INFERENCE FOR LARGE GAMES WITH EXCHANGEABLE

INFERENCE FOR LARGE GAMES WITH EXCHANGEABLE PLAYERS
KONRAD MENZEL†
Abstract. We develop a sampling theory for games with a large number of players that
are exchangeable from the econometrician’s perspective. We show that in the limit, heterogeneity can be separated into an individual and an aggregate component, and establish
conditional laws of large numbers and central limit theorems given the aggregate state of
the game. We then develop estimation procedures that eliminate heterogeneity in the aggregate through conditioning and invariance properties, which allow consistent estimation
and inference based on information from a finite number of games as the number of players grows large. Estimation is based on the marginal distribution of individual observable
characteristics and choices, where we treat the equilibrium selection rule as an unknown
nuisance parameter. We illustrate the applicability of our results for aggregative games.
JEL Classification: C12, C13, C14, C15
Keywords: Incomplete Models, Multiple Equilibria, Exchangeability, Large Games
The analysis of strategic interactions among individuals or firms has always been a central
theme in economics, and there has been a substantial effort to develop econometric tools for
estimation and inference for games. However, existing methods for estimation and inference
rely on the availability of data on a large number of independent realizations of a game
with a given number of players. In this paper, we develop a sampling theory for settings
with strategic interaction among a large number of agents, where players in the finite game
represent exchangeable draws from a common distribution.
Game-theoretic models pose several technical difficulties - strategic interdependence typically makes statistical objects of interest hard to compute and also induces statistical dependence among different agents’ choices. Furthermore economic theory does not make unique
predictions in the presence of multiple equilibria. We show that if agents are exchangeable,
it is possible to obtain conditional limiting results that allow us to address each of these
concerns at least for some relevant settings.
More specifically, exchangeability implies asymptotic conditional independence of players’
type-action profiles given their empirical distribution. This has three key consequences:
• In the many player limit of a game, only the marginal distribution of types and
actions is informative about economic parameters, whereas their joint distribution
does not provide additional identifying power.
Date: February 2012 - this version: August 2012.
†
NYU, Department of Economics, Email: [email protected].
1
2
KONRAD MENZEL
• Under many-player asymptotics we can use limit theory to derive conditional laws
of large numbers and central limit theorems which can be used to derive consistency
results and distribution theory for estimators and other statistics.
• Since the marginal distribution of type-action characters can be estimated consistently from a random subset of players in the game, estimation and inference for
incompletely observed games can be done using the same methods as for a complete
sample, with only minor adjustments or corrections.
Hence, in an asymptotic experiment which holds the number of games fixed but increases
the number of players, in the limit exchangeability restrictions lead to a separation of heterogeneity at the aggregate and individual level, respectively. In this paper we show that
individual heterogeneity can be dealt with using large-sample arguments like laws of large
numbers (LLN) and central limit theorems (CLT). Aggregate heterogeneity can arise even
in the many-player limit of a game e.g. from a high degree of strategic interdependence
between agents’ choices or multiplicity of equilibria.
Since economic theory and empirical evidence do not appear to make strong predictions
regarding selection of equilibria, structural estimation of games typically has to treat the
selection mechanism as an incidental parameter of the observed game. In this paper, we
argue that considering asymptotic approximations to the empirical distribution of typeaction profiles for a large number of players suggests several ways of eliminating this nuisance
parameter.
We argue that there are a number of empirically relevant settings in which the potential nondegeneracy of the distribution of aggregate states does not preclude inference on
structural payoff parameters. Instead, we consider conditional asymptotics which allows
us to exploit invariance properties, and discuss approaches to eliminate nuisance parameters characterizing the aggregate state through conditioning. Furthermore, in many cases
equilibrium distributions for the limiting experiment have a simpler structure than for the
finite-player game, so that the limiting distributions may serve as computationally attractive
approximations to the finite-player game.
Our analysis focuses on discrete aggregate games with complete information. Discrete
social interaction models with a large number of agents have been analyzed by Glaeser,
Sacerdote, and Scheinkman (1996), Brock and Durlauf (2001), Brock and Durlauf (2007),
Topa (2001), Liggett (2005) and Nakajima (2007) and others. However, this work assumes
that player types are private information, or that decisions are made sequentially by agents
that are not forward-looking.
Our paper derives a new result on convergence of large anonymous economies where agents’
types may be common knowledge among all players. Hildenbrand (1971) showed convergence
of competitive markets with random preferences to equilibrium points with respect to the
GAMES WITH EXCHANGEABLE PLAYERS
3
limit distribution. Brock and Durlauf (2001) analyze Bayes perfect Nash equilibria in a
discrete choice model with social interactions, where it is assumed that payoff shocks that are
unobserved by the econometrician are also private information among players. In contrast,
we explicitly allow for statistical dependence from strategic interaction when players have
knowledge about their opponents’ types.
Previous work in the theoretical literature on large games has focussed on purification
and approximation properties of distributional games. Kalai (2004) shows that in large
populations where types are private information, Bayes perfect Nash equilibrium is ex-post
Nash in ε best responses. For dynamic games, Weintraub, Benkard, and van Roy (2008)
show that oblivious equilibrium approximates a Markov-perfect dynamic equilibrium as the
number of players grows large. Their work aims at exploiting the computational advantages
of working with the large-player limit rather than the finite-player version of the game. In
contrast, our focus is on robustness with respect to equilibrium selection, which requires
that any finite-player equilibrium can be approximated by an appropriately chosen set of
equilibria in the limiting game. In this sense, our analysis complements existing theoretical
results on convergence of finite economies to continuous limiting games. However, it is
important to point out that the limiting game will typically remain easier to solve, so that
our approach is also in part motivated by computational considerations.
Limiting results for large economies of interacting agents have been derived under the
assumption that (a) agents are backward-looking and do not anticipate other players’ behavior (see e.g. Liggett (2005)), or (b) conditional on the common knowledge about types
among agents, assuming that other preference shocks are private information (see Brock and
Durlauf (2001)). Clearly, either assumption has its obvious merits especially for games with
large numbers of players, however we show that it is possible to obtain comparable results
for the complete information case.
Limits of distributions of exchangeable network models have been considered in the literature on random graphs (see e.g. Diaconis and Janson (2008) and Bickel, Chen, and Levina
(2011)), but their methods are not directly applicable to arrays that cannot be embedded
in an infinitely exchangeable sequence: The dependence properties of finitely exchangeable
arrays are generally different from those of infinitely exchangeable arrays, and since we are
interested in characterizations of equilibria in structural models of social interactions we
want to take those limits explicitly. While in general almost sure convergence results are
generally difficult to obtain for triangular arrays, we show that in our setup, there are reasonable conditions under which we can obtain the row-wise stability properties needed for
such a result.
Our treatment also differs from Andrews (2005) in that in this paper the conditioning
sigma field arises endogenously from strategic interaction among agents. In particular, we
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KONRAD MENZEL
are interested in structural inference, and hence the conditioning set is modeled explicitly
in terms of economic primitives and discuss conditions for (unconditional) consistency of
estimators of structural parameters.
Conditional convergence results for infinitely exchangeable arrays - e.g. Blum, Chernoff,
Rosenblatt, and Teicher (1958), or Andrews (2005) in econometrics - rely on de Finetti’s
theorem or related ideas. Our main focus is on coupling a triangular array of sequences
that are only finitely exchangeable, and to give the conditioning sigma-field a structural
interpretation. Identification for mixture models in the case of (finite player) discrete games
with multiple equilibria has been considered by Bajari, Hahn, Hong, and Ridder (2011). Our
results on aggregative games show some parallels with Shang and Lee (2011)’s conditional
analysis of the private information game with many players, and our analysis shows how
to adapt their results to the complete information case which does not assume that players
actions are i.i.d. conditional on observables.
In the following, we are going to describe the framework we use to model social interactions,
and section 3 shows how to establish the conditions for convergence from economic primitives
for aggregate games with complete information. Section 4 gives generic asymptotic results
for sample statistics from games with a large number of players, and section 5 concludes.
2. Setup
Sampling Experiment. We consider samples that are obtained from a small number of instances m = 1, . . . , M, possibly only one single realization, of a static game (”market”),
and we let nm denote the number of agents in the mth game. Each player i chooses an
action si ∈ S from a set of pure actions S that is the same for all players and known to
the researcher. There is heterogeneity across agents in that player i receives a public signal
tmi ∈ T which we refer to as player i’s type, where individual types tmi = (x′mi , ε′mi )′ are
i.i.d. draws from a common distribution Hm (t) = Hm (x, ε). Note that the distributions
of types may differ across markets, and in particular we also allow for market-level shocks
or covariates which often play an important role in the identification of interaction effects.
We also allow the dimension of types ti to depend on market size, e.g. for matching or
network formation problems it may be desirable to allow for pair-specific heterogeneity in
the propensity to form a link between any two agents in the mth market.
We do not assume that our data set for a given game is necessarily complete, but we
assume that we observe type-action characters ymi = (smi , x′mi )′ for a random sample (with
or without replacement) of Nm players in the markets m = 1, . . . , M, where the components
εmi of player i’s type are not observed by the econometrician but may be known to other
players. We also denote the action profile for the market by sm = (sm1 , . . . , smnm ) ∈ S nm ,
′
)′ .
and the type-action profile by ym = (y1′ , . . . , ym
n
GAMES WITH EXCHANGEABLE PLAYERS
5
Preferences and Equilibrium. For the purposes of this paper, we restrict our attention to
the case in which agents choose among finitely many discrete actions, S := s(1) , . . . , s(p)
so that S nm contains pnm different action profiles. For ease of exposition, we continue the
exposition in terms of a given realization of the market m, and we omit the first subscript
m in the following discussion whenever this does not lead to any ambiguities.
Preferences are given by utility functions un : S n × T → R, uni (s) := un (s, ti ; θ) that
depend on an unknown parameter θ. Since S is finite, the payoff function for player i can
n ′
also be represented by a vector ũni = ũn (ti ; θ) := uni (s(1) ), . . . , uni (s(p ) ) , where pn is the
dimension of the pure action space. Denote t = (t1 , . . . , tn ). We can then stack the vector of
payoff functions as u = u(t; θ) := (ũn (t1 ; θ)′ , . . . , ũn (tn ; θ)′ )′ and denote the payoff space for
n
the game by U ⊂ Rnp . We also let Σ denote the simplex of probability distributions over
Sn.
Most econometric problem considered in this paper concern estimation and inference regarding θ or a proper subvector of the parameter assuming that chosen actions si , i = 1, . . . , n
constitute an equilibrium according to a given solution concept for the market. More specifically, a solution concept Σ∗ is a set-valued mapping of payoff profiles to distributions of
strategy profiles, Σ∗ : U ⇒ Σ where Σ∗ (u) ⊆ Σ is usually defined in terms of inequality
conditions on agents’ payoffs which ensure that agents’ choices are strategically consistent
with their opponents’ play.
Example 2.1. (Binary Choice with Social Spillovers) Suppose there are n players,
each of whom can choose from two actions, si ∈ {0, 1}, and
payoffsare common
o
n players’
P
1
knowledge among all players and are given by ui (s, ti ; θ) = µ xi , n j6=i sj , θ + εi si for
i = 1, . . . , n, where µ(·) is a known function and θ ∈ Rk is an unknown parameter and
iid
εi ∼ N(0, 1).
As a solution concept for this game we assume that players actions are realizations of strategies that constitute a Nash equilibrium given players’ payoffs.
There are several interpretations for this very stylized game, e.g. actions could be firms’
market entry decisions with firm-specific costs and Cournot competition on product market,
or individual decisions in the presence of social contagion, stigma or peer effects. A private
information version of this problem has been analyzed by Brock and Durlauf (2001) who
analyze Bayes perfect Nash equilibria, where it is assumed that payoff shocks that are unobserved by the econometrician are also private information among players. Also, Nakajima
(2007) estimates a model of peer effects in youth smoking behavior based on the stationary
distribution of individual choices with adaptive behavior in which myopic agents make their
choices based on past play.
We now discuss how to relate a solution concept of a game to a set of distributions over
players actions. First, consider a discrete game with an action set S = {s(1) , . . . , s(p) }. Then
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KONRAD MENZEL
in the complete information game, a mixed strategy for player i is a measurable map
(
Tn → Σ
′
σi :
t
7→ σi (t) := σi (t, s(1) ), . . . , σi (t, s(p) )
where the dependence on t may be restricted under different assumptions on players’ information. Note that the individual subscripts indicate that strategies do not have to be
the same for all players of the same type, and we will maintain that level of generality
throughout the paper. Using standard notation, we let σ−i = (σ1 , . . . , σi−1 , σi+1 , . . . , σn ) denote the profile of strategies played by all players except i. With a slight abuse of notation,
we will denote player i’s expected payoff from choosing action si against the profile σ−i by
P
Q
∗
ui (si , σ−i ) = s−i ∈S n−1 ui (si , s−i ) j6=i σj (sj ). Then σ ∗ = σm
is a Nash equilibrium of the
∗
∗
′
∗
′
game if σi (s) > 0 implies that ui (s, σ−i ) ≥ ui (s , σ−i ) for all s ∈ S, and we denote the set of
the equilibrium strategy profiles by Σ∗m (t) = Σ∗m (um (t; θ)).
We can now parameterize the set of distributions over player types and actions that are
compatible with the equilibrium set with an equilibrium selection rule λ. A (possibly stochastic) equilibrium selection rule λ is a mapping from type-profiles to distributions over the
equilibrium strategy profiles σ ∗ ∈ Σ∗m (um (t; θ)). The parameter space for λm consists of the
mappings of types to the set of mixed Nash equilibria of the game
Λm := {λm : Θ × T nm → Σm |λm (σm ; θ, t) > 0 only if σm ∈ Σ∗m (um (t; θ))}
From the econometrician’s perspective we will treat the equilibrium selection rule λm in
the mth market as a random variable in a cross-section of one or several observed markets.
Some of our asymptotic arguments in the next sections require measurability of equilibrium
outcomes along a particular filtration, so that the equilibrium selection mechanism has to be
defined on a common probability space along sequences of games. However, in general this
does not necessarily lead to a restriction of outcome distributions for the game at a given
number of players.
Note that under these assumptions, observed outcomes are statistically independent across
markets, and the likelihood of ym = (y1 , . . . , ynm ) can be written as


Z Y
n  X

fm (y|θ, λ) =
λ(σ ∗ )σi∗ (t, si )Hm (dt)

 ∗ ∗
Tn
i=1
σ ∈Σ (t)
where Hm (t) is the marginal distribution of ti . The resulting family of distributions of y
is indexed by two parameters, the parameter of interest θ ∈ Θ which will be taken to be a
k-dimensional vector, and we also parameterize the model incompleteness with λ ∈ Λ(θ) a
stochastic equilibrium selection rule. Hence for any fixed value of λ we have a fully parametric
GAMES WITH EXCHANGEABLE PLAYERS
7
likelihood, i.e. there is a set of distributions that is indexed with (θ, λ),
y ∼ fY1 ,...,Ynm (y1 , . . . , ynm |θ0 , λm )
θ0 ∈ Θ, λm ∈ Λm
(2.1)
In particular, we assume that the observed data have been generated by one particular
equilibrium (possibly in mixed strategies), where the equilibrium selection rule is not known
to the econometrician. As we increase the number of players, the parameter space for the
equilibrium selection rule, Λ = Λn grows very fast in dimension, so that we can’t hope to
estimate λ0 consistently.
Exchangeability. One substantive simplification arises if we assume that agents are exchangeable: recall that a random sequence Z1 , . . . , Zn with joint distribution f (z1 , . . . , zn ) is exchangeable if for any permutation π ∈ Π(1, 2, . . . , n), f (z1 , . . . , zn ) = f (zπ(1) , . . . , zπ(n) ). We
will propose restrictions on the primitives of the game that yield distributions of type-action
characters y1 , . . . , yn that are exchangeable across agents. This restriction reflects a particular
kind of symmetry among different components of the observable data from the econometrician’s point of view. In particular, the identity of individual agents in a game may often be
unknown or irrelevant, so that the parametric family f (y|θ, λ), θ ∈ Θ and λ ∈ Λm , should
be invariant under permutations of, or within subsets of, the set of agents for each instance
of the game.1
Many previous papers in the literature on estimation of games have assumed that the
observed sample consists of i.i.d. draws of markets (see e.g. Ciliberto and Tamer (2009),
Galichon and Henry (2011) and Beresteanu, Molchanov, and Molinari (2009)), which implies exchangeability of observed instances of the game. If the econometric analysis doesn’t
distinguish between individuals players’ identities, then the likelihood is invariant with respect to permutations of agents within a given market. For example in the symmetric entry
game considered by Bresnahan and Reiss (1991), for every payoff profile u the number of
entrants is the same for all pure-strategy equilibria in Σ∗ (u). Since their analysis includes
only market-level but no firm-specific characteristics, the empirical distribution of entry decisions is therefore uniquely defined if only pure equilibria are considered, and inference does
not depend on the equilibrium selection mechanism.
However, for the our purposes the most important implication of exchangeability is that for
n large, the likelihood depends on the equilibrium selection rule λ0 only through the resulting
P
marginal distribution of type-action characters yi = (si , xi ), fY∗ (y|θ, λ) := n1 ni=1 fYi (yi |θ, λ) =
f (y1 |θ, λ), and we are going to develop a limit theory for relevant features of that distribution
as the number of players grows to infinity. For games with a small number of players, Menzel
1Note
that if the game has a spatial structure where interactions depend on some notion of distance, the
assumption of exchangeability would not be appropriate if the analysis is conditional on agents’ location.
However unconditional procedures can allow for spatial interaction if agent locations are endogenous or can
be modeled as part of their type.
8
KONRAD MENZEL
(2011) proposes inference based on the invariant likelihood with respect to permutations of
the set of players. However, the following classical result on sequences of finitely exchangeable random arrays suggests that a substantive simplification arises when the number of
players is large:
Theorem 2.1. (Theorem 3.1 in Kallenberg (2005)) Let Yn be an exchangeable sequence of
length n with empirical distribution Fn (y|θ, λn). If Fn (y|θ, λn) converges weakly to a (possibly
d
random) limit F∞ (y|θ, λ∞), then Yn → Y , an infinite exchangeable sequence of i.i.d. draws
from F∞ (y|θ, λ∞ ).
This results states that in the limit, agents’ actions can be statistically dependent only
through variation in the (stochastic) limiting distribution. Therefore exchangeability is a
useful notion for describing outcomes of social interactions because it allows us to separate
cross-sectional heterogeneity from aggregate uncertainty resulting from strategic interdependence or multiplicity of equilibria.
This insight also has implications for identification arguments: Some approaches to identification rely on joint variation in observed or unobserved heterogeneity across players (see
e.g. Tamer (2003) or Graham (2008)). However asymptotic conditional independence implies
that with exchangeability, equilibrium outcomes may become uninformative about underlying preferences as markets grow ”thick,” and in the limit any identifying information on
the parameter θ is contained in the marginal distribution of y1 .2 In particular arguments
that are based on “identification at infinity” with respect to all players except for one will
typically suffer from a curse of dimensionality with respect to the number of players and
break down in the limit.
While in general the joint distribution of ym contains more information about the parameter θ for any finite-player game than the empirical distribution of individual agents’
action-type profiles ymi , Theorem 2.1 suggests that under our exchangeability assumptions
this difference is negligible when the number of agents in the market is large.
Arguments based on exchangeability are also useful for the case in which only a random
sample of players (in general without replacement) from a large game is available to the
econometrician. Manski (1993) points out that the relationship between observations in a
sample of that type is in general different from that between players in the population from
which that sample was drawn. However since for large games the joint distribution can
be approximated by i.i.d. draws from the empirical distribution of player-level type-action
2For
example, Banerjee, Duflo, Ghatak, and Lafortune (2009) find that families placing marriage ads in an
Indian newspaper report a preference for marrying a spouse within their own caste over a potential spouse
from a different caste but with substantially higher income, whereas in the observed market outcome most
families find a suitable spouse with comparable income within their own caste.
GAMES WITH EXCHANGEABLE PLAYERS
9
characters, a sufficiently large subsample of players can be used to estimate the empirical distribution arising from the large-scale interaction model consistently even if it only represents
a small share of the agents interacting at the population level.
Empirical Distribution of Type-Action Characters. One general difficulty with structural
inference in large games is that the parameter space Λn of equilibrium selection rules which are functions of all the nm players’ types - grows very fast in dimension as we let the
number of players increase.3By Theorem 2.1, instead of considering the family of resulting
joint distributions of the type-action profile in the mth market,
f (ym1 , . . . , ymnm |θ, λm )
θ ∈ Θ, λm ∈ Λnm
in the many-player limit we can restrict our attention to the empirical distribution of the
exchangeable type-action characters which can be reparameterized as
∗
fm
(ym1 |θ, γm)
θ ∈ Θ, γm ∈ Γm
The parameter space Γm consists of mappings from Θ to the set of equilibrium points of
a properly defined limiting game. Most importantly, γ will not depend on the particular
realization of t. This parameter set will in general differ across m = 1, . . . , M if we allow
type distributions Hm (x, ε) to vary across markets, but does not depend on market size nm .
We are going to illustrate that in many cases this constitutes a great simplification relative
to the finite-player version of the game, especially when the set of equilibria in the limiting
game is finite.
For example in the symmetric entry game considered by Bresnahan and Reiss (1991),
for every payoff profile u the number of entrants is the same for each of the pure-strategy
equilibria in Σ∗ (u), so that under the assumption that entrants play a Nash equilibrium in
pure strategies, the empirical distribution for si is uniquely determined. Furthermore in their
analysis, there are market-level but no firm-specific characteristics and therefore inference
does not depend on any assumptions regarding equilibrium selection.
Example 2.2. Aggregate Games / Anonymous Interactions: In a finite action game, suppose that each player’s payoff only depends only on the fraction of players choosing each
action s ∈ {s(1) , . . . , s(p) }. Also let λm ∈ Λm denote the distributions over equilibrium strat∗
egy profiles σm
∈ Σ∗m . We then show in Section 3 that under regularity conditions, the
∗
empirical distribution fm
(Smi , Xmi |θ, λn ) of player’s actions Si and observable types Xi con∗
verges to a nonrandom limit fm
(S1 , X1 |θ, γ) as nm → ∞, where γ ∈ Γ maps θ into the
(finite) set of equilibrium points of the limiting average best-response correspondence.
3The
generic maximal number of pure Nash equilibria in an n-player discrete game with p strategies is pn−1
(see McLennan (1997)).
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KONRAD MENZEL
We will analyze this problem in more detail in the next section. We will show that under
regularity conditions, the limiting game has only a finite number of equilibrium distributions.
For a given value of θ, the resulting parameter space Γm for the empirical distributions spans a
finite-dimensional probability simplex whose dimension depends on the number of equilibria
in the limiting game corresponding to the marginal type distribution Hm (x1 , ε1 ) and θ.
Robust inference: In general, structural inference requires that we make statements about
the parameter θ that do not depend on the nuisance parameter γ which indexes the possible
limiting distributions for Ym1 = (Sm1 , Xm1 ). By conditional independence across agents
in the limit, it is sufficient that we consider statistics that depend only on the empirical
distribution of Ym1 , . . . , Ymnm . In the following we consider statistics at the market level that
are of the form
nm
1 X
m(Ynm i ; θ)
m̂nm (θ) :=
nm i=1
where m : Y × Θ → Rq is a vector-valued function of an agent’s type-action character.
Considering the limiting experiment, we also denote the expectation with respect to the
∗
distribution fm
(y|θ, γ) by
Z
∗
m0 (θ, γ) = Eγ [m(Y1 ; θ)] := m(y; θ)fm
(y|θ, γ)dy
We consider empirical restrictions of the following form: we will assume that for every market
m = 1, . . . , M, there exists a known set M0,m ⊂ Rq such that
m0 (θ0 , γ) ∈ M0,m
for all γ ∈ Γm
In other words we can characterize the expectation of the statistic under any possible limiting
distributions for the mth market.
Example 2.3. (Aggregate Games) We will show that under the assumptions in section 4,
the set of limiting distributions can be characterized as f∞ (y|θ0 , γ) where γ ∈ Γm the (finite)
set of values G∗ solving a fixed point problem of the form Φ0 (G∗ ) = G∗ . In particular, we
∂
log f∞ (y|θ, Ĝ∗n ) and M0,m = {0},
will show that we can use moment conditions m(yi ; θ) := ∂θ
where Ĝ∗n is a consistent estimator for G∗ .
Example 2.4. (Moment Inequalities) If strategic choices are a result of economic optimization, then we can often derive moment equalities or inequalities on payoffs or profit
functions, which under additional informational assumptions gives restrictions of the form
E[m(Yin , θ0 )] ≥ 0
in any equilibrium of the game (see e.g. Bajari, Benkard, and Levin (2007) or Pakes, Porter,
Ho, and Ishii (2006)), where for some s′ ∈ S m(Yin , θ0 ) := E[u(Sin , Xi ; θ) − u(s′ , Xi , θ)|Xi ] ⊗
GAMES WITH EXCHANGEABLE PLAYERS
11
ψ(Xi ), and ψ(x) is a q-dimensional vector of nonnegative functions of x. Then the empirical
restriction for the limiting game is of the form
m0 (θ0 , γ) ∈ M0,m := Rq+
the positive orthant in Rq .
Clearly, these restrictions only have identifying power if the “population” moments m0 (θ, γ)
vary with θ. Furthermore, the restrictions are uninformative against an alternative θ1 if the
data generating process puts zero probability on the event {m(θ1 , γ) ∈
/ M0,m } for a given
distribution for γ. In particular, identification will be more difficult if the set M0,m is very
rich due to a large number of equilibria. However we will show that in many large discrete
games, the parameter set Γm can often taken to be finite.
3. Aggregate Games
A class of normal form games with large numbers of players for which limiting results
can be obtained are aggregate games in which players interact strategically only through
a finite-dimensional aggregate state of the economy. In this section we consider a static,
full information discrete game with players i = 1, . . . , n where players’ types ti = (xi , εi )
consist of characteristics xi ∈ X that are observed by the econometrician, and unobserved
preference shocks εi . Player types ti are i.i.d. draws from the population distribution H(t)
and are common knowledge among all agents.
Each player selects an action from a finite set S = (s(1) , . . . , s(p) ), and we denote the
empirical distribution of actions in the market with Ĝn (σ) = (Ĝn (s(1) ; σ), . . . , Ĝn (s(p) , σ))′ ,
where
n
1X
σj (s), s ∈ S
Ĝn (s; σ) :=
n j=1
We also let ∆S denote the space of distributions over S represented by the (p−1) dimensional
probability simplex endowed with the Euclidean distance d(·, ·).
Interactions among players are anonymous in that we assume that payoffs are of the form
ui (si , σ−i , ti ) = u ti , si , Ĝn ((si , σ−i )); θ
where, with some abuse of notation, we let (si , σ−i ) denote the mixed strategy profile
(σ1 , . . . , σi−1 , δsi , σi+1 , . . . , σn ). In particular, we assume that each agent’s payoffs only depend on her own type, but not that of any other players. Note that with minor changes
to our notation, we can easily extend this framework to allow for preferences to depend on
the joint distribution of actions and indicators for a finite partition of the type space. For
example, we may want to allow for the respective number of male and female smokers to
affect a male or female player’s payoff to a different degree.
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KONRAD MENZEL
For finite type spaces, this model has been considered by Kalai (2004) and Al-Najjar
(2006) and includes the economies analyzed by Acemoglu and Jensen (2009) and Martimort
and Stole (2011). This setup differs from the binary choice model with social interactions
in Brock and Durlauf (2001) in that their analysis assumes that players’ types are private
information, however by the laws of large numbers derived in Kalai (2004) and this paper this
model becomes equivalent to a setting with common knowledge of types as the number of
players grows large.4 Brock and Durlauf (2007) establish point identification for the privateinformation binary choice game under conditional quantile restrictions on the distribution
of εi given xi .
In the following, we will denote
n
1
G−
σ
n−1
n−1
for σ, G ∈ ∆S. Given payoff functions, we also define the set of best responses to an aggregate
profile Gs as
G̃n,−1(G, σ) :=
ψ0 (G, t) := conv {δs |s ∈ S : ∀s′ ∈ S : u(s, G, t) ≥ u(s′, G, t) }
where δs is the sth unit vector in Rp−1. Since in the following we have to account for
multiplicity of best responses, we have to formulate the problem in terms of random sets
and correspondences, where for ease of exposition, we relegate the definition of the relevant
technical concepts to footnotes.
In order to solve for equilibrium distributions of actions Ĝ∗n (s), define the correspondence
of implicit best responses supporting an aggregate state G as5
n
o
ψn (G, t) := σ ∈ ∆S σ(s) > 0 only if s ∈ ψ0 (G̃n,−1(G, σ), t), and G̃n,−1(G, σ) ∈ ∆S
Notice that the first condition on σ requires that agents randomize only between best ren
1
sponses to the empirical distribution n−1
G− n−1
σ. If preferences are locally strictly monotone
in G, then ψn (G, t) will typically contain a finite (and usually odd) number of points for any
finite n.
We denote the projection of ψ(G, t) onto the sth dimension of the strategy space by
ψn (s; G, t) := {σ(s) : σ ∈ ψn (G, t)}
4Horst
and Scheinkman (2006) consider both global and local interaction among a fixed (but possibly infinite) set of agents, whereas Horst and Scheinkman (2009) extend that analysis to limits of finite-player
economies. Their results only consider invariant equilibria and assume a Lipschitz condition on individual
best response functions, which implies uniqueness of the equilibrium and excludes discrete action games with
full information.
5Recall that a correspondence from X to Y, denoted by Φ : X ⇒ Y is a mapping from elements x ∈ X to
subsets of Y
GAMES WITH EXCHANGEABLE PLAYERS
13
Also define the set of types for which s is a best response to G by
A(s, G) := {t ∈ T : u(s, G, t) ≥ u(s′ , G, t) for all s′ ∈ S}
Assumption 3.1. (i) The payoff functions are of the form u(s, G, t), where u is uniformly
continuous in G and t ∈ T for all s ∈ S, (ii) the collection
A := {A(s, G) : s ∈ S, G ∈ ∆S}
is a VC class of sets.
There are a number of empirically relevant examples under which the second part of
Assumption 3.1 is satisfied: Clearly, A is a VC class if the number of types t is finite as in
Kalai (2004). Also if payoffs have a linear index structure in ti for all G, i.e.
u(s, G, t) = v(t′ β(G), G)
then the sets A(s, G) are intersections of the type space T with linear half-spaces in Rdim(t) ,
and therefore a VC class with index less than or equal to dim(t) + 1. This example is a
generalization of the framework considered by Brock and Durlauf (2001).
For the following, notice that the sum of two correspondences Φ1 : X ⇒ Y and Φ2 : X ⇒ Y
is defined as the Minkowski sum of their images,
Φ1 (x) ⊕ Φ2 (x) := {y1 + y2 |y1 ∈ Φ1 (x) and y2 ∈ Φ2 (x) }
If all summands are singletons, Minkowski addition reduces to the standard notion of addition. We now define the aggregate response mapping
n
1M
Ψ̂n (G) :=
ψn (G, ti )
n i=1
and the population mean of the best-response correspondence
Ψ0 (G) := E [ψ0 (G; T )]
where E[·] is taken to denote the (Aumann) selection expectation of the random set ψ̃0 (G, T ) ⊂
∆S and T ∼ H0 .6 Figure 3 illustrates the construction of Ψ̂n (G) from the individual best
responses ψn (ti ; G) for a given realization of types in the binary action case.
The following Lemma characterizes the set of Nash equilibria of the n-player game as the
solutions to the inclusion G∗ ∈ Ψ̂n (G∗ ):
6The
Aumann selection expectation of a closed random set X : Ω → 2X is defined as
E[X] := closure{E[ξ] : ξ ∈ Sel(X)}
where Sel(X) denotes the set of measurable selections ξ(ω) ∈ X(ω) such that Ekξk1 < ∞. See Molchanov
(2005) for a full exposition.
KONRAD MENZEL
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
b5
Ψ
ψ̃(G, t, 5)
14
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0
0.2
0.4
0.6
G
0.8
1
0
0.2
0.4
0.6
0.8
1
G
Figure 1. The correspondence ψ5 (ti ; G) for five draws of ti in a five-player
binary action game (left), and the corresponding aggregate response mapping
Ψ̂5 (G) (right).
Lemma 3.1. Suppose Assumption 3.1 (i) holds. Then there exists a Nash equilibrium with
empirical distribution G∗ if and only if G∗ ∈ Ψ̂n (G∗ ), i.e. G∗ is a fixed point of the correspondence Ψ̂n . In particular, there exists a solution to the inclusion G∗ ∈ Ψ̂n (G∗ ) for
each n = 1, 2, . . . with probability 1, and any distribution G∗ that is supported by a Nash
equilibrium satisfies G∗ ∈ convΨ̂n (G∗ ), the convex hull of Ψ̂n (G∗ ).
Proof: Suppose that G∗ ∈
/ Ψ̂n (G∗ ). Then by definition of Ψ̂n , there exists at least
one player i such that σi ∈
/ ψn (G∗ , ti ), so that σi is not a best response to any profile σ−i
P
satisfying n1 nj=1 σj = G∗ . Conversely, if G∗ ∈ Ψ̂n (G∗ ), we can find σ1 , . . . , σn such that
P
σi ∈ ψn (G∗ , ti ) for all i = 1, . . . , n and n1 ni=1 σi = G∗ , so that G∗ is indeed generated by
a Nash equilibrium. Existence of an equilibrium distribution G∗ for finite n follows from
an existence theorem for Nash equilibria in finite games, see e.g. Proposition 8.D.2 in MasColell, Whinston, and Green (1995) Note that for the last part of Lemma 3.1 the converse doesn’t hold: there may be solutions
to G ∈ convΨ̂n (G) that do not correspond
to Nash equilibria.
Below we show that the set
n
o
∗
∗
∗
∗
of solutions to the inclusion Ĝn := G : G ∈ convΨ̂n (G ) approaches the set of solutions
to G ∈ Ψ̂0 (G), which implies that the set of Nash equilibria in the finite game approximates
a subset of G0∗ = {G∗ : G∗ ∈ Ψ0 (G∗ )}, but in general some solutions in G0∗ need not be
approximable by sequences of (strict) Nash equilibria.7
7One
approach in the literature has been to broaden the equilibrium concept to equilibria in ε-best responses,
as in Kalai (2004). In contrast, our Assumption 3.2 below imposes that all equilibria in G0∗ are regular points
of the limiting mapping, all of which are then shown to be approximable by (strict) Nash equilibria in the
proof of Theorem 3.1.
GAMES WITH EXCHANGEABLE PLAYERS
15
We will now give a condition that ensures that solutions to the limiting equilibrium mapping Ψ0 are locally unique, implying that the number of distinct equilibrium distributions is
finite. We also want to allow for the limiting equilibrium mapping to be set-valued, which
will in general be the case if the distribution of types is not continuous. For this we have
to define a graphical derivative of a correspondence:8 The contingent derivative of the correspondence Φ at (x′0 , y0 )′ ∈ gph Φ is a set-valued mapping DΦ(x0 , y0 ) : X ⇒ Y such that
for any u ∈ X
Φ(x0 + hu′ ) − y
v ∈ DF (x, y)(u) ⇔ lim inf
d v,
h↓0,u′ →u
h
where d(a, B) is taken to be the distance of a point a to a set B.9 Note that if the correspondence Φ is single-valued and differentiable, the contingent derivative is also single-valued
and equal to the derivative of the function Φ(x). The contingent derivative of Φ is surjective at x0 if the range of DΦ(x0 , y0 ) is equal to Y. In the following, let the mapping
Φ0 (G) := Ψ0 (G) − G.
Assumption 3.2. (Regular Economy): For every θ ∈ Θ and population distribution
H0 ∈ P, one of the following holds:
(i) Ψ0 (G) is single-valued for all G ∈ ∆S, and at every distribution G∗ solving G∗ ∈
Ψ0 (G∗ ), ∇G Φ0 (G∗ ) is defined, and ∇G Φ0 (G∗ ) has rank equal to S − 1, or
(ii) The contingent derivative of Φ0 (G) is lower semi-continuous and surjective for all G,
and satisfies
0 ∈ gph DΦ0 (G∗ , G∗ )(G) if and only if G = G∗
at every distribution G∗ solving G∗ ∈ Ψ0 (G∗ ).
Note that (i) is a special case of (ii) for which the conditions for local separation of equilibria are easier to interpret. In analogy with classical genericity analysis in general equilibrium
theory, it is possible to derive primitive conditions for this assumption in terms of the population distribution of types based on the restrictions on payoff functions in Assumption 3.1.
Furthermore, with payoffs depending on a parameter θ, this assumption imposes restrictions
on the parameter space Θ. Although violations of these generalized rank conditions are typically “non-generic” in the sense that they correspond to subsets of Θ of (Lebesgue) measure
zero, they will generally lead to unstable behavior of the economy for parameter values in
the neighborhood of those singular points. However, conditions of this type are standard in
econometric analysis of equilibrium models.10
8Recall
that the graph of a correspondence Φ : X ⇒ Y is the set gph Φ := {(x, y) ∈ X × Y : y ∈ Φ(x)}.
Definition 5.1.1 and Proposition 5.1.4 in Aubin and Frankowska (1990)
10See e.g. Assumption 2.1 in Hausman (1983) for the linear simultaneous equations model.
9See
16
KONRAD MENZEL
Figure 2. Schematic illustration of Assumption 3.2. The bottom three panels are examples of a failure of the transversality condition (left) and/or the
surjectivity condition (all three panels).
In order to understand regularity for the case of a limiting correspondence Ψ0 , consider
the case S = {0, 1}, so that G ∈ [0, 1]. If there is a fixed point G∗ such that Ψ0 (G∗ ) is
a convex set with nonempty interior, then condition (ii) of Assumption 3.2 is violated if
and only if there exists sequences Gn 6= G∗ with limn Gn = G∗ and Ψn ∈ Ψ0 (Gn ) such
Ψn −G∗
that limn |G
∗ = 0. Cases in which the regularity condition fails include if for all G in a
n −G |
one-sided neighborhood of G∗ , Ψ0 (G) is set valued and contains G∗ , or if in that one-sided
neighborhood Ψ0 (G) is single valued with limGn →G∗ Ψ0 (Gn ) = Ψ0 (G∗ ), and the one-sided
derivative of Ψ0 (G∗ ) = Ip−1 , the (p − 1) dimensional identity matrix. See also figure 3 for
graphical examples.
If the distribution of types Ti is continuous and payoff functions are strictly monotone
in each argument almost everywhere, then at every value of G players are indifferent between multiple actions with probability zero. Hence under Assumption 3.1, the limiting
correspondence Ψ0 is typically set-valued only if the type distributions has atoms.
Lemma 3.2. Under Assumption 3.2, the number of solutions of the inclusion G ∈ Ψ0 (G) is
finite.
Proof: Since the game is finite with |S| = p, the space of distributions ∆S is the (p − 1)
dimensional probability simplex, and therefore compact. Under Assumption 3.2, part (i),
GAMES WITH EXCHANGEABLE PLAYERS
17
each equilibrium G∗ is unique within an open neighborhood around G∗ by the inverse function theorem. If part (ii) of Assumption 3.2 holds, then each equilibrium G∗ is locally unique
by Proposition 5.4.8 in Aubin and Frankowska (1990). We can use these open neighborhoods
of the equilibrium points to cover ∆S, and by compactness, there exists a finite subcover
N1 , . . . , NJ , say. By construction, each set Nj , j = 1, . . . , J contains at most one equilibrium
point, so that the number of equilibria is at most J, and therefore finite This proof is very similar to the classical arguments for finiteness of the number of equilibria in regular market economies (e.g. Proposition 17.D.1 in Mas-Colell, Whinston, and
Green (1995)) or finite-player games (Theorem 1 in Harsanyi (1973)). We can now state our
main result on stochastic convergence of the equilibrium values of the aggregate Ĝn .
Proposition 3.1. Suppose Assumptions 3.1 and 3.2 hold, and that Ĝn is a sequence of
empirical distributions solving Ĝn ∈ Ψ̂n (Ĝn ). Then for G0∗ := {G∗ ∈ ∆S : G∗ ∈ Ψ0 (G∗ )}
a.s.
and as n → ∞, we have that (a) d(Ĝn , G0∗ ) → 0, and (b) with probability approaching 1,
for every G∗0 ∈ G0∗ and every neighborhood B(G∗0 ) of G∗0 we can find Gn ∈ B(G∗0 ) such that
Gn ∈ Ψ̂n (Gn ).
See the appendix for a proof for this proposition. Given the conclusion of 3.2, part (a) of
the conclusion implies that although the number of distinct equilibria may grow very fast
as the number of players increase, the implied distributions of actions become concentrated
around a finite set G0∗ . Taken together, parts (a) and (b) show that the set G0∗ of equilibrium
values for the population correspondence Φ0 is equal to the set of limiting points of the set
of equilibria in the n-player games. Note that the surjectivity condition in Assumption 3.2
is crucial in attaining the stronger conclusion (b).
Note that we can interpret the limiting game as the private information version of the
aggregate game, where players only know the distribution of opponents’ types, but not their
realizations. It is straightforward to check that the solutions of the fixed-point problem
G ∈ Φ0 (G) correspond to the Bayes perfect Nash equilibria (BPNE). Since by definition,
individual actions arising from a BPNE only depend on a player’s own type, from the individual agent’s perspective, the informational requirements for optimal play are much lower
in the limiting game than in the complete information version with finitely many players.
Grieco (2012) investigated identification of discrete games under different informational assumptions. Our results, together with those of Kalai (2004) suggest that those differences
appear to be less important in aggregate games with a large number of agents.
Some of the asymptotic arguments in the next section require that the joint distribution
of best responses varies continuously in the aggregate G or that the aggregate best response
correspondence Ψ0 (G) is single-valued. More specifically, we introduce the following set of
18
KONRAD MENZEL
additional assumptions on preferences and types for the results requiring a minimal degree
of smoothness:
Assumption 3.3. Types can be partitioned as ti = (t′1i , t′2i )′ such that (i) the conditional
distribution of t1i given t2i is tight and continuous with a uniformly bounded continuous p.d.f.,
and (ii) either of the following holds:
(a) (“Unordered Choice”) the subvector t1i ∈ Rp−1 , and for the payoff vector u(t, G) :=
u(s(1) , t, G), . . . , u(s(p) , t, G) , we have that ∇t1 u(t, G) has rank p − 1 for almost all
t∈T.
(b) (“Ordered Choice”) t1i ∈ R, and u(s(l) , t, G) − u(s(k) , t, G) is strictly monotone in
t1 for k 6= l and almost all t ∈ T .
(iii) xi is a subvector of t2i .
Note that most standard parametric random utility models for choice among discrete
alternatives meet either of these assumptions. Typically, specifications include additive
alternative-specific taste shifters that are continuously distributed and have unbounded support conditional on observable characteristics. While part (b) of the assumption does not
require a particular ordering of the alternative choices, classical specifications for ordered
choice models include a scalar source of heterogeneity that satisfies the monotonicity condition on utility differences. The crucial difference between the two sets of assumptions are
that part (a) does not impose any global conditions on the general shape of payoff functions
but requires rich heterogeneity in preferences. On the other hand, the increasing differences
requirement in part (b) allows to restrict the amount of heterogeneity needed to make the
problem sufficiently smooth. Part (iii) requires that the unobservable taste shifters are sufficiently rich such that the smoothness properties stated below also apply to the conditional
model given observable player characteristics, and will only be used for the derivation of
asymptotic results in section 4.
From a technical standpoint, Assumption 3.3 also greatly simplifies the analysis as the
following lemma shows.
Lemma 3.3. Suppose Assumption 3.3 (i) and (ii) hold. Then for almost all types ti , ψ0 (ti ; G)
is single-valued, and the aggregate best-response correspondence Ψ0 (G) is single-valued at all
values of G ∈ ∆S.
The proof for this lemma is in the appendix. This result allows us to work with the
simpler version of the transversality condition in Assumption 3.2 (i) and helps avoid tedious
case distinctions in the characterization of sequences that approach mixed strategy equilibria
in the limiting game when we turn to the construction of a coupling between games with
different numbers of players below.
GAMES WITH EXCHANGEABLE PLAYERS
19
4. Asymptotic Analysis
This section develops conditional laws of large numbers and central limit theorems for
statistic of the form
nm
1 X
m(Ynm i ; θ)
m̂nm (θ) :=
nm i=1
given an appropriately specified tail sigma field. The following analysis only considers one
single market, so that we can drop the m subscript without loss of generality. The results can
then be applied for each m = 1, . . . , M separately, so that information from different markets
can be combined for estimation or inference, where we can also allow for type distributions
and parameters to be market-specific.
In order to ensure that the finitely exchangeable sequences {Yni : 1 ≤ i ≤ n} are defined
on the same probability space for all n ≥ 1, we introduce auxiliary types νi that are not
payoff-relevant but determine choice among equilibria alongside with the payoff-relevant
types ti = (x′i , ε′i )′ . In a given market, types are assumed to be exchangeable random
variables with a fixed marginal distribution for payoff-relevant type:
Condition 4.1. (Types and Preferences) Preferences are given by utility functions
u : S × T → R, ui (s) := u(s, Xi , εi ; θ) that depend on an unknown parameter θ. The n
players’ payoff-relevant types Xi , εi are i.i.d. draws from a distribution H(x, ε), and there
is an augmented type Wi = (Xi′ , ε′i , νi′ ), such that W1 , W2 , . . . form an infinite exchangeable
sequence.
We will show below that we can typically allow νi to be rich enough to generate any mixture
over equilibrium distributions at any sample size n. In particular, the limiting empirical
distribution for the exchangeable sequence W1 , W2 , . . . will in general be non-deterministic.
We now introduce a coupling of the sequence of type-action characters determined in
equilibrium, Yn1, . . . , Ynn , to an infinitely exchangeable sequence Y1 , Y2 , . . . . We also define
a filtration {Fn }n≥1 generated by symmetric events in that random sequence: Given the
random sequence Y1 , Y2 , . . . we say that a random variable Zn is n-symmetric if it can
be written Zn = rn (Y1 , . . . , Yn , . . . ) for some function rn (·) satisfying r(y1, . . . , yn , . . . ) =
r(yπ(1) , . . . , yπ(n) , . . . ) for any permutation π of {1, . . . , n}. We then define Fn as the sigma
algebra defined by all n-symmetric random variables, we clearly have F1 ⊇ · · · ⊇ Fn ⊇
Fn+1 ⊇ · · · ⊇ F∞ . The tail field F∞ is the sigma algebra generated by the marginal
distributions of Y1 .
The following conditions on the equilibrium selection rule are needed to embed the observable n-player game into a sequence defined on a common probability space that allows
for meaningful conditional convergence results.
20
KONRAD MENZEL
Condition 4.2. (Equilibrium Selection)(i) The sequence of equilibrium selection rules
{λn }n≥1 , λn ∈ Λn is of the form λni = λn (wi , w1 , . . . , wn ) and consistent with the solution
concept based on payoff functions ui (s) := u(s, ti ; θ). (ii) There exists a set Γ and a family
of distributions {f (y1|θ, γ) : γ ∈ Γ} such that Y1 |γ̃ ∼ f (y1 |θ, γ̃) for some F∞ -measurable
random variable γ̃ such that γ̃ ∈ Γ almost surely, and (iii) there exists a deterministic null
sequence cn → 0 such that
|E [ m(Yn1 ; θ) − m(Y1 ; θ) |Fn ]| < cn a.s.
for all n.
Part (i) of Condition 4.2 can be understood as a row-wise stability condition for the
triangular array generated by equilibrium play, and its main purpose is to ensure that {Yin :
1 ≤ i ≤ n}n≥1 is adapted to a common filtration. It should also be noted that by restricting
λni (·) to depend only on players’ types, but not their identities, the solution concept of the
game has to treat players symmetrically, but for a given realization of ν1 , . . . , νn the realized
equilibrium need not be symmetric. The functional form of λn (·) is otherwise arbitrary,
so that apart from joint exchangeability of type-action characters, this condition does not
appear to be restrictive regarding the row-wise distributions of the triangular array. In
particular, we demonstrate that for aggregate games, we can generate any mixtures over not
necessarily symmetric equilibria even after conditioning on the payoff-relevant components
of types xi and εi .
Part (ii) requires that the rows of the triangular array are coupled in a way such that the
empirical distribution of type-action profiles converges almost surely to a well-defined limit.
In many applications, existence of this limit is not essential but one may allow the sequence
to oscillate among shrinking neighborhoods of a well-defined set of cumulation points. Hence
this part may in many cases be weakened but for clarity of the exposition we maintain this
slightly stronger condition. Note in particular that the (random) marginal distribution for
Y1 is F∞ -measurable, so that measurability of γ̃ does not impose any additional restrictions.
Below, we will illustrate how to establish Condition 4.2 from the primitive assumptions given
in section 3.
We maintain the following regularity conditions on the function m(·; θ) defining the moment function m̂n (θ) in conjunction with the type distribution H(x, ε) given in Condition
4.1:
Condition 4.3. (Uniform Integrability) (i) The family {m(y; θ) : θ ∈ Θ} is a VC class
of functions, and (ii) E|m((s, X1 ); θ)| and |E [m((s, X); θ) − m((s′ , X); θ)]| are bounded by a
constant for all s, s′ ∈ S and uniformly in θ ∈ Θ.
GAMES WITH EXCHANGEABLE PLAYERS
Denote the limit
m0 (θ, γ) := E[m(Y1 ; θ)|γ̃ = γ] =
Z
21
m(y; θ)f (y|θ, γ)dy
Since the distribution function F is measurable with respect to F∞ , strong convergence of
m̂n (θ) with respect to the filtration {Fn }n≥1 implies almost sure convergence to m0 (θ, γ).
This paper does not explicitly analyze point or partial identification for structural parameters of particular game-theoretic models, but we only assume that the model has empirical
implications of the form that any almost sure limits of the quantities m̂n (θ) belong to a
known, nontrivial subset of Rq .
Condition 4.4. (Empirical Restrictions) Let γ̃ be defined as in Condition 4.2. Then
there exists a set M0 ⊂ Rq such that m0 (θ, γ̃) ∈ M0 almost surely.
This condition primarily serves to characterize the relationship between moments of the
data and the structural parameter under the different tail events associated with different
values of γ ∈ Γ. While it is common practise to discuss identification for finite-sample
distributions and treat the asymptotic analysis for an estimation or inference procedure
as a separate problem, identification properties of game-theoretic models may well change
with the number of players. Since in this paper we describe the behavior of estimators or
tests around their limits, it is more useful to think about identifying power of econometric
restrictions in the many player limit than any finite size.
Condition 4.4 applies to a variety of estimation problems - e.g. the function m(Y ; θ) could
characterize a vector of moment inequalities derived individual optimization behavior, and
this restriction could be written in the form m(F ; θ) ≥ 0 a.s., so that M0 = Rq . We will also
give examples for which M0 consists of a finite set of points.
We are now going to state our first asymptotic result which is a fairly straightforward
adaptation of Birkhoff’s law of large numbers. Since this law of large numbers is uniform
with respect to θ, it can be used to derive consistency results for θ or parameter sets based
on Condition 4.4.
Theorem 4.1. (Conditional LLN) Suppose Conditions 4.1-4.3 hold. Then
n
1X
a.s.
m(Yin ; θ) → E[m(Y1 ; θ)|F∞ ]
n i=1
as n → ∞. Moreover, convergence is uniform in θ.
Proof: Note that the set of equilibria in the n-player game only depends on payoff
profiles which are determined by the values of (X1 , ε1), . . . , (Xn , εn ), so that by Conditions
4.1 and 4.2 (i), observable type-action characters Yni := (Sni , Xi′ )′ are exchangeable for every
n. By Condition 4.2, the column-wise limit of Yni as n → ∞ is well-defined with probability
22
KONRAD MENZEL
one for every i = 1, 2, . . . , so that n-exchangeability of Yn1 , . . . , Ynn implies that the limiting
sequence Y1 , Y2 , . . . is infinitely exchangeable.
Now a standard argument (see e.g. Kingman (1978)) yields that for any n-symmetric
event An ∈ Fn and any j = 1, . . . , n,
#
" n
1X
m(Yi ; θ)1lAn
E[m(Y1 ; θ)1lAn ] = E[m(Yj ; θ)1lAn ] = E
n i=1
P
Since n1 ni=1 m(Yi ; θ)1lAn is Fn -measurable, we get
#
"
n
n
X
1
1X
E[m(Y1 ; θ)|Fn ] = E
m(Yi ; θ) Fn =
m(Yi ; θ)
n i=1
n i=1
Hence by Condition 4.3, the sequence
n
Zn∗
n
1X
1X
m(Yni ; θ) +
[m(Yi ; θ) − m(Yni ; θ)]
:=
n i=1
n i=1
is a reverse martingale adapted to the filtration {Fn }∞
n=1 , so that by the reverse martingale
a.s.
theorem (e.g. Theorem 10.6.1 in Dudley (2002)), Zn∗ → E[m(Y1 ; θ)|F∞].
Since by Condition 4.2 (iii)
n
∗ 1X
m(Yni ; θ) = |E [m(Yi ; θ) − m(Yni ; θ)|Fn ]| < cn
Zn −
n
i=1
almost surely with cn → 0, we also have
n
1X
a.s.
m(Yni ; θ) → E[m(Y1 ; θ)|F∞ ]
n i=1
Since almost sure convergence is joint for any random vector satisfying the conditions of
this Theorem, uniformity in θ follows from Condition 4.3 following the same reasoning as
in the proof of the (almost sure) Glivenko-Cantelli theorem for the i.i.d. case, see van der
Vaart (1998), Theorems 19.1 and 19.4 The result can also be easily adapted to settings in which the econometrician only observes a random subsample of Nn < n players in the game. More specifically, let qi be an
indicator variable that equals one if agent i is included in the sample, and zero otherwise,
P
and qi is independent of Y1 , . . . , Yn . Note that the sample average N1n ni=1 qi m(Yin ; θ) is
i
h
P
Fn -measurable with E Nnn qi m(Yin ; θ) Fn = n1 ni=1 m(Yni ; θ). Then, if Nnn is bounded, the
conditions of Theorem 4.1 hold. so that we obtain the following conclusion:
GAMES WITH EXCHANGEABLE PLAYERS
23
Corollary 4.1. Suppose the assumptions of Theorem 4.1 hold, and we observe a random
sample of Nn ≤ n players’ type-action profiles, where Nnn is bounded by a constant. Then
uniformly in θ as n → ∞.
Nn
1 X
a.s.
m(Yni ; θ) → E[m(Y1 ; θ)|F∞]
Nn i=1
For some applications, it will also be useful to allow for moment functions of the form
m (Yi ; θ, η0 ) that also depend on an auxiliary parameter η0 ∈ H ⊂ Rl , where η0 is replaced
by an estimator η̂n based on the sampling distribution of player types. For example, a
restriction on m∗ (y; θ, η) may be derived from the scores corresponding to players’ best
response to the limiting equilibrium value of η0 := G∗0 . In that case, results should also be
sufficiently general to allow for the parameter η0 to be random even in the limit.
We find that such an extension of the LLN only requires strong consistency of η̂n mild
additional uniformity conditions with respect to η:
∗
Corollary 4.2. Suppose that η̂n → η0 a.s., where η0 is an F∞ -measurable random variable,
and m∗ ((s, Xi ); θ, η) is continuous in η a.e. with respect to the distributions of η0 and Xi .
Also let the assumptions of Theorem 4.1 hold for m(Yi ; θ) ≡ m∗ (Yi ; θ, η0 ), and Assumption
4.3 hold with constants and a VC index that is fixed with respect to η ∈ H. Then
n
1X ∗
a.s.
m (Yni ; θ, η̂n ) → E[m(Y1 ; θ, η0 )|F∞ ]
n i=1
uniformly in θ as n → ∞.
4.1. Construction of Coupling. We now discuss the construction of a coupling of the
type-action characters Yni of the finite-player aggregate game described in section 3 to an
infinitely exchangeable sequence Y1 , Y2, . . . that satisfies the requirements for the conditional
law of large numbers and central limit theorems in this section.
The construction of the coupling requires that we specify the distribution of the auxiliary
states ν1 , ν2 , . . . that are not payoff-relevant but determine which equilibria are chosen, and
a function λn (·) has to satisfy the symmetry restrictions given in Condition 4.2. Finally
we want the coupling to be flexible enough to allow to embed any possible cross-sectional
distribution over Nash equilibria in the n player game into a converging sequence for some
permissible choice of the functions λn .
Proposition 4.1. (Coupling for Aggregate Games) Suppose Assumptions 3.1-3.3 holds.
For the N-player game assume that the observed action profile is generated by a mixture over
the Nash equilibria for the type-profile T1 , . . . , TN with probability one, and let fN∗ (s, x|θ) be
the resulting unconditional distribution over type-action characters.
24
KONRAD MENZEL
Then there exists an i.i.d. sequence ν1 , ν2 , . . . of random variables in Rp , and a sequence
of functions λn : T n → S n such that fN (s, x|θ, λN ) = fN∗ (s, x|θ) for all s ∈ S n , and such
that Condition 4.2 holds.
The proof for this result is in the appendix. Since Assumptions 3.1-3.3 also imply Condition 4.1, we can apply the conditional LLN in Theorem 4.1 and obtain the following corollary:
Corollary 4.3. Suppose the assumptions of Proposition 4.1 and Condition 4.3 hold. Then
the average
n
1X
m(Yni ; θ) → E [m(Y1 ; θ)|F∞ ] a.s.
n i=1
4.2. Asymptotic Distribution. For inference and confidence statements, we next develop
a distribution theory that is conditional on the tail events in F∞ . To this end we are going to
consider mixing convergence in distribution: We say that for a real-valued random sequence
d
Zn and a random variable Z convergence in distribution, Zn → Z, is mixing relative to the
sigma-field F∞ if for all events A ∈ F∞ ,
lim P ({Zn ≤ z}|A) = P (Z ≤ z)
n
at all continuity points z of the c.d.f. of Z, see Hall and Heyde (1980). In general conditions
on the coupling to obtain a CLT with mixing will be more restrictive than what was needed
for a conditional LLN. We will focus on the case of aggregate games described in section 3,
and propose a fairly intuitive approach to incorporating equilibrium conditions as additional
estimating equations into a derivation of the asymptotic distribution of m̂n (θ).
Recall that any Nash equilibrium G∗n in the aggregate game satisfies the fixed-point inclusion
n
1M
ψ̃n (ti , G∗n ) − G∗n
0∈
n i=1
We can then stack the moment and fixed point conditions, and consider the joint distribution
of
"
#
"
#
n
′
′
M
m((Xi , ψn (ti , G)) ; θ)
m̂n (G, θ)
1
:=
n i=1
ψn (ti , G)
Ψ̂n (G)
Note that
m(Yi ; θ) = m ((Xi′ , ψ0 (ti , G0 ))′ ; θ) =: m0 (ti , G0 ; θ)
Similarly, we can write
m(Yni ; θ) = m (Xi′ , ψn (ti , Ĝn ))′ ; θ =: mn (ti , Ĝn ; θ)
√
We derive the asymptotic distribution of n(m̂n (θ) − m0 (θ)) conditional on F∞ by augmenting the estimating equations by a state condition which defines equilibrium in the
limiting game, but only approximates the fixed-point condition for the finite player game:
GAMES WITH EXCHANGEABLE PLAYERS
25
Given θ, we let the random variable G∗0 := E[Y1 |F∞ ], the limiting equilibrium value of G
conditional on F∞ . Also, m∗0 := E[m0 (ti , G∗0 ; θ)|F∞ ], and µ0 := (m∗0 , G∗0 )′ . Similarly, define
P
m∗n := n1 ni=1 mn (ti , Ĝn ; θ) and µ̂n := (m̂n , Ĝn ).
We can then stack the moment conditions defining µ̂n by defining the multi-valued function
"
#
mn (ti , G; θ) − m
rn (ti ; θ, µ) :=
ψn (ti ; G) − G
By inspection, holding θ ∈ Θ constant, µ̂n is a solution of the generalized equation
n
0 ∈ r̂n (θ, µ̂n ) :=
1M
rn (ti ; µ̂n )
n i=1
Now let ψ0∗ (ti ; G) ∈ ψ0 (ti ; G) be an arbitrary selection of the limiting best response correspondence, and let
"
#
m0 (ti , G; θ) − m
r(ti ; θ, µ) :=
ψ0∗ (ti ; G) − G
It follows that for a given value of θ and under the conditions of the theorem, µ0 is a solution
of
0 = r0 (θ, µ0 ) := E [r(ti ; θ, µ)|F∞]
Note that the general structure may be similar in other settings, e.g. dynamic games or
matching markets, although in these classes of games the equilibrium condition will typically
be infinite-dimensional.
For the derivation of the asymptotic distribution of m̂n (θ) we define the conditional covariance matrices
Ωmm′ := E[(m(Yi ; θ) − m∗0 )(m(Yi ; θ0 ) − m∗0 )′ |F∞ ]
Ωψψ′ := diag(G∗0 ) − G∗0 (G∗0 )′ ,
Ωmψ′ := E[(m(Yi ; θ0 ) − m∗0 )(ψ0 (ti ; G∗0 ) − G∗0 )′ |F∞ ]
We also let
Ω :=
"
Ωmm′ Ω′mψ′
Ωmψ′ Ωψψ′
#
Note that since the empirical distribution of {X1 , . . . , Xn } is Fn -measurable, conditioning
on F∞ already implies conditioning on the unordered sample {X1 , . . . , Xn } if the associated
conditional variance matrices converge in probability along the filtration {Fn }n≥1. Also let
ṀG := ∇G E [m((Xi′ , ψ0 (ti ; G∗0 ))′ ; θ0 )|F∞ ]
and

P ψ0 (s(1) , ti ; G∗0 ) = {1}


..
Ψ̇G := ∇G 

.
(p)
∗
P ψ0 (s , ti ; G0 ) = {1}

26
KONRAD MENZEL
We now impose standard regularity conditions on the problem that ensure a joint normal
asymptotic distribution for the aggregate state and the moment conditions m̂n (θ):
Assumption 4.1. (i) The equilibrium points G∗0 ∈ G0∗ are interior points of ∆S, (ii) the
class Ms := {m((Xi′ , s)′ ; θ) : θ ∈ Θ} is Donsker with respect to the distribution of Xi for
each s ∈ S with a square-integrable envelope function, and (iii) the eigenvalues of Ω are
bounded away from zero and infinity.
The first part of this assumption ensures that the aggregate state Ĝn is a regular parameter
in the sense of Bickel, Klaassen, Ritov, and Wellner (1993) conditional on any tail event in
F∞ . It is possible to show that this condition is satisfied e.g. if Assumption 3.3 (ii) is
strengthened to hold with all eigenvalues of ∇t1 u(t, G)∇t′1 u(t, G) bounded away from zero,
and if the support of t1 is equal to Rp−1 . Under those additional conditions, for every action
s ∈ S and G ∈ ∆S there is a positive mass of types such that s is a best response to G, so
that Φ0 (G) ∈ int ∆S, and therefore every fixed point has to be in the interior of ∆S. Parts
(ii) and (iii) of this assumption are fairly standard. Under these additional conditions, we
can obtain the following CLT:
Theorem 4.2. Suppose that Assumptions 3.1-3.3 and 4.1 hold. Then ṀG and (Ip − Ψ̇G )−1
exist and are well defined, and
√
d
n (m̂n (θ) − E[m(Y1 ; θ)|F∞]) → Z1 + ṀG (Ip − Ψ̇G )−1 Z2
(4.1)
mixing and uniformly in θ, where
"
#
Z1
∼N
Z2
"
0
0
# "
,
Ωmm′ Ωmψ′
Ωψm′ Ωψψ′
#!
This theorem is proven in the appendix. Note that despite the asymptotic conditional
independence of players’ actions by Theorem 2.1, we need to correct the asymptotic variance
for dependence after centering the statistic m̂n (θ) around its conditional limit. The crucial
difference to the case in which the unobserved characteristics εi are private information lies
in the necessity for adjusting the asymptotic variance of m̂n for endogeneity of Ĝn with
respect to the actual realizations of the unobserved types. The correction term is different
from the variance adjustment in Shang and Lee (2011)’s analysis of the private information
P
case, where the realized value of Ĝ∗n = n1 ni=1 si takes the role of a noisy measurement for
its expected value in equilibrium.
The result can also be easily adapted to settings in which the econometrician only observes
a random subsample of N < n players in the game. As before, we let qi be an indicator
variable that equals one if agent i is included in the sample, and zero otherwise, and q1 , . . . , qn
GAMES WITH EXCHANGEABLE PLAYERS
27
are independent of Y1 , . . . , Yn . We can then define
n
1 X
m̃N (θ) :=
qi m(Yi ; θ)
N i=1
and apply Theorem 4.2 to m̃N (θ) to obtain its asymptotic distribution.
Corollary 4.4. Suppose the assumptions of Theorem 4.2 hold, and we observe a random
sample of N ≤ n players’ type-action profiles, where Nn is bounded and limn Nn → α ∈ [0, 1].
Then
√
√
d
N (m̃N (θ) − E[m(Y1 ; θ)|F∞ ]) → Z1 + αṀG (Ip − Ψ̇G )−1 Z2
mixing and uniformly in θ, where Z1 and Z2 have the same properties as before.
This result also helps highlight the different roles played by the number of players in the
game and sample size in the asymptotic experiment. Specifically, the variance adjustment
accounting for the equilibrium condition is important only if the size of the observed sample
is of the same order of magnitude as the number of agents in the market.
We now give another extension of the CLT to deal with moment functions of the form
∗
m (Yi ; θ, η) where the equilibrium distribution of actions G takes the role of the nuisance
parameter introduced in Corollary 4.2. To be specific, we let the random variable η0 := G∗0 ,
which is F∞ -measurable, η̂n := Ĝn , and consider the sample moment
n
m̂∗n (θ, Ĝn )
1X ∗
m (Yni ; θ, Ĝn )
:=
n i=1
We further assume that the derivative Ṁη := ∇η E[m∗ (Yi ; θ, η)|F∞] is well-defined a.s. and
the assumptions of Theorem 4.2 hold. Then, noting that the total differential of r0 (µ) with
respect to G is given by ṀG + Ṁη , the following extension of Theorem 4.2 is immediate from
Corollary 4.2 and the proof of the original result:
Corollary 4.5. Suppose the assumptions of Theorem 4.2 hold for moment functions of the
form m∗ (Yi ; θ, η), where η ∈ ∆S, and we replace Assumption 4.1 (ii) with the requirement
that the class Ms := {m((Xi′ , s)′ ; θ, η) : θ ∈ Θ, η ∈ ∆S} be Donsker with respect to the
distribution of Xi for each s ∈ S with a square-integrable envelope function. Also assume
that the derivative Ṁη := ∇η E[m∗ (Yi ; θ, η)|F∞] exists a.s. and is continuous for η ∈ ∆S.
Then
h
i
−1
√ ∗
d
n m̂n (θ, Ĝn ) − E[m∗ (Y1 ; θ, G∗0 )|F∞ ] → Z1 + ṀG + Ṁη Ip − Ψ̇G
Z2
mixing and uniformly in θ, where Z1 and Z2 have the same properties as before.
4.3. Variance Estimation. We now turn to estimation of the conditional asymptotic variance for m̂n (θ): Given a sample y1 , . . . , yn we can estimate the Jacobians ṀG and Ψ̇G either
28
KONRAD MENZEL
ċ and Ψ
ḃ . For
parametrically given the distribution of ti , or nonparametrically to obtain M
G
G
semi-parametric index models, this derivative can be estimated at a root-n rate in the presence of a continuous observed covariate with a nonzero coefficient, e.g. using Powell, Stock,
and Stoker (1989)’s weighted average derivative estimator with constant weights (see their
Corollary 4.1), or the estimator proposed by Horowitz and Härdle (1996) if only a finite
number of markets are observed. We therefore state the following high-level assumption for
consistent variance estimation:
Assumption 4.2. (i) For every value of s ∈ S, the first two moments of |m((s, Xi′ )′ ; θ)| are
ċ and Ψ
ḃ
bounded by a constant for all θ ∈ Θ, and (ii) there exist consistent estimators M
G
G
for the Jacobians ṀG and Ψ̇G , respectively.
For primitive conditions for part (ii) of this assumption, see e.g. Powell, Stock, and
Stoker (1989) and Horowitz and Härdle (1996), where we can combine their consistency
arguments with the conditional law of large numbers in Corollary 4.3. Note in particular
that identification of the Jacobians using the arguments in Horowitz and Härdle (1996)
requires that we observe at least two markets with different limiting values of the aggregate
state, either due to differences in type distributions or equilibrium selection.
We can now define
ċ (I − Ψ
ḃ )−1 [e − Ĝ ]
v̂in (θ) := [m(yi ; θ) − m̂n (θ)] + M
G p
G
si
n
(4.2)
where ek ∈ {0, 1}p−1 is the kth unit vector if k ≤ p − 1 and the p − 1 dimensional null vector
if k = p. A consistent estimator for the asymptotic variance is
n
V̂n (θ) :=
1X
v̂i (θ)v̂i (θ)′
n i=1
We also let A := [Iq , (I − Ψ̇′G )−1 ṀG′ ]′ . Our findings are summarized in the following corollary.
Corollary 4.6. Suppose the conditions of Theorem 4.2 and furthermore Assumption 4.2
hold. Then
V̂n (θ) → A(θ)′ Ω(θ)A(θ) a.s.
Furthermore,
√
d
nV̂n (θ)−1/2 (m̂n (θ) − E[m(Y1 ; θ)|F∞]) → N(0, Iq ) (mixing)
See the appendix for a proof. Variance estimation for statistics based on a random subsample of agents as considered in Corollary 4.4 is completely analogous where vin (θ) is replaced
with
r
N ċ
ḃ )−1 [e − Ĝ ]
M G (Ip − Ψ
v̂iN (θ) := [m(yi ; θ) − m̂N (θ)] +
G
si
N
n
P
and ĜN := N1 ni=1 qi esi is the empirical distribution of actions in the observed subsample.
GAMES WITH EXCHANGEABLE PLAYERS
29
5. Simulation Study
For ease of exposition, we restrict attention to the case of binary choice with social interactions, so that the aggregate state G is a scalar. In order to obtain a parsimonious
simulation design that generates multiple equilibria, we model types as including both discrete and continuous components. To be specific, xi = (x1i , x2i ), where the characteristic
x1i is a discrete variable which takes values −k or k with probability p each, and zero with
probability 1 − 2p, and we set k = 5, p = 0.2, and σx2 = σε2 = 1. Furthermore, the market
is partitioned into subgroups of agents, where we allow the strength of social interactions
to be of different strength between individuals in the same group than across groups. The
component x2i is multinomial with g equally likely categories and serves as an identifier for
these subgroups. Finally, there is a characteristic εi ∼ N(0, σε2 ) that is i.i.d. across agents
and independent of xi . The component εi is not observed by the econometrician, but the
full types ti = (x′i , εi )′ are common knowledge among players.
We analyze two different setups, one in which interactions are anonymous,
!
n
X
1
sj + ε i
(5.1)
u(ti ; s−i ) := si x′i β + ∆0
n j=1
with a global interaction effect ∆0 . Our second setup allows for a global interaction effect
and a “local”, group-level interaction effect, with payoff functions
!
n
n
X
X
1
1
u(ti ; s−i ) := si xi + ∆0
sj + ∆1 g
sj 1l{x2j = x2i } + εi
(5.2)
n j=1
n j=1
Here ∆1 measures the group-level interaction effect. For the simulations, we chose parameter
values ∆0 = 8, ∆1 = 2, β = 1.
Note that since ∆0 and ∆1 are nonnegative, the game is one with strategic complementarities (see Milgrom and Roberts (1990)), and we can use a tâtonnement algorithm to find
the smallest and the largest equilibria (which are both in pure strategies) for any size of
the market. Since actions are binary, we can implement an algorithm that needs at most n
steps.
We first illustrate convergence of the best response correspondence underlying the basic
argument for conditional convergence for aggregate games in section 3. It is easy to verify
that due to the positive sign of the interaction effect ∆0 , ψn (G, t) is single-valued for all finite
n and values of G and t. Figure 5 shows the average response correspondence Ψ̂n (G) for a
single realization of a market with n = 5, 20 and 100 players, respectively together with the
limiting function Ψ0 (G). The limiting best response mapping Φ0 (G) has exactly three fixed
points, and for this particular simulation draw, each finite-player version of the game also
happens to have exactly three equilibrium values for Ĝn . In general there may be multiple
KONRAD MENZEL
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
b 20
Ψ
b5
Ψ
30
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0.6
0.8
1
G
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
Ψ0
b 100
Ψ
G
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0
0.2
0.4
0.6
G
0.8
1
0
0.2
0.4
G
Figure 3. Ψ̂n (G) for n = 5, 20, 100, and Ψ0 (G) (bottom right)
equilibria supporting the same value of Ĝn , and the probability that the number of distinct
equilibrium distributions in the finite player game coincides with the number of fixed points
of Φ0 (G) is strictly less than one.
We now turn to simulations of statistics of the type analyzed in section 4 and analyze the
quality of the asymptotic approximations. The Jacobians in the expression for the variance
in equation (4.1) can be obtained if we have an estimator for the conditional marginal effects
given Xi , q(x, G) := ∇G E [ψ0 (G; Ti )|Xi = x]. If the conditional distribution of εi given Xi
is continuous and fixed with respect to Xi with p.d.f. hε (z), the conditional model for
ψ0 (G; Ti ) given Xi has the linear index structure considered by Powell, Stock, and Stoker
(1989) and Horowitz and Härdle (1996), and we can obtain consistent estimators for the index
ˆ n and β̂n , and for the conditional p.d.f. ĥn (·). We can then form q̂n (x, G) :=
coefficients ∆
ˆ ĥε (x′ β̂n + ∆
ˆ n G). Since in our setup observable types are discrete and the number of
∆
markets is finite, the conditional marginal effects are in general not point-identified without
parametric assumptions on the conditional distribution of εi given Xi . We therefore use
GAMES WITH EXCHANGEABLE PLAYERS
31
the usual
estimator for q(x, G) based on the Probit specification, q̂(x, G) :=
′ parametric
ˆ nG
ˆn
x β̂n +∆
∆
, where ϕ(·) is the standard normal p.d.f..
ϕ
σ
σ
For the binary action case, let d(x; θ) := m((1, x′ )′ ; θ)−m((0, x′ )′ ; θ) so that we can express
ṀG = E [d(X; θ)q(X, G∗0 )|G∗0 ]
and Ψ̇G = E[q(X, G∗0 )|G∗0 ]
We can then obtain estimates of the Jacobians by replacing q(x, G) with its estimator q̂(x, G),
G∗0 with Ĝn , and expectations with sample averages over the observed values of Xi ,
n
ċ = 1 X d(X ; θ)q̂(X , Ĝ )
M
i
i
n
G
n i=1
n
ḃ = 1 X q̂(X , Ĝ )
and Ψ
G
i
n
n i=1
Given a consistent estimator q̂(x, G) for the conditional marginal effect, we can plug these
estimators into the expression for v̂in in equation (4.2) and obtain a consistent estimator for
the variance matrix of m̂n (θ).
Below we report simulation results for the value of the aggregate state Ĝn and the maxiˆ n for the interaction effect ∆0 based on the limiting distribution.
mum likelihood estimator ∆
Note that this estimator has a higher-order bias of order O(n−1 ) which becomes dominated
by the (asymptotically Gaussian) first-order term in the asymptotic expansion. We also report simulation results for the Jackknife MLE which removes part of the leading bias term,
where we use a parametric bootstrap bias correction for the standard errors that is asymptotically unbiased to second order. For results on the distribution of Ĝn , a given market picks
the lowest equilibrium with probability 0.3, and the highest equilibrium with probability 0.7.
The MLE is computed for samples of m = 16 markets with n players each, where in half the
markets the lowest equilibrium is chosen, and in the remaining games players coordinate on
the highest equilibrium. Note that for some realizations of types, especially for small values
of n, the equilibrium is unique so that the lowest and the highest equilibria coincide.
Figure 5 shows the p.m.f. for Ĝn for games of different sizes. Note that the two extremal
equilibria for games with strategic complementarities are always in pure strategies, so that
in our simulation design, the distribution of Ĝn is always discrete with values n1 , n2 , . . . , 1,
regardless whether types are discrete or continuous. The narrow spikes and gaps in the
histograms for Ĝn are “interference patterns” that are caused by the discreteness of the
variable and not a result of simulation error. These spikes average out between neighboring
values for Ĝn and appear to vanish for larger values of n. Figures 5 and 5 show kernel density
ˆ n and the corresponding t-ratios.
approximations to the simulated p.d.f.s for Ĝn and ∆
Finally, we simulate versions of the model in which agents also interact “locally.” In this
setting, each agent in each market belongs to one of g different groups and has preferences
given by (5.2). The interaction effects are held at ∆0 = 8 and ∆1 = 2, respectively, and
our simulation experiments consider the cases of g = 4 and g = 10 subgroups. We consider
a “fixed design” with respect to the marginal distribution of x2i in each market in that
32
KONRAD MENZEL
0.14
0.12
0.25
0.1
density
density
0.2
0.15
0.08
0.06
0.1
0.04
0.05
0.02
0
0.1
0.2
0.3
0.4
0.5
Ĝn
0.6
0.7
0.8
0.9
0
1
0.09
0.1
0.2
0.3
0.4
0.5
Ĝn
0.6
0.7
0.8
0.9
1
0.06
0.08
0.05
0.07
0.04
density
density
0.06
0.05
0.04
0.03
0.03
0.02
0.02
0.01
0.01
0
0
0.2
0.4
0.6
0.8
0
1
0
0.2
0.4
Ĝn
0.6
0.8
1
Ĝn
Figure 4. Distribution of Ĝn for n = 10, 20, 50, and n = 100, respectively
n=10
n=20
n=50
n=100
5.5
5
0.35
4.5
4
0.3
density
3.5
density
n=10
n=20
n=50
n=100
Gaussian
0.4
3
2.5
2
0.25
0.2
0.15
1.5
0.1
1
0.05
0.5
0.1
0.2
0.3
0.4
0.5
Ĝn
0.6
0.7
0.8
0.9
1
−4
−3
−2
−1
0
Ẑn
1
2
3
Figure 5. Simulated p.d.f. of Ĝn (left panel), and its studentization Ẑn =
√ Ĝn −G∗0
(right panel) for n = 10, 20, 50, 100.
n√
V̂G,n
4
5
GAMES WITH EXCHANGEABLE PLAYERS
33
0.4
0.7
n=50
n=100
n=500
Gaussian
0.6
n=50
n=100
n=500
Gaussian
0.35
0.3
0.5
density
density
0.25
0.4
0.2
0.3
0.15
0.2
0.1
0.1
0
0.05
−4
−3
−2
−1
0
Ẑn
1
2
3
4
5
−3
−2
−1
0
Ẑn
1
2
3
4
Figure 6. Simulated p.d.f. of the studentized MLE for ∆0 (left) from m = 12
markets with n = 50, 100, 500 players, and the (Jackknife) bias-corrected MLE
(right)
the proportion of players belonging to each group is held constant across replications of the
game.11 Fixing the marginal distribution for x2i allows us to reduce the relevant dimension of
the aggregate G from 2g−1 to g which speeds up the matrix inversions involved in computing
the asymptotic conditional variance of Ĝn . The results are reported in Figure 5, and there
appears to be a visible deterioration of the performance of asymptotic approximation as we
increase g.
6. Discussion
This paper establishes methods for conditional estimation and inference using data from
games with a large number of players. The argument relies crucially on the idea of exchangeability of individual agents, and addresses difficulties with stochastic dependence arising from
strategic behavior and multiplicity of equilibria. We also argue that distributions arising in
the many-player limit may also be computationally more tractable.
We discussed how to apply the asymptotic results to static aggregate games with discrete
actions. It is possible to extend some of the main ideas of this paper dynamic games, matching markets, network formation and other problems of non-anonymous strategic interaction.
However, this is beyond the scope of this study and will be left for future research.
Appendix A. Proofs for Results from Section 3
A.1. Proof of Proposition 3.1. We start by establishing claim (a), where the proof proceeds in the following steps: we first construct a convex-valued correspondence Ψ̃n (G) and
show uniform convergence in probability of the convex hull of Ψ̂n (G) to Ψ̃n (G). We then
11Note
that the empirical distribution of types is also Fn -measurable, so that conditioning on type distributions, rather than individual realizations, preserves exchangeability of type-action characters.
34
KONRAD MENZEL
12
n=50
n=100
n=200
n=500
10
n=50
n=100
n=200
n=500
Gaussian
0.35
0.3
0.25
density
density
8
6
0.2
0.15
4
0.1
2
0
0.05
0.1
0.2
0.3
0.4
0.5
Ĝn
0.6
0.7
0.8
0.9
1
−4
−3
−2
−1
0
Ẑn
1
2
3
4
5
−2
−1
0
Ẑn
1
2
3
4
5
0.4
n=50
n=100
n=200
n=500
n=50
n=100
n=200
n=500
Gaussian
0.35
10
0.3
8
0.25
density
density
12
6
0.2
0.15
4
0.1
2
0
0.05
0.1
0.2
0.3
0.4
0.5
Ĝn
0.6
0.7
0.8
0.9
1
−4
−3
Figure 7. Simulated p.d.f. of Ĝn (left panel), and its studentization Ẑn =
√ Ĝn −G∗0
(right panel) with g = 4 (top panels) and g = 10 (bottom) interacn√
V̂G,n
tion groups for n = 50, 100, 200, 500.
show graphical convergence of the correspondence Ψ̃n (G) to Ψ0 (G). Finally, we conclude
that any sequence of solutions Ĝ∗n of the fixed point inclusion Ĝ∗n ∈ Ψ̂n (Ĝ∗n ) approaches the
set G0∗ with probability one.
Recall that the correspondence
n
1M
Ψ̂n (G) :=
ψn (G, ti )
n i=1
where “
L
” denotes the Minkowski sum over the sets ψ̃n (s; G, ti ) for i = 1, . . . , n. Also let
Ψ̃n (G) := E [ψn (G, Ti )]
where the operator E[·] denotes the (Aumann) selection expectation with respect to the
random type Ti ∼ H0 . Note that from standard properties of Aumann expectations, Ψ̃n (G)
is convex at every value G ∈ ∆S.
GAMES WITH EXCHANGEABLE PLAYERS
35
Uniform Convergence. We will now show uniform convergence of the correspondence Ψ̂n to
Ψ̃n in probability. In the following, we denote the support function of the set Ψ̂n (G) by
ĥn (v; G, ε) :=
inf
G′ ∈Ψ̂n (G)
hv, G′i
for any v on the (p − 1) dimensional unit sphere in Rp , and we also let
h̃n (v; G, ε) :=
inf
G′ ∈Ψ̃
n (G)
hv, G′i
Since for the Minkowski sum of two convex sets K1 and K2 , the support function satisfies
h(a1 K1 ⊕ b2 K2 , v) = a1 h(K1 , v) + b2 h(K2 , v), we can rewrite
n
1X
̺n (v; G, ti )
ĥn (v; G) =
n i=1
where at a given value of G and type t, ̺n (v; G, t) := inf G′ ∈ψn (G,t) hv, G′ i = inf G′ ∈conv ψn (G,t) hv, G′i
is the support function of the set ψn (G, t).
In order to show that ĥn converges to h̃n uniformly, i.e.
p
sup ĥn (v; G) − h̃n (v; G) → 0
v,G
note first that the sets conv ψn (G, ti ) are convex polytopes whose vertices are equal to one
or zero, yielding at most 2p different sets for conv ψn (G, ti ; ε). Hence we can represent the
support function ̺n (v; G, t) as a simple function defined on sets that are p fold intersections
of the sets A(s, G) and their complements. Since the VC property is preserved under complements and finite intersections, Assumption 3.1 implies that ̺n (v; G, t) is a VC class of
functions indexed by (v, G) for every n with a VC index that does not depend of n.
Now by the same argument as in the proof of Theorem 3.1.21 in Molchanov (2005),
convergence of the support functions in probability implies convergence in probability of
P
the set conv n1 ni=1 ψn (G, ti ) with respect to the Hausdorff metric, where convergence is
uniform in G by boundedness of ̺n (v; G, t) and the VC property of the support functions
for ψn (G, ti ).
Graphical Convergence. Now fix δ > 0. In order to establish that Ψ̃n (G) = E [ψn (G, t)]
converges to Ψ0 (G) = E [ψ0 (G, t)] graphically, we show that for n large enough the two
δ
inclusions gphΨ0 ⊂ gphΨ̃n and gphΨ̃n ⊂ (gphΨ0 )δ hold:
First, consider a point G, Ψ such that Ψ ∈ Ψ0 (G), where Ψ = (Ψ(s(1) ), . . . , Ψ(s(p) )) ∈ ∆S.
From the definition of G it follows that P (t ∈ A(s, G)) ≥ Ψ(s) and P (t ∈
/ A(s, G)) ≥ 1−Ψ(s)
for all s. Now consider the extremal points Ψ∗p of Ψ0 (G1 ) of the form Ψ∗p (s(p) ) = P (t ∈
P
A(s(p) , G1 )) for p = 1, . . . , S −1, and s′ 6=s(p) Ψ∗p (s′ ) = 1−P (t ∈ A(s(p) , G1 )). It is immediate
that the values Ψ ∈ Ψ0 (G1 ) are convex combinations of Ψ∗p , p = 1, . . . , S − 1.
36
KONRAD MENZEL
Now let B be the closed δ/2 ball around G1 . Then for all sufficiently large n, the δ/2 blowup
S
of the set G∈B conv (Eψn (G, t)) includes the extremal points of Ψ0 (G1 ) and their convex
hull. In particular, we can find a point G̃1 ∈ B and Ψ̃1 ∈ Ψ̃n (Gn ) such that d(Ψ̃1 , Ψ1 ) < δ/2.
Hence for large n we can find (G̃1 , Ψ̃1 ) ∈ gph Ψ̃n such that d((G1, Ψ1 ), (G̃1 , Ψ̃1 )) < δ. Since
δ
this bound does not depend on G, Ψ, it follows that gphΨ0 ⊂ gphΨ̃n .
The argument for the second inclusion is completely analogous, so that we can conclude
that for n large enough, dH (gphΨ0 , gphΨ̃n ) < δ.
Convergence to Limiting Points. Given δ > 0 we define
η := inf {dH (G, Ψ0 (G)) : G ∈ ∆S : d(G, G0∗ ) ≥ δ}
By Lemma 3.2, the number of elements in G0∗ is finite, so that since ∆S is compact and
dH (G, Ψ0 (G)) is lower-semi-continuous in G, we have that η > 0, where existence of the
minimum follows e.g. by Theorem 1.9 in Rockafellar and Wets (1998).
Combining uniform convergence of Ψ̂n to Ψ̃n in probability, and graphical convergence of
Ψ̃n to Ψ0 via the triangle inequality, we can choose n large enough such that the probability
of dH (gphΨ̂n , gphΨ0 ) < min{δ, η} is arbitrarily close to one. Hence the probability that
there exists a solution of the inclusion G̃ ∈ Ψ̂n (G̃) with d(G̃, G0 ) > δ converges to zero as n
increases.
Finally, since any Ĝn solving Ĝn ∈ Ψ̂n (Ĝn ) also solves Ĝn ∈ Ψ̃n (Ĝn ) for n large enough,
we have that d(Ĝn , G0 ) < δ with probability approaching one, which establishes assertion
(a).
Achievability of Limiting Equilibria. We now turn to the proof of part (b). We proceed by
showing that for every equilibrium point G∗0 ∈ G0∗ , we can find a neighborhood of G∗0 such that
Ψ0 (G) is an inward map on that neighborhood. Then we show that by graphical convergence,
Ψ̂n (G) is also an inward map on all such neighborhoods with probability approaching one.
We then apply a fixed-point theorem for inward maps to a transformation of the problem
and conclude that with probability approaching 1, local solutions exist near every G∗0 ∈ G0∗ .
Construction of Neighborhoods. Let Γ0 (G) := Ψ0 (G) − G be defined as in the main text.
In the following, we say that Ψ0 is a (possibly multi-valued) inward map on a convex set
K ⊂ ∆S, if for all G ∈ K, Ψ(G) ∩ (G + TK (G)) 6= ∅, where TK (G) is the tangent cone of K
at G. We will now show that for every G∗0,j ∈ G0∗ , j = 1, . . . , |G0∗ |, and neighborhood U(G∗0,j )
of G∗0,j , we can find a compact convex set Kj ⊂ U(G∗0,j ) containing an open set around G∗0,j ,
and such that Ψ0 restricted to Kj is an inward map. To this end suppose without loss of
GAMES WITH EXCHANGEABLE PLAYERS
37
generality that for that every vector u 6= 0 and v ∈ DΓ0 (G∗0,j , G∗0,j )(u), we have that the
inner product u′ v < 0.12
We will prove this claim by contradiction: Suppose that for a sequence hn → 0 we can
construct a sequence of closed balls Kn of radius hn around the fixed point G∗0,j such that
for every n there exists Gn on the boundary of Kn such that Ψ0 (Gn ) ∩ (Gn + TKn (Gn )) = ∅.
In particular, Ψ0 (Gn ) − Gn ⊂ [TKn (Gn )]C , where the superscript C denotes the complement
of a set, and without loss of generality we will take Ψ0 (Gn ) to be single-valued.
Now, let Ψ̃0n (G) denote the least-squares projection of Ψ0 (G) onto Knhn , the hn -blowup of
C
C
Kn - note that if Ψ0 (G) − G ⊂ [TKn (G)]
, then also Ψ̃0n (G) − G⊂ [TKn (G)] . Now consider
Gn − G∗0,j , Ψ̃0n (Gn ) − Gn in Kn × Knhn . Since this
the sequence of vectors vn := h−1
n
sequence is contained in the closed ball of radius 2 - a compact set - there exists a converging
subsequence vn(ν) , ν = 1, 2, . . . and n(ν) → ∞, where limν→∞ vn(ν) =: v ≡ (vG , vΨ ). Since
0 ∈ Ψ̃0n (G∗0,j ) − G∗0,j by assumption, v is also an element of the graph of the contingent
derivative of Ψ0 (G) − G at (G∗ , G∗ ).
Furthermore, Kn is convex, so that G∗0,j − Gn ∈ TK (Gn ). Since Kn is a ball, at every
point Gn on the boundary of Kn TKn (Gn ) is an open half-space, implying that [TKn (Gn )]C =
−int(TKn (Gn )), where int(A) denotes the interior of a set A. Now, by construction Gn −
Ψ̃0n (Gn ) ∈ [TK (Gn )]C , so that the inner product (Gn − G∗0,j )′ (Ψ̃0n (Gn ) − Gn ) > 0 for all n.
′
Hence, taking limits we have vG
vΨ > 0. However, this contradicts that for the contingent
∗
∗
derivative of Ψ0 at (G0,j , G0,j ) we had u′ v < 0 whenever v ∈ DΓ0 (G∗0,j , G∗0,j )(u), so that Ψ0
has to be inward as claimed before.
Since Kn is compact, and Ψ0 (G)−G is continuous, we have that the length of the projection
of Ψ0 (G) − G onto TKn (G) is bounded away from zero as G varies over Kn . Noting that
the convex hull of Ψ̂n (G) is lower semi-continuous and converges to Ψ0 (G) graphically in
probability, we have that Ψ̂n is also inward on K1 , . . . , K|G0∗ | with probability approaching
one.
Local Existence of Fixed Points of Ψ̂n . In order to remove the non-convexities of the mapping
Ψ̂n (G), we consider a transformation of the graph of Ψ̂n on the set Kj under a continuous
one-to-one mapping Hj : Kj × Kj → G × G, [G, Ψ] 7→ [Hj (G, Ψ − G), Ψ]. We can choose
the mapping Hj such that Hj (G, Ψ − G) is strictly monotone in its second argument, and
12More
specifically, we can choose a diagonal matrix B with elements Bjj ∈ {−1, 1}, j = 1, . . . , p − 1, and
rewrite the inclusion as
G∗0,j − B(G − G∗0,j ) ∈ Ψ0 (G∗0,j − B(G − G∗0,j ))
for any such B, or equivalently,
G = G∗0,j + (G − G∗0,j ) ∈ Ψ0 (G∗0,j − B(G − G∗0,j )) + (I − B)(G − G∗0,j ) =: Ψ̃B (G)
Now if we let ΓB (G) := Ψ̃B (G) − G, by Assumption 3.2 (ii) the contingent derivative of ΓB is surjective, and
we can choose B such that there exists a vector vB ∈ gph DΓB (G∗0,j , G∗0,j )(u), the inner product u′ · vB ≤ 0.
The second part of Assumption 3.2 (ii) implies that this inequality is strict, and has the same sign for any
′
other vB
∈ gph DΓB (G∗0,j , G∗0,j )(u) by convexity of the contingent derivative.
38
KONRAD MENZEL
Hj (G, 0) = G, and the values of (Hj (G, Ψ̂n (G)), Ψ̂n (G)) are convex for every G ∈ ∆S. Note
that the two conditions on Hj (·) imply that G is a fixed point of Ψ̂n if and only if it is also
a fixed point of (Hj (G, Ψ̂n − G), (Ψ̂n (G)). Furthermore, since the transformations Hj map
any point on the boundary of Kj onto itself, they also preserve the inward mapping property
of Ψ̂n .
Since the transformed mapping is nonempty with closed and convex values, upper semicontinuous, and inward on Kj , existence of a fixed point in Kj follows from Theorem 3.2.5
in Aubin and Frankowska (1990). Since the number of sets K1 , . . . , Ks is finite, and the
set of fixed points of the transformed mapping is equal to that of Ψ̂n , it follows that with
probability approaching one, Ψ̂n has a fixed point in the neighborhood of each equilibrium
point G∗0,j ∈ G0∗ , establishing the conclusion in part (b) A.2. Proof of Lemma 3.3. We first consider case (a). Let t̃ := (t̃1 , t̃2 ) ∈ T1 ×T2 such that a
player of type t̃ is indifferent between two or more actions given the aggregate state G. Since
by Assumption 3.3, the type distribution is tight, for a fixed δ > 0, we can find a compact
subset K ⊂ T1 such that t1i ∈ K with conditional probability greater than 1−δ given t2i = t̃2 .
Since by Assumption 3.3 (ii), the Jacobian ∇t1 u(ti , G) has rank equal to p − 1, there exists
a neighborhood N(t̃1 ) of t̃1 such that for players of any type t = (t1 , t̃2 ), t1 ∈ N(t̃1 ) \ {t̃1 }
there is a unique best response give the aggregate value G. Without loss of generality we can
take the family of neighborhoods N(t̃1 ) to be a cover of K. Since K is compact, there is a
finite subcover, and since the N(t̃1 ) were constructed in a way such that each set contains at
most one value t1 such that agents with type (t′1 , t̃′2 )′ are indifferent between any two actions,
there exist only finitely many values of t1 ∈ K such that u((t′1 , t̃2 )′ , s, G) = u((t′1 , t̃2 )′ , s′ , G)
for some s 6= s′ . Since δ can be chosen arbitrarily small, by the law of total probability there
cannot be a strictly positive mass of types that are indifferent between any two actions at
G. Hence with probability 1, ψ0 (ti ; G) is a singleton, and the correspondence Ψ0 (G) is in
fact single-valued.
For case (b), it is sufficient to notice that due to strict monotonicity of the payoff differences
u(s(l) , t, G) − u(s(k) , t, G), for every value of t̃2 there is at most one value of t1 such that a
player of type (t1 , t̃′2 )′ is indifferent between actions s(k) and s(l) . Since the number of actions
is finite, the conditional probability of a tie given t2 = t̃2 is zero, and the conclusion follows
from the law of total probability Appendix B. Proofs for Results from Section 4
B.1. Proof of Corollary 4.2. Note that Condition 4.1 does not depend on the form of
the moment function, and the theorem already includes a strengthened version of Condition 4.3 that holds uniformly in η. Hence it only remains to be shown that 4.2 (iii)
holds for m∗ (Yi ; θ, η̂n ): Since m∗ ((s, Xi ); θ, η) is continuous in η for all s, we have that
GAMES WITH EXCHANGEABLE PLAYERS
39
m∗ ((Yni , Xi ); θ, η) is also continuous, and by the continuous mapping theorem, we can
find a null sequence dn → 0 such that |E[m∗ (Yni ; θ, η̂n ) − m∗ (Yni ; θ, η0 )|Fn ]| < dn a.s..
Hence, condition 4.2 (iii) for m∗ (Yi ; θ, η̂n ) follows from the corresponding restriction on
m(Yi ; θ) := m∗ (Yi ; θ, η0 ) and the triangle inequality.
Finally, note that since η0 is F∞ -measurable, it is also Fn -measurable. Hence
n
Zn∗
n
1X ∗
1X
:=
m (Yni ; θ, η̂n ) +
[m(Yi ; θ, η0 ) − m∗ (Yni ; θ, η̂n )]
n i=1
n i=1
is also a reverse martingale adapted to {Fn }n≥1 and satisfying the Conditions of Theorem
4.1, so that the claim of the corollary follows from the same steps as before B.2. Proof of Proposition 4.1. Let ν1 , ν2 , . . . be a sequence of i.i.d.n draws from the uni- o
P
form distribution on the (p−1)-dimensional probability simplex ∆S := π ∈ Rp+ : pq=1 πq = 1 ,
where p is the number of n
pure actions available to each player. Then (ν1′ ,o. . . , νn′ )′ is uniformly
n(p−1) Pp
distributed on (∆S)n = π ∈ R+
: q=1 πiq = 1 for all i = 1, . . . , n .
Now fix t ∈ T n and let λ∗ (t) be the distribution over equilibria in Σ∗ (t) that generates the
mixture fN∗ (s, x|θ). Without loss of generality we can assume that λ∗ (t) is symmetric with
respect to permutations among players with the same type ti . Then λn constitutes a partition
of (∆S)n that can be chosen to be symmetric with respect to the same permutations of indices
1, . . . , n, and therefore satisfies Condition 4.2 (i).13 In particular, the resulting distribution
of type-action characters, f (s, x|θ, λN ) = fN∗ (s, x|θ) for all s ∈ S.
We now prove parts (ii) and (iii) of Condition 4.2. By Proposition 3.1, we have that the
a.s.
equilibrium value of the aggregate d(Ĝ∗n , G0∗ ) → 0, where G0∗ is a finite set. We now show that
in order to characterize the the set of distributions of Y1 it is sufficient to consider a family
f (y1 |θ, µ), where the parameter space for µ is the |G0∗ |-dimensional probability simplex.
More specifically, fix n0 , ν1 , . . . , νn0 and t1 , . . . , tn0 , and let G̃∗ := arg minG∈G0∗ d(G, Ĝn ),
where Ĝn is the value of the equilibrium aggregate under the selection rule λn . Now fix ε > 0.
By Proposition 3.1 (b), there exists n0 large enough such that with probability 1 − ε we can
find an equilibrium that supports some Ĝ∗n for all n ≥ n0 such that d(Ĝ∗n , G̃∗ ) < rn for a
sequence rn → 0. Hence we can choose λn such that for the first n0 coordinates of ν1 , . . . , νn
and t1 , . . . , tn coinciding with ν1 , . . . , νn0 and t1 , . . . , tn0 , respectively, we have Ĝn = Ĝ∗n .
13Note
that we can identify the vertices of (∆S)n and elements of the partition containing them with profiles
in S by setting si = q if νiq = 1 for all i = 1, . . . , n and q = 1, . . . , p. Denoting ιn(p−1) = (1, 1, . . . , 1)′ an
n(p − 1)-dimensional vector of ones, we can now construct a partition by considering sections around the
central point π ∗ := p1 ιn(p−1) that are symmetric with respect to any permutation of individuals of the same
type, and such that for every strategy profile s ∈ S n the set containing s̄ has volume λ∗ (s̄) (possibly zero).
Finding such a partition is a finite-dimensional convex program with linear constraints so that existence and
uniqueness of the solution are not problematic.
n
40
KONRAD MENZEL
Now let M := supθ∈Θ maxs,s′∈S |E[m((s, x); θ) − m((s′ , x); θ)]|, which is finite by Condition
4.3. Now we can bound
E [ |m(Yn1 ; θ) − m(Y1 ; θ)|| Fn ] ≤ MP (Sn1 6= S1 |Fn )
Now notice that by Assumption 3.3 and Lemma 3.3, the set ψ0 (G̃∗ , ti ) is the singleton {Si }
with probability one. Therefore since t1 , . . . , tn are i.i.d. draws from H0 (t), we have
n
o
1X n
1l ψ0 (G̃∗ , ti ) 6= ψ0 (Ĝ∗n , ti ) ≤ 2P (ψ0 (G̃∗ , t1 ) 6= ψ0 (Ĝ∗n , t1 ))
P (Sn1 6= S1 |Fn ) =
n i=1
say, w.p.a.1 by the strong law of large numbers. Now notice that by Assumption 3.3,
P (ψ0 (G̃∗ , ti ) 6= ψ0 (G, ti )) is continuous in G so that, since Ĝ∗n → G̃∗ , we can make P (ψ0 (G̃∗ , ti ) 6=
ψ0 (Ĝ∗n , ti )) arbitrarily small by choosing n large enough, establishing Condition 4.2 part (iii)
B.3. Proof of Theorem 4.2. First, note that under Assumption 3.3, Ψ0 (G) is single-valued
at every point G ∈ ∆S by Lemma 3.3. Hence, the Aumann selection expectation defining
Ψ0 (G) coincides with the usual expectation of a vector-valued random variable, and under
Assumption 3.3 (i) the mapping G 7→ G − Ψ0 (G) is Fréchet differentiable with continuous
derivative Ip−1 − ∇G P (ψ0 (s, ti ; G) = {1}).
Similarly, by Assumption 3.3 (iii), the derivative of the conditional probability
ṖG (Xi ) := ∇G P ( ψ0 (s, ti ; G0 ) = {1}| Xi )
exists and is continuous in G a.s.. Hence we can interchange differentiation and integration,
so that by the law of iterated expectations
i
X h
Ṁ0 (G) := ∇G E[m0 (ti , G; θ)] :=
E m((Xi′ , s)′ ; θ)ṖG (Xi )
s∈S
Since by Assumption 4.1 (ii), the functions m((Xi′ , s)′ ; θ) are dominated by a square integrable envelope for all s ∈ S, we can use the Cauchy-Schwarz inequality to establish that the
elements of ∇G E[m0 (ti , G; θ)] are bounded and continuous in G in a neighborhood of G∗0 .
We will now verify the conditions for Theorem 3.3.1 in van der Vaart and Wellner (1996)
√
to derive the asymptotic distribution of n(µ̂ − µ0 ). Note that for notational simplicity, we
suppress the θ argument below whenever θ can be regarded as fixed.
We can decompose the difference between the maps r̂n (µ) := r̂n (θ, µ) and r0 (µ) := r0 (θ, µ)
by
!
n
n
M
X
√
1
[rn (ti ; µ0 ) − r(ti ; µ0 )] +
[r(ti ; µ0) − E [r(ti ; µ0 )|F∞ ]]
n(r̂n (µ0 ) − r0 (µ0 )) = √
n i=1
i=1
=: T1 + T2
GAMES WITH EXCHANGEABLE PLAYERS
41
We first show that the first term is asymptotically negligible: let rn∗ (ti ; µ0 ) be any mea
surable selection of rn (ti ; µ0 ). If we let Bn := G′ ∈ ∆S : kG0 − G′ k ≤ n1 and πn (µ0 ) :=
S
P ψ0 (ti ; G0 ) 6= G′ ∈Bn ψ0∗ (ti ; G′ ) , then we can bound
Var (rn∗ (ti ; µ0 ) − r(ti ; µ0 ))2 |Fn ≤ πn (µ0 )(1 − πn (µ0 ))
Similarly, we can bound
|E[rn∗ (ti ; µ0 ) − r(ti ; µ0)|F∞ ]| ≤ πn (µ0 )
Now notice that, as shown above Assumption 3.3 implies that at every value of G, there are
no atoms of ”switchers” in the type distribution. Since the density of the type distribution
is bounded from above, we therefore have that limn nπn (µ0 ) < ∞ for each µ0 . Since with
probability one, µ0 only takes finitely many values, the common upper bound for all µ0
is finite for all F∞ -measurable events. Therefore the variance and absolute value of the
L
expectation of the first term T1 := √1n ni=1 [rn∗ (ti ; µ0 ) − r(ti ; µ0 )] converge to zero almost
surely as n → ∞, so that applying Chebyshev’s inequality, T1 converges to zero in probability
conditional on F∞ .
Since ψ0 (ti , G∗0 ) is single-valued with probability 1, and ti are i.i.d. draws from the distribution h(t), we can apply a standard CLT and obtain
n
√ 1X
d
[r(ti ; µ0 ) − r0 (µ0 )] → N(0, Ω)
n
n i=1
(B.1)
Since conditional on F∞ , T1 → 0 in probability, convergence in distribution with mixing
follows from Slutsky’s theorem.
Next, consider the Jacobian of the (augmented) population moments
"
#
Iq
−Ṁ0 (G∗0 )
∇µ r0 (µ) = −
0 Ip−1 − Ψ̇0 (G∗0 )
By Assumption 3.2, the derivative Ip−1 − Ψ̇0 (G∗0 ) has full rank at every G∗0 ∈ G0∗ , and we
already showed above that Ṁ0 (G) and Ψ̇0 (G) are continuous in G. Due to the block-diagonal
structure it is therefore straightforward to verify that the inverse of ∇µ r0 (µ) is well-defined
and continuous in µ in a neighborhood around µ0 and for all values of θ ∈ Θ.
We will now check the conditions of Lemma 3.3.5 in van der Vaart and Wellner (1996).
By Assumption 3.1 (iii), the class T := {ψ0 (ti ; G) : G ∈ ∆S} is a VC subgraph class, and
therefore Donsker. Also, by Assumption 4.1, Ms := {m((x, s); θ), θ ∈ Θ} is a Donsker class
for each s ∈ S, and S is finite. Note that the class R := {r(ti , µ, θ), µ ∈ Rq × ∆S, θ ∈ Θ} is
a Lipschitz transformation of M1 , . . . , Mp and T with the same envelope function as M1 .
Hence, by Theorem 2.10.6 in van der Vaart and Wellner (1996), Z is also Donsker with
respect to the distribution of types.
42
KONRAD MENZEL
Note also that as shown before, ψ0 (s, ti ; G) is single-valued at every G except for a set of
types of measure zero. Since payoffs are continuous in t and G, and the p.d.f. of the type
distribution is continuous in t, it follows that Ekr(ti ; µ) − r(ti ; µ0 )k2 is also continuous in µ.
Finally, by Proposition 4.1, we can construct the coupling such that conditional on F∞ ,
Ĝn − G0 → 0 in outer probability, so that by Lemma 3.3.5 in van der Vaart and Wellner
(1996) we have that for the empirical process
√
kGn (r(·, µ) − r(·, µ0))k = o∗P 1 + nkµ − µ0 k
Hence we can apply Theorem 3.3.1 in van der Vaart and Wellner (1996) to obtain the
√
asymptotic distribution of n(µ̂n − µ0 ).
It remains to show uniformity with respect to θ. Note that since R is Donsker as θ varies
in Θ, convergence of r̂n (θ, µ) is also uniform in θ. Finally, since by Assumption 4.1 (ii), the
class Ms has a square-integrable envelope function, by the same arguments as before the
elements of the derivative ṀG are bounded so that the bounds in the previous arguments do
not depend on θ B.4. Proof of Corollary 4.6. By the same reasoning as in the proof of Theorem 4.2,
the matrices ṀG and (I − Ψ̇G )−1 are well-defined, so that by Assumption 4.2 (i), the first
P
two moments of |vi | are bounded. Therefore, Corollary 4.3 implies that n1 ni=1 vi vi′ →
E[vi vi′ |F∞ ] a.s.. Next, the covariance matrix of (m(Yi ; θ)′ , e′si )′ is given by Ωn (Ĝn ) where
Ωn (G) := Var(m((Xi′ , ψn∗ (ti ; G)′ )′ ; θ), ψn∗ (ti ; G)) for some selection ψn∗ (ti ; G) ∈ ψn (ti ; G). By
similar arguments as in the proof of Theorem 4.2, Ωn (G) → Ω(G) uniformly in G. Note that
Ω(G) is continuous in G and Ĝn → G∗0 a.s., so that Ω(Ĝn ) → Ω by the continuous mapping
theorem.
Since by Assumption 4.2, the estimated Jacobians converge to Ṁg and Ψ̇G , respectively,
we therefore have that the conditional expectation of the product vi vi′ converges to A′ ΩA.
By Assumption 4.2 (i) we therefore obtain V̂n → A′ ΩA almost surely, as claimed in the first
claim of the Corollary. The second part follows immediately from Theorem 4.2 and Slutsky’s
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