Modeling Infinitely Many Agents∗
Wei He†
Xiang Sun‡§
Yeneng Sun¶
May 27, 2013
Abstract
We propose to use the condition of “nowhere equivalence” to model the space
of agents. We show that this condition is more general than all of those special
conditions imposed on the spaces of agents to handle the failure of the classical
Lebesgue unit interval as an agent space. We also illustrate the optimality of this
condition by showing its necessity in deriving the existence of Nash equilibrium
for atomless games.
JEL classification: C7, D0, D5
Keywords: Agent spaces; Nowhere equivalence; Conditional atomlessness;
Relative saturation; Large game; Large game with traits; Existence of Nash equilibria.
∗
This paper has been presented at various places, including the fifth AEI-Four Joint Workshop in
Economics, Jeju, March 28, 2013, Clarence Tow Lecture at University of Iowa, May 1, 2013 (http:
//tippie.uiowa.edu/economics/tow/), and Mathematical Economics – Theory and Language: A conference in Honor of Ali Khan, May 4, 2013 (https://sites.google.com/site/alikhanconference/).
The authors thank Ali Khan, Satoru Takahashi and Nicholas C. Yannelis for encouragement.
†
Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076. E-mail: [email protected].
‡
Department of Mathematical Economics and Finance, Economics and Management School, Wuhan
University, Wuhan, Hubei, 430072, China. E-mail: [email protected].
§
Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076.
¶
Department of Economics, National University of Singapore, 1 Arts Link, Singapore 117570. Email: [email protected].
1
Contents
1 Introduction
3
2 Characterizations of the agent space
4
3 Application of nowhere equivalence in large games
8
4 Unification
4.1
10
Standard representation . . . . . . . . . . . . . . . . . . . . . . . .
10
4.1.1
Walrasian distributions in large economies . . . . . . . . . .
10
4.1.2
Games with many players . . . . . . . . . . . . . . . . . . .
11
4.2
Hyperfinite agent space . . . . . . . . . . . . . . . . . . . . . . . .
12
4.3
Saturated agent space . . . . . . . . . . . . . . . . . . . . . . . . .
13
4.4
Many more players than strategies . . . . . . . . . . . . . . . . . .
14
4.5
Many players of every type . . . . . . . . . . . . . . . . . . . . . .
15
5 Appendix: Proof of Theorem 1
17
5.1
Sufficiency part . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
5.2
Necessity part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2
1
Introduction
Every economic model involves economic agents. When a model considers a fixed
finite number of agents, the most natural agent space is the set {1, 2, . . . , n} for
some positive integer n. In a vast literature in economics, one also needs to model
the interaction of many agents in order to discover mass phenomena that do not
necessarily occur in the case of a fixed finite number of agents.1 A well-known
example is the Edgeworth conjecture that the set of core allocations will shrink to
the set of competitive equilibria as the number of agents goes to infinity though
the former set is in general strictly bigger than the latter set for an economy with
a fixed finite number of agents.2
To avoid complicated combinatorial arguments that may involve multiple steps
of approximations for a large but finite number of agents, it is natural to consider
economic models with an infinite number of agents. The mathematical abstraction
of an atomless (countably-additive) measure space of agents provides a convenient
idealization for a large but finite number of agents. The archetype space in such
a setting is the classical Lebesgue unit interval. That is why a general atomless
measure space of agents is often referred to as a continuum of agents in a large
economics literature.
However, it has also been found that the Lebesgue unit interval does not have
a number of desirable properties in various situations as an agent space. First, it
is pointed out in general equilibirum theory that large economies with the same
distribution may not have the same set of Walrasian equilibrium distributions.
Second, pure-strategy Nash equilibria may not exist in the large game with uncountably many actions. Third, the dissonance between a large game and its
discretized versions has also been found. In addition, from a mathematical point
of view, some regularity properties (such as convexity, compactness, purification,
and upper-semicontinuity) of the distribution of correspondences and of the integration of correspondences in an infinite-dimensional setting fail to hold when the
underlying measure space is the Lebesgue unit interval. To resolve those problems, different types of measure spaces have been proposed, such as the standard
representation of a distribution on the characteristics, hyperfinite agent spaces,
1
As pointed out by von Neumann and Morgenstern (1953), “When the number of participants
becomes really great, some hope emerges that the influence of every particular participant will become
negligible, and that the above difficulties may recede and a more conventional theory become possible.
Indeed, this was the starting point of much of what is best in economic theory.” For more discussions
of mass phenomena in economics, see Khan and Sun (2002).
2
See Debreu and Scarf (1963).
3
saturated probability spaces, and agents spaces with the condition of “many more
agents than strategies” or “many agents of every type”.
A key point in this paper is to separate the concept of an agent space with
the concept of the characteristics type space which is generated by the mapping
of agents’ characteristics. The “nowhere equivalence” condition proposed here
requires that the agent space is strictly richer than the characteristics type space
on any nontrivial collection of agents. We will show that this condition is more
general than all the special conditions mentioned in the end of the last paragraph.
We will also show that it can be used to handle the failure of the Lebesgue unit
interval as discussed above. In addition, we illustrate the optimality of the nowhere
equivalence condition by showing that it is necessary and sufficient for deriving
certain game-theoretic results.3
This paper is organized as follows. Section 2 presents the condition of nowhere
equivalence for agent spaces, and shows that it is equivalent to three other conditions. In Section 3, we prove that the condition of nowhere equivalence is necessary
and sufficient for the existence of pure-strategy Nash equilibria in games with many
agents. In Section 4, we show that the condition of nowhere equivalence is more
general than all of those special conditions imposed on the spaces of agents to
handle the failure of the classical Lebesgue unit interval. Some technical proofs
are given in Section 5.
2
Characterizations of the agent space
A typical economic model starts with an agent space. Each agent is described
by some characteristics, such as strategy/action set, payoff, preference, endowment/income, information, social or bio-logical traits and etc. The mapping for
the characteristics of all the agents will generate a sub-σ-algebra on the agent
space. Thus it is natural to restrict our attention to a sub-σ-algebra that is to be
used in modeling agents’ characteristics. The corresponding restricted probability
space on such a sub-σ-algebra will be called the characteristics type space. In this
section, we will introduce several conditions on these two probability spaces and
show their equivalence.
Let (Ω, F, P ) be an atomless probability space with a complete countablyadditive probability measure P .4 Let G be a sub-σ-algebra of F. The probability
3
4
He and Sun (2013) provide parallel results of finite-player games with incomplete information.
A probability space (Ω, F, P ) (or its σ-algebra) is atomless if for any nonnegligible subset E ∈ F ,
4
spaces (Ω, F, P ) and (Ω, G, P ) will be used to model the agent space and the
characteristics type space respectively. For any nonnegligible subset D ∈ F, the
restricted probability space (D, G D , P D ) is defined as follows: G D is the σ-algebra
{D ∩ D′ : D′ ∈ G} and P D the probability measure re-scaled from the restriction
of P to G D . Furthermore, (D, F D , P D ) can be defined similarly.
Let X, Y denote Polish spaces (complete separable metrizable topological spaces),
and M(X) the space of all Borel probability measures on X with the weak topology. We recall that M(X) is again a Polish space. For any µ ∈ M(X × Y ), let
µX be the marginal of µ on X.
Now we are ready to present the following definition.
Definition 1. (1) F is said to be nowhere equivalent to G if for every D ∈ F
with P (D) > 0, there exists an F-measurable subset D0 of D such that
P (D0 △D1 ) > 0 for any D1 ∈ G D .
(2) F is conditional atomless over G if for every D ∈ F with P (D) > 0, there
exists an F-measurable subset D0 of D such that on some set of positive
probability,
0 < P (D0 | G) < P (D | G).
(3) F is said to be relatively saturated with respect to G if for any Polish spaces
X and Y , any measure µ ∈ M(X × Y ), and any G-measurable mapping f
from Ω to X with P ◦ f −1 = µX , there exists an F-measurable mapping g
from Ω to Y such that µ = P ◦ (f, g)−1 .
(4) G admits an atomless independent supplement in F if there exists another sub-σ-algebra H of F such that (Ω, H, P ) is atomless, and for any
C1 ∈ G and C2 ∈ H, P (C1 ∩ C2 ) = P (C1 )P (C2 ).
When (Ω, F, P ) and (Ω, G, P ) model the respective spaces of agents and characteristics types. The condition that F is nowhere equivalent to G implies that for
any nonnegligible set D of agents, F D is always essentially richer than G D .5
These four conditions characterize the relation between F and G from different
aspects, and the following proposition shows that they are equivalent given that
G is countably generated. Note that a probability space (or its σ-algebra) is said
to be countably generated if its σ-algebra can be generated by countably many
measurable subsets together with the null sets.
there is a F-measurable subset E ′ of E such that 0 < P (E ′ ) < P (E).
5
Condition (2) is simply called “F is atomless over G” in Definition 4.3 of Hoover and Keisler
(1984). The concept of “relative saturation” refines the concept of “saturation” used in Corollary 4.5(i)
of Hoover and Keisler (1984).
5
Proposition 1. Let (Ω, F, P ) be an atomless probability space, and G a sub-σalgebra of F. If G is countably generated,6 then the following statements are equivalent.
(i) F is nowhere equivalent to G.
(ii) F is conditional atomless over G.
(iii) F is relatively saturated with respect to G.
(iv) G admits an atomless independent supplement in F.
Proof. “(i)⇒(ii)”: Suppose that F is not conditional atomless over G, then there
exists a set D ∈ F with P (D) > 0 such that for any F-measurable subset D0 of
D, we have P (D0 | G) = 0 or P (D0 | G) = P (D | G) for P -almost all ω ∈ Ω. For
such an F-measurable set D0 , let E = {ω : P (D0 | G) = P (D | G)}. Then we have
E ∈ G and P (D0 | G) = P (D | G)1E = P (D ∩ E | G) for P -almost all ω ∈ Ω,
where 1E is the indicator function of E. Hence we can obtain
∫
∫
∫
P (D0 ) =
1D0 dP =
P (D0 | G) dP =
P (D ∩ E | G) dP
Ω
Ω
∫Ω
=
1D∩E dP = P (D ∩ E).
Ω
Next, we have
∫
P (D0 ∩ E ) =
c
∫
P (D0 ∩ E | G) dP =
1E c P (D0 | G) dP
c
∫Ω
Ω
1E c P (D | G)1E dP = 0.
=
Ω
Let D0′ = D0 ∩E c . Then D0′ is a null set and (D0 \D0′ ) ⊆ D∩E, hence P (D0 \D0′ ) =
P (D ∩E). Therefore, P (D0 △(D ∩E)) = 0, which contradicts with the assumption
that F is nowhere equivalent to G.
“(ii)⇒(i)”: Suppose that F is not nowhere equivalent to G. Then there exists
a set D ∈ F with P (D) > 0, for any F-measurable subset D0 of D, there exists a
set E ∈ G such that P (D0 △(E ∩ D)) = 0. Hence we have P (D0 | G) = P (E ∩ D |
G) = 1E P (D | G) for P -almost all ω ∈ Ω, which contradicts with the assumption
that F is conditional atomless over G.
6
The implication “(iii)⇒(iv)” may not be true without the condition that “G is countably generated”,
for example, if G is saturated and F = G, the statement (iii) holds while the statement (iv) is certainly
false. Other implications are still true even though G is not countably generated.
6
“(i)⇒(iii)”: Suppose that F is nowhere equivalent to G. To prove the relative
saturation as in Definition 1 (3), we note that F is also nowhere equivalent to
σ(f ). Thus F is conditional atomless over σ(f ). The claim then holds by referring
to the proof of Corollary 4.5 (i) in Hoover and Keisler (1984) (f and g here are x1
and x2 therein).
“(iii)⇒(iv)”: Since G is countably generated, there exists a mapping f from
Ω to [0, 1] such that the σ-algebra G is generated by f . Let η be the Lebesgue
measure on [0, 1], and µ = (P ◦ f −1 ) ⊗ η. Since F is relatively saturated with
respect to G, there exists an F-measurable mapping g from Ω to [0, 1] such that
P ◦ (f, g)−1 = µ. It is clear that g is independent of f and generates an atomless
σ-algebra. Therefore, the σ-algebra generated by g is atomless and independent
of G.
“(iv)⇒(ii)”: This is exactly Lemma 4.4 (iv) of Hoover and Keisler (1984).
The following lemma shows that for any atomless measure space, we can always find an atomless sub-σ-algebra, to which the original σ-algebra is nowhere
equivalent.
Lemma 1. Let (Ω, F, P ) be an atomless probability space. Then there exists a
sub-σ-algebra G ⊆ F such that G is atomless, countably generated, and admits an
atomless independent supplement in F.
Proof. Consider the product space (I × I, B ⊗ B, η ⊗ η) of two Lebesgue unit
intervals, where I = [0, 1], B is the Borel σ-algebra, and η is the Lebesgue measure.
Since (Ω, F, P ) is atomless, there exists a measurable mapping h = (f, g) from Ω
to [0, 1] × [0, 1] which induces η ⊗ η. Then f and g are independent and generate
atomless sub-σ-algebras H and G of F respectively. It is clear that G is countably
generated, and admits an atomless independent supplement H in F.
The next lemma shows that if F is nowhere equivalent to G under a probability
measure P , then F will be nowhere equivalent to G under any measure which is
absolutely continuous with respect to P .
Lemma 2. Let (Ω, F, P ) be a probability space, and P ′ a probability measure on
(Ω, F) which is absolutely continuous with respect to P . If F is nowhere equivalent
to G under the probability measure P , then F is also nowhere equivalent to G under
the probability measure P ′ .
7
Proof. Suppose F is not nowhere equivalent to G under P ′ . Thus one can find a
set D′ ∈ F with P ′ (D′ ) > 0, such that for any F-measurable subset D0′ of D′ ,
′
there exists a set D1′ ∈ G D with P ′ (D0′ △D1′ ) = 0. Let f be the Radon-Nikodym
derivative of P ′ with respect to P , and E0 the set {ω ∈ Ω : f (ω) > 0}.
Let D = D′ ∩ E0 . Then we have
∫
∫
′
′
f dP =
0 < P (D ) =
D′
D′ ∩E0
∫
f dP =
f dP,
D
and hence P (D) > 0. For any P ′ -null set B ⊆ D (⊆ E0 ), we have 0 = P ′ (B) =
∫
B f dP , and hence P (B) = 0.
Any F-measurable subset D0 of D is also a subset of D′ . Hence there exists a
set E1 ∈ G such that P ′ (D0 △(D′ ∩ E1 )) = 0. Let E2 = D ∩ E1 . Then E2 ∈ G D .
We have
P ′ (E2 \ D0 ) = P ′ ((D ∩ E1 ) \ D0 ) ≤ P ′ ((D′ ∩ E1 ) \ D0 ) = 0,
and
P ′ (D0 \ E2 ) = P ′ (D0 \ (D ∩ E1 ))
≤ P ′ (D0 \ (D′ ∩ E1 )) + P ′ (D0 ∩ (D′ \ D)) = 0.
Thus P ′ (D0 △E2 ) = 0, which implies that P (D0 △E2 ) = 0. However, this contradicts the statement that F is nowhere equivalent to G under P .
3
Application of nowhere equivalence in large
games
In this section, we consider games with many players. The agent space is modeled
by an atomless probability space (Ω, F, P ). Let A be a compact metric space
which is the common action space for all the players, and M(A) the space of all
Borel probability measures on A which is the space of societal responses. Let U
be the space of continous real-valued function on A × M(A). The payoff function
for an agent is an element in U, which means that the agent’s payoff continuously
depends on her own action and the societal response. If the agents take actions
as specified in an F-measurable function g : Ω → A, then the societal response is
represented by the distribution P ◦g −1 . Now we are ready to present the definition
8
of a large game and its Nash equilibria.
Definition 2. A large game with an agent space (Ω, F, P ) and a common action
space A is an measurable mapping G from Ω to U.
A Nash equilibrium of a game G is an measurable function g from Ω to A such
that for P -almost all ω ∈ Ω,
G(ω)(g(ω), P ◦ g −1 ) ≥ G(ω)(a, P ◦ g −1 ) for all a ∈ A.
In addition to the agent space (Ω, F, P ) and the common action space A, Khan
et al. (2013) work with a complete separable metrizable (Polish) space T representing the traits of agents, which is endowed with a probability Borel measure ρ.
A large game with traits and its Nash equilibria are then built up from these four
basic objects. Let M(T × A) be the space of Borel probability distributions on
T ×A, and Mρ (M ×A) the subspace of M(T ×A) such that for any ν ∈ Mρ (T ×A),
its marginal probability νT on T is ρ. Mρ (T × A) will be the space of societal
responses. The space of agents’ payoffs V is the space of all continuous functions
on the product space A × Mρ (T × A), endowed with its sup-norm topology and
the resulting Borel σ-algebra.
Definition 3. A large game with traits with the agents space (Ω, F, P ), the common action space A and the trait space T with a probability measure ρ is a measurable function G from Ω to T × V such that P ◦ G−1
1 = ρ, where Gi is the projection
of G on its i-th coordinate, i = 1, 2.
A Nash equilibrium of a game G is a measurable function g : Ω → A such
that for P -almost all ω ∈ Ω, and with vω abbreviated for G2 (ω), and α : Ω → T
abbreviated for G1 ,
vω (g(ω), P ◦ (α, g)−1 ) ≥ vω (a, P ◦ (α, g)−1 ) for all a ∈ A.
Note that a large game can be viewed as a large game with traits where the trait
space is a singleton. The next theorem shows that if we distinguish the agent space
(Ω, F, P ) and the characteristics type space (Ω, G, P ) as in the previous section,
then we will be able to characterize the existence of Nash equilibira for large
games with traits via nowhere equivalence. Here we assume that G is countably
generated. The proof will be given in Section 5.
Theorem 1. Any G-measurable game with traits has an F-measurable Nash equilibrium if and only if F is nowhere equivalent to G.
9
The following corollary states a corresponding result for large games.
Corollary 1. Any G-measurable game has an F-measurable Nash equilibrium if
and only if F is nowhere equivalent to G.
4
Unification
As discussed in the introduction, the ‘nowhere equivalence’ condition provides a
unification for various conditions on the agent space. In this section, we shall
discuss the unification in details.
4.1
4.1.1
Standard representation
Walrasian distributions in large economies
Let Rl+ be the commodity space, P the set of preference relations ≿ on Rl+ , and
Pmo the set of monotonic preference relations (i.e., x ≥ y and x ̸= y implies
x ≻ y). We endow the set Pmo with the metric of closed-convergence; for details,
see Hildenbrand (1974). A distribution on Pmo × Rl+ then means a probability
measure on the σ-algebra B(Pmo × Rl+ ) of Borel sets in Pmo × Rl+ . A large
economy is a measurable mapping E from an atomless agent space (Ω, F, P ) to
Pmo × Rl+ . For agent ω ∈ Ω, let E (ω) = (≿ω , e(ω)).
Definition 4. An integrable function f : Ω → Rl+ is called a Walras allocation for
the economy E if there is a price vector p ∈ Rl+ , p ̸= 0 such that
(i) for P -almost all ω ∈ Ω, f (ω) is a maximal element for ≿ω in the budge set
{x ∈ Rl+ | p · x ≤ p · e(ω)};
∫
∫
(ii) Ω f dP = Ω e dP .
The set of all Walras allocations of the economy E is denoted by W (E ). Let
DW (E ) denote the set of distributions of all Walras allocations W (E ). A basic
observation in Hildenbrand (1974) is that the set of distributions of Walras equilibrium allocations is not completely determined by the distribution of agents’ characteristics; i.e., for two atomless economies E1 and E2 such that P ◦E1−1 = P ◦E2−1 ,
DW (E1 ) and DW (E2 ) could be different.
To solve this issue, Hildenbrand (1974) modeled the agent space as the product
space of the space of characteristics and Lebesgue unit interval (I, B, η). Suppose
10
ϱ is a distribution of agents’ characteristics. The atomless measure space of agents
is given by (Pmo × Rl+ ) × [0, 1] with the product measure P = ϱ ⊗ η, where η is the
Lebesgue measure on [0, 1]. The mapping E ϱ is the projection: E ϱ (≾, e, i) = (≾, e)
for every (≾, e, i) ∈ Pmo × Rl+ × [0, 1]. The economy E ϱ is the so-called “standard
representation” of ϱ. They show that DW (E1 ) and DW (E2 ) are identical when
the standard representation is used.
We shall resolve the above issue by using the nowhere equivalence condition.
Lemma 3. Let G be a sub-σ-algebra of F, which is nowhere equivalent to F.
Suppose E1 and E2 are G-measurable with the same distribution. Then DW (E1 )
and DW (E2 ) are same.
Proof. Let f1 be an equilibrium of E1 . Since P ◦ E1−1 = P ◦ E2−1 , by relative
saturation, there exists an F-measurable mapping f2 : Ω → Rl+ such that P ◦
(E1 , f1 )−1 = P ◦ (E2 , f2 )−1 . Therefore, f2 is an equilibrium of E2 with distribution
P ◦ f1−1 .
Note that the result on the standard representation is actually special case
of Lemma 3. In the construction of the standard representation, the σ-algebra
induced by E ϱ is B(Pmo × Rl+ ) ⊗ {I, ∅}, which admits an atomless independent
supplement {Pmo × Rl+ , ∅} ⊗ B in B(Pmo × Rl+ ) ⊗ B. Hence B(Pmo × Rl+ ) ⊗ {I, ∅}
is nowhere equivalent to B(Pmo × Rl+ ) ⊗ B. By Proposition 1 and Lemma 3,
DW (E1 ) and DW (E2 ) are identical.7
4.1.2
Games with many players
Large games are defined in Section 3. Instead of working with measurable mappings, here we work with their distributions. We shall follow the notations there.
Definition 5. A measure game with the action space A is a probability measure
ι ∈ M(U).
A Nash equilibrium distribution of a measure game ι is a probability measure τ ∈ M(U × A) such that τU = ι, and
τ {(u, x) : u(x, τA ) ≥ u(a, τA ), ∀a ∈ A} = 1.
7
We plan to investigate further the application of the nowhere equivalence condition in general
equilibrium theory in a forthcoming project.
11
We have the following lemma, which is a direct consequence of relative saturation.
Lemma 4. Let ι ∈ M(U) be a measure game, τ ∈ M(U × A) its Nash equilibrium
distribution.8 Let G : Ω → U be a G-measurable mapping with distribution P ◦
G−1 = ι. If F is nowhere equivalent to G, then there exists an F-measurable Nash
equilibirum g : Ω → A such that P ◦ (G, g)−1 = τ .
Here is a related result provided by Noguchi (2009) (Corollary 1) using the
idea of standard representation. π is the projection from U × [0, 1] to U with the
distribution ι, where U is endowed with the Borel σ-algebra B(U) and [0, 1] is
endowed with the Borel σ-algebra B and the Lebesgue measure η. It is claimed in
Corollary 1 of Noguchi (2009) that there exists a pure-strategy Nash equilibrium
f from U × [0, 1] to A such that ι ⊗ η(π, f )−1 = τ . Note that this result is a special
case of Lemma 4 by taking Ω = U × [0, 1], F = B(U) ⊗ B and G = B(U) × {∅, I}.
4.2
Hyperfinite agent space
In a finite-agent model, {1, 2, . . . , n} is used to label the agents with the counting
probability. A Loeb counting probability space as introduced in Loeb (1975) can be
viewed as an equivalence class of the sequence of finite counting probability spaces,
which is the head-counting measure in the infinite setting. By its construction,
Loeb counting probability spaces have the property of asymptotic implementability, which means that one can go back and forth between exact results on Loeb
counting probability spaces and approximate results for the asymptotic large finite
case. The Lebesgue unit interval does not have this property. In fact, economists
argued that the interest in an ideal economic model is proportional to how much
new information can be derived for the asymptotic large but finite case. Thus,
a Loeb counting probability space is a more appropriate agent space than the
Lebesgue space, which is the main theme of Khan and Sun (1999).9
Loeb counting probability spaces have the following homogeneity property;
see Proposity 9.2 of Keisler (1984). A probability space (Ω, F, P ) is said to be
homogeneous if any two random variables x and y on Ω with the same distribution,
there is a P -almost surely bijection h from Ω to Ω which preserves F-measurability
and P -measures, such that x(ω) = y(h(ω)) for P -almost all ω ∈ Ω. Based on the
homogeneity property, we can prove the following simple lemma.
8
See Mas-Colell (1984) for the existence.
For some other early references on the use of hyperfinite agent spaces, see Brown and Robinson
(1975), Brown and Loeb (1976), Anderson (1988).
9
12
Lemma 5. Let (Ω, F, P ) be a Loeb counting probability space, and G a countablygenerated sub-σ-algebra of F. Then G admits an atomless independent supplement
H in F.
Proof. Since G is countably generated, there exists a mapping g : Ω → ([0, 1], B, µ)
which generates G, where B is the Borel σ-algebra on [0, 1] and µ = P ◦ g −1 .
It is clear that there exists a mapping (g ′ , f ′ ) : Ω → [0, 1] × [0, 1] such that P ◦
(g ′ , f ′ )−1 = µ ⊗ η, where η is the Lebesgue measure. Note that g and g ′ share the
same distribution on F. By the homogeneity property, there is an F-measurable
measure-preserving bijection h on Ω such that g = g ′ ◦ h. Let f = f ′ ◦ h. Then
(g, f ) induces the distribution µ ⊗ η, and hence the σ-algebra H generated by f is
independent of G.
4.3
Saturated agent space
In Keisler and Sun (2009), the saturation property is used to characterize a class
of probability spaces that can handle the failure of the Lebesgue unit interval.
It is usually easy to transfer a result on Loeb spaces to a result on saturated
probability spaces via saturation property.10 On the other hand, one can also
obtain the necessity of saturation in various contexts.
Definition 6. An atomless probability space (S, S, Q) is said to have the saturation property for a probability measure µ on the product of Polish spaces
X × Y if for every random variable f : S → X which induces the distribution as
the marginal measure of µ over X, then there is a random variable g : T → Y such
that the induced distribution of the pair (f, g) on (S, S, Q) is µ.
(S, S, Q) is said to be saturated if it has the saturation property for every
probability measure µ on every product of Polish spaces.
The following is an obvious corollary of Proposition 1.
Corollary 2. Let (Ω, F, P ) be a atomless probability space. Then the following
are equivalent.11
1. (Ω, F, P ) is saturated.
2. F is nowhere equivalent to any countably-generated sub-σ-algebra.
10
Any atomless Loeb space is saturated; see Hoover and Keisler (1984).
Condition (3) is called ℵ1 -atomless in Hoover and Keisler (1984), and the equivalence between (1)
and (3) is shown in Corollary 4.5(i) therein. For additional equivalent conditions, see Fact 2.5 in Keisler
and Sun (2009).
11
13
3. F is conditional atomless over any countably-generated sub-σ-algebra.
4. F is relatively saturated with respect to any countably-generated sub-σ-algebra.
5. Any countably-generated σ-algebra admits an atomless independent supplement in F.
The following corollary shows that if (Ω, F, P ) is saturated under a probability
measure P , then (Ω, F, P ′ ) is also saturated under any measure which is absolutely continuous with respect to P . It is a simple consequence of Lemma 2 and
Corollary 2.
Corollary 3. Suppose P ′ is a probability measure on (Ω, F), which is absolutely
continuous with respect to P on (Ω, F). If (Ω, F, P ) is saturated, then so is
(Ω, F, P ′ ).
The following result is a simple corollary of Theorem 1 and Corollary 1.12
Corollary 4. Let (Ω, F, P ) be an atomless agent space. Any large game with/without
traits G has a Nash equilibrium if and only if (Ω, F, P ) is saturated.
It is well-known that the Lebesgue unit interval is countably-generated, and
hence not saturated. However, one can extend the Lebesgue unit interval (I, B, η)
into a saturated probability space (I, F, η ′ ).13 Since B is countably generated, B
admits an atomless independent countably-generated σ-algebra H in F. Thus, for
any B-measurable large game, there exists a σ(B ∪ H)-measurable Nash equilibrium. Note that σ(B ∪ H) is countably generated. To handle the nonexistence of
Nash equilibrium for the game in Example 3 of Rath et al. (1995), Khan and Zhang
(2012) present a countably-generated Lebesgue extension as the agent space. Their
result is a special case of Corollary 1 since their countably-generated Lebesgue extension provides an atomless independent supplement to B.
4.4
Many more players than strategies
Rustichini and Yannelis (1991) proposed the following ‘many more players than
strategies’ condition which aims to solve the convexity problem of the Bochner
integral of a correspondence from a finite measure space to an infinite dimensional
12
For such results, see Keisler and Sun (2009) and Khan et al. (2013) in the setting of large games and
large games with traits respectively. He and Sun (2013) generalize the results about correspondences
in Keisler and Sun (2009) to the setting on nowhere equivalence.
13
See Kakutani (1944).
14
Banach space,14 and Yannelis (2009) used this condition to prove the existence of
equilibrium.
For any given atomless agent space (Ω, F, P ), let L∞ (P ) denote the Banach
space of all essentially bounded functions endowed with the norm ∥ · ∥∞ , and
∞
L∞
E (P ) the subspace of L (P ) with the elements which vanishes off E. card(K)
denotes the cardinality of the set K.
Assumption (Many more players than strategies). (Ω, F, P ) is a measure space
and Z is an infinite-dimensional Banach space. For each E ∈ F with µ(E) > 0,
dim(L∞
E (P )) > dim(Z).
As discussed in Remark 4 of Rustichini and Yannelis (1991), with the continuum hypothesis and the fact that an infinite-dimensional Banach space cannot
have a countable Hamel basis, this condition implies that for each E ∈ F with
µ(E) > 0, dim(L∞
E (P )) > c, thus (Ω, F, P ) is a nowhere countably-generated
space. Therefore, all results hold on saturated probability spaces will also hold
given the condition of “many more players than strategies”.
4.5
Many players of every type
The following is the standard definition of conditional probabilities.15
Definition 7. Let (Ω, F, P ) be a probability space, and X a Polish space. A
system of regular conditional probabilities {Px : x ∈ X} is said to be generated by
a measurable mapping G from Ω to X if
1. for every x ∈ X, Px is a probability measure on (Ω, F);
2. for every D ∈ F , P (·, D) is measurable with respect to B(X);
∫
3. P (G−1 (E) ∩ D) = E Px (D) dκ for each D ∈ F and E ∈ B(X), where
κ = P ◦ G−1 .
The family of conditional probabilities {Px : x ∈ X} is called proper if Px is
concentrated on G−1 ({x}) for κ-almost all x ∈ X, and atomless if Px is atomless
for κ-almost all x ∈ X.
The following lemma shows that given a measurable mapping G, if the family of
conditional probabilities generated by G is well behaved (i.e., proper and atomless),
then the nowhere equivalence condition is satisfied.
14
15
See also Tourky and Yannelis (2001).
See Definition 10.4.2 in Bogachev (2007).
15
Lemma 6. Following the notations in Definition 7. Suppose that F is countably
generated and the family of conditional probabilities {Px : x ∈ X} generated by G
is proper and atomless. Then the σ-algebra G generated by G admits an atomless
independent supplement in F.16
Proof. Since F is countably generated, based on Theorem 6.5.5 in Bogachev
(2007), there is a measurable mapping ϕ from (Ω, F) to (I, B) such that ϕ could
generate the σ-algebra F. Define a mapping f : X × I → [0, 1] as f (x, i) =
Px ◦ ϕ−1 ([0, i]). For each x ∈ X, denote fx (·) = f (x, ·), and hence fx is the distribution function of the probability measure Px ◦ ϕ−1 on I. For κ-almost all x ∈ X,
the atomlessness of Px on F implies that Px ◦ ϕ−1 ({i}) = 0 for all i ∈ I. Thus the
distribution function fx (·) is continuous on I for each x ∈ X.
Let g(ω) = f (G(ω), ϕ(ω)) for each ω ∈ Ω. We claim that g is independent
of G and P ◦ g −1 is the Lebesgue measure η on I. Firstly, by condition (3)
(
)
∫
of Definition 7, we have P (G−1 (E) ∩ g −1 ([a, b])) = E Px g −1 ([a, b]) dκ for any
E ∈ B(X) and 0 ≤ a < b ≤ 1. Secondly, the condition that {Px : x ∈ X} is
(
)
proper implies that Px is concentrated on G−1 ({x}), and hence Px g −1 ([a, b]) =
(
)
Px (fx ◦ ϕ)−1 ([a, b]) for κ-almost all x ∈ X. Thirdly, for κ-almost all x ∈ X, fx is
the continuous distribution function of the probability measure Px ◦ ϕ−1 , it then
follows by Example 3.6.2 of Bogachev (2007) that Px ◦ ϕ−1 ◦ fx−1 = η. Thus we
have P (G−1 (E) ∩ g −1 ([a, b])) = κ(E)η([a, b]).
Therefore, g is independent of G and induces an atomless σ-algebra, which
yields the assertion.
Remark 1. The converse direction of Lemma 6 may not be true. Let (I, I, η) be a
countably generated Lebesgue extension of the Lebesgue unit interval (I, B, η) such
that B admits an atomless independent supplement in I.17 Let (Ω, F, P ) = (I, I, η)
and X = I. Take G as an identity mapping; i.e., G(i) = i for all i ∈ I. Then
G = B, and G admits an atomless independent supplement in F. It is easy to see
that {Pi = δ{i} : i ∈ I} is the family of conditional probabilities generated by G.
For almost all i ∈ I, Pi is atomic.
Noguchi (2009) obtained the existence of pure strategy Nash equilibrium of
large games based on the conditions of Lemma 6, which is called ‘many players of
16
This lemma is essentially the same as Theorem 10.8.3 of Bogachev (2007). A stronger conclusion
is obtained there under stronger conditions.
17
See the last paragraph of Subsection 4.3 for the existence of such an extension.
16
every type’ therein.18 It is clear that this result follows directly Corollary 1 and
Lemma 6.
5
Appendix: Proof of Theorem 1
5.1
Sufficiency part
Suppose that F is nowhere equivalent to G and G = (α, v) : Ω → T × V is a
G-measurable game with traits. Pick any saturated probability space (S, S, Q).
Then there exists an S-measurable game with traits F = (β, u) from S to T × V
such that Q ◦ F −1 = P ◦ G−1 . Since (S, S, Q) is saturated, Theorem 1 of Khan
et al. (2013) implies that F has a Nash equilibrium f from S to A. Define ν =
Q◦(F, f )−1 ∈ M(T ×V ×A). If F is nowhere equivalent to G, and hence relatively
saturated to G, then there exists an F-measurable mapping g from Ω to A such
that P ◦ (G, g)−1 = ν.
We shall show that g is a Nash equilibrium of G. Towards this end, let ξ be
the marginal of ν on T × A, and {an : n ∈ N} a countable dense subset of A.
For any j ∈ N, define a function ψj : A × V → R as ψj (a, ϕ) = ϕ(a, ξ) − ϕ(aj , ξ).
It is clear that ψj is continuous. Next, define two more functions h1j : S → R
and h2j : Ω → R as follows: h1j (s) = ψj (f (s), u(s)) and h2j (ω) = ψj (g(ω), v(ω)).
−1
Since P ◦ (v, g)−1 = Q ◦ (u, f )−1 , we have P ◦ h−1
2j = Q ◦ h1j . Since f is a Nash
equilibrium of the game F , h1j (s) ≥ 0 for Q-almost all s ∈ S, and hence h2j (ω) ≥ 0
for P -almost all ω ∈ Ω.
Finally, by grouping countably many P -null sets together, we obtain that for
P -almost all ω, v(ω)(g(ω), ξ) ≥ v(ω)(aj , ξ) for all j ∈ N, which implies that
v(ω)(g(ω), ξ) ≥ v(ω)(a, ξ) for all a ∈ A by the continuity of v(ω). Therefore, g is
a Nash equilibrium of the game G.
In Noguchi (2009), (Ω, F, P ) is the agent space, U = X is the space of continuous payoff functions,
and G : Ω → U is a large game. The conditions therein are slightly different. In particular, the family of
conditional probabilities {Pu : u ∈ U} is assumed to be atomless on a countably generated sub-σ-algebra
T where T is essentially the same as F modulo null sets. However, these two settings are equivalent.
One can obtain a T -measurable game G′ by modifying the game G on a null set. Then the result
follows by applying Lemma 6 and Corollary 1 to the modified game G′ with the agent space (Ω, T , P ).
For other applications of the conditions of this type, see Section 10 of Bogachev (2007) and Podczeck
(1997).
18
17
5.2
Necessity part
The result in the following lemma is well-known.19 Here we give a simple and
direct proof.
Lemma 7. If (Ξ, Σ, Λ) is an atomless probability space and Σ is countably generated, then there exists a measure-preserving mapping ψ from (Ξ, Σ, Λ) to the
Lebesgue unit interval (I, B, η) such that for any E ∈ Σ, there exists a set E ′ ∈ B
such that Λ(E△ψ −1 (E ′ )) = 0.
Proof. Since Σ is countably generated, based on Theorem 6.5.5 in Bogachev (2007),
there is a measurable mapping ψ1 from Ξ to I such that ψ1 could generate the
σ-algebra Σ. Since (Ξ, Σ, Λ) is atomless, the induced measure Λ ◦ ψ1−1 on I is
atomless. Moreover, by Theorem 16 (p. 409) in Royden (1988), (I, B, Λ ◦ ψ1−1 ) is
isomorphic to the Lebesgue unit interval (I, B, η); denote this isomorphism by ψ2 .
Let ψ = ψ2 ◦ ψ1 , then ψ satisfies the requirement.
Note that the probability space (Ω, G, P ) may not be atomless. There exist
disjoint G-measurable subsets Ω1 and Ω2 such that Ω1 ∪ Ω2 = Ω, and P |Ω1 is the
atomless part of P , while P |Ω2 is the purely atomic part of P . Let P (Ω1 ) = γ.
We shall assume 0 < γ ≤ 1. The lemma above shows that there exists a measurepreserving mapping ϕ : (Ω1 , G Ω1 , P ) → ([0, γ), B1 , η1 ) such that for any E ∈ G Ω1 ,
there exists a set E ′ ∈ B1 such that P (E△ϕ−1 (E ′ )) = 0, where B1 is B [0,γ) and η1
is the Lebesgue measure on B1 .
Let A1 = [0, 1], A2 = {0, 1, · · · , n − 1} (n ≥ 2), and A0 = A1 × A2 with
the induced standard metric in R2 . Let d(·, ·) be the Prohorov metric on M(A0 ).
Define a probability measure η 2 on A0 as follows. For any Borel measurable set E ⊆
A1 and any j ∈ A2 , η 2 (E × {j}) =
1
nγ η(E ∩ [0, γ)).
Let η 1 be a convex combination
of η 2 and the Dirac measure concentrated at (1, n − 1): η 1 = γη 2 + (1 − γ)δ(1,n−1) .
Let f : A0 × [0, 1] → R be defined as follow. For any a1 ∈ A1 , a2 ∈ A2 and
b ∈ [0, 1],
=

0,
f ((a1 , a2 ), b)
if b = 0 or a1 = kb
(δ ({a }) − 1 ) min{a − (nk + j)b, (nk + j + 1)b − a },
j
2
1
1
2
if a1 ∈ ((nk + j)b, (nk + j + 1)b)
for some k ∈ N and j ∈ A2 . The function f is continuous on A0 × [0, 1].
19
This result plays a key role in obtaining the necessity of saturation in Keisler and Sun (2009). See
Fremlin for a general result of this kind.
18
Define a mapping u from [0, γ) to the space of continuous functions on A0 ×
M(A0 ) as follows.20 For any i ∈ [0, γ), a1 ∈ A1 , a2 ∈ A2 and ν ∈ M(A0 ),
u(i)((a1 , a2 ), ν) = f ((a1 , a2 ), f0 (ν)) − |i − a1 |,
where f0 (ν) =
1
n
d(η 1 , ν).
Example 1. Fix a natural number n ≥ 2. Let (Ω, F, P ) be the agent space, and
A = A0 the action space. Define a G-measurable large game G : Ω → U as follows:

f ((a1 , a2 ), f0 (ν)) − |ϕ(ω) − a1 |,
G(ω)((a1 , a2 ), ν) =
|a | + |a |,
1
2
if ω ∈ Ω1 ,
if ω ∈ Ω2 ,
for any (a1 , a2 ) ∈ A and ν ∈ M(A).
Lemma 8. Suppose that there exists an F-measurable Nash equilibrium g for the
above large game G. Then there exists an F-measurable partition {E0 , . . . , En−1 }
of Ω1 such that for each j = 0, . . . , n − 1, (1) P Ω1 (Ej ) = n1 , (2) Ej is independent
of G Ω1 under the probability measure P Ω1 .
Proof. Since u(i) is a continuous function on A × M(A) for any i ∈ I. Moreover,
it is clear that u is a continuous mapping from [0, γ) to U under the supremum
norm, and G is a constant function on Ω2 , hence G is a G-measurable game.
Suppose that g = (g1 , g2 ) is an F-measurable Nash equilibrium of G. Let
ϑ = P ◦ g −1 . We shall first consider ϑ ̸= η 1 . Then d(ϑ, η 1 ) ∈ (0, 1]. Denote
b0 =
1
n
d(ϑ, η 1 ); then b0 ≤ n1 .
For each agent ω ∈ Ω2 , since the payoff function is |a1 | + |a2 |, it is obvious that
g(ω) = (1, n − 1). Now, we consider the best response (a∗1 , a∗2 ) for agent ω ∈ Ω1
with ϕ(ω) ̸= kb0 for any k ∈ N. For any (x, j) ∈ A with x ̸= ϕ(ω),21
G(ω)((x, j), ϑ) − G(ω)((ϕ(ω), j), ϑ)
= f ((x, j), f0 (ϑ)) − |ϕ(ω) − x| − f ((ϕ(ω), j), f0 (ϑ)) < 0.
Thus g1 (ω) = a∗1 = ϕ(ω) regardless of the value of a∗2 . There is a unique pair (k, j ′ )
with k ∈ N and j ′ ∈ A2 such that ϕ(ω) ∈ ((nk + j ′ )b0 , (nk + j ′ + 1)b0 ). For any
j ∈ A2 , G(ω)((ϕ(ω), j), ϑ) = f ((ϕ(ω), j), f0 (ϑ)). Its value is positive only if j = j ′ ,
and thus g2 (ω) = a∗2 = j ′ . It is obvious that ϑ coincides with η 1 on [γ, 1] × A2 .
20
21
The payoff function used here is motivated from Example 3 in Rath et al. (1995).
When a2 and ν are fixed, f is a Lipschitz function in terms of a1 with the Lipschitz constant 12 .
19
We will show that d(ϑ, η 1 ) is at most (n − 1)b0 . Fix ϵ = (n − 1)b0 .
For j = 0, · · · , n − 2, let W × {j} be the support of ϑ on [0, γ) × {j}. The
set W should be the union of finite disjoint intervals; denote them by W1 , . . . , Wm
in the increasing order.22 The distance between Wℓ and Wℓ+1 is (n − 1)b0 for
ℓ = 1, . . . , m − 1. It is clear that the length of Wℓ is at most b0 , ℓ = 1, . . . , m.
Take a Borel set E ⊆ [0, γ). Without loss of generality we may assume E does
not contain any endpoint of these subintervals. For 1 ≤ ℓ ≤ m−1, let Eℓ = Wℓ ∩E,
then Eℓ , Eℓ +b0 , . . . , Eℓ +(n−1)b0 are all disjoint, and (Eℓ +tb0 )×{j} is included in
(E × {j})ϵ for any t = 0, . . . , n − 1, where (E × {j})ϵ is ϵ-neighborhood of E × {j}.
Since η(Em ) ≤ b0 and η 1 (D × {j}) = n1 η(D) for any Borel set D ⊆ [0, γ), we have
ϑ(E × {j})
=
=
m−1
∑
ϑ(Eℓ × {j}) + ϑ(Em × {j}) =
ℓ=1
m−1
∑(
m−1
∑
η(Eℓ ) + η(Em ) ≤ n
ℓ=1
m−1
∑
η 1 (Eℓ × {j}) + b0
ℓ=1
)
η 1 (Eℓ × {j}) + η 1 ((Eℓ + b0 ) × {j}) + · · · + η 1 ((Eℓ + (n − 1)b0 ) × {j}) + b0
ℓ=1
(
)
≤ η 1 (E × {j})ϵ + b0 .
For j = n − 1, let W ′ × {n − 1} be the support of ϑ on [0, γ) × {n − 1}. The set
′
W ′ should be the union of finite disjoint intervals; denote them by W1′ , . . . , Wm
′
in the increasing order. The distance between {0} and W1′ , and Wℓ′ and Wℓ+1
are all (n − 1)b0 for ℓ = 1, . . . , m − 1. It is clear that the length of Wℓ′ is at
most b0 , ℓ = 1, . . . , m. Take a Borel set E ⊆ [0, γ). Without loss of generality
we may assume E does not contain any endpoint of the subintervals above. For
1 ≤ ℓ ≤ m, let Eℓ = Wℓ′ ∩ E, then Eℓ , Eℓ − b0 , . . . , Eℓ − (n − 1)b0 are all disjoint,
and (Eℓ − tb0 ) × {n − 1} is included in (E × {n − 1})ϵ for any t = 0, . . . , n − 1. We
have
ϑ(E × {n − 1})
=
m
∑
ℓ=1
=
ϑ(Eℓ × {n − 1}) =
m (
∑
m
∑
η(Eℓ ) = n
ℓ=1
m
∑
η 1 (Eℓ × {n − 1})
ℓ=1
)
η 1 (Eℓ × {n − 1}) + η 1 ((Eℓ − b0 ) × {n − 1}) + · · · + η 1 ((Eℓ − (n − 1)b0 ) × {n − 1})
ℓ=1
(
)
≤ η 1 (E × {n − 1})ϵ \ {(1, n − 1)} .
22
For each ℓ = 1, . . . , m, Wℓ is in the form of W ∩ ((nk + j)b0 , (nk + j + 1)b0 ) for some k.
20
Given any Borel set C ⊆ A. Suppose that C = ∪0≤k≤n−1 Ck × {k}, where
C0 , · · · , Cn−1 ⊆ A1 . Let Ck1 = Ck ∩ [0, γ) and Ck2 = Ck ∩ [γ, 1] for k = 1, · · · , n − 1.
We have
ϑ(C) =
=
≤
n−1
∑
k=0
n−1
∑
ϑ(Ck × {k})
ϑ(Ck1 × {k}) +
n−1
∑
ϑ(Ck2 × {k})
k=0
k=0
n−2
∑
{η
1
(
(Ck1
× {k})
ϵ
)
+ b0 } + η
1
(
1
(Cn−1
)
× {n − 1}) \ {(1, n − 1)} +
ϵ
k=0
n−1
∑
η 1 (Ck2 × {k})
k=0
≤ η (C ) + (n − 1)b0 = η (C ) + ϵ.
1
ϵ
1
Hence d(ϑ, η 1 ) ≤ (n − 1)b0 =
ϵ
n−1
n
d(ϑ, η 1 ), which is a contradiction.
Therefore, the only possibility is that ϑ = η 1 . Then f (a, f0 (ϑ)) = 0 for any
a ∈ A and G(ω)((a1 , a2 ), ϑ) = −|ϕ(ω) − a1 | for any ω ∈ Ω1 . The best response
correspondence is H(ω) = {(ϕ(ω), 0), · · · , (ϕ(ω), n − 1)} for each ω ∈ Ω1 and
{(1, n − 1)} for each ω ∈ Ω2 . By the definition of Nash equilibria, g(ω) ∈ H(ω) for
P -almost all ω ∈ Ω.
For any C ∈ G Ω1 , by Lemma 7, there exists an C1 ∈ B1 such that P (C△ϕ−1 (C1 )) =
0. Define Dj = {ω ∈ Ω1 : g(ω) = (ϕ(ω), j)} for j ∈ A2 . Thus we have
P (Dj ∩ C) = P (Dj ∩ ϕ−1 (C1 )) = P (g ∈ (C1 × {j}))
1
1
1
= η 1 (C1 × {j}) = η(C1 ) = P (ϕ ∈ C1 ) = P (C),
n
n
n
and hence P Ω1 (Dj ) = n1 . Therefore, Dj is independent of G Ω1 under P Ω1 for any
j ∈ A2 .
Now we are ready to prove the necessity part of Theorem 1.
Proof. First assume γ = 0, which means that G is purely atomic. Since F is
atomless, F is nowhere equivalent to G. Therefore we onlu focus on the case when
0 < γ ≤ 1 as considered in the previous lemma.
Step 1. Firstly we will show the existence of an F-measurable partition {E0 , . . . , En−1 }
of Ω1 such that for each j = 0, . . . , n − 1, (1) P Ω1 (Ej ) = n1 , (2) Ej is independent
of G Ω1 under the probability measure P Ω1 .
21
Consider the game G in Example 1. Define a game with traits G′ = (α, v) from
Ω to T × V, where T is a singleton, α is a constant mapping, and v(ω)(a, ν) =
G(ω)(a, νA ) for ω ∈ Ω. Then G′ is G-measurable. For the new game G′ , there
exists an F-measurable mapping g ′ which is a Nash equilibrium of G′ . It is easy to
see that g ′ is also a Nash equilibrium of the game G because of the construction of
the payoff function v. Therefore, Lemma 8 implies the existence of a F-measurable
partition {E0 , . . . , En−1 } of Ω1 such that for each j = 0, . . . , n−1, (1) P Ω1 (Ej ) = n1 ,
(2) Ej is independent of G Ω1 under the probability measure P Ω1 .
Step 2. In the following, we will complete the proof by contradiction. Suppose
that F is not nowhere equivalent to G, then there exists a nonnegligible set D ∈ F
such that for any L1 ∈ F D , there exists a set L2 ∈ F D , P (L1 △L2 ) = 0. If
P (Ω2 ∩ D) > 0, then G Ω2 ∩D is purely atomic while F Ω2 ∩D is atomless., which
is a contradiction. Thus we can assume that D is a subset of Ω1 without loss
of generality. Choose a natural number n which is sufficiently large so that
1 Ω1
(D).
2P
1
n
<
Assume that E1 , . . . , En are the n sets obtained in Step 1.
Now we focus on the subset Ω1 . For P Ω1 -almost all ω ∈ Ω1 , we have
P Ω1 (Ej ∩D | G Ω1 ) ≤ P Ω1 (Ej | G Ω1 ) = P Ω1 (Ej ) =
1
1
< P Ω1 (D), for j = 1, . . . , n.
n
2
Denote E = {ω : P Ω1 (D | G Ω1 ) > 12 P Ω1 (D)}. Then it is clear that E ∈ G Ω1 and
P Ω1 (E) > 0. For each j, there exists a set Cj ∈ G Ω1 such that P ((Ej ∩ D)△(Cj ∩ D)) =
0. Thus for P Ω1 -almost all ω ∈ Ω1 , we have
1
1Cj P Ω1 (D | G Ω1 ) = P Ω1 (Cj ∩ D | G Ω1 ) = P Ω1 (Ej ∩ D | G Ω1 ) < P Ω1 (D),
2
which implies P Ω1 (Cj ∩ E) = 0.
Next we have,
∫
P
Ω1
(D ∩ (∪j Cj )) =
P Ω1 (D ∩ (∪j Cj ) | G Ω1 ) dP Ω1
Ω1
∫
P Ω1 (D ∩ (∪j Ej ) | G Ω1 ) dP Ω1
=
∫Ω1
P Ω1 (D | G Ω1 ) dP Ω1 = P Ω1 (D),
=
Ω1
22
and hence P Ω1 (D \ (∪j Cj ) = 0. Moreover,
∫
∫
P
Ω1
(D ∩ E) =
P
Ω1
(D ∩ E | G
Ω1
) dP
Ω1
1E P Ω1 (D | G Ω1 ) dP Ω1
=
Ω1
Ω1
1
> P Ω1 (D)P Ω1 (E) > 0.
2
Thus we have P Ω1 ((∪j Cj ) ∩ E) > 0, which contradicts that P Ω1 (Cj ∩ E) = 0 for
j = 1, . . . , n. Therefore, F is nowhere equivalent to G.
Remark 2. Note that the necessity part of Corollary 1 follows from Lemma 8 and
Step 2 in the proof of the necessity part of Theorem 1.
Next we present another large game with traits F such that the action space
is finite and the trait space is uncountable. We shall show that Step 1 in the proof
of necessity part of Theorem 1 can also be proved via this game.
Example 2. Fix a natural number n ≥ 2. Let (Ω, F, P ) be the agent space,
T = A1 the trait space with the measure ρ, and A = A2 the action space. The
restricted measure ρ on [0, γ) is the Lebesgue measure, and the restricted measure
ρ on [γ, 1] is (1 − γ)δn−1 . The large game with traits F = (α, v) is defined as
follows:
α(ω) =
and,
v(ω)(a, ν) =

ϕ(ω),
if ω ∈ Ω1
1,
if ω ∈ Ω2

f (ϕ(ω), a, f0 (ν)),
if ω ∈ Ω1
|a|,
if ω ∈ Ω2
for any a ∈ A and ν ∈ Mρ (T × A).
Lemma 9. Suppose that there exists an F-measurable Nash equilibrium f for F .
Then there exists a F-measurable partition {E0 , . . . , En−1 } of Ω1 such that for
each j = 0, . . . , n − 1, (1) P Ω1 (Ej ) =
1
n,
(2) Ej is independent of G Ω1 under the
probability measure P Ω1 .
Proof. Suppose f is an F-measurable Nash equilibrium for the G-measurable large
game with traits F . Let ϑ = P ◦ (α, f )−1 and b0 =
1
n
d(ϑ, η 1 ).
Suppose b0 > 0. Note that for agent ω ∈ Ω1 such that ϕ(ω) ̸= kb0 for any k ∈ N
and s ∈ A, the best response must be j ′ such that ϕ(ω) ∈ ((nk+j ′ )b0 , (nk+j ′ +1)b0 )
for some k ∈ N. (α, f ) = g for P -almost all ω, where g is the best response in the
proof of the case ‘ϑ ̸= η 1 ’ in Lemma 8, which is impossible.
23
Thus ϑ = η 1 , which means that f0 (ϑ) = 0. Then for agent ω ∈ Ω1 , any a ∈ A
is a best response, while for agent ω ∈ Ω2 , 1 is the best response. We regard (α, f )
as the function g in the proof of Lemma 8 for the case ‘ϑ = η 1 ’. The rest is clear.
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