Approximation and Characterization of Nash
Equilibria of Large Games
Guilherme Carmona∗
University of Cambridge and Universidade Nova de Lisboa
Konrad Podczeck†
Universität Wien
January 15, 2010
Abstract
We characterize Nash equilibria of games with a continuum of players in
terms of approximate equilibria of large finite games. This characterization
precisely describes the relationship between equilibrium sets of the two classes
of games.
Further implications of the characterization we provide are: First, it allow
us to obtain, in an unified way, limit results for equilibria of games with a finite
number of players and asymptotic properties for Nash equilibria of games with
a continuum of players. Second, it yield several approximation results for Nash
equilibria of games with a continuum of players, which roughly state that all
finite-player games that are sufficiently close to a given game with continuum of
players have approximate equilibria that are close to a given Nash equilibrium
of the non-atomic game.
∗
Address: University of Cambridge, Faculty of Economics, Sidgwick Avenue, Cambridge, CB3
9DD, UK; Phone: (44) 122 333 5225; email: [email protected].
†
Address: Institut für Wirtschaftswisenschaften, Universität Wien, Hohenstaufengrasse 9, A-1010
Wien, Austria. Email: [email protected]
1
Journal of Economic Literature Classification Numbers: C72
Keywords: Nash equilibrium; non-atomic games; large games; approximation.
1
Introduction
Games with a continuum of players are viewed as a tractable idealization of games
with a large but finite number of players. This view has been justified by studies
that establish (a) limit results for sequences of equilibria of games with a large but
finite number of players (e.g. Green (1984)) and (b) asymptotic properties of Nash
equilibria of games with a continuum of players, i.e., properties of Nash equilibria
of games with a continuum of players that hold for approximate equilibria of games
with a sufficiently large number of players (e.g., Rashid (1983)).
These results make clear that there is a relationship between equilibria of games
with a continuum of players and of games with a large but finite number of players.
Furthermore, recent work by Khan and Sun (1999), Al-Najjar (2008) and Carmona
and Podczeck (2009) establish equivalence results for equilibria of non-atomic games
and approximate equilibria of games with a large finite number of players. However,
these equivalence results was obtained either for a special class of non-atomic games
(games with a Loeb space of players in Khan and Sun (1999) and games with a
countable set of players endowed with a finitely additive distribution in Al-Najjar
(2008)) or regarding a particular property (namely, the existence of pure strategy
equilibria in Carmona and Podczeck (2009)).
As a consequence of the special nature of the above results, the precise relationship
between equilibria of games with a continuum of players and of games with a large but
finite number of players remains to be established. Thus, in this paper we ask to what
exactly do Nash equilibria of general games with a continuum of players correspond to
in terms of approximate equilibria of games with a large but finite number of players?
We answer this question by providing complete characterizations of Nash equilibria
of general games with a continuum of players in terms of approximate equilibria of
close by finite-player games. In this way, we provide a description of the exact extent
2
of the relationship between the equilibrium sets of these two classes of games.
Our first characterization result reveals that a strategy is a Nash equilibrium of a
game with a continuum of players if and only if there exist a sequence of finite-player
games with a number of players converging to infinity and a corresponding sequence
of strategies, inducing a sequence of distributions over payoff functions and actions
converging to that induced in the game with a continuum of players, such that each
strategy in the sequence is an approximate equilibrium with a level of approximation
converging to zero. Our second characterization result shows that strategy is a Nash
equilibrium of a game with a continuum of players if and only if for all sequences
of finite-player games with a number of players converging to infinity and strategies
such that their distribution over payoff functions and actions converges to that of the
game with a continuum of players, there exist a sequence of levels of approximations
converging to zero, such that the strategies are approximate equilibria relative to
those levels.
Besides answering the above conceptual question, the importance of our characterization results lies on the following two aspects. First, they allow us to obtain
simultaneously, in an unified way, limit results for equilibria of games with a finite
number of players and asymptotic properties for Nash equilibria of games with a continuum of players. This shows that these results are simply two aspects of the same
problem, namely the relationship between Nash equilibria of games with a continuum
of players and approximate equilibria of finite-player games.
Second, our characterization results yield several approximation results for Nash
equilibria of games with a continuum of players, according to which all finite-player
games that are sufficiently close to a given game with continuum of players have
approximate equilibria that are close to a given Nash equilibrium of the non-atomic
game. Compared with the few approximation results available, which to the best of
our knowledge consist only of those of Housman (1988) and Yang (2009), our results
hold under general assumptions.
Furthermore, our approximation results provide an useful tool to obtain asymptotic properties of Nash equilibria of non-atomic games. This is illustrated by pro3
viding a short proof of the existence of pure strategy approximate equilibrium in all
games with a sufficiently large finite set of players.
Our results are for games where players have a common (compact, metric) action
space and each player’s payoff function depends on his action and on the distribution
of actions chosen by all players. This provides us a sufficiently general setting where
our questions can be addressed and our results can be established using (somewhat)
elementary arguments. It is likely that these arguments can be extended to obtain
more general results, for instance, by allowing the action space to differs across players
or by allowing each player’s payoff function to depend on the choice of the others
in a more general way. Such goal may also be obtained by means of the different
but related techniques used in Balder (2007) to address the approximation of mixed
strategy equilibria of a fixed game.
The paper is organized as follows. In Section 2, we introduce our notation and
basic definitions. We present our characterization results in Section 3 and our approximation results in Section 4. Some auxiliary results are in the Appendix.
2
Notation and definitions
We consider games where all players have the same action space S and where each
player’s payoff depends on his choice and on the distribution of actions induced on S
by the choices of all players. The formal setup of the model is as follows.
The action space S common to all players is a compact metric space. We let M (S)
denote the set of Borel probability measures on S endowed with the narrow topology,1
and C the space of bounded, continuous, real-valued functions on S × M (S) endowed
with the sup-norm. Note that since S is a compact metric space, M (S) is compact
and metrizable, and hence C is a complete separable metric space.
1
Recall that if Z is a metric space, the narrow topology on the space M (Z) of Borel probability
R
measures γ on Z is the coarsest topology on M (Z) making the map γ 7→ γdf continuous for every
bounded continuous real-valued function f on Z; if Z is a compact metric space, then M (Z) is
compact and metrizable for the narrow topology.
4
The space of players is described by a probability space (T, Σ, ϕ). A game G is
then a triple G = ((T, Σ, ϕ), V, S) where (T, Σ, ϕ) is the probability space of players,
S is the action space and V is a measurable function from T to C; V (t) is the payoff
function of player t, with the interpretation that V (t)(s, γ) is player t’s payoff when
he plays action s and faces a distribution γ in M (S) induced by the actions of all
players.
We will consider only games G = ((T, Σ, ϕ), V, S) where either (T, Σ, ϕ) is atomless
and complete, or T is finite, Σ = 2T and ϕ is the uniform distribution on T (i.e.,
ϕ({t}) = 1/|T | for all t ∈ T ). The former case will be referred to as a non-atomic
game, and the latter as a finite-player game.
By a strategy f in a game G = ((T, Σ, ϕ), V, S) we mean a measurable function
f : T → S. Measurability of a strategy f ensures that the distribution of f is
defined in M (S), so that f can be evaluated by players’ payoff functions. Of course,
measurability does not impose any restriction on strategies if G is a game with finitely
many players. Note that, in any case, if f : T → S is measurable, then so is any
function f 0 : T → S which differs from f in only one point of S, so the notion of
strategy we employ captures individual deviations from any given strategy.
In the sequel, given any game G = ((T, Σ, ϕ), V, S) and any strategy f in G, the
distribution of f is denoted by ϕ ◦ f −1 , and player t’s payoff as
U (t)(f ) = V (t)(f (t), ϕ ◦ f −1 ).
(1)
Further, f \ t s denotes the strategy obtained if player t changes his choice from f (t)
to s. Thus, f \ t s is the strategy defined by setting f \ t s(t) = s, and f \ t s(t0 ) = f (t0 )
for all t0 6= t. Note that if G is non-atomic, then ϕ ◦ (f \ t s)−1 = ϕ ◦ f −1 , i.e., in a
non-atomic game no player has any impact on the distributions of actions.
For all ε ≥ 0, the set {t ∈ T : U (t)(f ) ≥ sups∈S U (t)(f \ t s) − ε} is measurable. Indeed, this is clear for a game with finitely many players. If the space
(T, Σ, ϕ) of players is non-atomic then, by the previous paragraph, this set is just
(V, f )−1 ({(u, y) ∈ C × S : u(y, ϕ ◦ f −1 ) ≥ u(x, ϕ ◦ f −1 ) − ε for all x ∈ S}) and therefore measurable because {(u, y) ∈ C×S : u(y, ϕ◦f −1 ) ≥ u(x, ϕ◦f −1 )−ε for all x ∈ S}
5
is closed in C × S and (V, f ) is (jointly) measurable (the latter holding because S and
C are separable metric spaces).
For all real numbers ε, η ≥ 0, we say that a strategy f is an (ε, η)-equilibrium of
the game G = ((T, Σ, ϕ), V, S) if
µ
¶
ϕ {t ∈ T : U (t)(f ) ≥ sup U (t)(f \ t s) − ε} ≥ 1 − η.
(2)
s∈S
Thus, in an (ε, η)-equilibrium, only a fraction of players smaller than η can gain more
than ε by deviating from f . A strategy f is an ε-equilibrium of G if it is an (ε, η)equilibrium for η = 0. Finally, a strategy f is a Nash equilibrium of G if it is an
ε-equilibrium of G for ε = 0.
If G = ((T, Σ, ϕ), V, S) is a game and f a strategy in G, ϕ ◦ (V, f )−1 denotes the
(joint) distribution of the function (V, f ), i.e. the Borel probability measure on C × S
defined by setting ϕ ◦ (V, f )−1 (B) = ϕ((V, f )−1 (B)) for every Borel set B in C × S.
Note that since C and S are separable metric spaces, (V, f ) is jointly measurable,
so ϕ ◦ (V, f )−1 is well-defined. Convergence in M (C × S) is always understood with
respect to the narrow topology on M (C × S), writing M (C × S) for the space of all
Borel measures on C × S. Let ρ denote the Prohorov metric on M (C × S) and note
that it induces the narrow topology. We abuse notation by also using ρ to denote the
Prohorov metric on M (C) and on M (S).
Let X be a metric space. The Borel σ-algebra of X is denoted by B(X). For all
x ∈ X, 1x denotes the Dirac measure at x, i.e., 1x (B) = 1 if x ∈ B and 1x (B) = 0 if
x 6∈ B, for all B ∈ B(X). If Y is a metric space and τ ∈ M (X × Y ), τX (resp. τY )
denotes the marginal distribution of τ on X (resp. Y ).
3
Characterization of Nash equilibria of non-atomic
games
In this section, we provide two characterizations of Nash equilibria of games with a
continuum of players. Both of the characterizations are in terms of (ε, η)-equilibria of
games with finitely many players which are close to a given non-atomic game. In our
6
characterizations, the notion of closeness is defined in terms of the number of players
of the finite-player game and of the distance of the distributions of payoff functions
and actions induced by the strategies and players’ payoff functions.
The two characterizations are presented in Corollary 1, which combines Theorems
1 and 2. Theorem 2 is a limit result for approximate equilibria of large finite-player
games whereas Theorem 1 is an asymptotic implementation of equilibria of games
with a continuum of players. Alternatively, we can describe Theorem 1 as seeking
properties that hold for all finite-player games that are close to a given non-atomic
game and Theorem 2 as seeking properties that hold for at one sequence of finiteplayer games converging to the given non-atomic game.
More precisely, Theorem 1 shows that for all sequences of games with a sufficiently large finite number of players and approximate equilibria in these games that
approximate in distribution a non-atomic game and a Nash equilibrium in this game
there is a vanishing level of approximation vanishes such that the strategy in each
finite-player game is an approximate equilibrium relative to that approximation level.
Theorem 1 Let G = ((T, Σ, ϕ), V, S) be a non-atomic game and f a Nash equilibrium of G. Then, for all sequences {Gn }n and {fn }n , where Gn = ((Tn , Σn , ϕn ), Vn , S)
is a finite-player game and fn is a strategy in Gn for all n ∈ N such that |Tn | → ∞
and ϕn ◦ (Vn , fn )−1 → ϕ ◦ (V, f )−1 , there exists a sequence {εn }n in R+ such that
εn → 0 and fn is an (εn , εn )-equilibrium of Gn for all sufficiently large n.
Proof. For all n ∈ N, let εn ∈ R+ ∪ {∞} be defined by
εn = inf{ε ≥ 0 : ϕn ({t ∈ Tn : Un (t)(fn ) ≥ sup Un (t)(fn \ t s) − ε}) ≥ 1 − ε}.
s∈S
We need to show that limn εn = 0. It suffices to show that for all ε > 0, limn ϕn ({t ∈
Tn : Un (t)(fn ) ≤ sups∈S Un (t)(fn \ t s) − ε}) = 0.
Set γ = ϕ ◦ f −1 , and for each n ∈ N, set γn = ϕ ◦ fn−1 . By hypothesis, γn → γ.
For each n ∈ N, let Bn = {γ 0 ∈ M (S) : ρ(γn , γ 0 ) ≤ 1/|Tn |}. Then for each n ∈ N,
Bn is compact and for each t ∈ Tn and s ∈ S, we have ϕ ◦ fn \ t s ∈ Bn , by definition
of the Prohorov metric. Note also that since |Tn | → ∞, we have γn0 → γ whenever
{γn0 }n is a sequence with γn0 ∈ Bn for each n ∈ N.
7
Since S and the sets Bn are compact, we can define continuous functions h and
hn from C × S to R+ by setting h(u, x) = max{u(y, γ) : y ∈ S} − u(x, γ) and
hn (u, x) = max{u(y, γ 0 ) : y ∈ S, γ 0 ∈ Bn } − u(x, γn ). Using the compactness of S and
the fact that γn0 → γ whenever γn0 ∈ Bn for each n ∈ N, it is straightforward to check
that hn → h uniformly on compact subsets of C × S.
Set τ = ϕ ◦ (V, f )−1 , and for each n ∈ N, set τn = ϕn ◦ (Vn , fn )−1 . Since f is a
Nash equilibrium of the non-atomic game G, we have
τ ({(u, s) ∈ C × S : h((u, s)) 6= 0}) = ϕ({t ∈ T : h ◦ (V, f )(t) 6= 0}) = 0
and hence
R
h dτ = 0.
On the other hand, note that for each n ∈ N and each t ∈ Tn , we have
sup Un (t)(fn \ t s) = sup Vn (t)(s, ϕn ◦ fn \ t s) ≤
s∈S
s∈S
Vn (t)(s, γ 0 )
sup
s∈S,γ 0 ∈B
n
because ϕ ◦ fn \ t s ∈ Bn for each s ∈ S. Consequently, given ² > 0, for each n ∈ N we
have
¡
¢
ϕn {t ∈ Tn : Un (t)(fn ) ≤ sup Un (t)(fn \ t s) − ε}
¡
s∈S
≤ ϕn {t ∈ Tn : Vn (t)(fn (t), γn ) ≤
sup
s∈S,γ 0 ∈Bn
Vn (t)(s, γ) − ε}
¢
¡
¢
= ϕn {t ∈ Tn : hn ◦ (Vn , fn )(t) ≥ ε}
¢
= τn ({(u, s) ∈ C × S : hn (u, s) ≥ ε} .
Since τn → τ by hypothesis, the conclusion follows by Lemma 1 in the Appendix.
Theorem 2 shows that if a non-atomic game and a strategy in this game can be
approximated in distribution by a sequence of games with finitely many players and
approximate equilibria in these games such that the level of approximation vanishes,
then the strategy in the non-atomic game is a Nash equilibrium.
Theorem 2 Let G = ((T, Σ, ϕ), V, S) be a non-atomic game and f a strategy in G.
If there are sequences {εn }n and {ηn }n in R+ , and sequences {Gn }n and {fn }n , where
Gn = ((Tn , Σn , ϕn ), Vn , S) is a finite-player game and fn is an (εn , ηn )-equilibrium of
Gn for each n ∈ N, such that |Tn | → ∞, ϕn ◦ (Vn , fn )−1 → ϕ ◦ (V, f )−1 , εn → 0 and
ηn → 0, then f is a Nash equilibrium of G.
8
Proof. Set τ = ϕ ◦ (V, f )−1 and γ = ϕ ◦ f −1 . It suffices to show that whenever
(u, x) ∈ supp(τ ), then u(x, γ) = maxy∈S u(y, γ).
To this end, for each n ∈ N, set τn = ϕn ◦ (Vn , fn )−1 and γn = ϕ ◦ fn−1 . Let
Sn ⊆ Tn be given as Sn = {t ∈ Tn : Un (t)(fn ) < sups∈S Un (t)(fn \ t s) − εn }. Set
An = (Vn , fn )(Sn ) and note that τn (An ) = ϕn (Sn ) for each n ∈ N. Thus τn (An ) → 0
by hypothesis.
Consider any (u, x) ∈ supp(τ ). Since τn → τ by hypothesis, by Lemma 6 in
the Appendix we may find a subsequence {τnk }k of {τn }n and, for each k ∈ N, a
point (uk , xk ) ∈ supp(τnk ) \ Ank so that (uk , xk ) → (u, x). In particular, then, for
each k ∈ N, (uk , xk ) = (Vnk , fnk )(tk ) for some tk ∈ Tnk \ Snk . Pick any y ∈ S and,
for each k ∈ N, let γk0 = ϕnk ◦ (fnk \ tk y)−1 . Then for each k ∈ N we must have
uk (xk , γnk ) ≥ uk (y, γk0 ) − εnk because tk ∈ Tnk \ Snk . Note also that γnk → γ and
hence, since 1/|Tn | → 0, γk0 → γ as well. Since εnk → 0 by the hypotheses of the
lemma, it follows that u(x, γ) ≥ u(y, γ). As y ∈ S was chosen arbitrarily, we may
conclude that u(x, γ) = maxy∈S u(y, γ).
Combining Theorems 1 and 2, we obtain the following characterizations for Nash
equilibria of non-atomic games.
Corollary 1 Let G = ((T, Σ, ϕ), V ) be a non-atomic game and f a strategy in G.
Then the following are equivalent.
1. f is a Nash equilibrium of G.
2. There are a sequence {εn }n in R+ and sequences {Gn }n and {fn }n , where Gn =
((Tn , Σn , ϕn ), Vn , S) is a finite-player game and fn is an (εn , εn )-equilibrium of
Gn for each n ∈ N, such that |Tn | → ∞, ϕn ◦ (Vn , fn )−1 → ϕ ◦ (V, f )−1 and
εn → 0.
3. If {Gn }n and {fn }n are sequences, where Gn = ((Tn , Σn , ϕn ), Vn , S) is a finiteplayer game and fn is a strategy of Gn for each n ∈ N such that |Tn | → ∞ and
ϕn ◦ (Vn , fn )−1 → ϕ ◦ (V, f )−1 , then there is a sequence {εn }n in R+ such that
εn → 0 and fn is an (εn , εn )-equilibrium of Gn for all sufficiently large n.
9
Proof. That 2 implies 1 and that 1 implies 3 follows directly from Theorem 2
and Theorem 1, respectively.
As for the other implications, recall the standard fact that if G is a non-atomic
game and f a strategy in G, then a sequence {Gn }n = {((Tn , Σn , ϕn ), Vn , S)}n of
finite-player games together with a sequence {fn }n of strategies for the Gn ’s such
that ϕn ◦ (Vn , fn )−1 → ϕ ◦ (V, f )−1 and |Tn | → ∞ do exist.2 Combining this fact with
Theorem 1, we obtain that 1 implies 2, and combining it with Theorem 2, it follows
that 3 implies 1.
4
Approximation of Nash equilibria of non-atomic
games
In this section, we present several approximation results for Nash equilibria of games
with a continuum of players. The motivation for these results arises from the characterization of Nash equilibria of non-atomic games presented in Corollary 1, which can
roughly be described as establishing the approximate continuity of the equilibrium
correspondence in the following sense. The equilibrium correspondence maps games,
represented by their distribution over payoff functions and number of players, to the
distributions over payoffs and actions induced by the game and its Nash equilibria.
Continuity of the equilibrium correspondence means, as usual, its upper and lower
hemicontinuity. It is well-know that the equilibrium correspondence is upper hemicontinuous (see Green (1984) and Housman (1988), and also Theorem 2 above). Here,
we focus on the lower hemicontinuity of the equilibrium correspondence.
Lower hemicontinuity of the equilibrium correspondence at a given non-atomic
game requires, in particular, all finite-player games, with a sufficiently large number
of players and a distribution over payoff functions sufficiently close to the one of the
given non-atomic game, to have a Nash equilibrium such that the distribution induced
2
This fact is a straightforward consequence of the fact that if τ is a Borel probability measure
on separable metric space Z, there is an equidistributed sequence in Z for τ , i.e. a sequence {zi }i
Pn
of points of Z such that limn→∞ n1 i=1 1zi = τ .
10
by it and by the players’ payoff function is close to the one induced by the given Nash
equilibrium of the non-atomic game. Although this property does not hold in general,
Theorem 3 shows that it holds for (ε, ε)-equilibria in general games. Furthermore,
Theorems 4 to 6 show that it also holds for ε-equilibrium when special assumptions,
such as convexity, compactness and equicontinuity, are added.
Theorem 3 Let G = ((T, Σ, ϕ), V, S) be a non-atomic game and f be a Nash equilibrium of G. Then, for all η, ε > 0, there is δ > 0 such that if G0 = ((T 0 , Σ0 , ϕ0 ), V 0 , S)
is a finite-player game satisfying ρ(ϕ0 ◦ (V 0 )−1 , ϕ ◦ V −1 ) < δ and 1/|T 0 | < δ, there
exists an (ε, ε)-equilibrium f 0 of G0 such that ρ(ϕ0 ◦ (V 0 , f 0 )−1 , ϕ ◦ (V, f )−1 ) < η.
Proof. Fix η, ε > 0. By Theorem 1 there is a number δ1 > 0 such that if
G0 = ((T 0 , Σ0 , ϕ0 ), V 0 , S) is a finite-player game satisfying 1/|T 0 | < δ1 and f 0 is a
strategy of G0 with ρ(ϕ0 ◦ (V 0 , f 0 )−1 , ϕ ◦ (V, f )−1 ) < δ1 , then f 0 is an (ε, ε)-equilibrium
of G0 . By Lemma 5 in the Appendix, there is a number δ2 > 0 such that if G0 =
((T 0 , Σ0 , ϕ0 ), V 0 , S) is a finite-player game satisfying ρ(ϕ0 ◦ (V 0 )−1 , ϕ ◦ V −1 ) < δ2 and
1/|T 0 | < δ2 , there is a strategy f 0 for G0 such that ρ(ϕ0 ◦ (V 0 , f 0 )−1 , ϕ ◦ (V, f )−1 ) <
min{η, δ 1 }. Then, δ = min{δ1 , δ2 } is a number as desired.
4.1
Equicontinuous and compact games
In this subsection, we consider the case of non-atomic and finite-player games whose
players’ payoff functions belong to a given equicontinuous set of payoff functions. As
Theorem 4 below shows, this allow us to strengthen the conclusion of Theorem 3
from (ε, ε)-equilibrium to ε-equilibrium. The assumption of equicontinuous payoff
functions allows us to change the actions of those players who are not ε-optimizing
in the (ε, ε)-equilibrium obtained via Theorem 3. In fact, since the fraction of these
players is small, such change has a small impact on the distribution of actions, and
due to equicontinuity, on players’ payoffs.
Theorem 4 Let K ⊆ C be equicontinuous, G = ((T, Σ, ϕ), V, S) a non-atomic game
with V (T ) ⊆ K, and f a Nash equilibrium of G. Then, for all η, ε > 0, there is δ > 0
11
such that for all finite-player games G0 = ((T 0 , Σ0 , ϕ0 ), V 0 , S) satisfying V 0 (T 0 ) ⊆ K,
ρ(ϕ0 ◦ (V 0 )−1 , ϕ ◦ V −1 ) < δ and 1/|T 0 | < δ, there exists an ε-equilibrium f 0 of G0 such
that ρ(ϕ0 ◦ (V 0 , f 0 )−1 , ϕ ◦ (V, f )−1 ) < η.
Proof. Fix η, ε > 0. By the equicontinuity of K, we can choose a number θ > 0
such that |v(s, τ ) − v(s0 , τ 0 )| < ε/4 whenever v ∈ K, d(s, s0 ) ≤ θ for s, s0 ∈ S and
ρ(τ, τ 0 ) ≤ θ for τ, τ 0 ∈ M (S). Clearly, we can choose θ so as to have in addition
θ < min{ε/2, η/2}.
By Theorem 3, there is a 0 < δ < θ such that if G0 = ((T 0 , Σ0 , ϕ0 ), V 0 , S) is a
finite-player game satisfying ρ(ϕ0 ◦ (V 0 )−1 , ϕ ◦ V −1 ) < δ and 1/|T 0 | < δ then G0 has an
(θ, θ)-equilibrium fˆ satisfying ρ(ϕ0 ◦ (V 0 , fˆ)−1 , ϕ ◦ (V, f )−1 ) < θ. Given such a game
V 0 and (θ, θ)-equilibrium fˆ of G0 , suppose V 0 (T 0 ) ⊆ K and define a strategy f 0 in G0
by letting f 0 (t) be a solution to maxs∈S V (t)(s, ϕ0 ◦ fˆ−1 ) in case V (t)(fˆ(t), ϕ0 ◦ fˆ−1 ) <
sups∈S V (t)(s, ϕ0 ◦ (fˆ\ t s)−1 ) − θ, and f 0 (t) = fˆ(t) otherwise.
As fˆ is an (θ, θ)-equilibrium of G0 , the fraction of players t in G0 for which fˆ(t)
is not within θ of a best reply to ϕ0 ◦ fˆ−1 is smaller than θ. Thus the fraction
of players t in G0 for which fˆ(t) differs from f 0 (t) is smaller than θ. This implies
ρ(ϕ0 ◦ f 0−1 , ϕ0 ◦ fˆ−1 ) ≤ θ and that ρ(ϕ0 ◦ (f 0 \ t s)−1 , ϕ0 ◦ (fˆ\ t s)−1 ) ≤ θ for all t ∈ T 0
and s ∈ S. Note also that ρ(ϕ0 ◦ fˆ−1 , ϕ0 ◦ (fˆ\t s)−1 ) < θ since 1/|T 0 | < θ. These facts,
together with the fact that V 0 (T 0 ) ⊆ K, imply that f 0 is an ε-equilibrium. Indeed, if
t ∈ T 0 is such that V (t)(fˆ(t), ϕ0 ◦ fˆ−1 ) < sups∈S V (t)(s, ϕ0 ◦ (fˆ\t s)−1 ) − θ, then for all
s ∈ S, V (t)(f 0 (t), ϕ0 ◦ f 0−1 ) > V (t)(f 0 (t), ϕ0 ◦ fˆ−1 ) − ε/4 ≥ V (t)(s, ϕ0 ◦ fˆ−1 ) − ε/4 >
V (t)(s, ϕ0 ◦ (fˆ\ t s)−1 ) − 2ε/4 > V (t)(s, ϕ0 ◦ (f 0 \ t s)−1 ) − 3ε/4. Otherwise, i.e., if
ˆ t s)−1 ) − θ, then for all s ∈ S we have that
V (t)(fˆ(t), ϕ0 ◦ fˆ−1 ) ≥ sups∈S V (t)(s, ϕ0 ◦ (f\
f 0 (t) = fˆ(t) and V (t)(f 0 (t), ϕ0 ◦ f 0−1 ) > V (t)(fˆ(t), ϕ0 ◦ fˆ−1 ) − ε/4 > V (t)(s, ϕ0 ◦ (fˆ\
−1
t s) )
− ε/4 − θ > V (t)(s, ϕ0 ◦ (f 0 \ t s)−1 ) − 2ε/4 − θ > V (t)(s, ϕ0 ◦ (f 0 \ t s)−1 ) − ε.
Finally, note that ρ(ϕ0 ◦ (V 0 , fˆ)−1 , ϕ0 ◦ (V 0 , f 0 )−1 ) ≤ θ and so ρ(ϕ0 ◦ (V 0 , f 0 )−1 , ϕ ◦
(V, f )−1 ) < 2θ = η.
Theorem 5 below improves over Theorem 4 by providing an uniformity over the
non-atomic games with V (T ) ⊆ K and its Nash equilibria. This is achieved by
assuming that players’ payoff functions are contained in a set in C which is not only
12
equicontinuous but also bounded.
Theorem 5 Let K be a compact subset of C. Then, for all ε > 0, there is a δ > 0
such that if G = ((T, Σ, ϕ), V, S) is a non-atomic game with V (T ) ⊆ K, and f a Nash
equilibrium of G, then every finite-player game G0 = ((T 0 , Σ0 , ϕ0 ), V 0 , S) satisfying
V 0 (T 0 ) ⊆ K, 1/|T 0 | < δ and ρ(ϕ0 ◦ (V 0 )−1 , ϕ ◦ V −1 ) < δ has an ε-equilibrium f 0 such
that ρ(ϕ0 ◦ (V 0 , f 0 )−1 , ϕ ◦ (V, f )−1 ) < ε.
Proof. Let
©
Z = τ ∈ M (K × S) : there is a non-atomic game G = (T, Σ, ϕ), V, S)
ª
with V (T ) ⊆ K and a Nash equilibrium f of G such that τ = ϕ ◦ (V, f )−1 .
Using the fact that every τ ∈ M (K × S) can be represented as τ = ϕ ◦ (V, f )−1 for
some mapping (V, f ) from an atomless probability space (T, Σ, ϕ) to K ×S, arguments
analogous to that in the proof of Theorem 2 show that Z is closed in M (K ×S), hence
compact because compactness of K × S implies that M (K × S) is compact.
Fix ε > 0. Noting that K, being compact, is equicontinuous by the Ascoli-Arzela
Theorem, for each τ = ϕ ◦ (V, f )−1 ∈ Z choose a number δτ > 0 corresponding
to G = ((T, Σ, ϕ), V, S), f , ε and η = ε/2 according to Theorem 4, and then let
Uτ = {τ 0 ∈ Z : ρ(τK0 , τK ) < δτ /2, ρ(τ 0 , τ ) < ε/2)}. Then, for each τ ∈ Z, Uτ is
an open neighborhood of τ in Z, and by the choice of δτ and the triangle inequality
it is plain that if Ĝ = ((T̂ , Σ̂, ϕ̂), V̂ , S) is a finite-player game satisfying V̂ (T̂ ) ⊆ K,
ρ(ϕ̂ ◦ V̂ −1 , τK0 ) < δτ /2 for some τ 0 ∈ Uτ and 1/|T̂ | < δτ , there is an ε-equilibrium fˆ of
Ĝ such that ρ(ϕ̂ ◦ (V̂ , fˆ)−1 , τ 0 ) < ε.
Being compact, Z can be covered by finitely many of the sets Uτ , say Uτ1 , . . . Uτn ,
and setting δ = min{δτi /2 : i = 1, . . . , n}, we have a δ > 0 as desired.
4.2
Equicontinuous and convex games
Additional assumptions are needed to strengthen the conclusion of Theorem 3 from
(ε, ε)-equilibrium to ε-equilibrium. In this subsection, we consider the case of nonatomic games whose players have payoff functions belonging to an equicontinuous set
13
and of finite-player games that are convex, in the sense that players’ action spaces are
a convex subset of a vector space and each player’s payoff function is quasiconcave
in his own choice. These assumptions will allow us to change the actions of those
players who are not ε-optimizing in the (ε, ε)-equilibrium obtained via Theorem 3.
Roughly, those players will play a Nash equilibrium of the game obtained when we
fixed the strategy of the other players.
Theorem 6 obtains, in a simple way, the approximation result (Theorem 7) in
Housman (1988) when specialized to the case of games with finitely many players
approximating a non-atomic game. As described in the previous paragraph, the
simplification comes from the use of Theorem 3 to obtain an (ε, ε)-equilibrium, which
is then modified following the approach in Housman (1988) to obtain an ε-equilibrium.
The notion of a convex game are defined as follows. A game G = ((T, Σ, ϕ), V, S)
is convex if S is convex and x 7→ V (t)(x, (1 − 1/|T |)π + 1x /|T |) is quasiconcave for
all t ∈ T and π ∈ M (S).
Theorem 6 Let G = ((T, Σ, ϕ), V, S) be an equicontinuous non-atomic game. Suppose that S is a convex subset of a vector space and that f is a Nash equilibrium of G.
Then for all η, ε > 0, there is δ > 0 such that if G0 = ((T 0 , Σ0 , ϕ0 ), V 0 , S) is a convex
finite-player game satisfying ρ(ϕ0 ◦ (V 0 )−1 , ϕ ◦ V −1 ) < δ and 1/|T 0 | < δ, there exists
an ε-equilibrium f 0 of G0 such that ρ(ϕ0 ◦ (V 0 , f 0 )−1 , ϕ ◦ (V, f )−1 ) < η.
Proof. Let ε, η > 0 be given. Let 0 < θ < ε/6 be given and let 0 < ζ < η/2 be
such that |v(s, τ ) − v(s0 , τ 0 )| < ε/4 whenever d(s, s0 ) < ζ and ρ(τ, τ 0 ) < ζ, s, s0 ∈ S,
τ, τ 0 ∈ M (S) and v ∈ V (T ).
Let Bθ (V (T )) = {u ∈ C : there exists v ∈ V (T ) such that ||u − v|| < θ}. Let
0 < β < ζ/2 be given and let δ 0 > 0 be such that ϕ0 ◦ (V 0 )−1 (Bθ (V (T ))) > 1 − β for all
finite-player games G0 satisfying ρ(ϕ0 ◦ (V 0 )−1 , ϕ ◦ V −1 ) < δ 0 . Let 0 < α < min{θ, ζ/2}
and let 0 < δ < δ 0 be given by Theorem 3 corresponding to ε = α and η/2.
Let G0 be a finite-player game satisfying ρ(ϕ0 ◦(V 0 )−1 , ϕ◦V −1 ) < δ and 1/|T 0 | < δ.
Then, G0 has an (α, α)-equilibrium fˆ satisfying ρ(ϕ0 ◦ (V 0 , fˆ)−1 , ϕ ◦ (V, f )−1 ) < η/2.
Define a strategy f¯ by changing the actions of those players with payoff functions
14
in Bθ (V (T )) that are not within α of their best-replies to best-replies to ϕ0 ◦ fˆ−1 .
ˆ t s)−1 ) − α} and,
Formally, let T̄ = {t ∈ T 0 : V (t)(fˆ, ϕ0 ◦ fˆ−1 ) ≤ maxs∈S V (t)(s, ϕ0 ◦ (f\
for all t ∈ T̄ , g(t) be such that V (t)(g(t), ϕ0 ◦ fˆ−1 ) ≥ V (t)(s, ϕ0 ◦ fˆ−1 ). Then, for all
t ∈ T 0 , let f¯(t) = g(t) if t ∈ (V 0 )−1 (Bθ (V (T ))) ∩ T̄ and f¯(t) = fˆ(t) otherwise.
Consider the game played by the players with payoff functions outside Bθ (V (T ))
with the action of the others fixed at f¯: formally, G̃ = ((T̃ , Σ̃, ϕ̃), Ṽ , S) is defined
by T̃ = (V 0 )−1 (Bθ (V (T ))c ), Σ̃ = 2T̃ , ϕ̃ is the uniform probability measure on T̃ and
Ṽ (t)(s, τ ) = V 0 (t)(s, h(τ )), where, for all Borel measurable subset B of S, h(τ )(B) =
(|{t ∈ T 0 \ T̃ : f¯(t) ∈ B}| + |T̃ |τ (B))/|T 0 |. Since G0 is convex, so is G̃. Hence, G̃ has
a Nash equilibrium f˜.
Define f 0 by setting f 0 (t) = f¯(t) if V 0 (t) ∈ Bθ (V (T )) and f 0 (t) = f˜(t) if V 0 (t) ∈
Bθ (V (T ))c . We claim that f 0 is an ε-equilibrium. Note first that ρ(ϕ0 ◦ (fˆ\t s)−1 , ϕ0 ◦
(f¯\ t s)−1 ) ≤ α < ζ and that ρ(ϕ0 ◦ (f¯\ t s)−1 , ϕ0 ◦ (f 0 \ t s)−1 ) ≤ β < ζ for all t ∈ T 0 and
s ∈ S. Note also that h(ϕ0 ◦ f˜−1 ) = ϕ0 ◦ f 0−1 and h(ϕ0 ◦ f˜\ t s−1 ) = ϕ0 ◦ f 0 \ t s−1 for all
t ∈ T̃ and s ∈ S. Thus, if t ∈ T̃ , then V (t)(f 0 (t), ϕ0 ◦ f 0−1 ) = Ṽ (t)(f˜(t), ϕ0 ◦ f˜−1 ) ≥
Ṽ (t)(s, ϕ0 ◦ (f˜\ t s)−1 ) = V (t)(s, ϕ0 ◦ (f 0 \ t s)−1 ).
Finally, consider t ∈ T 0 \ T̃ . Then, there exists u ∈ V (t) such that u ∈ Bθ (V (t)).
If t ∈ T 0\T̄ , then f¯(t) = fˆ(t) and, for all s ∈ S, u(f¯(t), ϕ0 ◦ f¯−1 ) > u(fˆ, ϕ ◦ fˆ−1 ) − θ >
V (t)(fˆ, ϕ ◦ fˆ−1 ) − 2θ > V (t)(s, ϕ ◦ (fˆ \ t s)−1 ) − 3θ > u(s, ϕ ◦ (fˆ \ t s)−1 ) − 4θ >
¯ t s)−1 ) − 5θ. If t ∈ T̄ , then f¯(t) = g(t) and, for all s ∈ S, u(f¯(t), ϕ0 ◦ f¯−1 ) >
u(s, ϕ ◦ (f\
u(g(t), ϕ0 ◦ fˆ−1 ) − θ > V (t)(g(t), ϕ0 ◦ fˆ−1 ) − 2θ ≥ V (t)(s, ϕ0 ◦ fˆ−1 ) − 2θ > u(s, ϕ0 ◦
fˆ−1 ) − 3θ > u(s, ϕ0 ◦ f¯−1 ) − 4θ > u(s, ϕ0 ◦ (f¯\ t s)−1 ) − 5θ. Thus, for all t ∈ T 0 \ T̃ , it
follows that V (t)(f 0 (t), ϕ0 ◦ (f 0 )−1 ) > u(f 0 (t), ϕ0 ◦ (f 0 )−1 ) − θ > u(f¯(t), ϕ0 ◦ f¯−1 ) − 2θ >
u(s, ϕ0 ◦ (f¯\t s)−1 ) − 7θ > u(s, ϕ0 ◦ (f 0 \t s)−1 ) − 8θ > V (t)(s, ϕ0 ◦ (f 0 \t s)−1 ) − 9θ. Since
9θ < ε, then f˜ is an ε-equilibrium.
To conclude, we have that ρ(ϕ0 ◦ (V 0 , f 0 )−1 , ϕ0 ◦ (V 0 , fˆ)−1 ) ≤ α + β < ζ < η/2 and
so ρ(ϕ0 ◦ (V 0 , f 0 )−1 , ϕ ◦ (V, f )−1 ) < η.
15
4.3
Existence of equilibrium in large games
Our approximation results allow us to obtain easily asymptotic result regarding the
existence and purification of approximate equilibria from analogous results for nonatomic games.
Regarding existence of equilibrium, Theorem 7 combines Theorem 4 with Theorem 1 in Mas-Colell (1984) to provide a short proof of the asymptotic existence result
in Carmona and Podczeck (2009) in the case where the players’ payoff functions
belong to a compact subset of C.
Theorem 7 For all compact subsets K of C and all ε > 0, there exists N ∈ N such
that every finite-player game G = ((T, Σ, ϕ), V, S) satisfying V (T ) ⊆ K and |T | ≥ N
has an ε-equilibrium.
Proof. Let ε > 0 be given. Let δ > 0 be as in Theorem 5 (corresponding to
K and ε) and let N ∈ N be such that 1/N < δ. Let G = ((T, Σ, ϕ), V, S) be a
game with |T | ≥ N and V (T ) ⊆ K. By Mas-Colell (1984, Theorem 1) there is a
τ ∈ M (C × S) such that τ is an equilibrium distribution for ϕ ◦ V −1 . By a standard
fact, we can represent τ as τ = ϕ̂ ◦ (V̂ , fˆ)−1 , where (V̂ , fˆ) is a measurable mapping
from an atomless probability space (T̂ , Σ̂, ϕ̂) to C × S. Then, fˆ is a Nash equilibrium
of the non-atomic game Ĝ = ((T̂ , Σ̂, ϕ̂), V̂ , S). Since ϕ̂ ◦ V̂ −1 = ϕ ◦ V −1 , V (T ) ⊆ K
and 1/|T | < δ, it follows from Theorem 5 that G has an ε-equilibrium.
A
Appendix
In this appendix, we collect the lemmas that are used in the proof of our main
results. As before, convergence of measures on metric spaces is always understood
for the narrow topology.
Lemma 1 Let Z be a metric space, let {τn } be a sequence in M (Z) converging to
some τ , and let {hn } be a sequence of continuous functions from Z to R+ . Suppose
R
there is a continuous function h : Z → R+ with h dτ = 0 such that hn → h uniformly
16
on compact subsets of Z. Then, for every ε > 0,
lim τn ({z ∈ Z : hn (z) ≥ ²}) = 0.
n→∞
Proof. By Hildenbrand (1974, p. 51, 38), the convergence hypotheses imply that
−1
τn ◦ h−1
n → τ ◦ h . In particular, for every ² > 0, we must have
−1
lim τn ◦ h−1
n ({r ∈ R+ : r ≥ ²}) ≤ τ ◦ h ({r ∈ R+ : r ≥ ²}),
whence lim τn ({z ∈ Z : hn (z) ≥ ²}) ≤ τ ({z ∈ Z : h(z) ≥ ²}). On the other hand,
R
since h dτ = 0, we have τ ({z ∈ Z : h(z) ≥ ²}) = 0 for every ² > 0.
For finite set, the consequences of convergence of probability measures are easy to
understand because it implies that the probabilities of each point in the set converge to
the corresponding limit probability. This property does not hold for general separable
metric spaces. However, Lemma 2 shows that, for every probability measure, the
space can be partitioned into a countable collection of measurable subsets with a
small diameter such that the probabilities of those sets converge to the corresponding
limit probability.
Lemma 2 Let X be a separable metric space, and µ a Borel probability measure on
X. Then given any ε > 0 there is a countable partition (Ei )i∈N of X into Borel sets,
each with diameter less than ε, such that whenever {µn }n is sequence in M (X) with
µn → µ, then µn (Ei ) → µ(Ei ) for each i ∈ N.
Proof. Recall the following facts, denoting by ∂A the boundary of a subset A of
a topological space Z.
(a) If Z is a metric space and µ a probability measure on Z, then the set of open
sets O with µ(∂O) = 0 is a base for the topology of Z; see Fremlin (2003,
411G(i)).
(b) If Z is a metric space and T is any base for the topology of Z, then give ε > 0
there is a set T 0 ⊆ T such that each element of T 0 has diameter less than ε and
T 0 is still a base. (To see this, let O be an open subset of Z and x ∈ O. There is
17
a real number r > 0 with 0 < r < ε/2 such that Br (x) ⊆ O where Br (x) is the
open r-ball around x. As T is a base, there is V ∈ T such that x ∈ V ⊆ Br (x),
and such set V must have diameter less than ε.)
(c) If Z is any topological space and A, B are subsets of Z, then ∂(B\A) ⊆ ∂B ∪∂A;
S
S
if A0 , . . . , An are finitely many subsets of Z, then ∂( ni=0 Ai ) ⊆ ni=0 ∂Ai .
Let µ be a Borel probability measure on X and fix ε > 0. Since X is second
countable, using (a) and (b) it follows that there is a countable family (Bi )i∈N of
S
open subsets of X with ∞
i=0 Bi = X such that, for each i ∈ N, the diameter of Bi
is less than ε and µ(∂Bi ) = 0. Define a family (Ei )i∈N of Borel sets of X by setting
S
Ei = Bi \ i−1
j=0 Bj for each i ∈ N. Use (c) and the fact that the union of finitely many
null sets is a null set to see that µ(∂Ei ) = 0 for each i ∈ N. Hence, the conclusion
follows from the Portmanteau Theorem.
Lemma 3 considers a sequence of functions converging in distribution and shows
that both the limit function and the ones in the sequence can be closely approximated
by functions having a finite range. By finite probability space we mean a probability
space (T, Σ, ϕ) where T is finite, Σ = 2T , and ϕ is the uniform distribution.
Lemma 3 Let (X, dX ) be a separable metric space, (T, Σ, ϕ) a probability space and
g : T → X a measurable mapping. Further, let {(Tn , Σn , ϕn )}n∈N be a sequence of
finite probability spaces together with mappings gn : Tn → X, n ∈ N, such that ϕn ◦
gn−1 → ϕ ◦ g −1 . Then, for all ε > 0, there is a finite set F ⊆ X and mappings
ḡ : T → F and ḡn : Tn → F , for all n ∈ N, such that ϕn ◦ ḡn−1 → ϕ ◦ ḡ −1 ,
ϕ({t ∈ T : dX (ḡ(t), g(t)) ≤ ε}) > 1 − ε,
and
ϕn ({t ∈ Tn : dX (ḡn (t), gn (t)) ≤ ε}) > 1 − ε
for all sufficiently large n.
Proof. Fix ε > 0 and let (Ei )i∈N be a partition of X chosen with respect to ϕ◦g −1
S
and ε according to Lemma 2. We can find an ī such that ϕ ◦ g −1 ( i≥ī Ei )) < ε. For
each i ≤ ī, pick a point xi ∈ Ei and let F = {xi : 1 ≤ i ≤ ī}.
18
Define ḡ : T → F by setting ḡ(t) = xi if g(t) ∈ Ei for i < ī and ḡ(t) = xī if
S
g(t) ∈ i≥ī Ei . Analogously, for each n ∈ N, define ḡn : Tn → F by setting ḡn (t) = xi
S
if gn (t) ∈ Ei for i < ī, and ḡn (t) = xī if gn (t) ∈ i≥ī Ei .
S
Sī−1
−1
By choice of (Ei )i∈N , we have ϕn ◦ gn−1 ( ī−1
i=0 Ei ) → ϕ ◦ g ( i=0 Ei ) and hence
S
S
S
ϕn ◦ gn−1 ( i≥ī Ei ) → ϕ ◦ g −1 ( i≥ī Ei ). Consequently ϕn ◦ gn−1 ( i≥ī Ei ) < ε for all
sufficiently large n, whence, since the diameter of Ei is at most ε by choice of (Ei )i∈N ,
we have
ϕn ({t ∈ Tn : dX (ḡn (t), gn (t)) ≤ ε}) > 1 − ε
for all sufficiently large n. Similarly, we have ϕ({t ∈ T : dX (ḡ(t), g(t)) ≤ ε}) > 1 − ε.
Finally note that we have
ϕn ◦ ḡn−1 ({xi }) = ϕn ◦ gn−1 (Ei ) → ϕ ◦ g −1 (Ei ) = ϕ ◦ ḡ −1 ({xi })
for each 1 ≤ i < ī, and therefore also ϕn ◦ ḡn−1 ({xī }) → ϕ̄ ◦ g −1 ({xī }). Thus, since
ϕ◦ ḡ −1 and all the measures ϕn ◦ ḡn−1 have a support in the same finite set {x1 , . . . , xī },
we have ϕn ◦ ḡn−1 → ϕ ◦ ḡ −1 .
Lemma 5 considers a lower hemicontinuity property of the correspondence that
assigns to each game the of strategies available to players. The lower hemicontinuity
property considered is analogous to the one considered in our approximation results,
in the sense that it is only established for non-atomic games and only finite-player
games are considered as approximations. Lemma 4 is used in its proof and, although
stated abstractly, it considers a special case of Lemma 5, namely that of non-atomic
games with finitely many actions and payoff functions.
Lemma 4 Let X and Y be finite sets, and τ a probability measure on X × Y . If
{(Tn , Σn , ϕn )}n∈N is a sequence of finite probability spaces with |Tn | → ∞, and for
each n ∈ N, gn : Tn → X is such that ϕn ◦ gn−1 → τX , then there is a mapping
fn : Tn → Y for each n ∈ N such that ϕn ◦ (gn , fn )−1 → τ .
Proof. Let X = {x1 , . . . , xk } and Y = {y1 , . . . , ym }. For each 1 ≤ i ≤ k, 1 ≤ j ≤
m, and n ∈ N, let µi = τX ({xi }), τi,j = τ ({(xi , yj )}), and Tin = {t ∈ Tn : gn (t) = xi }.
P
Note that for each 1 ≤ i ≤ k, |Tin |/|Tn | → µi by hypothesis and that m
j=1 τi,j = µi .
19
n
Making the convention 0/0 = 0, for each 1 ≤ i ≤ k and n ∈ N, define θi,j
=
P
n
n
= |Tin | − m−1
max{θ ∈ N : θ ≤ |Tin |τi,j /µi } for 1 ≤ j ≤ m − 1, and θi,m
j=1 θi,j . We can
then choose a mapping fn : Tn → Y for each n ∈ N such that
¯
¯
n
¯{t ∈ Tn : (gn (t), fn (t)) = (xi , yj )}¯ = θi,j
for all 1 ≤ i ≤ k and 1 ≤ j ≤ m. Note that for each 0 ≤ i ≤ k and 1 ≤ j ≤ m − 1,
n
θi,j
1
|Tin |τi,j
|Tin |τi,j
−
<
≤
|Tn |µi
|Tn |
|Tn |
|Tn |µi
n
(again, 0/0 = 0), showing that θi,j
/|Tn | → τi,j for such i and j because |Tin |/|Tn | → µi
P
n
and 1/|Tn | → 0. Consequently, we also have θi,m
/|Tn | → µi − m−1
j=1 τi,j = τi,m for
each 1 ≤ i ≤ k. It is now plain that ϕn ◦ (gn , fn )−1 → τ .
Lemma 5 Let G = ((T, Σ, ϕ), V, S) be a non-atomic game and f a strategy in G.
Then for each ε > 0 there exists δ > 0 such that for all finite-player games G0 =
((T 0 , Σ0 , ϕ0 ), V 0 , S) satisfying ρ(ϕ0 ◦ (V 0 )−1 , ϕ ◦ V −1 ) < δ and 1/|T 0 | < δ there exists a
strategy f 0 in G0 such that ρ(ϕ0 ◦ (V 0 , f 0 )−1 , ϕ ◦ (V, f )−1 ) < ε.
Proof. It suffices to show that, given ε > 0, if {Gn }n = {((Tn , Σn , ϕn ), Vn , S)}n
is any sequence of games with finitely many players satisfying both |Tn | → ∞ and
ϕn ◦ Vn−1 → ϕ ◦ V −1 , then for all sufficiently large n there are strategies fn : Tn → S
such that ρ(ϕn ◦ (Vn , fn )−1 , ϕ ◦ (V, f )−1 ) < ε.
Fix ε > 0 and let {Gn }n be a sequence of games with finitely many players
satisfying the above requirements. By Lemma 3, there are a finite set F ⊆ C and
measurable mappings V̄ : T → F and V̄n : Tn → F , for all n ∈ N, such that the
following hold (a) ϕn ◦ V̄n−1 → ϕ ◦ V̄ , (b) ϕ({t ∈ T : ||V̄ (t) − V (t)|| ≤ ε/4}) > 1 − ε/4,
and (c) for all sufficiently large n, ϕn ({t ∈ Tn : ||V̄n (t) − Vn (t)|| ≤ ε/4}) > 1 − ε/4.
Since S is compact, we can choose a finite partition {Bl }m
l=1 of S into Borel sets,
each with diameter less than or equal to ε/4. For each 1 ≤ l ≤ m pick a point sl ∈ Bl .
Set Y = {s1 , . . . , sm } and let f¯ : T → Y be defined by f¯(t) = sl if f (t) ∈ Bl . Then,
from (b) above, ρ(ϕ ◦ (V, f )−1 , ϕ ◦ (V̄ , f¯)−1 ) ≤ ε/4 (for an appropriate product metric
on C × S).
20
As F and Y are finite, by (a) we can use Lemma 4 to find mappings fn : Tn → Y ,
n ∈ N, such that ϕn ◦(V̄n , fn )−1 → ϕ◦(V̄ , f¯)−1 , and so ρ(ϕ◦(V̄ , f¯)−1 , ϕn ◦(V̄n , fn )−1 ) <
ε/4 for all sufficiently large n. Note that from (c) we must have ρ(ϕn ◦ (V̄n , fn )−1 , ϕn ◦
(Vn , fn )−1 ) ≤ ε/4 for all sufficiently large n. Using the triangle inequality, it follows
that ρ(ϕ ◦ (V, f )−1 , ϕn ◦ (Vn , fn )−1 ) < ε for all sufficiently large n, as was to be shown.
Recall that every Borel probability measure on a complete separable metric space
has a support. Lemma 6 below considers a converging sequence of measures and
establishes a property of the support of the limit measure. It appeared before as
Lemma 12 in Carmona and Podczeck (2009), and so its proof is omitted.
Lemma 6 Let Z be a complete separable metric space, {τk }∞
k=1 be a sequence in
M (Z) converging to τ ∈ M (Z), and {Ak }∞
k=1 be a sequence of Borel subsets of Z with
limk τk (Ak ) = 0. Then, for all z ∈ supp(τ ), there exists a subsequence {τkj }∞
j=1 of
{τk }∞
k=1 and an element zj ∈ supp(τkj ) \ Akj for all j ∈ N such that limj zj = z.
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