Pure-strategy Nash equilibria in nonatomic games with infinite-dimensional action spaces

Transcription

1 Pure-strategy Nash equilibria in nonatomic games with infinite-dimensional action spaces Xiang Sun Yongchao Zhang Submitted: August 28, 22. Revised and Resubmitted: December 27, 22 Abstract his paper studies the existence of pure-strategy Nash equilibria for nonatomic games where players take actions in an infinite-dimensional Banach space. For any infinitedimensional Banach space, we construct a nonatomic game with actions in this space which has no pure-strategy Nash equilibria, provided that the player space is modeled by the usual Lebesgue unit interval. We also show that if the player space is modeled by a saturated probability space, every nonatomic game has a pure-strategy Nash equilibrium. On the other hand, if every game with a fixed nonatomic player space and a fixed infinitedimensional action space has a pure-strategy Nash equilibrium, then the underlying player space must be saturated. ( words) JEL classification: C62, C72. Keywords: Nonatomic game; Pure-strategy Nash equilibrium; nfinite-dimensional Banach space; Saturated probability space. he authors are very grateful to Yi-Chun Chen, Wei He, M. Ali Khan, Xiao Luo, Kali P. Rath, Yeneng Sun, Nicholas C. Yannelis, Haomiao Yu and two anonymous referees for helpful comments. he work of Yongchao Zhang is supported by Shanghai Leading Academic Discipline Project (B8) and National Natural Science Foundation of China (2283). his paper was presented at the 2th SAE (Society for the Advancement of Economic heory) conference held at Brisbane, Australia, June 29 July 2, 22. Department of Mathematics, National University of Singapore, Lower Kent Ridge Road, 976, Singapore. xiangsun@nus.edu.sg. School of Economics, Shanghai University of Finance and Economics, 777 Guoding Road, Shanghai, 2433, China. yongchao@mail.shufe.edu.cn. Key Laboratory of Mathematical Economics (SUFE), Ministry of Education, Shanghai, 2433, China.

2 Contents ntroduction 3 2 Basics 5 3 A counterexample 6 4 Saturation and games 9 4. Saturated probability space he sufficiency result he necessity result Discussion 3 6 Appendix 3 6. Proofs of results in Section Proof of heorem Proof of Proposition References 2 2

3 ntroduction t is common sense that pure-strategy Nash equilibrium may not exist in general noncooperative games. However, it is important from a game-theoretical point of view to know when pure-strategy Nash equilibria exist. For a finite-player game, the existence of purestrategy Nash equilibria follows from certain conditions on the payoff functions and strategy spaces. For games with a nonatomic measure-theoretic structure that models the space of players or information, a general purification principle due to Dvoretsky, Wald and Wolfowitz (95a) guarantees that one can always obtain a pure-strategy Nash equilibrium from a mixed-strategy Nash equilibrium, when the action space is finite; see Dvoretsky, Wald and Wolfowitz (95b), Khan, Rath and Sun (26) and their references. For games with countable actions, similar results on pure-strategy Nash equilibria can be found in Khan and Sun (995). For games with a nonatomic measure-theoretical structure and an uncountable compact metric action space, when the players payoffs depend on their own actions and the action distribution of other players, there are several subtle possibilities. First, when the space of players or information is modeled by the Lebesgue unit interval, counterexamples are constructed to show the nonexistence of pure-strategy Nash equilibria; see Khan, Rath and Sun (997, 999). Second, when the Lebesgue unit interval is replaced by nonatomic Loeb spaces, positive results on pure-strategy Nash equilibria are shown in Khan and Sun (999). hird, for a fixed nonatomic player space, it is shown in Keisler and Sun (29) that any game with the given player space has a pure-strategy Nash equilibrium if and only if the underlying player space is saturated in the sense that any subspace is not essentially countably-generated. 2 he purpose of this paper is to consider the pure-strategy Nash equilibria for games with a nonatomic player space and an uncountable compact action set in an infinite-dimensional Banach space, where players payoffs depend on their own actions and the average action of other players. 3 As shown in Khan, Rath and Sun (997), when the player space is the Lebesgue unit interval and the action space is an uncountable compact subset of the Hilbert space l 2 the space of square-summable real-valued sequences, pure-strategy Nash equilibria may not exist. Since various infinite-dimensional Banach spaces are widely used in the economics literature, 4 a natural question is whether we could find a right infinitedimensional Banach space rather than l 2 to deliver a positive result on the existence of the pure-strategy Nash equilibria. We show that this is impossible as long as the player space is he supermodular game is a good example; see opkis (979). Here is another example, when each player s payoff function is quasi-concave and continuous, and the strategy space is a convex and compact subset in a finite-dimensional Euclidean space, there exists a pure-strategy Nash equilibrium by Kakutani s fixed-point theorem. 2 Similar results hold for finite-player games with nonatomic information spaces, see Khan and Zhang (22b). See Fu (27) for the relation between the games with nonatomic player spaces and finite-player games with nonatomic information spaces. 3 When players payoff functions are linear in their own actions, see Khan (986) for the existence of the approximate pure-strategy Nash equilibria for nonatomic games with infinite-dimensional action spaces; see also Rustichini and Yannelis (99) for the existence of exact pure-strategy Nash equilibria for nonatomic games with infinite-dimensional action spaces, provided that the game has many more players than actions. 4 See, for example, Bewley (972), Yannelis (29) and the books Khan and Yannelis (99) and Stokey, Lucas and Prescott (989). 3

4 the Lebesgue unit interval. n particular, given any infinite-dimensional Banach space, there always exists a nonatomic game (denoted by G ) with actions in an uncountable compact subset of this Banach space such that it does not have a pure-strategy Nash equilibrium, provided that the player space is the Lebesgue unit interval. Nevertheless, if the player space is not the Lebesgue unit interval, it is possible to deliver a positive result on pure-strategy Nash equilibria for nonatomic games with infinitedimensional action spaces. Khan and Sun (999) show that when the Lebesgue unit interval is replaced by a nonatomic Loeb space, there exists a pure-strategy Nash equilibrium for any nonatomic game with any uncountable compact action set of an infinite-dimensional Banach space. 5 t follows from the existence result in Khan and Sun (999) and general saturation property 6 that the existence result of pure-strategy Nash equilibria still holds when the player space is modeled by a saturated probability space. A further and more interesting question is whether the converse of the above result is true. n other words, whether the necessity result of the saturation property in Keisler and Sun (29) can be established in the context of nonatomic games with infinite-dimensional action spaces. We provide an answer in the affirmative. n particular, we show that given a nonatomic player space and a fixed infinite-dimensional Banach space, if every game with actions in a compact subset of this Banach space has a pure-strategy Nash equilibrium, then the underlying player space must be a saturated probability space. 7 Put differently, if the player space is a non-saturated probability space, then one can always construct a nonatomic game with this player space where players take actions from a given infinite-dimensional Banach space such that it has no pure-strategy Nash equilibrium. t is worthwhile to note that the proof of our necessity result is not implied by, actually far more involved than, the proof of the necessity part of Keisler and Sun (29, heorem 4.6). o summarize, to deliver a positive result on pure-strategy Nash equilibria for nonatomic games with actions in infinite-dimensional spaces, the measure-theoretic structure of the player space plays a fundamental role. t is worth noting that as far as the above counterexample G is concerned, to guarantee the existence of pure-strategy Nash equilibria, one is not necessary to turn to saturated probability spaces, a simple extension of the Lebesgue unit interval does serve this purpose. 8 his paper is organized as follows. Section 2 presents basic definitions and notations on the Banach spaces and on the nonatomic games. n Section 3, nonatomic games are constructed on the Lebesgue unit interval taking actions in any infinite-dimensional Banach space such that there does not have a pure-strategy Nash equilibrium for each game. n Section 4, we show that the player space being a saturated probability space is a sufficient and necessary condition to establish a general theory on the existence of pure-strategy Nash equilibria in such nonatomic games based on this player space. We conclude in Section 5. he technical proofs are collected in Section 6. 5 For nonatomic games with a compact action set in a finite-dimensional space, the existence of pure-strategy Nash equilibria is shown in Rath (992). See also Khan et al. (997) for the positive results on pure-strategy Nash equilibria for games with countable actions in an infinite-dimensional space. 6 See Remark 3 below. 7 See heorem 2 below. 8 See Proposition 2 below. 4

5 2 Basics Let (,, µ) be a nonatomic probability space. n this paper, we use the convention that a probability space is always a complete, countably-additive probability space. he probability space (,, µ) is nonatomic (or atomless) if for any -measurable set S with positive measure µ(s), there is a -measurable subset S of S such that < µ(s ) < µ(s). hroughout this paper, the Lebesgue unit interval is denoted by (, L, η), i.e., a probability space on the unit interval = [, ] endowed with the Lebesgue σ-algebra L and the Lebesgue measure η. Moreover, let N be the set of all nonnegative integers. Let (X, ) be an infinite-dimensional Banach space with norm. Denote by d(, ) the distance operator on X defined as d(x, y) = x y for any x, y X. Let X be the dual space of (X, ). t is a well-known result that there exists a biorthogonal system on X X, denoted by {(x n, x n ): x n X, x n X and n N} such that x m(x n ) = for any distinct m, n N and x n(x n ) = for all n N. 9 We next review some concepts and notations on the measurability and integrability for functions defined from the probability space (,, µ) to the Banach space (X, ). A function f : X is said to be -measurable if there exists a sequence of simple functions {f n } n N such that lim f n(t) f(t) = for µ-almost all t. A -measurable n function f : X is called Bochner integrable if there exists a sequence of simple functions {f n } n N such that lim n f n f dµ = ; accordingly, the Bochner integral of f, denoted by f dµ, is defined to be lim n f n dµ. Similarly, for any S, f dµ is defined to be S lim n S f n dµ. Let F be a correspondence from (,, µ) to X, denoted it by F : X. A selection of F is a -measurable function f : X such that f(t) F (t) for all t. he Bochner integral of F, denoted by F dµ, is { } F dµ = f dµ: f is a Bochner integrable selection of F. () Let A be a nonempty norm-compact (henceforth compact) subset of X, and con(a) the closed convex hull of A. Note that con(a) is also a compact set, moreover, { con(a) } f dη(t): f is a Bochner integrable function from to A. (2) Let U A be the space of norm-continuous (henceforth continuous) real-valued functions on A con(a) endowed with the sup-norm topology and its corresponding Borel σ-algebra which is generated by this topology. Finally, we specify some components of the nonatomic games with actions in Banach spaces. he player space is modeled by a nonatomic probability space (,, µ). t is worthwhile noting that we use the phrase player space instead of the set of players or the set of players names because we want to emphasize the importance of the probability structure in the modeling. A compact set A X will be the common action set available for all players, and the corresponding set con(a), the closed convex hull of A, serves as the 9 See Lindenstrauss and zafriri (977, Proposition.f.3). For more details, see Diestel and Uhl (977, Chapter 2). his is Mazur s heorem; see Diestel and Uhl (977, heorem 2). 5

6 space of all the possible societal responses. he payoff function for any individual player is a continuous function on A con(a), i.e., an element in U A. Put differently, for any player, his or her payoff continuously depends on his or her own action, and the societal response the average of the actions taken by all players. Suppose that all players in the game take actions as specified in a -measurable function f : A, the the average responses of the society or the societal response can be written as the Bochner integral of f, i.e., f dµ. t follows from Eq. (2) that f dµ con(a). Now we are ready to present the definitions of the game and the pure-strategy Nash equilibrium. Definition. A nonatomic game with the player space (,, µ) and the action space A in an infinite-dimensional Banach space (X, ) is a -measurable function G from to U A. A -measurable function f : A is called a pure-strategy Nash equilibrium of G if for µ-almost all t, ( ) ( ) G(t) f(t), f(t) dµ(t) G(t) a, f(t) dµ(t) where the integral is the Bochner integral. for all a A, 3 A counterexample n this section, we will present a nonatomic game G where the player space is modeled by the Lebesgue unit interval (, L, η) and there does not exist a pure-strategy Nash equilibrium in G, where the players can take actions from any infinite-dimensional Banach space. Before going to the construction of the game, we need to make some preparations. n below, we first review the Walsh system defined on the Lebesgue unit interval, then based on the Walsh system, we will construct a function ψ : X; this function will play an important role in the construction of the payoff functions. We first review the Walsh system {W n } n N defined on the Lebesgue unit interval (, L, η). Here W. For any n, let the binary representation be n = n + 2n a n a where n k is either or for k a 2 and n a =. he n-th Walsh function W n is defined as follows. For any t [, ], denote the binary representation by t = t 2 + t tk +, where each t 2 k k is either or, W n (t) = ( ) n t +n t + +n a t a. (3) t is well-known that {W n (t)} n= is a complete orthogonal basis of the space of squareintegrable functions on the Lebesgue unit interval. 2 For each n N, let E n = {t : W n (t) = }. t is clear that E = and E n is a union of some subintervals in. Moreover, for each integer n, W and W n are orthogonal, i.e., W W n dη =, we have that η(e n ) = /2. Based on a fixed biorthogonal system {(x n, x n ): n N} in Section 2 and the Walsh 2 See Walsh (923). 6

7 system above, we define a mapping ψ : X as follows, for any t, ψ(t) = n= x n 2 n x n W n(t). (4) t is easy to see that ψ is a Bochner integrable function. 3 Let e X be the value of integral of ψ on the Lebesgue unit interval, i.e., e = ψ(t) dη(t) = x x. (5) Moreover, let Ψ: X be the correspondence defined below, for any t, Ψ(t) = {, ψ(t)}. (6) Note that both and e in Eq. (5) belong to the Bochner integral of the correspondence Ψ. However, we have the following negative result, which plays a central rule in our construction in this paper. he proof is given in Section 6. Lemma. e 2 / Ψ dη. Remark. We can define the following vector measure G: L X from the X-valued function ψ in Eq. (4). For any E L, G(E) = ψ dη. t is clear that the range of E the vector measure, G(L) = {G(E): E L}, is identical to the Bochner integral of the correspondence Ψ, Ψ dη. One straightforward implication of Lemma is that the integral of Ψ is not a convex set. As a result, it implies that the range of the vector measure G(L) is not convex. aking the infinite-dimensional Banach space X to be l 2, the space of squaresummable sequences endowed with the following norm, for any x = (x, x, x 2, ) l 2, x = n= x2 n, the statement in Lemma for this special case is exactly the famous Lyapunov example on the range of vector measures (see p. 262 of Diestel and Uhl, 977). 4 Next, we specify a set A, which will serve as the common action set for all the players in our game G. Let A be any compact set in X which contains Ψ dη, the integral of the correspondence Ψ: X. 5 Let M = max{ x : x A}. t is clear that M e = since e A. Moreover, for any b con(a), b M and b e/2 b + e/2 < 2M. Let β = /(2M). Note that βd (b, e/2) = 2M b e/2, where d(, ) is the distance operator in X. Recall that U A denotes the space of the continuous functions on A con(a). Define a function V : U A as follows. For any t, a A and b con(a), V (t)(a, b) = h(t, a, ψ(t), βd(b, e/2)) a a ψ(t), (7) where h: X X R + R + is a function to be defined as below. For any t = 3 ake s m(t) = m lim m n= ψ(t) sm(t) dη(t) =. x n 2 n x n Wn(t). hen {sm(t)} m= is a sequence of Bochner integrable simple functions, and 4 For more discussion on the Lyapunov example on the range of vector measures, see Khan and Zhang (22a). 5 Note that the integral of Ψ, Ψ dη, is contained in { n= anxn : an 2 n }, which is a compact set, hence such a set A can certainly be found. 7

8 [, ], x, y X and l, l sin t l h(t, x, y, l) = π ( x + ( ) [ l] ) ( t x y + + ( ) [ l] ) t, if l > ;, if l =. (8) Applying the same argument as in the proof of Lemma 3 of Khan, Rath and Sun (997), V : U A is a L-measurable function. Now we are ready to specify our nonatomic game G : U A. he player space is modeled by the Lebesgue unit interval (, L, η). he set A X above, which contains the integral of the correspondence Ψ, will serve as the common action set of all the players in this game. Meanwhile, the closed convex hull of A, con(a) will serve as the set of societal responses. Next we present the construction of the payoff functions. For any player t, suppose that his or her own action is a A and the other players take actions as specified in a L-measurable function f : A, the payoff of player t is defined as follows, ( G (t) a, ) ( f dη = V (t) a, ) f dη, (9) where V is defined in Eq. (7). he following is our main result in this section. he proof is given in Section 6. heorem. here does not exist a pure-strategy Nash equilibrium in the game G. Remark 2. he key of the construction of the game G, as well as the example in Khan, Rath and Sun (997, Section 4), is as follows. Based on the particular payoff functions in Eq. (9), the societal response induced by a pure-strategy Nash equilibrium of the game G, if there exists an equilibrium, must be equal to e/2. Meanwhile, when facing this special societal response e/2, the best response for each player t [, ] is either or ψ(t). n other words, if there exists a pure-strategy Nash equilibrium of the game G, it must be a L-measurable selection of the correspondence Ψ in Eq. (6) and its Bochner integral is e/2. However, such a selection does not exist according to Lemma. t is worthwhile to note that such an idea to extend the counterexample for l 2 to a counterexample for arbitrary infinite-dimensional Banach spaces is also roughly mentioned in Khan, Rath and Sun (997). 6 For any s (, ], we next consider another nonatomic game G s : ( = [, ], L, η) U A such that there is no pure-strategy Nash equilibrium either. For any player t, the payoff function of this player is defined to be a function G s (t) : A con(a) R as follows, for every a A, every societal response b con(a), ( G t ( ) s) a, b s, if t [, s] and b s ( con(a); G s (t) (a, b) = G t s) (a, c b), if t [, s] and b s / con(a); a, if t (s, ]. 6 See p. 33 of Khan, Rath and Sun (997), by an appealing to Diestel and Uhl (977, Corollary 6, p. 265), one should hopefully be able to set a version of the counterexample in any arbitrary infinite dimensional Banach space. () 8

9 where G ( ) is the payoff function of in the game G, c b is the intersection between the boundary of con(a) and the ray from to b. Notice that for any t, G s (t) is a continuous function on A con(a). And it is easy to check that G s is a Lebesgue measurable function from the Lebesgue unit interval to U A. Corollary. For any s (, ], there is no pure-strategy Nash equilibrium in G s. 4 Saturation and games We know from Section 3 that if the player space is modeled by the Lebesgue unit interval and the common action set is in an arbitrary infinite-dimensional Banach space, then there is a corresponding nonatomic game such that there is no pure-strategy Nash equilibrium. n contrast, Khan and Sun (999) find that such a general theory can be established if the player space is modeled by a nonatomic hyperfinite Loeb space (see heorem 2 therein). he purpose of this section is to answer the question that which property of the Lebesgue unit interval (or of the nonatomic Loeb spaces) is responsible to the failure (or success) here. We find that the underlying player space being a saturated probability space is a sufficient and necessary condition for the existence of pure-strategy Nash equilibria in such nonatomic games. Section 4. introduces the definition of the saturated probability space; the sufficiency and necessity results are presented in Sections 4.2 and 4.3 respectively. 4. Saturated probability space Given a nonatomic probability space (,, µ), for any subset S with µ(s) >, denote by (S, S, µ S ) the probability space restricted to S. Here, S is the σ-algebra {S S : S } and µ S is the probability measure re-scaled from the restriction of µ to S. Definition 2. A probability space (,, µ) is called countably-generated (modulo the null sets) or essentially countably-generated if there is a countable set {A n : n N} such that for any S, there is a set S in the σ-algebra generated by {A n : n N} with µ(s S ) =, where denotes the symmetric difference in. A probability space (,, µ) is called saturated if for any subset S with µ(s) >, the re-scaled probability space (S, S, µ S ) is not essentially countably-generated. t is well-known that the Lebesgue unit interval is essentially countably-generated, and hence not saturated. n contrast, any nonatomic Loeb space is saturated. One can also extend the Lebesgue unit interval into a saturated probability space, see Kakutani (944), Podczeck (28, Section 6) and Sun and Zhang (29). 7 Remark 3. n literature, saturated probability spaces are firstly considered by Hoover and Keisler (984). hey introduce the saturation property as follows: a nonatomic probability space (,, µ) is said to have the saturation property for a probability measure ν on the product of Polish spaces X Y if for every random variable f : X which induces the 7 n Sun and Zhang (29), the construction of a saturated extension of the usual Lebesgue unit interval is not an issue, while the key is to construct a rich Fubini extension based on this extended Lebesgue interval. 9

10 distribution as the marginal measure of ν over X, then there is a random variable g : Y such that the induced distribution of the pair (f, g) on (,, µ) is ν; (,, µ) is said to be saturated if it has the saturation property for every probability measure ν on every product of Polish spaces. By Maharam s heorem (see Maharam (942)), the saturated probability space in Definition 2 is different but equivalent to some existing concepts such as superatomless space (see Podczeck (28)) and nowhere (essentially) countably-generated space, i.e., its Maharam spectrum is a set of uncountable cardinals. t has been shown that the saturated probability space defined in Hoover and Keisler (984) is equivalent to that in Definition 2; see e.g., Fajardo and Keisler (22, heorem 3B.7, p. 47). As a result, for the sake of briefness, we re-define saturated probability spaces as in Definition he sufficiency result n this section, we assume that (X, ) is an infinite-dimensional Banach space, and A is a nonempty compact subset of it. As before, let U A be the space of continuous real-valued functions on A con(a) endowed with both the sup-norm topology and the corresponding Borel σ-algebra (generated by this topology). Proposition. f G is a measurable map from to U A, where (,, µ) is a saturated space, then the game has a pure-strategy Nash equilibrium. 8 he following three remarks provide three straightforward but different proofs of Proposition based on the previous results. Remark 4. Proposition can be proved by transferring the existing results on nonatomic Loeb spaces (see heorem 2 in Khan and Sun (999)) via the saturation property in Remark 3. 9 For the game G on the saturated probability space (,, µ), we have a game F on a nonatomic Loeb space such that F has the same distribution with G. By heorem 2 in Khan and Sun (999), there is a pure-strategy Nash equilibrium f of the game F. he saturation property implies the existence of g : A, such that the joint distribution of F and f is same as the joint distribution of G and g. t is not difficult to see g is a pure-strategy Nash equilibrium of the game G. Remark 5. his proposition can also be proved via the existing result on nonatomic games where the societal response is formulated as the distribution. Let ÛA be the space of realvalued continuous functions on A M(A) endowed with the sup-norm topology, where M(A) is the space of all Borel probability measures on A with Prohorov metric. A new game Ĝ : (,, µ) ÛA can be defined as follows: for each player t, a A and τ M(A), ( ) Ĝ(t)(a, τ) = G(t) a, d A dτ, A where d A is the identity map on A. t follows from Keisler and Sun (29, heorem 4.6) that this new game Ĝ has a pure-strategy Nash equilibrium f : A; see also Corollary 4(4) Carmona and Podczeck (29). t is also a pure-strategy Nash equilibrium of the 8 A result similar to Proposition is also obtained independently by Yu (22), see Lemma 3 therein. 9 For more discussion on this transferring technique, see Keisler and Sun (29, Section 5).

11 original game G because A d A d ( µ f ) = f dµ, where the equation holds according to the substitution of variables. Remark 6. One can also show Proposition by applying the fixed-point theorem of Fan (952) and Glicksberg (952). One needs to check the convexity, compactness and preservation of upper hemi-continuity for the integration of Banach space-valued correspondences on general saturated probability spaces. Here is a brief discussion of related papers on the integration theory of the Banach-valued correspondences. For example, Khan and Majumdar (986) and Yannelis (99) consider the approximate versions of Fatou s lemma. However, one should note that Proposition needs the exact version of Fatou s lemma which guarantees the preservation of upper hemicontinuity. Rustichini and Yannelis (99) show properties of convexity, compactness and preservation of upper hemi-continuity for the Bochner integral of a correspondence from a probability space (,, µ) to a fixed weakly compact set in an infinite-dimensional Banach space under the condition that for each nonnegligible E, the cardinality of L E (µ) is larger than the continuum. Sun (997) proves those properties for the integration of general correspondences over nonatomic Loeb spaces. he convexity and compactness results over general saturated probability spaces are shown in Podczeck (28). Based on the compactness result, Yannelis (99) points out preservation of upper hemi-continuity for the Bochner integral of correspondences can be proved directly; see heorem 3. and Remark 3. therein. 2 Sun and Yannelis (28) point out that all the above desired properties for the integration of correspondences over general saturated property spaces follow straightforwardly from the results over nonatomic Loeb spaces via the saturation property; see Proposition therein. Remark 7. Other different models of nonatomic game with infinite-dimensional actions are also considered in Khan, Rath and Sun (997) and Khan and Sun (999). First, instead of taking actions from a norm-compact subset, the players in the nonatomic game in Khan and Sun (999, heorem 2) can take actions from a weakly compact subset from a separable Banach space and this nonatomic game is a weakly continuous function. Note that a normcompact set is also a weakly compact set in a Banach space and norm continuous functions on norm-compact sets are weakly continuous. As a result, the model in Khan and Sun (999) is more general than the model in this paper. Nevertheless, our Proposition above still holds for this more general model, provided that the Banach space is separable. 2 t is in this sense that we claim that our Proposition generalizes heorem 2 of Khan and Sun (999). Second, Khan, Rath and Sun (997, Section 6) also consider a setting of nonatomic games where players taking actions from a weak* compact subset in the dual of a separable Banach space. n this setting, the societal response is represented by the Gel fand integral of the strategy profile and a nonatomic game is a weak* continuous function. his modeling 2 We are grateful for one referee to point out this fact. 2 his claim follows from the Fan-Glicksberg s fixed-point argument and the results in Sun and Yannelis (28) on the integration of weakly compact Banach-valued correspondences over the saturated spaces; the corresponding convexity and the compactness results are also proved by Podczeck (28). t is worthwhile to note that the separability is necessary, since the weak topology on a weakly compact set in a separable Banach space is metrizable; otherwise, the preservation of upper hemi-continuity may fail.

12 is closely related to the nonatomic games with actions in compact metric spaces, and the societal response is represented as the action distribution of the strategy profile; see Khan, Rath and Sun (997) for more discussion. t follows from the standard Fan-Glicksberg fixedpoint argument, together with the results in Sun and Yannelis (28) on the integration of weakly* compact correspondences over the saturated spaces, that there exists a purestrategy Nash equilibrium for such nonatomic games; the corresponding convexity and the compactness results are also proved by Podczeck (28). Remark 8. n Yu and Zhu (25), a nonatomic game model is presented where the players payoffs depend on their own actions and the average of certain transformation of the strategy profile for all the other players, where the transformation is a continuous map transforming actions into some statistics data in a finite-dimensional space. According to Proposition, the results in Yu and Zhu (25) can be generalized to infinite-dimensional Banach spaces, provided that the player space is modeled by a saturated probability space. 4.3 he necessity result n this section, (X, ) is any infinite-dimensional Banach space, and A is fixed to be the compact subset of X which contains the integral of the correspondence Ψ: X as in Eq. (6). We now present the following result. heorem 2. Fix the player space (,, µ), a nonatomic probability space and the common action set A. f there exists a pure-strategy Nash equilibrium for any nonatomic game with the above player space and the action set, then (,, µ) is a saturated probability space. Here is a claim equivalent to this heorem. f the player space is modeled by a nonsaturated probability space, then there is a nonatomic game based on this player space such that there does not exist a pure-strategy Nash equilibrium. he following is a sketch for the construction of the nonatomic game in this situation. First, note that the player space is modeled by a non-saturated probability space (,, µ), by Definition 2, there is a non-negligible subset, denote it by S with measure s = µ(s), such that the restricted subspace over S is an essentially countably-generated probability space. As a result of Maharam s theorem (see Maharam, 942), the measure algebra of the restricted subspace over S is isomorphic to that of the Lebesgue unit interval, thus that of the Lebesgue subinterval [, s]. t is a well-known result that a measurable map q can be constructed from the probability space (,, µ) to the Lebesgue unit interval such that it maps S to [, s] and the restriction of q over S can induce the measure-algebra isomorphism mentioned above; see Fremlin (989, heorem 4.2, p. 937). Next, a nonatomic game can be obtained by a composition of the nonatomic game G s in Section 3 and the above mapping q; denote the new nonatomic game by G s, where the player space is (,, µ). Note that the game G s is defined on the usual Lebesgue unit interval and there is no pure-strategy Nash equilibrium in such a game; see Corollary. he construction of the mapping q can ensure us to transfer the non-existence result from the nonatomic game G s to the non-existence in the game G s. n particular, suppose that there is a pure-strategy Nash equilibrium in G s, then one can construct a pure-strategy Nash equilibrium in G s. t contradicts the statement in Corollary. We thus show by contradiction that there is no pure-strategy Nash equilibrium in G s. 2

13 Remark 9. n heorem 2, we show that the underlying player space being a saturated probability space serves a necessity condition for the existence of pure-strategy Nash equilibrium for nonatomic games with actions in infinite-dimensional Banach spaces. A similar necessity result is presented in Keisler and Sun (29, heorem 4.6) in the context of nonatomic games with actions in a common uncountable compact metric space and there the societal response is modeled by the distribution of actions for all the players. From a technical point of view, the results in Keisler and Sun (29) are built on the theory of distribution for compact space-valued correspondences, while our resluts on the theory of integration for Banach space-valued correspondences. As discussed in Remark 5, Proposition is a straightforward result of the sufficiency part of Keisler and Sun (29, heorem 4.6). However, the proof of heorem 2 is not implied by, actually far more involved than, the proof of the necessity part of Keisler and Sun (29, heorem 4.6). 5 Discussion n this paper, we find that the player space being a saturated probability space is a sufficient and necessary condition for the existence of pure-strategy Nash equilibria in nonatomic games taking actions in infinite-dimensional Banach spaces. 22 his property is sufficient in the sense that if the player space is a saturated probability space, then every such a nonatomic game has a pure-strategy Nash equilibrium, and it is necessary that if every nonatomic game has a pure-strategy Nash equilibrium, then the player space must be a saturated space. his finding answers why one can not establish a general result on the existence of pure-strategy Nash equilibria in nonatomic games on the Lebesgue unit interval as in heorem but one can on nonatomic Loeb space as in Khan and Sun (999). t is simply because that the Lebesgue unit interval is not saturated but nonatomic Loeb spaces are. However, as far as the counterexample in heorem is concerned, to guarantee the existence of a pure-strategy Nash equilibrium in this nonatomic game G, it is not necessary to turn to saturated probability spaces, while a simpler extension of the Lebesgue unit interval in the sense of Khan and Zhang (22a, Section 2.2) does serve this purpose. n this section, we fix (,, λ) to denote the extended Lebesgue interval constructed in Khan and Zhang (22a, Section 2.). We have the following positive result. Proposition 2. here exists a pure-strategy Nash equilibrium in G if the player space is modeled by the extended Lebesgue interval (,, λ). 6 Appendix 6. Proofs of results in Section 3 Proof of Lemma. We prove this result by contradiction. Suppose that e/2 Ψ(t) dη(t), that is, there is a selection f : X of Ψ such that f dη = e/2. Here f is a L-measurable 22 n Khan and Zhang (22b), sufficient and necessary results on the property of saturation are established in the context of finite-player games with diffused private information. 3

14 function and for any t, f(t) is either or ψ(t). Let E = {t : f(t) }. t is clear that E L and f dη = E ψ dη.. As a result e/2 = ψ(t) dη(t), that is, E x 2 x = x n 2 n W n (t) dη(t). () x n E n= Apply x on the both sides of Eq. (), we will have η(e) = /2. hen for each integer n, apply x n on the both sides of Eq. (), we will have η(e E n ) = η(e E c n), where E c n is the complement of E n. Let g = χ E χ E c, where χ E is the indicator function of E. hen W n(t)g(t) dη(t) = for each integer n. Because that the Walsh system is a complete orthogonal basis of the space of all square integrable functions on the Lebesgue unit interval, g =, hence it is a contradiction to the definition of g. herefore e/2 / Ψ dη. Before we offer a proof of heorem, we shall need three additional inequalities. Lemma 2. For α > and any real numbers b, b 2 and c, with b < b 2, we have Proof. Routine. b2 Lemma 3. For l >, and any integer n, Proof. t is obvious that b ( ) [ t+c l ] dη(t) l. ( ) [ l] t Wn (t) dη(t) min{, 2nl}. ( ) [ l] t Wn (t) dη(t). We next show the other part. Consider the binary representation of n, n = n + 2n n a n a, where n a = and n, n,..., n a 2 are either or. For any t ( ) s 2, s a 2, where s is an a integer from {, 2,..., 2 a }, it is clear that the first a numbers in the binary representation of t are fixed. By definition, W n (t) = ( ) cs over ( ) s 2, s a 2, where a cs is a constant integer. hen we have a ( ) [ 2 l] t Wn (t) dη(t) s= s 2 a ( ) [ t s 2 a a 2 l] Wn (t) dη(t) = s= s 2 a s 2 a ( ) [ t+csl l ] dη(t). 4

15 t follows from Lemma 2 that a ( ) [ 2 l] t Wn (t) dη(t) l = 2 a l 2nl, where the last equality holds because of the binary representation of n. s= Lemma 4. For l >, n= 2 n+ ( ) [ l] t Wn (t) dη(t) < 2l. Proof. ake m = [/(2l)], then m /l and m + > /(2l). By Lemma 3, we will have n= 2 n+ ( ) [ l] t Wn (t) dη(t) min{, 2nl} 2n+ n= m 2 n+ 2nl + = l n= = 2l ( 2 m m n=m+ ) + 2 m+ 2 n+ 2(m + 2)l 2 m+ < 2l. Proof of heorem. We will prove this result by contradiction. Suppose that there exists a pure-strategy Nash equilibrium in the game G, and assume that the L-measurable function f : A is a pure-strategy Nash equilibrium. We next prove that the following two cases can not happen, () f(t) dη(t) = e/2 and (2) f(t) dη(t) e/2, where f(t) dη(t) is the societal response when players take actions as in the Nash equilibrium f and e = x / x as in Eq. (5). herefore we obtain a contradiction. Next we discuss these two cases separately by dividing our arguments into the following two parts. Part. Suppose that f(t) dη(t) = e/2. hen for any player t, the payoff function of player t reduces to the following form, for any a A, ) u t (a, f dη = u t (a, e/2) = a a ψ(t), (2) As a result, the best response of player t is the doubleton set {, ψ(t)} for any t. Hence the pure-strategy Nash equilibrium f, as a function from the Lebesgue unit interval to A, is a measurable selection of the correspondence Ψ in Eq. (6). However, Lemma ensures that there does not exist a Lebesgue measurable selection from this correspondence Ψ whose Bochner integral is e/2. Hence it is a contradiction that f(t) dη(t) = e/2. Part 2. Suppose that f(t) dη(t) e/2. hat is, in the pure-strategy Nash equilibrium f : A, the societal response is no longer e/2. As a result, in the first item h, see Eq. (8), 5

16 of the payoff function for any player t, the fourth argument is no longer zero. Let ( l = βd f(t) dη(t), e ). (3) 2 Certainly, < l. Divide the unit interval into intervals of length l, so that we obtain ( n N [nl, (n + )l ) ), where N is the set of all nonnegative integers. We first characterize the set of best responses for any player when other players take actions in the Nash equilibrium f, i.e., the societal response is f dη. On the one hand, we first fix a player t (nl, (n + )l ). f n is even, then the integer part of t/l, [t/l ], is even, as a result the payoff function of this player t (see Eq. (9)) reduces to the following form, for any a A, ) u t (a, f dη = l sin t π l a ( a ψ(t) + 2) a a ψ(t). Note that the value of u t (a, f dη) for any a A, and the value is if and only if a =. As a result, u t (a, f dη) takes the maximum value only at a =. hat is, the best response for this player t, when facing the societal response f dη, is the singleton set {}. Similarly, for player t (nl, (n + )l ), if n is an odd natural number, so is the integer part [t/l ]. As a result, the payoff function of this player t reduces to the following form, for any a A, ) u t (a, f dη = l sin t π l ( a + 2) a ψ(t) a a ψ(t). Using the similar argument as above as n is even, we can obtain that the best response for this player t, when facing the societal response f dη, is the singleton set {ψ(t)}. On the other hand, we consider the player t = nl for some n N. n this case, the payoff function of player t (see Eq. (9)) reduces to the following form, for any a A, ) u t (a, f dη = a a ψ(t). t is clear that the set of best responses for this player t, when facing the societal response f dη, is a doubleton set {, ψ(t)}. o summarize, except for the players in the set {t = nl : n N}, the best response for any other player is a singleton set. Note also that {t = nl : n N} is a η-null set in the Lebesgue interval. As a result, the pure-strategy Nash equilibrium f : A is of the following form, for almost all t, f(t) = ( ) 2 [ ] t l ψ(t). (4) Next we calculate the distance between the societal response f dη and e/2, where the 6

17 pure-strategy Nash equilibrium f is determined in Eq. (4) above. ( d f(t) dη(t), e ) = 2 f(t) dη(t) e 2 = f(t) dη(t) ψ(t) dη(t) 2 [ = t l ( ) ]ψ(t) dη(t) 2 [ ] t x l n = ( ) 2 2 n x n= n W n(t) dη(t) [ ] = n= 2 ( ) t x l n 2 n x n W n(t) dη(t) [ ] t x l n 2 ( ) n= 2 n x n W n(t) dη(t) [ t l 2 n+ ( ) ]W n (t) dη(t) n= By virtue of the inequality in Lemma 4, we have ( d f(t) dη(t), e ) < 2 n= 2 n+ 2l = 2l. (5) Notice that l = βd( f, e/2), where β = /(2M) and M = max{ a : a A}. Note that M, it follows that β /2. t follows from Eq. (5) that d( f dη, e/2) < 2l = 2βd( f dη, e/2). his implies that β > /2. t is a contradiction with β /2. herefore we complete the proof of Part 2 that there does not exist a pure-strategy Nash equilibrium f : A with f dη e/2. Proof of Corollary. We show it by contradiction. Suppose that f : [, ] A is a purestrategy Nash equilibrium in the game G s. Note that for any player t (s, ], the best response is. As a result, in this Nash equilibrium f, the actions for players in (s, ] do not affect the societal response. n particular, f dη = s s s f dη = s f dη [,s], which is exactly an element in con(a). Moreover, for any player t [, s], if the other players follow actions as in this Nash equilibrium f, for any a A, ( ) ( ) ( t G s (t) a, f dη = G a, s s s ) f dη. For η-almost all player t [, s], since f is a pure-strategy Nash equilibrium, f(t) is a best response for this player t given others players follow f. hat is, for any a A G ( t s ) ( f(t), s s ) ( ) ( t f dη G a, s s s ) f dη. (6) 7

18 Let g : [, ] A defined by g(t ) = f(t s) for η-almost all t [, ]. t follows from substitution of variables that s s f(t) dη(t) = g(t ) dη(t ). According to Eq. (6), for all t [, ] and a A ( G (t ) g(t ), ) ( g dη G (t ) a, ) g dη. Hence g is a pure-strategy Nash equilibrium for the nonatomic game G. t is a contradiction with heorem. 6.2 Proof of heorem 2 We prove this theorem by contradiction. Suppose that (,, µ) is not a saturated probability space, we will construct a game with (,, µ) being the player space and A the set of common actions such that there does not exist a pure-strategy Nash equilibrium in this game. Before going to the construction of the game, we take a particular look at some subspace of (,, µ) which is essentially countably-generated. Since (,, µ) is not a saturated probability space, by Definition 2, there exists a nonnegligible subset S, such that the restricted probability space (S, S, µ S ) is an essentially countably-generated probability space, where S = {S E : E } and µ S (E S) = µ(e S)/µ(S) for any E. Let µ(s) = s. Since S is nonnegligible and (,, µ) itself is not an essentially countablygenerated probability space, < s. As a result of Maharam s theorem (see Maharam, 942), the measure algebra of the Lebesgue subinterval, ([, s], L [,s], η [,s] ) is isomorphic to the measure algebra of (S, S, µ S ). t is a known result that this isomorphism can be realized by a measure preserving map q from (S, S, µ S ) to ([, s], L [,s], η [,s] ); see Fremlin (989, heorem 4.2, p. 937). We now consider the measure space restricted to \S, the complementary set for S in. Since it is also a nonatomic measure space, by Keisler and Sun (29, Lemma 2.), there exists a measurable map q 2 : \S [s, ] such that the induced distribution of q 2 on [s, ] is the Lebesgue measure on [s, ]. Let q be a map from (,, µ) to the Lebesgue unit interval defined as follows, q (t), if t S; q(t) = (7) q 2 (t), if t / S. t is clear that q is a -measurable map and the induced distribution of q is the Lebesgue measure η. We are now ready to construct a game with the player space (,, µ) such that there does not exist pure-strategy Nash equilibrium, denote it by G s. his game can be obtained from the game G s in Corollary. n particular, for any player t, suppose that his or her own action is a A and the other players take actions following the action profile g : (,, µ) A, the payoff function of player t is defined as follows, ( ) ( ) G s(t) a, g dµ = G s (q(t)) a, g dµ = G s (q (t)) ( a, g dµ), if t S; a, if t / S. (8) 8

19 where G s is the game defined on the Lebesgue unit interval in Corollary (see Eq. ()), q is the map from the player space (,, µ) to the Lebesgue unit interval in Step, and moreover q is the measure-preserving map from (S, S, µ S ) to the Lebesgue subinterval ([, s], L [,s], µ [,s] ). Finally, we show by contradiction that there does not exist pure-strategy Nash equilibrium in G s. Suppose that g : A is a pure-strategy Nash equilibrium for the game G s. According to the payoff function of this game, it is clear that for any player t / S, the best response is. As a result, such players does not affect the societal response. n particular, g dµ = g dµ = g dµ S con(a). s s S S As a result, for every player t S, the payoff function is, for all a A, ( ) ( ) ( ) ( ) G s(t) q (t) a, g dµ = G s (q (t)) a, g dµ = G a, g dµ S, (9) s S where the second equation follows from Eq. (). Since g is a pure-strategy Nash equilibrium in the game G s, it follows from the proof of heorem (see, e.g., Eqs. (2) and (4)) that for µ-almost all player t S, the best response g(t) is either ψ (q (t)/s) or. Let S denote the set of players in S whose best response is non-zero, i.e., S = {t S : g(t) }. Since g is -measurable, S. Note that q induces a measure-algebra isomorphism over subspaces restricted to S and [, s]. According to Corollary of Khan and Zhang (22a), there exists a Lebesgue subset E [, s] such that q (E), a -measurable subset, and S differ up to a µ-null set, i.e., µ[s q (E)] = where is the symmetric difference operator in. Now define g : [, ] A as follows, ( ) t ψ g (t s, if t E; ) =, otherwise. t is clear that g is a L-measurable map and g (q (t)) = g(t) for µ-almost all t S. Moreover, g dµ = g dµ = g (q ) dµ = g dη = g dη, (2) S q (E) E where the second equation follows from µ[s q (E)] =, the third from substitution of variables and µq = η, and the first and last from the fact that g and g are almost elsewhere. Because g is a pure-strategy Nash equilibrium in the game G s, then for µ-almost all t S, ( ) ( ) G s (q (t)) g(t), g dµ G s (q (t)) a, g dµ, for all a A. According to Eqs. (2) and g (q (t)) = g(t) for µ-almost all t S, it follows that for (2) 9

20 µ-almost all t S, ( G s (q (t)) g (q (t)), ) ( g dη G s (q (t)) a, ) g dη, for all a A. Since q induces a measure-algebra isomorphism over subspaces restricted to S and [, s], it follows that for η-almost all t [, s], ( G s (t ) g (t ), ) ( g dη G s (t ) a, ) g dη, for all a A. Note that when t (s, ], by the definition of g, g (t ) = ; is already the best response for this player. herefore g is a pure-strategy Nash equilibrium for the nonatomic game G s. A contradiction to Corollary. 6.3 Proof of Proposition 2 n the essentially countably-generated extended Lebesgue interval (,, λ) constructed in Khan and Zhang (22a, Section 2.), by Lemma 2 of Khan and Zhang (22a), there exists a subset S such that for any t [, ], λ([, t] S) = t/2. Recall that for any nonnegative integer n N, E n is the set of points where the value of W n is ; by the definition of the Walsh system, see p. 6, E = [, ] and for n, E n is a union of some subintervals of the Lebesgue unit interval with η(e n ) = /2. As a result, λ(e n S) = λ(e n )/2 for all n N. Since that (,, λ) is the extended Lebesgue interval, E n and λ(e n ) = µ(e n ) for all n. o summarize, we have the following equations, Now define a function f : (,, λ) X as follows, 2, if n = ; λ(e n S) = 4, if n. (22) ψ(t), if t S; f(t) =, if t / S. Note that f is an -measurable function since ψ : X is a L-measurable function and (,, λ) is an extended Lebesgue interval. t follows from Eq. (22) that f dλ = e/2. We complete the proof by showing that the function f defined above is a pure-strategy Nash equilibrium for the game G if the player space is upgraded by the extended Lebesgue interval (,, λ). Note that f dλ = e/2, so for any player t [, ], his or her payoff function G (t) as defined in Eq. (9) is reduced to the following form: for any a X, ( ) G (t) a, f dλ = G (t) (a, e/2) = a a ψ(t). Notice that for this player t, f(t) is either or ψ(t) by the definition of f, hence f(t) is a best response against e/2. herefore f is a pure-strategy Nash equilibrium for the game G if the player space is modeled by the extended Lebesgue interval (,, λ). 2