � �i;

where we assume that �i is unobserved and is independent of the observed covariate Xi and

follows a standard normal distribution. As we saw in Section 2.3.2, we have in this case

Bi;�(ti) =j�2jpNs � 1

r�12log � � B�; say.

Then, with �Y�i;s = 1Ns�1

PNsj=1;j 6=i Yj;

�i;U(1) = ���>1Xi + �2 �Y�i;s +B�

�and

�i;L(1) = 1� ���>1Xi + �2 �Y�i;s �B�

�:

ECONOMETRIC INFERENCE ON A LARGE GAME 19

For each i 2 Ns, let us �nd bounds �ij;U(1) and �ij;L(1). For each a < b and � 2 R, de�ne

z(�; a; b) � argmaxz2[a;b] j� (z +�)� � (z)j

= 1f�� < aga+ 1fa � �2� � bg(�2�) + 1fb � �2�gb:

Also, we de�ne if �2 � 0;

zi = z

��2

Ns � 1; �>1Xi +B� +

�2Ns � 1

; �>1Xi +B�

�and if �2 < 0;

zi = z

��2

Ns � 1; �>1Xi +B�; �

>1Xi +B� +

�2Ns � 1

�:

Then we obtain that �ij;U(1) = �ij;L(1) = �ij, where

(3.6) �ij =1

(1� ��)2S ���

�zi +

�2Ns � 1

�� � (zi)

�2gij(XI(i))

2,

where �� = �1fj�2j > 0g. Therefore, the inequality in Theorem 2 holds with

wj;U(� ; 1) = wj;L(� ; 1) =

vuut�12

NXi=1

�2ij log� �2J

�:

When Ns = n for all s = 1; � � �; S (so that all the information groups are of same size n)we can check that B� = O(n�1=2) and w(�) = O((nS)�1=2) = O(N�1=2). As n ! 1, B�and w(�) become asymptotically negligible. But when S !1 only with n �xed, only w(�)

becomes asymptotically negligible.

When �2 = 0; so that there is no strategic interaction among the players. Then, we obtain

that �i;U(1) = �i;U(1) = ���>1Xi

�and with P = 1 and gij(x) = gj(x) for all i for some

nonnegative real function gj,

1

N

NXi=1

����>1Xi

�� P fYi = 1jXig

�gj(Xi) = 0:

Thus as for the sample version of the moment restrictions, we use this restriction to derive

that1

N

NXi=1

fYi � ���>1Xi

�ggj(Xi) =

1

N

NXi=1

fYi � P fYi = 1jXigggj(Xi):

The asymptotic analysis of the sample moments now can be done using the standard argu-

ments. The tuning parameters � and � do not have any role here.

20 SONG

However, when �2 is away from zero, B� and w(�) increase, and the testable implications

from Theorem 2 become weaker, yielding wider con�dence intervals for the structural para-

meters. This issue becomes alleviated when the sample size (i.e., the number of the players)

is large.

3.3. Bootstrap Inference. For inference, we compare the actual actions of all the playersand their predicted actions conditional on X. Let P0 be the collection of distributions Pthat the Nature uses for drawing types for individual players. Given each pure strategy

equilibrium y, let Py � fP y : P 2 P0g, i.e., the collection of the joint distributions of(y(T ); T ), as the distribution of T runs in P0. To accommodate multiple equilibria, we let Pbe the collection of the joint distributions of (y(T ); T ), as the distribution of T runs in P0 andy runs in the set of pure strategy Bayesian Nash equilibria (according to some heterogeneous

beliefs Q = (Q1; � � �; QN)). We search for an inference procedure that is robust against anychoice of distributions in P, that is, against any choice of the probability by the Nature, anyequilibrium among the multiple equilibria, and any beliefs formed by the individual players

(within the boundary set by Assumptions 1 and 2). From now on, we do not distinguish

between P y and P and simply write P to denote a generic joint distribution of (y(T ); T ) (or

the corresponding marginal distribution of T ) from the collection P.First, we de�ne

ri;U(ai; �) � 1fYI(i) = aig ��1� 1

1� �i� �i;L(ai)

�and(3.7)

ri;L(ai; �) � 1fYI(i) = aig �1

1� �i� �i;U(ai):

Let, for a 2 D,

Lj;U(a; �) � 1

N

NXi=1

ri;U(ai; �)gj(XI(i)) and(3.8)

Lj;L(a; �) � 1

N

NXi=1

ri;L(ai; �)gj(XI(i)):

Using the sums Lj;U(a; �) and Lj;L(a; �), we take the following as our test statistic:

(3.9) T (�) =1

J jDjXa2D

JXj=1

�[Lj;U(a; �)� wU(�)]+ + [Lj;L(a; �) + wL(�)]�

�2;

where [x]+ = maxfx; 0g and [x]� = maxf�x; 0g; x 2 R.Although the test statistic takes a similar form as many researches in the literature of

moment inequalities (e.g. Rosen (2008), Andrews and Soares (2010), and Andrews and Shi

(2010) among others), the sample moments Lj;U(a; �) and Lj;L(a; �) here are not necessarily

ECONOMETRIC INFERENCE ON A LARGE GAME 21

the sum of independent or conditionally independent random variables as typically assumed

in the literature. The summands ri;U(a; �)gj(XI(i)) and ri;L(a; �)gj(XI(i)) involve Wi whose

subvector is Y�i for each i. Since Y�i are obviously dependent across i�s, the summands

ri;U(a; �)gj(XI(i)) and ri;L(a; �)gj(XI(i)) are dependent across i�s in a complicated manner.

We use the testable implications in Theorem 2 to deal with this issue. First, we write

(3.10) Lj;U(a; �) = �j(a) +1

N

NXi=1

ei;U(a)gj(XI(i)).

where �j(a) �PN

i=1 r�i (a)gj(XI(i)) and

r�i (a) � 1fYI(i) = aig � PfYI(i) = aijXI(i)g:

The �rst sum �j(a) is the sum of the mean zero random variables, and the second sum in

(3.10) is the sum of the terms that depend onWi for each i, rendering its asymptotic analysis

complex. Using the similar inequality for LLj (a; �) and applying Theorem 2, we deduce that

with probability at least greater than 1� � ,

[Lj;U(a; �)� wU(�)]+ + [Lj;L(a; �) + wL(�)]�(3.11)

� [�j(a)]+ + [�j(a)]� = j�j(a)j:

We base the inference on the asymptotic distribution of �j(a). By Assumption 6(i) and the

condition that the acquaintance groups are not overlapping, we can easily show that �j(a)

is a sum of martingale di¤erence arrays. Under some regularity conditions, the martingale

central limit theorem gives us the following: for a 2 Aq (as N !1)

(3.12) rN�j(a)D! �j(a)Zj(a);

jointly over j�s and a�s, where rN ! 1 is an appropriate normalizing sequence, Zj(a)is a random variable distributed as N (0; 1) and �2j(a) is a nonnegative random variable

independent of Zj(a).It remains to obtain an approximate distribution of �j(a)Zj(a) in large samples. Andrews

(2005) and Kuersteiner and Prucha (2012) used random norming to pivotize the test statistic.

However, this is not possible in our case for two reasons. First, the quantities �j(a) involve

PfYI(i) = aijXI(i)g and the nonparametric functions PfYI(i) = aijXI(i) = �g are heteroge-neous across i�s unless we assume a symmetric equilibrium where y1 = � � � = yN . Second, the

test is on inequality restrictions rather than equality restrictions. Thus, we cannot pivotize

the test, for example, by using an inverse covariance matrix.

To deal with this situation, we �rst propose a benchmark method of constructing bootstrap

critical values that are shown later to be asymptotically valid, and then develop a way to

22 SONG

improve the inference when all the players have hindsight regrets that are asymptotically

negligible.

3.3.1. A Benchmark Method. We adapt the cluster wild bootstrap method proposed by

Cameron, Gelbach, and Miller (2008) to our inference environment. We �rst relabel the

acquaintance groups as

(3.13) fIm : m = 1; � � �; N0g = fI(i) : i 2 Ng:

Then we �rst take su¢ ciently large integer B and draw "m;b�s with m = 1; � � �; N0; andb = 1; � � �; B, from a common distribution with mean zero and variance one (independently

and identically distributed across m�s, and b�s.) (For each (m; b), the random numbers "m;bare taken to be common across j = 1; � � �; J .) We consider the following bootstrap test

statistic:

T �b �Xa2Aq

JXj=1

1

N

NXi=1

Zij(ai)"m(i);b

!2; b = 1; � � �; B;

where "m(i);b = "m;b if and only if I(i) = Im, and

Zij(a) � 1fYI(i) = aiggj(XI(i))�1

N

NXk=1

1fYI(k) = akggj(XI(k)).

Let c�1��;B be the (1��+�(�))-th percentile of the bootstrap test statistics T �b , b = 1; 2; ���; B,where

(3.14) �(�) =�

2

�1�j��U j > 0

+ 1

�j��Lj > 0

�:

The con�dence set for � 2 � is de�ned to be

(3.15) CB =�� 2 � : T (�) � maxfc�1��;B; "g

;

where " > 0 is a �xed small number such as 0.001. The maximum in the critical value in CBis introduced to ensure the uniform validity of the bootstrap con�dence set even when the

test statistic becomes degenerate.

Conveniently, the critical value c�1��;B depends on � 2 � only through �(�), not throughthe bootstrap test statistic T �b . This expedites the computation of the con�dence set sub-

stantially. This con�dence set is asymptotically valid as shown in Theorem 3 below.

Theorem 3: Suppose that Assumptions 1-6 hold and that there exist c > 0 and � > 0 such

that for all s = 1; � � �; S; and i 2 Ns,

(3.16) infP2P

P�E�jjgj(XI(i))jj4+� jC

�< c= 1:

ECONOMETRIC INFERENCE ON A LARGE GAME 23

Then

liminfN0!1

infP2P

P f�0 2 C1g � 1� �.

The condition (3.16) is the moment condition for gj(XI(i)). This condition is satis�ed if one

take indicator functions or some polynomials of XI(i) up to r degree, and E[jjXI(i)jjr(4+v)jC]is �nite almost everywhere.

To see the intuition of why the bootstrap method works, �rst note that for each � 2 �,the distribution of T (�) is �rst order stochastically dominated by that of

(3.17)Xa2Aq

JXj=1

�[�j(a)]+ + [�j(a)]�

�2=Xa2Aq

JXj=1

�2j(a);

by (3.11). By comparing the variances, one can show that the asymptotic distribution of the

last quantity is again �rst order stochastically dominated by the asymptotic distribution of

(3.18)Xa2Aq

JXj=1

1

N

NXi=1

Zij(ai)

!2;

where

Zij(ai) � 1fYI(i) = aiggj(XI(i))� E�1fYI(i) = aiggj(XI(i))jF

�;

and F is a certain sigma �eld. The last sum is approximated by the conditional distributionof T �b given (Y;X) when N and B are su¢ ciently large. Since

PfT �b > maxfc�1��;B; "gjY;Xg � �;

by the de�nition of c�1��;B, we also have

PfT (�) > maxfc�1��;B; "gjFg - �;

where - denotes inequality that holds in the limit. Hence taking the expectation on both

sides, we �nd that the bootstrap test is still asymptotically valid. In the proof of Theorem

3 in the Appendix, we make this approximation rigorous using a Berry-Esseen bound for

martingale di¤erence arrays, to ensure validity that is uniform over P 2 P.Since we allow for the types within each acquaintance group to be correlated in an arbi-

trary way, the e¤ective sample size among theN players is in fact the number of acquaintance

groups N0. It is worth noting that the asymptotic validity of the con�dence sets in Theorem

3 does not require any particular way the ratio N0=N behaves asymptotically as N ! 1.See Hansen (2007) for a similar observation in the context of estimating a robust covari-

ance matrix in linear panel models. More recently Ibragimov and Müller (2010) suggested

24 SONG

a method of very general robust inference based on t-statistic. Their method is directly

applicable here, unless the dimension J is one and the game is a binary action game.

3.3.2. Modi�ed Bootstrap with Asymptotically Negligible Hindsight Regrets. We assume that

the hindsight regrets are asymptotically negligible as Ns !1 for each s 2 S. Many modelsof large games with semianonymous payo¤s exhibit belief-free hindsight regrets that are

asymptotically negligible. Typically when the belief-free hindsight regrets are asymptotically

negligible as the number of the players goes to in�nity, we can show that ��U and ��L are

asymptotically negligible, so that the bounds wU(�) and wL(�) become also asymptotically

negligible. In this section, we propose an alternative inference method that utilizes this fact.

We consider the following bootstrap test statistic:

TMod�b � 1

J jDjXa2D

JXj=1

��L�j;U;b(a; �)

�++�L�j;L;b(a; �)

��

�2; b = 1; � � �; B;

where

L�j;U;b(a; �) =1

N

NXi=1

ri;U(ai; �)gj(XI(i))"m(i);b and

L�j;L;b(a; �) =1

N

NXi=1

ri;L(ai; �)gj(XI(i))"m(i);b:

We take cMod�1��;B(�) to be the (1��+�(�))-th percentile of the bootstrap test statistics TMod�

b ,

b = 1; 2; � � �; B, and de�ne the con�dence set for � 2 � to be

CModB =

�� 2 � : T (�) � maxfcMod�

1��;B(�); "g:

To see how this method achieves validity, we note that

L�j;U;b(a; �) =1

N

NXi=1

ri;U(ai; �)gj(XI(i))"m(i);b

=1

N

NXi=1

r�i (ai)gj(XI(i))"m(i);b +1

N

NXi=1

ei;U(ai)gj(XI(i))"m(i);b:

The conditional variance of the last term given (Y;X) is equal to�N0N2

�� 1N0

N0Xm=1

Xi2Im

ei;U(ai)gj(XI(i))

!2:

We can show that when the hindsight regrets are asymptotically negligible (or more speci�-

cally, � ij in Theorem 2 is asymptotically negligible uniformly over i 2 N), we have the last

ECONOMETRIC INFERENCE ON A LARGE GAME 25

term also asymptotically negligible. This means that we have

(3.19) L�j;U;b(a; �) �1

N

NXi=1

r�i (ai)gj(XI(i))"m(i);b;

and similarly, using the fact that ��L is also asymptotically negligible,

(3.20) L�j;L;b(a; �) �1

N

NXi=1

r�i (ai)gj(XI(i))"m(i);b:

The bootstrap distribution of the last sum approximates the distribution of �j(a). From

(3.11), we conclude that this modi�ed bootstrap is asymptotically valid. We formally state

the result as follows.

Theorem 4: Suppose that the conditions of Theorem 3 hold. Furthermore assume that as

N !1, we have maxi2N � ij ! 0 for all j = 1; � � �; J . Then

liminfN!1

infP2P

P��0 2 CMod

1� 1� �.

While the con�dence set CModB is less conservative than CB, it is asymptotically valid only

when the hindsight regrets of all the players are asymptotically negligible.

It is worth noting that the wild bootstrap procedure preserves both the cross-sectional

dependence due to the acquaintance group structure and the cross-sectional dependence due

to the public signals. Cross-sectional dependence due to the acquaintance group structure

is captured by using the same "m;b for individuals in the same acquaintance group. Cross-

sectional dependence due to the public signals is preserved by the wild bootstrap procedure.

To see this latter fact, assume for simplicity a private information game with a single infor-

mation group with a public signal C, and take J = 1 and A = f0; 1g so that we have in thiscase

N � TMod�b �

1pN

NXi=1

r�i (1)gj(Xi)"i;b

!2; b = 1; � � �; B:

The conditional variance of the sum is equal to

1

N

NXi=1

r�2i (1)g2j (Xi):

By the law of the large numbers applied to conditionally independent sums, the last sum is

approximated by1

N

NXi=1

E�r�2i (1)g

2j (Xi)jC

�;

26 SONG

which is also the conditional variance of �1(1) given C. Hence the cross-sectional dependence

through the presence of public signal is captured by the wild bootstrap in this case.

As mentioned before, the cluster bootstrap method that this paper relies on was originally

proposed by Cameron, Gelbach, and Miller (2008) for linear regression models to deal with

within-cluster dependence. Their wild bootstrap procedure draws regression residuals mul-

tiplied by random variables with a two-point distribution. The latter random variables are

chosen to be cluster-speci�c, to preserve the cross-sectional dependence structure. The main

di¤erence here is that we have two layers of cross-sectional dependence one with acquaintance

groups and the other with information groups. While both groups can be viewed as clusters,

our bootstrap method controls only the acquaintance group structure. The main reason is

that the cross-sectional dependence within information group is due to some public signals,

and can be captured by the wild bootstrap procedure as we saw before. Note that this

cross-sectional dependence will not be captured by simply resampling (with replacement)

from acquaintance groups in general.

3.4. Partial Observation of the Players. When there are many players in a game, itis not rare that the econometrician does not observe all the players. The issue that the

econometrician faces in the case of partial observation of the players is distinct from the

well-known issue of missing values in the econometrics literature, because the payo¤ function

that the econometrician uses should depend on the actions of all the players (including those

unobserved by the econometrician). Under certain conditions, the hindsight regret approach

delivers an asymptotically valid inference method even when the econometrician observes

only part of the players.

Condition 1: (i) The game is a private information game consisting of several or manyinformation groups indexed by s = 1; � � �; S, and players from di¤erent information groups

are payo¤ disjoint (Assumption 3).

(ii) In each large game, the payo¤ di¤erential is additive in the proportion of the actions of

the other players in his payo¤ group, i.e.,

�umk(a�i; ti) = vmk(ti) + � � (�am � �ak) �1

Ns � 1

NsXj 6=i

aj:

Condition 2: The outcomes observed by the econometrician are realizations from a pure

strategy Bayesian Nash equilibrium, where within each information group, the strategies are

identical among the players, and the types are drawn from the same distribution by the

Nature.

ECONOMETRIC INFERENCE ON A LARGE GAME 27

Condition 1(ii) tells us that the expected payo¤di¤erential does not depend on the number

or identity of the other players. Condition 2 has the consequence that the observed outcomes

have the identical distribution across the players within each information group.

When Conditions 1-2 are satis�ed, we have for all m; k = 1; � � �; K;

E�u�mk(Y�i;Ti)jTi

�= vmk(Ti) + � � (�am � �ak)E[YjjTi],

where player j belongs to player i�s information group. The last conditional expectation

E[YjjTi] does not depend on the number of the players in the large game, so one can proceedwith only a subsample of the players. More speci�cally, suppose that the set of players in

information group s is a proper subset N1s of Ns. Then the ex post version of the equilibriumconstraints can be formulated using the players in N1s, where one uses (N1

s�1)�1P

j2N1snfig Yj,

instead of E[YjjTi]. To address the bias caused by this replacement, one uses the hindsightregret bound in Theorem 1 which becomes in this case Bi;�(ti) as in Section 2.3.1, except

that we use N1s now instead of Ns. Using N1

s instead of Ns makes the hindsight regret

bound larger. Hence the moment inequalities in Theorem 2 still hold for the players in the

subsample N1s. Therefore, the asymptotic validity of the inference procedure is not a¤ected,when one uses only part of the moment inequalities, as long as the number of the players is

still large.

4. Monte Carlo Simulation Studies

4.1. Basic Data Generating Processes. We consider S number of private informationBayesian games, where each game s is populated by Ns number of players. The action space

for each player is f0; 1g. The i-th player in game s = 1; � � �; S has the following form of a

payo¤ di¤erential:

ui(1; a�i;Ti)� ui(0; a�i;Ti) = Xi;s�0 + �0

1

Ns � 1

NsXj=1;j 6=i

aj

!+ �i;s;

where Xi;s and Wi;s are observable characteristics of individual i, and �is is an unobservable

characteristic. We have speci�ed

(4.1) Xi;s = Zi;s=3 + 1;0C1;s � 0:2:

where Zi;s is an idiosyncratic component and C1;s is a public signal that is speci�c to group s.

The random variables �i;s, Z1;i;s; Z2;i;s; C1;s and C2;s are drawn independently from N(0; 1).

The variables, Z1;i;s�s and Z2;i;s�s are independent across i�s and s�s and C1;s�s and C2;s�s are

independent across s�s.

28 SONG

The utility speci�cation is similar to the one that is often used in the literature of social

interactions. The parameter �0 measures the presence of social interactions. Throughout

the exercise, we set �0 to be 1.

To generate observed outcomes, we assume private information game, i.e., I(i) = fig; andequilibrium strategies are identical within each subgroups and �nd any ps for each game s

such that

ps = E [� (Xi;s�0 � 0:2 + �0 � ps)] :Then, we generate

Yi;s = 1�Xi;s�0 + �0 � ps + �i;s

:

The parameter �0 is set to be 1. For the simulations, we assume that each game has the

same number of players equal to Ns = n. Then, the belief-free hindsight regret for player i

in game s (denoted by Bi;s;�(Ti)) is simpli�ed as

(4.2) Bi;s;�(Ti) =

s� �20Ns � 1

log(�)

2:

Throughout the simulation study, we have set � = 0:01: For the inference, we set the Monte

Carlo simulation number to be 1,000, and the bootstrap Monte Carlo simulation number

(B) to be 1,000.6 For the construction of moment inequalities, we used the following:7

g1(Xi;s) = 1; g2(Xi;s) = jXi;sj; and g3(Xi;s) = 2 � 1fXi;s � 0g.

Throughout the simulation studies, we have chosen � = 0:01 and � = 0:01.

4.2. Finite Sample Coverage Probabilities of Bootstrap Tests. We �rst investigatethe �nite sample validity of the con�dence intervals. For this study, we have �xed �0 = 1

and vary �0 in f0; 0:5g. The main focus is on the role of belief-free hindsight regret. Sincethe belief-free hindsight regret in (4.2) is increasing in �0, we expect that the con�dence set

will become more conservative as �0 moves away from zero. This is analogous to a situation

where all the moment inequalities are away from being binding.

The �rst simulation study investigate the �nite sample coverage probabilities. The results

are shown in Table 1. In Table 1, we observe that the benchmark bootstrap con�dence set

gives conservative �nite sample coverage probabilities even when �0 = 0, i.e. there is no

hindsight regret. This is because the termP

a2AqPJ

j=1 �2j(a) in (3.17) is stochastically dom-

inated by the termP

a2AqPJ

j=1(1N

PNi=1 Zij(ai))

2 in (3.18). Recall that the bootstrap test

6The simulation study in the current version used �i = 1f� i > 0g instead of using �i = � � (c� ii=(1 + c� ii))for the test statistic, and the modi�ed bootstrap procedure.7An alternative choice of g3(Xi;s) = X2

i;s was tried initially. This choice leads to a very stable coverageproperties, but the power of the test for �0 = � and �0 = � was very weak. Hence the choice of thenonnegative map gj(�) is crucial for sharp inference results.

ECONOMETRIC INFERENCE ON A LARGE GAME 29

relies on the latter quantity, because there is no reasonable way to simulate the distribution

of the former quantity, when there exists nonnegligible hindsight regrets and the equilibrium

is asymmetric with heterogeneous beliefs. This stochastic dominance is not removed even

when there is no hindsight regret.

Table 1: Finite Sample Coverage Probabilities at 95%

Benchmark Modi�ed

�0= 0:0 �0= 0:5 �0= 0:0 �0= 0:5

Ns= 100 0.983 1.000 0.942 1.000

S = 3 Ns= 200 0.979 1.000 0.941 1.000

Ns= 500 0.991 1.000 0.942 1.000

Ns= 100 0.988 1.000 0.956 1.000

S = 10 Ns= 200 0.990 1.000 0.946 1.000

Ns= 500 0.991 1.000 0.942 1.000

Ns= 100 0.986 1.000 0.935 1.000

S = 50 Ns= 200 0.991 1.000 0.939 1.000

Ns= 500 0.992 1.000 0.949 1.000

The con�dence sets from the modi�ed bootstrap assuming asymptotically negligible hind-

sight regrets are shown to produce nonconservative coverage probabilities, when there is

no hindsight regret (�0 = 0): This is because when the hindsight regrets are asymptotically

negligible, one can simulate the distributionP

a2AqPJ

j=1 �2j(a) directly as we saw previously.

Hence as compared to the benchmark method, the modi�ed method is much less conserva-

tive. However, this modi�ed method becomes conservative as �0 moves away from zero. This

is the situation where we are further into the interior of the moment inequality restrictions,

and hence the coverage probabilities show conservativeness.

One might think that when the hindsight regrets are asymptotically negligible, we may

ignore the hindsight regrets altogether and consider a test statistic of the following:

1

J jDjXa2D

JXj=1

L2j(a; �);

where Lj(a; �) is equal to Lj;U(a; �) = Lj;L(a; �) with �i = 0, and Bij;� is replaced by zero,

and one may use the bootstrap critical values using

1

J jDjXa2D

JXj=1

L�2j (a; �);

30 SONG

where L�j(a; �) is the same as L�j;U(a; �) except that Bij;� is set to be zero. This inference

is tantamount to assuming that the Bayesian Nash equilibrium that the econometrician

observes is in fact its re�nement as an ex post Nash equilibrium, where the equilibrium

strategies remain an equilibrium even after all the types are revealed to the players. However,

as we will see in Table 2 below, this leads to invalid inference in general.

Table 2: Finite Sample Coverage Probabilities at 90% When the Hindsight Regrets are

Ignored

Benchmark Modi�ed

�0= �1:0 �0= �3:0 �0= �1:0 �0= �3:0Ns= 100 0.894 0.757 0.837 0.723

S = 3 Ns= 200 0.899 0.741 0.828 0.699

Ns= 500 0.909 0.757 0.831 0.702

Ns= 100 0.895 0.752 0.830 0.731

S = 10 Ns= 200 0.887 0.739 0.814 0.694

Ns= 500 0.877 0.712 0.797 0.654

Ns= 100 0.905 0.739 0.840 0.705

S = 50 Ns= 200 0.900 0.706 0.839 0.662

Ns= 500 0.872 0.576 0.784 0.509

Table 2 reports the �nite sample coverage probabilities of the bootstrap tests when the

hindsight regrets are entirely ignored in the construction of the test statistic. Here we set

0;1 = 0 and �0 = 1. The table shows that the �nite sample coverage probabilities deterio-

rate when the hindsight regrets are ignored. (The benchmark bootstrap procedure and the

modi�ed bootstrap procedure that properly take into account hindsight regrets as proposed

in this paper produced the coverage probability of 1 in this set-up.) The deterioration be-

comes severe as the sample size becomes larger. Therefore, when the hindsight regrets are

asymptotically negligible, one cannot simply eliminate the hindsight regret component Bij;�and �i in the test statistic.

8

4.3. Finite Sample Power of the Bootstrap Tests for Parameter Values. From the

coverage probability results that are conservative, especially when there is nonzero hindsight

regret, one may be concerned about the small sample power properties of the bootstrap

method. Hence we have investigated the small sample power properties of the bootstrap

method.

8When we took �0 to be a positive number, the coverage probability deterioration did not arise.

ECONOMETRIC INFERENCE ON A LARGE GAME 31

Figure 1. The Empirical Power of the Test: H0 : �0 = � vs. H1 : �0 6= �at 5%.with S = 10: The x-axis represents the true value of �0. The testis constructed using the Bonferroni approach to construct a con�dence setfor �0. As expected, as �0 is away from zero, the hindsight regret becomeslarger, and hence it becomes harder to distinguish the null hypothesis fromthe alternatives. Note the conspicuous improvement in power by the modi�edbootstrap.

We �rst consider the test for �0. The null hypothesis and the alternative hypothesis is

given by

H0 : �0 = � vs: H1 : �0 6= �:

The nominal level of the test is set to be at 5%, and the number of the information groups

are set to be 10. The test is performed by constructing a Bonferroni bootstrap con�dence

set for �0 and checking whether the hypothesized value � belongs to the the con�dence set.

The results are shown in Figure 1. The x-axis represents the hypothesized value of � under

the null hypothesis. The upper row panels use the true value of �0 = 0:0; 0:5; and 1:0, with

the group size Ns = 100. And the lower row panels use the same values of �0, but with a

larger group size Ns = 300:

From Figure 1, the bootstrap test has a higher power when �0 = 0, but the power becomes

lower as �0 is away from zero. However, it is clear that even when the empirical coverage

probability for the bootstrap is highly conservative (i.e., when �0 = 0:5 and �0 = 1:0),

32 SONG

Figure 2. The Empirical Power of the Test: H0 : �0 = � vs. H1 : �0 6= � at5%. .with S = 10: The x-axis represents the value of �0.under the alternativehypothesis. The test is constructed using the Bonferroni approach to constructa con�dence set for �0. As expected, as �0 is away from zero, the hindsightregret becomes larger, and hence it becomes harder to distinguish the nullhypothesis from the alternatives. Note the conspicuous improvement in powerby the modi�ed bootstrap.

this does not necessarily mean that the test has negligible power for the range of values �

considered here.

The power improvement by the modi�ed bootstrap procedure appears conspicuous. Again,

even when the coverage probabilities are both 1 almost always for the case of �0 = 0:5 and

�0 = 1, the power improvement achieved by the modi�ed bootstrap looks outstanding. Also,

as expected, the increased information group size increases the power of the test, apparently

by reducing the size of hindsight regrets.

One can expect that the conservativeness of the bootstrap test also a¤ects the estimation

of �0, depending on the size of the hindsight regrets. To investigate this, we consider the

test for �0 now:

H0 : �0 = � vs: H1 : �0 6= �

ECONOMETRIC INFERENCE ON A LARGE GAME 33

with di¤erent values of �0. Again, the test is performed by constructing a Bonferroni boot-

strap con�dence set for �0 and checking whether the hypothesized value � belongs to the

the con�dence set.

Similarly as in the case of testing �0 = �, the test shows decrease in power as we move �0away from zero, due to higher hindsight regrets. Also, note that the power improvement by

the modi�ed bootstrap method appears conspicuous throughout all the cases.

One might be concerned about the lower power properties when �0 is away from zero.

While one may investigate further and produce a method with better power properties, the

low power with larger hindsight regrets is general a phenomenon not with the method, but

inherent in the inference problem, because inference in the presence of belief heterogeneity

and its larger role in the players�strategic decisions will obviously make the inference problem

harder.

Lastly, the bootstrap test performs very well in testing the null hypothesis of no strategic

interaction H0 : �0 = 0 vs. H1 : �0 6= 0. The test performs quite well because the null

hypothesis corresponds to the case with no hindsight regret.

4.4. Bootstrap Validity in the Presence of Public Signals. Previously, we emphasizedthat the modi�ed bootstrap test is designed to capture both layers of cross-sectional depen-

dence, one due to the acquaintance groups and the other due to the information groups. Since

we are focusing on the private information game here, the focus now is on the cross-sectional

dependence due to the public signals.

The magnitude of the public signals is determined by the parameter 0;1. When 0;1 = 0,

there is no public signal, but when 0;1 is large, there is a large role for the public signal in

the game.

We investigate the �nite sample coverage probabilities by using di¤erent values of 0;1.

The results are shown in Table 3. The number of the information groups (S) was set to be

10. We focus on the case of no hindsight regret (�0 = 0). The parameter 0;1 was chosen

from f0; 1=3; 1g.Table 3 shows that the benchmark bootstrap method becomes more and more conservative

as the public signals become larger. However, the modi�ed bootstrap method gives stable

coverage probabilities over di¤erent values of 0;1. This supports our observation that the

modi�ed bootstrap captures the cross-sectional dependence structure well.

Here are a summary of the results from the Monte Carlo study. First, as expected,

the inference becomes more conservative, as the strategic interaction (among unacquainted

players) becomes larger (i.e. �0 becomes away from zero.) In particular, the coverage

probabilities becomes 1 mostly when �0 = 0:5 or larger.

34 SONG

Table 3: Coverage Probabilities with Various Magnitudes of Public Signals:

�0 = 0; �0 = 1, and S = 10

90% 95%

1;0= 0 1;0= 1=3 1;0= 1 1;0= 0 1;0= 1=3 1;0= 1

Ns= 100 0.957 0.972 0.988 0.989 0.988 0.995

Benchmark Ns= 200 0.968 0.969 0.985 0.987 0.990 0.995

Ns= 500 0.967 0.965 0.992 0.987 0.991 0.997

Ns= 100 0.885 0.909 0.912 0.940 0.956 0.949

Modi�ed Ns= 200 0.902 0.904 0.893 0.944 0.946 0.958

Ns= 500 0.906 0.897 0.903 0.956 0.942 0.946

Second, the power properties re�ect the same phenomenon. As �0 becomes away from

zero, the empirical power of the tests for �0 or for �0 becomes lower, as expected. However,

it should be noted that even when the coverage probability becomes 1, the test still shows

reasonable power properties. In particular, the performance of modi�ed bootstrap method

shows outstanding improvement over the benchmark method, and hence this method is rec-

ommendable, when all the players are thought to have asymptotically negligible hindsight

regrets. It is also shown that even when the hindsight regrets are thought to be asymptot-

ically negligible, one should not remove them from the construction of the test statistic, as

it invalidates the inference procedure.

Finally, the presence of public signals tend to make the benchmark bootstrap test more

conservative. In contrast, the modi�ed bootstrap method is not a¤ected by the presence

the public signals. This re�ects that the modi�ed bootstrap captures the cross-sectional

dependence characterized by the public signals.

5. Conclusion

This paper proposes a large game perspective for empirical researches on many agents

interacting with each other. For this, the paper develops a general inference framework for

a large Bayesian game where the econometrician observes a Nash equilibrium pro�le. The

development proceeds by fully utilizing the structure of cross-sectional dependence that is

derived from the structural primitives of the game through the notion of hindsight regret.

The hindsight regret approach has the merit of not relying on the assumptions about the

way individual players form their beliefs about other players� types. Hence the approach

immediately accommodates the situation where players have heterogeneous beliefs. In this

framework, nontrivial inference is still possible when the players�payo¤s are impacted as-

ymptotically negligibly by a deviation by another player outside his acquaintance group.

ECONOMETRIC INFERENCE ON A LARGE GAME 35

6. Appendix

The following is a conditional version of McDiarmid�s inequality (McDiarmid (1989)) which

is used in the proof of Theorem 1.

Lemma A1 (McDiarmid�s Inequality): Let X = (X1; ���; Xm) be a vector of random vec-

tors taking values in a subset X of a Euclidean space and Xi�s are conditionally independent

given a random vector Z. Then for any f : Xm ! R, and all " > 0,

P ff(X)� E [f(X)jZ] � "jZg � exp�� 2"2Pm

i=1 Vi(f)2

�;

where Vi(f) is the maximal variation of f along the i-th coordinate.

Proof of Theorem 1: Given a pure strategy equilibrium y, let ui(a) � ui(a; Y�i;TI(i)).

For each s = 1; ����; S; let Ns be the set of players in information group s. The �nite collectionfTI(l) : l 2 Nsg is conditionally independent given Cs by Assumption 1(i). (Also recall thatTi for each i 2 Ns already contains Cs as its subvector.)For any � > 0; a 2 AnfYjg, i 2Ms, and j 2 I(i); we write

Qyi�uj(Yj)� uj(a) � ��jTI(i)

(6.1)

= Qyi

(uj(Yj)� uj(a)� Eyi

�uj(Yj)� uj(a)jTI(i)

�� ��� Eyi

�uj(Yj)� uj(a)jTI(i)

� jTI(i)

)� Qyi

�uj(Yj)� uj(a)� Eyi

�uj(Yj)� uj(a)jTI(i)

�� ��jTI(i)

;

where the last inequality holds everywhere. We write

uj(Yj)� uj(a)� Eyi�uj(Yj)� ui(a)jTI(i)

�= vj(a; T )� Eyi

�vj(a; T )jTI(i)

�;

where vj(a; t) � uj(yj(tj); y�j(t); tI(j)) � uj(a; y�j(t); tI(j)) and y�j(t) = (yk(tI(k)))Nk=1;k 6=j.

When we choose l 2 Ns=I(i) and perturb player l�s type tl, this alters vj(a; t) through itse¤ect on the actions of all players k who observe the type of player l, i.e., for all k 2 I(l).

Hence this perturbation a¤ects vj(a; t) by not more thanXk2I(l)

�jk(tI(j); a):

(We do not need to consider players outside Ns due to the assumption of payo¤disjointednessacross the information groups.) We replace � by Bij;�(TI(j); a) in (6.1) and use Lemma A1

36 SONG

to bound the last probability in (6.1) by

(6.2) exp

0B@� 2B2ij;�(TI(j); a)P

l2MsnI(i)

�Pk2I(l)�jk(TI(j); a)

�21CA = exp

��2B2

ij;�(TI(j); a)

ij(TI(j); a)

�;

where the inequality follows by the de�nition of j(Tj). Now

P�uj(Yj)� uj(a) � �Bij;�(TI(j); a) for some (a; j) 2 (AnfYjg)� I(i)jTI(i)

� (K � 1)

Xj2I(i)

exp

��2B2

ij;�(TI(j); a)

ij(TI(j); a)

�= �;

where we used the de�nition of Bij;�(TI(j)) and the fact that jAnfYjgj = K�1. We concludethat

Qyi�uj(Yj)� uj(a) > �Bij;�(TI(j); a), 8a 2 AnfYjg and 8j 2 I(i)jTI(i)

> 1� �;

which is the desired result. Q.E.D.

Proof of Theorem 2: Given a pure strategy equilibrium y, let ui(aI(i)) � ui(aI(i); Y�I(i);TI(i))

as before. De�ne the events:

Si;U(aI(i)) �(uj(aI(j))� uj(c; aI(j)nfjg) � �Bij;�(TI(j); c)

for all c 2 A and all j 2 I(i)

).

By the de�nition of Bij;�(TI(j)); Assumption 1(ii), and Theorem 1, we have (everywhere)

(6.3) P (Si;U(YI(i))jTI(i)) � 1� �:

Now, observe that

(6.4) P (Si;U(YI(i))jTI(i)) =X

aI(i)2AI(i)

P (Si;U(aI(i))jTI(i))1�YI(i) = aI(i)

� 1� �:

From this, we deduce that

(6.5) 1�YI(i) = aI(i)

� 1

nP ( ~Si;U(aI(i))jTI(i)) � 1� �

o;

where ~Si;U(aI(i)) � Si;U(aI(i)) \�YI(i) = aI(i)

. Similarly also from (6.4), we have

1�YI(i) 6= aI(i)

� 1

8<: XdI(i)2AI(i)nfaI(i)g

P (Si;U(dI(i))jTI(i))1�YI(i) = dI(i)

� 1� �

9=;= 1

8<: XdI(i)2AI(i)nfaI(i)g

P (Si;U(dI(i)) \�YI(i) = dI(i)

jTI(i)) � 1� �

9=; :

ECONOMETRIC INFERENCE ON A LARGE GAME 37

Since Si;U(dI(i)) \�YI(i) = dI(i)

is disjoint across dI(i)�s, we conclude that

1�YI(i) 6= aI(i)

� 1

nP�~Si;L(aI(i))jTI(i)

�� 1� �

o;

where~Si;L(aI(i)) �

[dI(i)2AI(i)nfaI(i)g

Si;U(dI(i)) \�YI(i) = dI(i)

:

Recall the de�nition Wi = (XI(i); Y�I(i); C). Taking conditional expectation on both sides

on (6.5) and using Markov�s inequality, we �nd that

P�YI(i) = aI(i)jXI(i)

� 1

1� �EhP ( ~Si;U(aI(i))jTI(i))jXI(i)

i(6.6)

=1

1� �EhP ( ~Si;U(aI(i))jWi)jXI(i)

i=

1

1� �P ( ~Si;U(aI(i))jWi) +

1

1� �Ri;U(aI(i));

where Ri;U(aI(i)) � P ( ~Si;U(aI(i))jWi)� P ( ~Si;U(aI(i))jXI(i)): Similarly,

(6.7) P�YI(i) 6= aI(i)jXI(i)

� 1

1� �P ( ~Si;L(aI(i))jWi) +

1

1� �Ri;L(aI(i));

where Ri;L(aI(i)) � P ( ~Si;L(aI(i))jWi)� P ( ~Si;L(aI(i))jXI(i)):

Given a = (a1; � � �; aJ) 2 Aq, let ai = (a1; � � �; aI(i)) 2 AjI(i)j: Since gj(XI(i)) � 0, we

multiply both ends of (6.6) by gj(XI(i)) and summing them up over i = 1; � � �; N , we �ndthat for a = (a1; � � �; aJ) 2 Aq,

1

N

NXi=1

P�YI(i) = aijXI(i)

gj(XI(i))(6.8)

� 1

N

NXi=1

1

1� �iP ( ~Si;U(ai)jWi)gj(XI(i)) + vj;U(a);

where vj;U(a) � 1N

PNi=1

11��i

Ri;U(ai)gj(XI(i)). Similarly, from (6.7), we also �nd that

1

N

NXi=1

P�YI(i) = aijXI(i)

gj(XI(i))(6.9)

� 1

N

NXi=1

�1� 1

1� �P ( ~Si;L(ai)jWi)

�gj(XI(i))� vj;L(a);

where vj;L(a) � 1N

PNi=1

11��i

Ri;L(a)gj(XI(i)).

38 SONG

Now it su¢ ces to control vj;U(a) and vj;L(a). We de�ne

ei;U(ai) � P�YI(i) = aijXI(i)

� 1

1� �P ( ~Si;U(ai)jWi)gj(XI(i))

and

ei;L(ai) � P�YI(i) = aijXI(i)

��1� 1

1� �P ( ~Si;L(ai)jWi)

�gj(XI(i)):

We write for a given vector of nonnegative constants wU = (wj;U(a))Jj=1;a2Aq and wL =

(wj;L(a))Jj=1;a2Aq ,

(6.10) M(w) =ML(wL) \MU(wU);

where w = (wU ; wL),

ML(wL) =

(Xa2Aq

JXj=1

"1

N

NXi=1

ei;L(ai)gj(XI(i)) + wj;L(a)

#�

= 0

)and

MU(wU) =

(Xa2Aq

JXj=1

"1

N

NXi=1

ei;U(ai)gj(XI(i))� wj;U(a)

#+

= 0

):

By (6.8) and (6.9),

(6.11) P (ML(vL)jX;C) = 1 and P (MU (vU) jX;C) = 1;

where vL = (vj;L(a))Jj=1;a2Aq and vU = (vj;U(a))Jj=1;a2Aq .

Fix nonnegative constants wU and wL and de�ne the event

MA = fvj;U(a) � wU and vj;L(a) � wL; 8a 2 Aq and j = 1; � � �; Jg :

We write

(6.12) P (M (w) jX;C) = P (M (w) \MAjX;C) + P (M (w) \McAjX;C) :

Note that �rst probability is increasing in w. Hence by the de�nition of MA; the �rst

probability is bounded from below by

P (ML (vL) \MU (vU) \MAjX;C) = P (MAjX;C) ;

where the last equality follows by (6.11). Also, we write the last probability in (6.12) as

P (McAjX;C)� P (M (w)c \Mc

AjX;C) :

From (6.12), we conclude that

(6.13) P (M (w) jX;C) � 1� P (McAjX;C) :

ECONOMETRIC INFERENCE ON A LARGE GAME 39

Now, it su¢ ces to obtain a bound for the last probability. For this, �rst note that

P [McAjX;C] � P

�max

a2Aq ;j=1;���;Jvj;U(a) > wU or max

a2Aq ;j=1;���;Jvj;L(a) > wLjX;C

��

Xa2Aq

JXj=1

P fvj;U(a) > wU jX;Cg+Xa2Aq

JXj=1

P fvj;L(a) > wLjX;Cg ;

because wL; wU > 0. We analyze the �rst probability only. The second probability can

be analyzed similarly. Note that essentially vj;U(a) is a random variable as a real valued

function of Y�i. Note that vj;U(a) and vj;L(a) are nonstochastic functions of (�;X), where

� = (�i)i2N and X = (Xi)i2N. By Assumption 6(ii), �i�s are conditionally independent given

X = (Xi)i2N and C:

We write

vj;U(a) = fj;U(Y;X; a) = ~fj;U(�;X; a) and

vj;L(a) = fj;L(Y;X; a) = ~fj;L(�;X; a);

for some functions fj;U(�; a) and fj;L(�; a), where

~fj;U(�;X; a) = fj;U(y(�;X); X; a) and ~fj;L(�;X; a) = fj;L(y(�;X); X; a),

using the fact that Y = y(T ) and T = (�;X), with y(T ) = (y1(TI(1)); � � �; yN(TI(N))).We use McDiarmid�s inequality to deduce that

P fvj;U(a) > wU jX;Cg � exp

� 2w2UPN

i=1 V2i (~fj;U(�; X; a))

!

� exp

� 2w2UPN

i=1maxk2H(i) V2k (fj;U(�; X; a))

!

� exp

� 2w2UPN

i=1 �2ij;U(X; a)

!;

where the second inequality follows because the variations in �i a¤ects fj;U only through the

pure strategies yk(�I(k); XI(k)) of players k who observe �i. Similarly, we also obtain

P fvj;L(a) > wLjX;Cg � exp � 2w2LPN

i=1 �2ij;L(X; a)

!:

Using the de�nitions of wU(�) and wL(�) in Theorem 2, we have

P fvj;U(a) > wU(�)jX;Cg � �

2and

P fvj;L(a) > wL(�)jX;Cg � �

2:

40 SONG

By applying this to (6.13), we obtain the desired inequality:

P (M (w(�)) jX;C) � 1� � :

Q.E.D.

Proof of Theorem 3: We consider C1 �rst. Since �I(i) is conditionally independent of

(Y�I(i); C) given XI(i) for each i 2 N by Assumption 6(ii), we can writeZHi;U (aIm ;XIm )

dGs;�(�ImjXIm) = P��Im 2 Hi;U(aIm ;XIm)jWi

:

On the other hand, because Y is from a pure strategy equilibrium y (Assumption 4(i)), we

write P�YIm = a[m]jWi

as

P�yIm(TIm) = a[m]jXIm ; Y�Im ; C

= P

�yIm(TIm) = a[m]jXIm ; C

= P

�yIm(TIm) = a[m]jXIm

= P

�YIm = a[m]jXIm

;

where yIm(TIm) = (yj(TI(j)))j2Im and the equalities are due to Assumptions 5(ii) and 4(ii).

Similarly as in (3.10), we decompose the last sum and apply the same decomposition to

Lj;L(a; �) to obtain that

Lj;U(a; �) =1

N

NXi=1

� ij(ai) +1

N

NXi=1

ei;U(ai)gj(XI(i)) and

Lj;L(a; �) =1

N

NXi=1

� ij(ai) +1

N

NXi=1

ei;L(ai)gj(XI(i));

where � ij(ai) = r�i (ai)gj(XI(i)); r�i (ai) = 1fYI(i) = aig � PfYI(i) = aijFm(i)�1g, and

~ei;U(ai) = PfYI(i) = aijFm(i)�1g �1

1� �i� �i;U (ai) and

~ei;L(ai) = PfYI(i) = aijFm(i)�1g ��1� 1

1� �i� �i;L (ai)

�;

and m(i) = m if and only if i 2 Im.By Theorem 2, we have for all �a 2 A,

(6.14) Lj;U(a; �)� wU(�) �1

N

NXi=1

� ij(ai) � Lj;L(a; �) + wL(�),

with probability at least 1� � .

Recall the de�nition

Zij(ai) � 1fYI(i) = aiggj(XI(i))�1

NPfYI(i) = aijFm(i)�1g:

ECONOMETRIC INFERENCE ON A LARGE GAME 41

It is convenient if we stack up � ij(a), Zij(a), and Zij(a) as vectors with j = 1; � � �; J and arunning in Aq. To do this, we de�ne � i(a) Zi(a), and Zi(a) to be the J-dimensional columnvector whose j-th entry is given by � ij(a), Zij(a), and Zij(a). Then let � i; Zi, and Zi be

J jAqj-dimensional vector constructed by stacking up � i(a); Zi(a), and Zi(a) starting fromthe �rst member of Aq to the last (e.g. by a lexicographic order).Therefore from (6.14), we can bound

(6.15)Xa2Aq

JXj=1

[Lj;U(a; �)� wj;U(� ; a)]++ [Lj;L(a; �)� wj;L(� ; a)]�

!2� 1N

NXi=1

� i

2

;

with probability at least 1� � . Now, our asymptotic analysis focuses on jj 1N

PNi=1 � ijj2.

We de�ne Fm � [mk=1Ik and write

1

N

NXi=1

� i =1

N

N0Xm=1

Xi2Im

� i =N0N

1

N0

N0Xm=1

�m;S;

where �m;S =P

i2Im � i. Similarly we de�ne Z"m;S =

Pi2Im Zi"m;b:

To deal with the case of asymptotically degenerate distribution in a way that is uniform

over P 2 P, we use arguments similar to those in the proof of Theorem 2 of Lee, Song and

Whang (2013). First, �x a small number 0 < � < 1=2 and let f�i;� : i 2 Ng be a sequenceof i.i.d. J jAqj-dimensional random vectors such that the entries are distributed i.i.d. as

uniform [��; �] and f�i;�g1i=1 is independent of f(Yi; Xi; C)g1i=1. We de�ne

�m;S;� �Xi2Im

� i +

p3�i;�pjImj

!and

Z"m;S;� �Xi2Im

Zi +

p3�i;�pjImj

!"m;b:

Let C � (Cs)Ss=1 and for each j = 1; 2; � � �; let

Fm;� � ��f(YIk ; XIk)

mk=1; XIm+1 ; (vIk;�)

mk=1; C

�;

where vIk;� = (vi;�)i2Ik . De�ne

F� �1\m=1

Fm;�:

42 SONG

And

F[�]� (tjF�) � P

( 1pN0

N0Xm=1

�m;S;�

� tjF�

)and

F[�]Z (tjY;X) � P

( 1pN0

N0Xm=1

Z"m;S;�

� tjY;X):

Let F�(tjF�) and FZ(tjY;X) be F [�]� (tjF�) and F[�]Z (tjY;X) with �m;S;� and Z"m;S;� replaced

by �m;S and Z"m;S, respectively. De�ne for a large number N0 > 0 and let aN0 be a positive

sequence such that aN0 !1 and aN0=pN0 ! 0 as N0 !1. For � 2 (0; 1=2) and " >

p�;

let

hN0(�) �aN0 jlog aN0 � logN0j

12K(�)+3

pN0

;

where K(�) � � log(�2)= log(J jAjq). Note that hN0(�) does not depend on P 2 P.Recall the sequence aN0 ! 1 in the de�nition of hN0(�; "). We introduce truncated

versions of �m;S;� and Z"m;S;� :

��m;S;� �Xi2Im

� i1� g(XI(i))

� aN0+

p3�i;�pjImj

!and

�Z"m;S;� �Xi2Im

Zi1

� g(XI(i)) � aN0

+

p3�i;�pjImj

!"m;b:

Let

�F[�]� (tjF�) � P

( 1pN0

N0Xm=1

��m;S;�

� tjF�

)and

�F[�]Z (tjY;X) � P

( 1pN0

N0Xm=1

�Z"m;S;�

� tjY;X):

Also, we de�ne

V 2� � 1

N0

N0Xm=1

Eh��2m;S;�jFm�1;�

iand

W 2� � 1

N0

N0Xm=1

E��Z2m;S;�jGm�1;�

�;

where Gm;� = ��("m(i);b)i2Fm ; (�i;�)i2Fm ; Y;X

�.

First, we show the following claims.

ECONOMETRIC INFERENCE ON A LARGE GAME 43

Claim 1: There exists c1 > 0 such that for each � 2 (0; 1=2) and " > 0;

P

�supt�0

��� �F [�]� (tjF�)� �F[�]�;1(tjF�)

��� � c1hN0(�)

�= 1;

where �F [�]�;1(tjF�) = PfV�Z � tjF�g:

Claim 2: There exists c2 > 0 such that for each � 2 (0; 1=2) and " > 0;

P

�supt�"

��� �F [�]Z (tjY;X)� �F [�]1 (tjY;X)��� � c2hN0(�)

�= 1;

where F [�]Z;1(tjY;X) = PfW�Z � tjY;Xg:

Claim 3: For any choice of cN0 > 0 and �N0 > 0, we have

Pn�F[�]�;1(tjF�)� �F

[�]Z;1(tjY;X) � �cN0

o� 1�

�3J jAqj �M2a2N0�N0cN0

pN0

+C1�N0cN0�

2

�;

where C1 is the maximum value of �2 density with degree of freedom equal to J jAqj.

Claim 4: For C = maxi2N supP2P EP�jjg(XI(i))jj4

�, we have (as N0 !1)

Pn��� �F [�]� (tjF�)� F

[�]� (tjF�)

��� � C �Ma� 23

N0+ '

[1]N0(�; ") + c1hN0(�)

o= 1 and

Pn��� �F [�]Z (tjF�)� F

[�]Z (tjF�)

��� � C �Ma� 23

N0+ '

[1]N0(�; ") + ~hN0(�)

o! 1;

where '[1]N0(�; ") is a nonstochastic map that does not depend on P such that limN0!1'[1]N0(�; ")

for all � > 0; " > 0, and ~hN0(�) � C �Ma�2=3N0

=(N0 logN0) + c2hN0(�):

Claim 5: For each � > 0 and " > 0, there exists a positive number '[2]N0(�; ") > 0 such thatlim�!0 limN0!1 '

[2]N0(�; ") = 0 for each " > 0 and

supP2P

supt�"

���F�(t)� F[�]� (t)

��� � '[2]N0(�; ") + � and

P

�supt�"

���FZ(tjY;X)� F[�]Z (tjY;X)

��� � '[2]N0(�; ") + �

�= 1:

By chaining Claims 1-5. we conclude that

(6.16) infP2P

P

�supt�"jF�(t)� FZ(tjY;X)j � (c1 + c2)hN0(�; ")

�! 0;

where

hN0(�; ") = 2('[2]N0(�; ") + �) + 2fC �Ma

� 23

N0+ '

[1]N0(�; ") + 2c1hN0(�) + 2

~hN0(�)g:

Note that as N0 !1, and then �!1, we have hN0(�; ")! 0.

44 SONG

By the de�nition of c�1��;1

1� �+ �(�) � P

8<: 1N

NXi=1

Zi"i;b

2

� c�1��;1 _ "jY;X

9=;= P

8<: 1p

N0

NXi=1

Zi"i;b

2

�N2(c�1��;1 _ ")

N0jY;X

9=; :

The last term is bounded by

P

8<: 1p

N0

NXi=1

� i

2

�N2(c�1��;1 _ ")

N0jF�

9=;+ oP (1);

where oP (1) is obtained by applying (6.16) and sending N0 ! 0 and then �! 0, uniformly

over P 2 P. The leading probability is bounded by

P�T (�) � c�1��;1 _ "jF�

+ oP (1);

uniformly over P 2 P. Hence, we obtain the desired result.

Proof of Claim 1: We note that f��m;S;� : m = 1; � � �; N0g is a martingale di¤erence arraywith respect to the �ltration fFm;�g1m=1. First, de�ne

(6.17) V� �1

N0

N0Xm=1

Eh��m;S;�

��>m;S;�jFm�1;�

i=1

N0

N0Xm=1

Eh��m;S;�

��>m;S;�jC

i;

because ��m;S;��s across m�s are conditionally independent given C. By the de�nition of �i;�,

for any N0 and ! 2 ;

(6.18) V�(!) =1

N0

N0Xm=1

E��m;S�

>m;SjFm�1;�

�(!) + �2I � �2I:

Now we apply the Berry-Esseen bound for Hilbert-valued martingale di¤erence arrays in

Rackauskas (1991) (Theorem on page 345). (Instead of the unconditional probability there,

we use the conditional probability Pf�jF�g.) For this we prove the conditions in the theorem.We focus on the following sum:

1pN0

N0Xm=1

��m;S;�:

By (6.17), Conditions (i) and (ii) in Theorem of Rackauskas (1991) are satis�ed trivially

with �2ii = 1 there. As for Condition (iii) in the theorem, note that

max1�m�N0 ��m;S;� p

N0� aN0p

N0:

ECONOMETRIC INFERENCE ON A LARGE GAME 45

As for Condition (iv), the eigen values of V�(!) are bounded from below by �2 from (6.18).

Hence Condition (iv) in Theorem of Rackauskas (1991) is satis�ed with constant K in the

theorem equals 1. Thus, we apply Theorem of Rackauskas (1991) (p.345) to deduce that for

some nonstochastic c1 > 0;

supt�0

��� �F [�]� (tjF�)� �F[�]�;1(tjF�)

��� � c1aN0 jlog aN0 � logN0j12K(�)+3

pN0

.

Proof of Claim 2: We �nd that E��Z"m;S;�jGm�1;�

�is equal to

E

"Xi2Im

Zi1

� g(XI(i)) � aN0

+

p3�i;�pjImj

!"m;bjGm�1;�

#= 0:

Furthermore, �Z"m;S;� is Gm;�-measurable for each m = 1; � � �; N0. Hence f �Z"m;S;� : m =

1; � � �; N0g is a martingale di¤erence array with respect to Gm�1;�. We �rst de�ne

W� �1

N0

N0Xm=1

E��Zm;S;� �Z

>m;S;�jGm�1;�

�=1

N0

N0Xm=1

Zm;SZ>m;S + �2I:

Observe that

max1�m�N0

���� �Zm;S;�pN0

���� � aN0pN0

:

Similarly as before, we use Theorem of Rackauskas (1991) to obtain the desired result.

Proof of Claim 3: Let Rm;S =P

i2Im Ri;S and Ri;S is a J jAjq-vector with

1fYI(i) = aiggj(XI(i))1

�maxi2Im

g(XI(i)) � aN0

�stacked up with j�s running j = 1; � � �; J and a running in jAj. Note that

��m;S = Rm;S � E [Rm;SjFm�1] :

As for V� and W�, note that

1

N0

N0Xm=1

Eh��m;S��

>m;SjFm�1

i=1

N0

N0Xm=1

�E�Rm;SR

>m;SjFm�1

���E [Rm;SjFm�1]E

�R>m;SjFm�1

���;

and1

N0

N0Xm=1

Zm;SZ>m;S =

1

N0

N0Xm=1

Rm;SR>m;S �

1

N0

N0Xm=1

Rm;S

! 1

N0

N0Xm=1

R>m;S

!:

Using the similar arguments as before, we can show that

1

N0

N0Xm=1

Zm;SZ>m;S = W + rN0 ;

46 SONG

where rN0 = �1;N0 + �2;N0 ;

�1;N0 =1

N0

N0Xm=1

�Rm;SR

>m;S � E

�Rm;SR

>m;SjFm�1

���2;N0 =

1

N0

N0Xm=1

E [Rm;SjFm�1]!

1

N0

N0Xm=1

E�R>m;SjFm�1

�!

� 1

N0

N0Xm=1

Rm;S

! 1

N0

N0Xm=1

R>m;S

!;

and

~W =1

N0

N0Xm=1

E�Rm;SR

>m;SjFm�1

�� 1

N0

N0Xm=1

E [Rm;SjFm�1]!

1

N0

N0Xm=1

E�R>m;SjFm�1

�!:

Let ~W� = ~W + �2I: Hence we have (everywhere)

~W� � V� =1

N0

N0Xm=1

E [Rm;SjFm�1]E�R>m;SjFm�1

�� 1

N0

N0Xm=1

E [Rm;SjFm�1]!

1

N0

N0Xm=1

E�R>m;SjFm�1

�!:

We conclude that ~W� � V� everywhere.

We consider ���P n ~W 1=2� Z

� tjY;Xo� P

n W 1=2� Z

� tjY;Xo���(6.19)

� P

�t2 ��� �

W 1=2� Z

2 � t2 +��jY;X�;

where �� = jjj ~W 1=2� Zjj2 � jjW 1=2

� Zjj2j: Note that

�� =���tr � ~W�ZZ>

�� tr

�W�ZZ>

���� = ���tr �( ~W� �W�)ZZ>���� :

Choose � > 0 and bound the last probability in (6.19) from below by

P�t2=�2 ���=�

2 � kZk2 � t2=�2 +��=�2jY;X

� P

�t2=�2 � �=�2 � kZk2 � t2=�2 + �=�2jY;X

+ P f�� > �jY;Xg :

ECONOMETRIC INFERENCE ON A LARGE GAME 47

The �rst probability is bounded by C1��=�2

�, where C1 is the maximum value of �2 distri-

bution with degree of freedom equal to J jAqj. The second probability is bounded by1

�Eh���tr �( ~W� �W�)ZZ>

���� jY;Xi =1

�Ehtr�(W� � ~W�)E

�ZZ>

��jY;X

i� J jAqj

�Ehtr(W� � ~W�)jY;X

i=

J jAqj krN0k�

:

First, note that �1;N0 is a sum of uncorrelated matrices, and that jjRm;Sjj � �MaN0. hence

Eh �1;N0 2 jF�i � �M4a4N0

N0

and similarly, Eh �2;N0 2 jF�i � 2 �M4a4N0=N0 after some algebra. We conclude that

E [jrN0j jF�] �3 �M2a2N0p

N0:

From (6.19), we �nd that

Eh���P n ~W 1=2

� Z � tjY;X

o� P

n W 1=2� Z

� tjY;Xo��� jF�i(6.20)

�3J jAqj �M2a2N0

�pN0

+C1�

�2:

Now, we take cN0 > 0 and write

Pn�F[�]�;1(tjF�)� �F

[�]Z;1(tjY;X) � �cN0

o� P

nPn V 1=2

� Z � tjY;X

o� P

n ~W 1=2� Z

� tjY;Xo� �cN0 +DN0

o;

where

DN0 =���P n ~W 1=2

� Z � tjY;X

o� P

n W 1=2� Z

� tjY;Xo��� :

We bound the last probability from below by

PnPn V 1=2

� Z � tjF�

o� P

n ~W 1=2� Z

� tjY;Xo� 0o� P fDN0 > cN0g

= 1� P fDN0 > cN0g :

By Markov�s inequality and (6.20), the last probability is bounded by

3J jAqj �M2a2N0�cN0

pN0

+C1�

cN0�2 :

Hence we obtain the desired result.

48 SONG

Proof of Claim 4: De�ne

D[�]� =

1pN0

N0Xm=1

(��m;S;� � �m;S;�) and

D[�]Z =

1pN0

N0Xm=1

( �Zm;S;� � Zm;S;�)"m;b:

Note that f��m;S;� � �m;S;�gN0m=1 is a (multi-dimensional) martingale di¤erence array withrespect to fFm;�gN0m=1 and f �Zm;S;� � Zm;S;�gN0m=1 is also a martingale di¤erence array withrespect to fGm;�gN0m=1. Hence

E

� D[�]�

2 jF�� � 1

N0

N0Xm=1

Eh ��m;S;� � �m;S;�

2 jF�i(6.21)

� 1

N0

N0Xm=1

P

(Xi2Im

jjg(XI(i))jj > aN0jF�

)

� 1

a2N0N0

N0Xm=1

E

24 Xi2Im

jjg(XI(i))jj!235 � C �M

a2N0:

Similarly, we also deduce that The last term vanishes uniformly in P 2 P, as N !1. Thismeans that j �F [�]� (tjF�)� F

[�]� (tjF�)j is bounded by

P

(����� 1p

N0

N0Xm=1

��m;S;�

� t

����� � D[�]�

jF�)(6.22)

= P

(����� 1p

N0

N0Xm=1

��m;S;�

� t

����� � D[�]�

and D[�]�

� a� 23

N0jF�

)

+P

(����� 1p

N0

N0Xm=1

��m;S;�

� t

����� � D[�]�

and D[�]�

> a� 23

N0jF�

):

The �rst probability on the right hand side of (6.22) is bounded by

P

(����� 1p

N0

N0Xm=1

��m;S;�

� t

����� � a� 23

N0jF�

)� P

n��� V 1=2� Z

� t��� � a

� 23

N0jF�o+ 2�1(F�);

where

�1(F�) � supt�"

��� �F [�]� (tjF�)� F[�]� (tjF�)

��� :As for the leading probability, note that it is written as

Pnt� a

� 23

N0� V 1=2

� Z � t+ a

� 23

N0jF�o:

ECONOMETRIC INFERENCE ON A LARGE GAME 49

Since t � "; aN0 ! 1 and jjV 1=2� jj � C �M; it is bounded by some function '[1]N0(�; ") that

does not depend on P such that

limN0!0

'[1]N0(�; ") = 0:

On the other hand, the last probability in (6.22) is bounded by

a43N0E

��D[�]�

�2jF���a43N0C �M

a2N0= C �Ma

� 23

N0:

Thus we conclude that

(6.23)��� �F [�]� (tjF�)� F

[�]� (tjF�)

��� � C �Ma� 23

N0+ '

[1]N0(�; ") + 2�1(F�):

As for the second statement of Claim 4, we note that

E

� D[�]Z

2 jY;X� � 1

N0

N0Xm=1

1

(Xi2Im

jjg(XI(i))jj > aN0

)

=1

N0

N0Xm=1

1

(Xi2Im

jjg(XI(i))jj > aN0

)� P

(Xi2Im

jjg(XI(i))jj > aN0jF�

)!

+1

N0

N0Xm=1

P

(Xi2Im

jjg(XI(i))jj > aN0 jF�

);

where the last sum is bounded by

1

a2N0N0

N0Xm=1

E

24 Xi2Im

jjg(XI(i))jj!2jF�

35 � C �M

a2N0;

similarly as in (6.21). Hence using the same arguments that led to (6.23), we obtain the

following inequality:��� �F [�]Z (tjY;X)� F[�]Z (tjY;X)

��� � (C �M + 8��1)a� 23

N0+ 2�2(Y;X)

+�3(Y;X)a43N0;

where �2(Y;X) � supt�" j �F[�]� (tjY;X)� F

[�]Z (tjY;X)j and

�3(Y;X) �1

N0

N0Xm=1

1

(Xi2Im

jjg(XI(i))jj > aN0

)� P

(Xi2Im

jjg(XI(i))jj > aN0 jF�

)!:

50 SONG

Note that E [�23(Y;X)jF�] � C �M=(N0a

2N0): Hence

P

8<:��� �F [�]Z (tjY;X)� F[�]Z (tjY;X)

��� � C �Ma� 23

N0+ '

[1]N0(�; ") + 2�2(Y;X) +

C �Ma� 23

N0

N0 logN0

9=;� P

8<:��� �F [�]Z (tjY;X)� F

[�]Z (tjY;X)

��� � C �Ma� 23

N0+ '

[1]N0(�; ") + 2�2(Y;X) +�3(Y;X)a

43N0

and �3(Y;X) � C �MN0 logN0a2N0

9=; :

The last probability is bounded from below by

1� P

��3(Y;X) >

C �M

N0 logN0a2N0

�� 1� C �M

N0 logN0a2N0! 1;

as N0 !1. Using Claims 1 and 2, we obtain the desired result.

Proof of Claim 5: First, we use Claim 4 to �nd that���F�(t)� F[�]� (t)

��� � P

(����� 1pN0

N0Xm=1

�m;S;� � t

����� � ��

)

� P

(����� 1pN0

N0Xm=1

��m;S;� � t

����� � ��

)+ (C �M + '

[1]N0(�; "))a

� 23

N0+ hN0(�);

where

�� =

����� 1pN0

N0Xm=1

p3pjImj

Xi2Im

�i;�

����� :Using the previous arguments,

P

(����� 1pN0

N0Xm=1

��m;S;� � t

����� � ��

)� P fjV�Z� tj � ��g+ hN0(�):

We write the last probability as

PnjV�Z� tj � �� and �� �

p�o

(6.24)

+PnjV�Z� tj � �� and �� >

p�o:

The probability is written as

Pnt��� � V�Z � �� + t and �� �

p�o

� Pnt�

p� � V�Z �

p�+ t

o:

Since V� 2 [�;C �M ] and t � "; there exists a positive function '[2]N0(�; ") that does not depend

on P such that the last probability is bounded by '[2]N0(�; ") and lim�!0 limN0!1 '[2]N0(�; ") =

0.

ECONOMETRIC INFERENCE ON A LARGE GAME 51

The second probability in (6.24) is bounded by (for some c1 > 0)

1

�E

24 1pN0

N0Xm=1

p3pjImj

Xi2Im

�i;�

!235 � �:

We conclude that ���F�(t)� F[�]� (t)

��� � '[2]N0(�; ") + �:

Using the same arguments as previously, we also conclude that with P = 1;���FZ(tjY;X)� F[�]Z (tjY;X)

��� � '[2]N0(�; ") + �:

Thus we obtain the desired result. Q.E.D.

Proof of Theorem 4: It su¢ ces to show that

E

"����� 1pN0

NXi=1

ei;U(a)gj(XI(i))"m(i);b

����� jY;X#and

E

"����� 1pN0

NXi=1

ei;L(a)gj(XI(i))"m(i);b

����� jY;X#

are asymptotically negligible conditional on (Y;X). We only deal with the �rst conditional

expectation. Note that it is bounded by

C

vuut 1

N0

N0Xm=1

Xi2Im

ei;U(a)gj(XI(i))

!2:

Since all the players are asymptotically negligible, we have

maxi2N

ei;U(a)! 0 and maxi2N

ei;L(a)! 0

as N !1. Hence we obtain the desired result. Q.E.D.

52 SONG

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Department of Economics, University of British Columbia, 997 - 1873 East Mall, Van-

couver, BC, V6T 1Z1, Canada

E-mail address: kysong@mail.ubc.ca