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Noncompact qualitative games with application to equilibria
Nonlinear Analysis 65 (2006) 593–600 www.elsevier.com/locate/na
Noncompact qualitative games with application to equilibria Shiow-Yu Chang ∗ Department of Mathematics, Soochow University, Taipei, Taiwan, ROC Received 27 August 2005; accepted 26 September 2005
Abstract In 2003, S.Y. Chang [Maximal elements in noncompact spaces with application to equilibria, Proc. Amer. Math. Soc. 132 (2) (2003) 535–541] obtained a maximal theorem for L S -majorized correspondences in noncompact spaces. In this paper, we extend this result to obtain new existence theorems for qualitative games. Applying it, we have more refined existence theorems for both abstract economies and generalized n-person games than before. c 2005 Elsevier Ltd. All rights reserved. MSC: primary 91A13; secondary 52A07, 91B50 Keywords: L S -majorized correspondences; Qualitative game; Abstract economy; Generalized n-person game
1. Introduction In Section 2, we give notations and definitions. In Section 3, we first prove one useful lemma. Next, by combining it and the maximal theorem for L S -majorized correspondences in noncompact spaces in [2], we obtain a new existence theorem for qualitative games. Also, by proving the L S -majorized property of upper semicontinuous preference correspondences, we have another existence theorem for qualitative games. In Section 4, we establish more refined existence theorems for both abstract economies and generalized n-person games than before— see, for example, [1–9]. ∗ Tel.: +886 2 288 194716705; fax: +886 2 288 11 526.
E-mail address: [email protected]. c 2005 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2005.09.041
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2. Notations and definitions 1. 2. 3. 4. 5. 6. 7.
2 A denotes the set of all subsets of A; co A denotes the convex hull of the set A; Conv A denotes the convex closure of the set A; A/B denotes the difference of sets A and B; cl A, int A denotes the closure and interior of the set A; φ : X → 2Y is a correspondence, that is, φ(x) is a subset of Y for each x ∈ X; Let X = i∈I X i and π j : X → X j be called the projection of X onto X j , if π j (x) = x j for each x = (x i )i∈I ∈ X.
Definition 1. Let I be a (possibly uncountable) set. For each agent i ∈ I , let its choice set or strategy set X i be a nonempty set in a topological vector space. Let X = j ∈I X j . Following [6], Γ = (X i , Pi )i∈Iis a qualitative game if, for each player i ∈ I , X i is the strategy set of player i , and Pi : X = j ∈I X j → 2 X i is a preference correspondence of player i which is irreflexive [i.e. πi (x) ∈ Pi (x) for each x ∈ X]; also, a point x˜ ∈ X is said to be an equilibrium point of the game Γ = (X i , Pi )i∈I if Pi (x) ˜ = ∅ for each i ∈ I . A generalized model is as follows: Definition 2. Let I denote the set of agents. For each i ∈ I , let X i be a nonempty set and X = j ∈I X j . Following [3], an abstract economy or (generalized game) G = (X i , A i , Bi , Pi )i∈I is defined as a family of quadruples (X i , Ai , Bi , Pi ), where Ai , Bi : X → 2 X i are feasible correspondences of agent i and Pi : X → 2 X i are preference correspondences of agent i . An equilibrium point for G is an x˜ ∈ X, where x˜ = (x˜i ) satisfies 1. x˜i ∈ cl Bi (x); ˜ 2. Pi (x) ˜ ∩ Ai (x) ˜ =∅ for each i ∈ I . When Ai = Bi , for each i ∈ I our definition coincides with the standard definition, e.g. in [1] or in [8]. Definition 3. Let X be a topological space, Y a nonempty subset of a vector space E, θ : X → E a map, and φ : X → 2Y a correspondence. The following notions were introduced in [1–9]. 1. φ is said to be upper semicontinuous (u.s.c.) at x if, for each open set V in Y with φ(x) ⊂ V , there exists an open neighborhood U of x in X such that φ(x ) ⊂ V for each x ∈ U . 2. A set G ⊂ X is said to be compactly open in X if, for each compact set K in X, the set G ∩ K is open in K . 3. φ is said to have compactly open lower sections in X if, for each y ∈ Y , the set φ −1 (y) = {x ∈ X | y ∈ φ(x)} is compactly open in X. 4. φ is said to have locally compactly open lower section at x if there is an open set Wx containing x such that, for each y ∈ Y , the set φ −1 (y) ∩ Wx is compactly open in X. 5. φ is said to be of class L θ,S if, for every x ∈ X, θ (x) ∈ co φ(x), and φ has compactly open lower sections in X. 6. A correspondence φx : X → 2Y is said to be an L θ,S -majorant of φ at x ∈ X if there exists an open neighborhood Nx of x in X such that (a) for each z ∈ Nx , φ(z) ⊂ φx (z) and θ (z) ∈ co φx (z), (b) for each y ∈ Y , φx−1 (y) is compactly open in X. 7. Suppose X ⊂ X. φ is L θ,S -majorized in X if, for each x ∈ X with φ(x) = ∅, there exists an L θ,S -majorant of φ at x in X.
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In this paper, we deal mainly with the case (I) X = Y which is a nonempty convex subset of the topological vector space E and θ = I X , the identity map on X, or the case (II) X = i∈I X i and θ = π j : X → X j is the projection of X onto X j and Y = X j is a nonempty convex subset of a topological vector space. In both cases (I) and (II), we write L S in place of L θ,S . 3. Noncompact qualitative games The results of this paper are to extend the following in [2]. Theorem 1. Let X be a convex subset of a Hausdorff topological vector space E and let U : X → 2 X be L S -majorized. Suppose there exists a compact set D of X such that, for each finite subset S of X, there is a compact convex set K containing S and satisfying U −1 (x) ⊂ D. K/ x∈K
Then {x ∈ X | U (x) = ∅} is a nonempty subset of D. We shall need the following simple result. Lemma 1. Let Γ = (X i , Pi )i∈I be a qualitative game, X = i∈I X i , and D be a compact subset of X. For each i ∈ I , X i is a nonempty convex subset of a Hausdorff topological vector space and Pi : X → 2 X i is L S -majorized. Suppose, for each x ∈ D, there is some i ∈ I such that Pi (x) = ∅. Then there are finite compactly open cover O1 , . . . , On of D, sets S1 , . . . , Sn , and L S -majorized correspondences P˜i1 , . . . , P˜in such that, for each j = 1, . . . , n 1. Gr Pi j ⊂ Gr P˜i j ; 2. S j is a finite subset of X i j ; 3. if x ∈ O j then P˜i j (x) ∩ S j = ∅. Proof. Fix x ∈ D, there is some i ∈ I such that Pi (x) = ∅. Then there is φx an L S -majorant of Pi in an open neighborhood W of x. Choose an open neighborhood U of x such that cl U ⊂ W and x i ∈ Pi (x). Let Vx = U ∩ φx−1 (x i ). Let Pi : X → 2 X i be defined by φ (z), if z ∈ cl Vx ; Pi (z) = x Pi (z), if z ∈ [X/cl Vx ]. Then Gr Pi ⊂ Gr Pi , Pi is everywhere L S -majorized, and Pi (z) = ∅ for each z ∈ Vx . Suppose D/Vx = ∅. Fix y ∈ D/Vx . Continue the above process with {P j | j ∈ I /{i }} ∪ {Pi }. Since D is compact, after continuing finite times, there are finite compactly open cover O1 , . . . , On of D, sets S1 , . . . , Sn , and L S -majorized correspondences P˜i1 , . . . , P˜in such that for each j = 1, . . . , n 1. Gr Pi j ⊂ Gr P˜i j ; 2. S j is a finite subset of X i j ; 3. if x ∈ O j then P˜i j (x) ∩ S j = ∅. By the separation property, we yield that an upper semicontinuous preference correspondence is L S -majorized in the following: Lemma 2. Let X be as in Lemma 1, X i be a convex subset of a locally convex space E i , x˜ ∈ X, and Pi : X → 2 X i . If Pi (x) ˜ = ∅, x˜i ∈ Conv Pi (x), ˜ and Pi is u.s.c. at x, ˜ then Pi is L S -majorized at x. ˜
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Proof. Since Pi (x) ˜ = ∅, x˜i ∈ Conv Pi (x), ˜ and E i is a locally convex space, there is an open convex neighborhood W of 0 in E i such that [x˜i + W ] ∩ [Conv Pi (x) ˜ + W ] = ∅. Since both Pi and πi are u.s.c. at x, ˜ there is an open neighborhood V of 0 in Pi (x˜ + V ) ⊂ [Conv Pi (x) ˜ + W ],
i∈I
(*) E i such that
πi (x˜ + V ) ⊂ [x˜i + W ].
(**)
˜ + W ]} = ∅. {[x˜ + V ] × [x˜i + W ]} ∩ {[x˜ + V ] × [Conv Pi (x)
(***)
By (*), we have Define P˜i : X → 2 X i by ˜ + W, if x ∈ x˜ + V ; Conv Pi (x) ˜ Pi (x) = ∅, otherwise. ˜ Thus Pi is Then P˜i has an open graph. By (**) and (***), P˜i is an L S -majorant of Pi at x. ˜ L S -majorized at x. By Theorem 1 and Lemma 1, we shall present a noncompact form of an existence theorem for qualitative games in the following: Theorem 2. Let Γ = (X i , Pi )i∈I be a qualitative game such that, for each i ∈ I , 1. X i is a convex subset of a Hausdorff topological vector space; 2. Pi : X → 2 X i is L S -majorized. Suppose there exists a compact set D of X such that, for each finite subset S of X, there is a compact convex set K = i∈I K i containing S and satisfying K/ Pi−1 (x i ) ⊂ D. i∈I,x i ∈K i
Then Γ has an equilibrium point, i.e. there exists a point x˜ ∈ X such that Pi (x) ˜ = ∅ for all i ∈ I. Proof. Suppose that the conclusion is false. Then, for each x ∈ D, there is some i ∈ I such that Pi (x) = ∅. From Lemma 1, there are finite compactly open cover O1 , . . . , On of D, sets S1 , . . . , Sn , and L S -majorized correspondences P˜i1 , . . . , P˜in such that, for each j = 1, . . . , n 1. Gr Pi j ⊂ Gr P˜i j ; 2. S j is a finite subset of X i j ; 3. if x ∈ O j then P˜i j (x) ∩ S j = ∅. Consider the extended game Γ˜ = (X i , P˜i ). By the hypothesis, there is a compact convex set K = i∈I K i such that K i j contains S j for each j = 1, . . . , n and satisfying K/ (*) P˜i−1 (x i ) ⊂ D. i∈I,x i ∈K j
By our assumption, we see that K / i∈I,xi ∈K j P˜i−1 (x i ) = ∅. That is, for each x ∈ K , there is some i such that P˜i (x) ∩ K i = ∅. By Lemma 1 again, there are finite compactly open cover G 1 , . . . , G m of K , sets T1 , . . . , Tn , and L S -majorized correspondences Ui1 , . . . , Uim such that, for each j = 1, . . . , m
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1. Gr P˜i j ⊂ Gr Ui j ; 2. T j is a finite subset of K i j ; 3. if x ∈ G j then Ui j (x) ∩ T j = ∅. For each t ∈ I , define Ut : K → 2 K by Ut (x) = πt−1 ( P˜t (x)) ∩ K if t = i j and Ut (x) = πt−1 (Ut (x)) ∩ K if t = i j for some j ∈ {1, . . . , m}. Foreach x ∈ K , define I (x) = {t ∈ I | Ut (x) = ∅}. Define U : X → 2 X by U (x) = i∈I (x) Ui (x) for each x ∈ X. Suppose U (x) = ∅. Then there is i ∈ I (x) and an open neighborhood G j of x such that Ui j (x ) = ∅ for all x ∈ G j . Since Ui j is L S -majorized, there exists an open neighborhood Wx of x with Wx ⊂ G j and φx : X → 2 X such that (1) for each x ∈ Wx , Ui j (x ) ⊂ φx (x ),
x ∈ co φx (x ) and (2) for each y ∈ X, φx−1 (y) is open. Then U is majorized by φx on G x . This implies that U is L S -majorized in X. By Theorem 1, there exists x˜ ∈ K such that U (x) ˜ = ∅. Then U (x) ˜ = ∅ and Ui j (x) ˜ ∩ Ki j = ∅ for each j = 1, . . . , m. It is a contradiction and this completes the proof. Restrict Γ˜ on K . From Lemma 2, there is x 0 ∈ K such that Pi (x 0 ) ∩ K i = ∅ for all i ∈ I . From (*), x 0 ∈ D. Hence x 0 ∈ O j and P˜i j (x 0 ) ∩ S j = ∅ for some j . Thus P˜i j (x 0 ) ∩ K i j = ∅. It is a contradiction and this completes the proof. By Theorem 2, we shall obtain a new compact qualitative game in the following: Corollary 1. Let Γ = (X i , Pi )i∈I be a qualitative game such that, for each i ∈ I , 1. X i is a nonempty compact convex subset of a Hausdorff topological vector space; 2. Pi : X → 2 X i is L S -majorized. Then Γ has an equilibrium point, i.e. there exists a point x˜ ∈ X such that Pi (x) ˜ = ∅ for all i ∈ I. Remark 1. Corollary 1 improves Theorem 2.4 of Toussaint [7, p. 101] and Theorem 3.4 of Kim and Yuan [5, p. 175] without assuming that the set G i = {x ∈ X | Pi (x) = ∅} is open for each i ∈ I . Theorem 2 improves Theorem 3 of Ding and Tan [3, p. 233], Theorem 2.5 of Yuan [9, p. 60], and Theorems 3.6 and 3.7 of Kim and Yuan [5, p. 176–7]. By Theorem 2 and Lemma 2, we shall present the other form of an existence theorem for qualitative games in the following: Theorem 3. Let Γ = (X i , Pi )i∈I be a qualitative game such that, for each i ∈ I , 1. X i is a convex subset of a locally convex topological vector space; 2. Pi : X → 2 X i such that Pi is u.s.c. at x and x i ∈ Conv Pi (x), whenever Pi (x) = ∅. Suppose there exists a compact set D of X such that, for each finite subset S of X, there is a compact convex set K = i∈I K i containing S and satisfying K/ Pi−1 (x i ) ⊂ D. i∈I,x i ∈K i
Then Γ has an equilibrium point, i.e. there exists a point x˜ ∈ X such that Pi (x) ˜ = ∅ for all i ∈ I. Remark 2. In Theorem 3, Pi may not be u.s.c. at x if Pi (x) = ∅.
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4. Abstract economies and generalized N -person games For the first application of Theorem 2, we prove the following existence theorem of equilibrium for an abstract economy with an infinite number of agents in noncompact topological vector spaces. Theorem 4. Let (X i , Ai , Bi , Pi )i∈I be an abstract economy and D be a compact subset of X = i∈I X i . Suppose the following conditions are satisfied: 1. 2. 3. 4. 5. 6.
X i is a nonempty convex subset of a Hausdorff topological vector space for each i ∈ I ; for each x ∈ X, Ai (x) is nonempty and co Ai (x) ⊂ Bi (x) for each i ∈ I ; i := {x ∈ X | x i ∈ cl Bi (x)} is closed in X for each i ∈ I ; Ai : X → 2 X i has compactly open lower sections for each i ∈ I ; Ai ∩ Pi : X → 2 X i is L S -majorized in i for each i ∈ I ; Moreover, for each finite set S ⊂ X, there exists a compact convex set K = i∈I K i containing S such that: for each x ∈ [K /D], there is some i ∈ I such that (Pi ∩ Ai )(x) ∩ K i = ∅.
Then an equilibrium point for the game exists. Proof. Define Ui : X → 2 X i by [Pi (x) ∩ Ai (x)], if x ∈ i ; Ui (x) = Ai (x), otherwise. If x ∈ i , then it is obvious that Ui is L S -majorant at x. If x ∈ i and Pi (x) ∩ Ai (x) = ∅, there exists an open neighborhood G x of x and ψx : X → 2 X i such that (1) for each x ∈ G x , Pi (x ) ∩ Ai (x) ⊂ ψx (x ), x ∈ co ψx (x ) and (2) for each y ∈ X, ψx−1 (y) is compactly open. Define φx : X → 2iX by [ψx (z) ∩ Ai (z)], if z ∈ i ; φx (z) = Ai (z), otherwise. Then φx is an L S -majorant of Ui at x. Thus Ui is L S -majorized for each i . Moreover, for each finite set S ⊂ X, there exists a compact convex set K = i∈I K i containing S such that, for each x ∈ K /D, there is some i ∈ I such that Ai (x) ∩ Pi (x) ∩ K i = ∅. Thus K/ U −1 j (x j ) ⊂ D. j ∈I,x j ∈K j
Then, by Theorem 2, there exists a point x˜ ∈ X such that Ui (x) ˜ = ∅ for all i ∈ I . Then x˜ is an equilibrium point for the game. Corollary 2. Let X = i∈I X i and (X i , Ai , Bi , Pi )i∈I be an abstract economy. Suppose the following conditions are satisfied: 1. 2. 3. 4. 5.
X i is a nonempty convex subset of a Hausdorff topological vector space E i for each i ∈ I ; for each x ∈ X, Ai (x) is nonempty and co Ai (x) ⊂ Bi (x) for each i ∈ I ; i := {x ∈ X | x i ∈ cl Bi (x)} is closed in X for each i ∈ I ; Ai : X → 2 X i has compactly open lower sections for each i ∈ I ; Ai ∩ Pi : X → 2 X i is L S -majorized in i for each i ∈ I ;
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6. there exist a nonempty compact convex subset K 0 = i∈I K i of X and a nonempty compact subset D of X such that, for each y ∈ X/D, there is an x ∈ co(K 0 ∪ {y}) with x i ∈ co(Ai (y) ∩ Pi (y)) for all i ∈ I . Then an equilibrium point for the game exists. Remark 3. Assumptions 3, 5 and 6 in Theorem 4 improve the relative assumptions in Theorem 2.2 of Hou [4]. Also, Theorem 4 improves Theorem 3 of Chang [2] without i∈I {x ∈ i | (Ai ∩ Pi )(x) = ∅} being closed. For the other application of Theorem 2, we prove the following existence theorem of equilibrium for a generalized n-person in noncompact topological vector spaces. Let Γ = {Λ, (X i )i∈Λ , ( pi )i∈Λ } be a generalized n-person game in normal form, where Λ is the set of different time or conditions agents (may be uncountable), X i is a nonempty compact convex subset of a Hausdorff topologicalvector space, X i is the set of strategies of the i th agent, pi is a function from the product X = i∈Λ X i into the real numbersR, and pi represents the utility (payoff or cost) function of i th agent. For each i , let X −i = j =i X j . For x ∈ X and i ∈ Λ, x = (x i , x −i ). We say that a vector x is a pure-strategy Nash equilibrium if, for each i ∈ Λ, x i ∈ X i and for, each t ∈ X i , pi (x i , x −i ) ≤ pi (t, x −i ). For each i , define a correspondence Pi : X → 2 X i by Pi (x) = {yi ∈ X i | pi (yi , x −i ) < pi (x)}. If pi (., x −i ) is quasiconvex on X i , then Pi (x) is convex and x i ∈ Pi (x) for all x ∈ X. If pi is continuous, then Pi has an open graph. Thus we can replace preference correspondences {Pi }i∈Λ in Theorem 2 by cost functions { pi }i∈Λ and have the following: Theorem 5. Let X = i∈Λ X i and {Λ, (X i )i∈Λ , ( pi )i∈Λ } be a generalized n-person game. Suppose the following conditions are satisfied for each i ∈ Λ: 1. X i is a nonempty convex subset of a Hausdorff topological vector space E i ; 2. the function pi is lower semicontinuous on each compact subset of X and, for each fixed yi ∈ X i , the function pi (yi , .) is upper semicontinuous on each compact subset of X −i ; 3. the function pi (., x −i ) is quasiconvex on X i for each x; and 4. suppose there exists a compact set D of X such that, for each finite subset S of X, there is a compact convex set K = i∈I K i containing S and satisfying, for each x ∈ K /D, inf inf ( pi (z i , x −i ) − pi (x)) < 0.
i∈Λ z∈K
Then the game admits a pure-strategy Nash equilibrium. References [1] A. Borglin, H. Keiding, Existence of equilibrium actions of equilibrium: A note on the ‘new’ existence theorems, J. Math. Econom. 3 (1976) 313–316. [2] S.Y. Chang, Maximal elements in noncompact spaces with application to equilibria, Proc. Amer. Math. Soc 132 (2) (2003) 535–541. [3] X.P. Ding, K.K. Tan, On equilibria of noncompact generalized games, J. Math. Anal. Appl. 177 (1993) 226–238. [4] J.-C. Hou, Existence of equilibria for generalized games without para-compactness, Nonlinear Anal. 56 (2004) 625–632. [5] W.K. Kim, G.X.-Z. Yuan, Existence of equilibria for generalized games and generalized social systems with coordination, Nonlinear Anal. 45 (2001) 169–188.
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[6] D. Gale, A. Mass-Colell, On the role of complete, transitive preferences in equilibrium theory, in: G. Schw¨odiauer (Ed.), Equilibrium and Disequlibrium in Economic Theory, Reidel, Dordrecht, 1978, pp. 7–14, 243–251. [7] S. Toussaint, On the existence of equlibria in economies with infinitely many commodities and without ordered preferences, J. Econom. Theory 33 (1984) 98–115. [8] N.C. Yannelis, N.D. Prabhakar, Existence of maximal elements and equilibria in linear topological spaces, J. Math. Econom. 12 (1983) 233–245. [9] G.X.-Z. Yuan, The existence of equilibria for noncompact generalized games, Appl. Math. Lett. 13 (2000) 57–63.
Noncompact qualitative games with application to equilibria Shiow-Yu Chang ∗ Department of Mathematics, Soochow University, Taipei, Taiwan, ROC Received 27 August 2005; accepted 26 September 2005
Abstract In 2003, S.Y. Chang [Maximal elements in noncompact spaces with application to equilibria, Proc. Amer. Math. Soc. 132 (2) (2003) 535–541] obtained a maximal theorem for L S -majorized correspondences in noncompact spaces. In this paper, we extend this result to obtain new existence theorems for qualitative games. Applying it, we have more refined existence theorems for both abstract economies and generalized n-person games than before. c 2005 Elsevier Ltd. All rights reserved. MSC: primary 91A13; secondary 52A07, 91B50 Keywords: L S -majorized correspondences; Qualitative game; Abstract economy; Generalized n-person game
1. Introduction In Section 2, we give notations and definitions. In Section 3, we first prove one useful lemma. Next, by combining it and the maximal theorem for L S -majorized correspondences in noncompact spaces in [2], we obtain a new existence theorem for qualitative games. Also, by proving the L S -majorized property of upper semicontinuous preference correspondences, we have another existence theorem for qualitative games. In Section 4, we establish more refined existence theorems for both abstract economies and generalized n-person games than before— see, for example, [1–9]. ∗ Tel.: +886 2 288 194716705; fax: +886 2 288 11 526.
E-mail address: [email protected]. c 2005 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2005.09.041
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S.-Y. Chang / Nonlinear Analysis 65 (2006) 593–600
2. Notations and definitions 1. 2. 3. 4. 5. 6. 7.
2 A denotes the set of all subsets of A; co A denotes the convex hull of the set A; Conv A denotes the convex closure of the set A; A/B denotes the difference of sets A and B; cl A, int A denotes the closure and interior of the set A; φ : X → 2Y is a correspondence, that is, φ(x) is a subset of Y for each x ∈ X; Let X = i∈I X i and π j : X → X j be called the projection of X onto X j , if π j (x) = x j for each x = (x i )i∈I ∈ X.
Definition 1. Let I be a (possibly uncountable) set. For each agent i ∈ I , let its choice set or strategy set X i be a nonempty set in a topological vector space. Let X = j ∈I X j . Following [6], Γ = (X i , Pi )i∈Iis a qualitative game if, for each player i ∈ I , X i is the strategy set of player i , and Pi : X = j ∈I X j → 2 X i is a preference correspondence of player i which is irreflexive [i.e. πi (x) ∈ Pi (x) for each x ∈ X]; also, a point x˜ ∈ X is said to be an equilibrium point of the game Γ = (X i , Pi )i∈I if Pi (x) ˜ = ∅ for each i ∈ I . A generalized model is as follows: Definition 2. Let I denote the set of agents. For each i ∈ I , let X i be a nonempty set and X = j ∈I X j . Following [3], an abstract economy or (generalized game) G = (X i , A i , Bi , Pi )i∈I is defined as a family of quadruples (X i , Ai , Bi , Pi ), where Ai , Bi : X → 2 X i are feasible correspondences of agent i and Pi : X → 2 X i are preference correspondences of agent i . An equilibrium point for G is an x˜ ∈ X, where x˜ = (x˜i ) satisfies 1. x˜i ∈ cl Bi (x); ˜ 2. Pi (x) ˜ ∩ Ai (x) ˜ =∅ for each i ∈ I . When Ai = Bi , for each i ∈ I our definition coincides with the standard definition, e.g. in [1] or in [8]. Definition 3. Let X be a topological space, Y a nonempty subset of a vector space E, θ : X → E a map, and φ : X → 2Y a correspondence. The following notions were introduced in [1–9]. 1. φ is said to be upper semicontinuous (u.s.c.) at x if, for each open set V in Y with φ(x) ⊂ V , there exists an open neighborhood U of x in X such that φ(x ) ⊂ V for each x ∈ U . 2. A set G ⊂ X is said to be compactly open in X if, for each compact set K in X, the set G ∩ K is open in K . 3. φ is said to have compactly open lower sections in X if, for each y ∈ Y , the set φ −1 (y) = {x ∈ X | y ∈ φ(x)} is compactly open in X. 4. φ is said to have locally compactly open lower section at x if there is an open set Wx containing x such that, for each y ∈ Y , the set φ −1 (y) ∩ Wx is compactly open in X. 5. φ is said to be of class L θ,S if, for every x ∈ X, θ (x) ∈ co φ(x), and φ has compactly open lower sections in X. 6. A correspondence φx : X → 2Y is said to be an L θ,S -majorant of φ at x ∈ X if there exists an open neighborhood Nx of x in X such that (a) for each z ∈ Nx , φ(z) ⊂ φx (z) and θ (z) ∈ co φx (z), (b) for each y ∈ Y , φx−1 (y) is compactly open in X. 7. Suppose X ⊂ X. φ is L θ,S -majorized in X if, for each x ∈ X with φ(x) = ∅, there exists an L θ,S -majorant of φ at x in X.
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In this paper, we deal mainly with the case (I) X = Y which is a nonempty convex subset of the topological vector space E and θ = I X , the identity map on X, or the case (II) X = i∈I X i and θ = π j : X → X j is the projection of X onto X j and Y = X j is a nonempty convex subset of a topological vector space. In both cases (I) and (II), we write L S in place of L θ,S . 3. Noncompact qualitative games The results of this paper are to extend the following in [2]. Theorem 1. Let X be a convex subset of a Hausdorff topological vector space E and let U : X → 2 X be L S -majorized. Suppose there exists a compact set D of X such that, for each finite subset S of X, there is a compact convex set K containing S and satisfying U −1 (x) ⊂ D. K/ x∈K
Then {x ∈ X | U (x) = ∅} is a nonempty subset of D. We shall need the following simple result. Lemma 1. Let Γ = (X i , Pi )i∈I be a qualitative game, X = i∈I X i , and D be a compact subset of X. For each i ∈ I , X i is a nonempty convex subset of a Hausdorff topological vector space and Pi : X → 2 X i is L S -majorized. Suppose, for each x ∈ D, there is some i ∈ I such that Pi (x) = ∅. Then there are finite compactly open cover O1 , . . . , On of D, sets S1 , . . . , Sn , and L S -majorized correspondences P˜i1 , . . . , P˜in such that, for each j = 1, . . . , n 1. Gr Pi j ⊂ Gr P˜i j ; 2. S j is a finite subset of X i j ; 3. if x ∈ O j then P˜i j (x) ∩ S j = ∅. Proof. Fix x ∈ D, there is some i ∈ I such that Pi (x) = ∅. Then there is φx an L S -majorant of Pi in an open neighborhood W of x. Choose an open neighborhood U of x such that cl U ⊂ W and x i ∈ Pi (x). Let Vx = U ∩ φx−1 (x i ). Let Pi : X → 2 X i be defined by φ (z), if z ∈ cl Vx ; Pi (z) = x Pi (z), if z ∈ [X/cl Vx ]. Then Gr Pi ⊂ Gr Pi , Pi is everywhere L S -majorized, and Pi (z) = ∅ for each z ∈ Vx . Suppose D/Vx = ∅. Fix y ∈ D/Vx . Continue the above process with {P j | j ∈ I /{i }} ∪ {Pi }. Since D is compact, after continuing finite times, there are finite compactly open cover O1 , . . . , On of D, sets S1 , . . . , Sn , and L S -majorized correspondences P˜i1 , . . . , P˜in such that for each j = 1, . . . , n 1. Gr Pi j ⊂ Gr P˜i j ; 2. S j is a finite subset of X i j ; 3. if x ∈ O j then P˜i j (x) ∩ S j = ∅. By the separation property, we yield that an upper semicontinuous preference correspondence is L S -majorized in the following: Lemma 2. Let X be as in Lemma 1, X i be a convex subset of a locally convex space E i , x˜ ∈ X, and Pi : X → 2 X i . If Pi (x) ˜ = ∅, x˜i ∈ Conv Pi (x), ˜ and Pi is u.s.c. at x, ˜ then Pi is L S -majorized at x. ˜
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Proof. Since Pi (x) ˜ = ∅, x˜i ∈ Conv Pi (x), ˜ and E i is a locally convex space, there is an open convex neighborhood W of 0 in E i such that [x˜i + W ] ∩ [Conv Pi (x) ˜ + W ] = ∅. Since both Pi and πi are u.s.c. at x, ˜ there is an open neighborhood V of 0 in Pi (x˜ + V ) ⊂ [Conv Pi (x) ˜ + W ],
i∈I
(*) E i such that
πi (x˜ + V ) ⊂ [x˜i + W ].
(**)
˜ + W ]} = ∅. {[x˜ + V ] × [x˜i + W ]} ∩ {[x˜ + V ] × [Conv Pi (x)
(***)
By (*), we have Define P˜i : X → 2 X i by ˜ + W, if x ∈ x˜ + V ; Conv Pi (x) ˜ Pi (x) = ∅, otherwise. ˜ Thus Pi is Then P˜i has an open graph. By (**) and (***), P˜i is an L S -majorant of Pi at x. ˜ L S -majorized at x. By Theorem 1 and Lemma 1, we shall present a noncompact form of an existence theorem for qualitative games in the following: Theorem 2. Let Γ = (X i , Pi )i∈I be a qualitative game such that, for each i ∈ I , 1. X i is a convex subset of a Hausdorff topological vector space; 2. Pi : X → 2 X i is L S -majorized. Suppose there exists a compact set D of X such that, for each finite subset S of X, there is a compact convex set K = i∈I K i containing S and satisfying K/ Pi−1 (x i ) ⊂ D. i∈I,x i ∈K i
Then Γ has an equilibrium point, i.e. there exists a point x˜ ∈ X such that Pi (x) ˜ = ∅ for all i ∈ I. Proof. Suppose that the conclusion is false. Then, for each x ∈ D, there is some i ∈ I such that Pi (x) = ∅. From Lemma 1, there are finite compactly open cover O1 , . . . , On of D, sets S1 , . . . , Sn , and L S -majorized correspondences P˜i1 , . . . , P˜in such that, for each j = 1, . . . , n 1. Gr Pi j ⊂ Gr P˜i j ; 2. S j is a finite subset of X i j ; 3. if x ∈ O j then P˜i j (x) ∩ S j = ∅. Consider the extended game Γ˜ = (X i , P˜i ). By the hypothesis, there is a compact convex set K = i∈I K i such that K i j contains S j for each j = 1, . . . , n and satisfying K/ (*) P˜i−1 (x i ) ⊂ D. i∈I,x i ∈K j
By our assumption, we see that K / i∈I,xi ∈K j P˜i−1 (x i ) = ∅. That is, for each x ∈ K , there is some i such that P˜i (x) ∩ K i = ∅. By Lemma 1 again, there are finite compactly open cover G 1 , . . . , G m of K , sets T1 , . . . , Tn , and L S -majorized correspondences Ui1 , . . . , Uim such that, for each j = 1, . . . , m
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1. Gr P˜i j ⊂ Gr Ui j ; 2. T j is a finite subset of K i j ; 3. if x ∈ G j then Ui j (x) ∩ T j = ∅. For each t ∈ I , define Ut : K → 2 K by Ut (x) = πt−1 ( P˜t (x)) ∩ K if t = i j and Ut (x) = πt−1 (Ut (x)) ∩ K if t = i j for some j ∈ {1, . . . , m}. Foreach x ∈ K , define I (x) = {t ∈ I | Ut (x) = ∅}. Define U : X → 2 X by U (x) = i∈I (x) Ui (x) for each x ∈ X. Suppose U (x) = ∅. Then there is i ∈ I (x) and an open neighborhood G j of x such that Ui j (x ) = ∅ for all x ∈ G j . Since Ui j is L S -majorized, there exists an open neighborhood Wx of x with Wx ⊂ G j and φx : X → 2 X such that (1) for each x ∈ Wx , Ui j (x ) ⊂ φx (x ),
x ∈ co φx (x ) and (2) for each y ∈ X, φx−1 (y) is open. Then U is majorized by φx on G x . This implies that U is L S -majorized in X. By Theorem 1, there exists x˜ ∈ K such that U (x) ˜ = ∅. Then U (x) ˜ = ∅ and Ui j (x) ˜ ∩ Ki j = ∅ for each j = 1, . . . , m. It is a contradiction and this completes the proof. Restrict Γ˜ on K . From Lemma 2, there is x 0 ∈ K such that Pi (x 0 ) ∩ K i = ∅ for all i ∈ I . From (*), x 0 ∈ D. Hence x 0 ∈ O j and P˜i j (x 0 ) ∩ S j = ∅ for some j . Thus P˜i j (x 0 ) ∩ K i j = ∅. It is a contradiction and this completes the proof. By Theorem 2, we shall obtain a new compact qualitative game in the following: Corollary 1. Let Γ = (X i , Pi )i∈I be a qualitative game such that, for each i ∈ I , 1. X i is a nonempty compact convex subset of a Hausdorff topological vector space; 2. Pi : X → 2 X i is L S -majorized. Then Γ has an equilibrium point, i.e. there exists a point x˜ ∈ X such that Pi (x) ˜ = ∅ for all i ∈ I. Remark 1. Corollary 1 improves Theorem 2.4 of Toussaint [7, p. 101] and Theorem 3.4 of Kim and Yuan [5, p. 175] without assuming that the set G i = {x ∈ X | Pi (x) = ∅} is open for each i ∈ I . Theorem 2 improves Theorem 3 of Ding and Tan [3, p. 233], Theorem 2.5 of Yuan [9, p. 60], and Theorems 3.6 and 3.7 of Kim and Yuan [5, p. 176–7]. By Theorem 2 and Lemma 2, we shall present the other form of an existence theorem for qualitative games in the following: Theorem 3. Let Γ = (X i , Pi )i∈I be a qualitative game such that, for each i ∈ I , 1. X i is a convex subset of a locally convex topological vector space; 2. Pi : X → 2 X i such that Pi is u.s.c. at x and x i ∈ Conv Pi (x), whenever Pi (x) = ∅. Suppose there exists a compact set D of X such that, for each finite subset S of X, there is a compact convex set K = i∈I K i containing S and satisfying K/ Pi−1 (x i ) ⊂ D. i∈I,x i ∈K i
Then Γ has an equilibrium point, i.e. there exists a point x˜ ∈ X such that Pi (x) ˜ = ∅ for all i ∈ I. Remark 2. In Theorem 3, Pi may not be u.s.c. at x if Pi (x) = ∅.
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4. Abstract economies and generalized N -person games For the first application of Theorem 2, we prove the following existence theorem of equilibrium for an abstract economy with an infinite number of agents in noncompact topological vector spaces. Theorem 4. Let (X i , Ai , Bi , Pi )i∈I be an abstract economy and D be a compact subset of X = i∈I X i . Suppose the following conditions are satisfied: 1. 2. 3. 4. 5. 6.
X i is a nonempty convex subset of a Hausdorff topological vector space for each i ∈ I ; for each x ∈ X, Ai (x) is nonempty and co Ai (x) ⊂ Bi (x) for each i ∈ I ; i := {x ∈ X | x i ∈ cl Bi (x)} is closed in X for each i ∈ I ; Ai : X → 2 X i has compactly open lower sections for each i ∈ I ; Ai ∩ Pi : X → 2 X i is L S -majorized in i for each i ∈ I ; Moreover, for each finite set S ⊂ X, there exists a compact convex set K = i∈I K i containing S such that: for each x ∈ [K /D], there is some i ∈ I such that (Pi ∩ Ai )(x) ∩ K i = ∅.
Then an equilibrium point for the game exists. Proof. Define Ui : X → 2 X i by [Pi (x) ∩ Ai (x)], if x ∈ i ; Ui (x) = Ai (x), otherwise. If x ∈ i , then it is obvious that Ui is L S -majorant at x. If x ∈ i and Pi (x) ∩ Ai (x) = ∅, there exists an open neighborhood G x of x and ψx : X → 2 X i such that (1) for each x ∈ G x , Pi (x ) ∩ Ai (x) ⊂ ψx (x ), x ∈ co ψx (x ) and (2) for each y ∈ X, ψx−1 (y) is compactly open. Define φx : X → 2iX by [ψx (z) ∩ Ai (z)], if z ∈ i ; φx (z) = Ai (z), otherwise. Then φx is an L S -majorant of Ui at x. Thus Ui is L S -majorized for each i . Moreover, for each finite set S ⊂ X, there exists a compact convex set K = i∈I K i containing S such that, for each x ∈ K /D, there is some i ∈ I such that Ai (x) ∩ Pi (x) ∩ K i = ∅. Thus K/ U −1 j (x j ) ⊂ D. j ∈I,x j ∈K j
Then, by Theorem 2, there exists a point x˜ ∈ X such that Ui (x) ˜ = ∅ for all i ∈ I . Then x˜ is an equilibrium point for the game. Corollary 2. Let X = i∈I X i and (X i , Ai , Bi , Pi )i∈I be an abstract economy. Suppose the following conditions are satisfied: 1. 2. 3. 4. 5.
X i is a nonempty convex subset of a Hausdorff topological vector space E i for each i ∈ I ; for each x ∈ X, Ai (x) is nonempty and co Ai (x) ⊂ Bi (x) for each i ∈ I ; i := {x ∈ X | x i ∈ cl Bi (x)} is closed in X for each i ∈ I ; Ai : X → 2 X i has compactly open lower sections for each i ∈ I ; Ai ∩ Pi : X → 2 X i is L S -majorized in i for each i ∈ I ;
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6. there exist a nonempty compact convex subset K 0 = i∈I K i of X and a nonempty compact subset D of X such that, for each y ∈ X/D, there is an x ∈ co(K 0 ∪ {y}) with x i ∈ co(Ai (y) ∩ Pi (y)) for all i ∈ I . Then an equilibrium point for the game exists. Remark 3. Assumptions 3, 5 and 6 in Theorem 4 improve the relative assumptions in Theorem 2.2 of Hou [4]. Also, Theorem 4 improves Theorem 3 of Chang [2] without i∈I {x ∈ i | (Ai ∩ Pi )(x) = ∅} being closed. For the other application of Theorem 2, we prove the following existence theorem of equilibrium for a generalized n-person in noncompact topological vector spaces. Let Γ = {Λ, (X i )i∈Λ , ( pi )i∈Λ } be a generalized n-person game in normal form, where Λ is the set of different time or conditions agents (may be uncountable), X i is a nonempty compact convex subset of a Hausdorff topologicalvector space, X i is the set of strategies of the i th agent, pi is a function from the product X = i∈Λ X i into the real numbersR, and pi represents the utility (payoff or cost) function of i th agent. For each i , let X −i = j =i X j . For x ∈ X and i ∈ Λ, x = (x i , x −i ). We say that a vector x is a pure-strategy Nash equilibrium if, for each i ∈ Λ, x i ∈ X i and for, each t ∈ X i , pi (x i , x −i ) ≤ pi (t, x −i ). For each i , define a correspondence Pi : X → 2 X i by Pi (x) = {yi ∈ X i | pi (yi , x −i ) < pi (x)}. If pi (., x −i ) is quasiconvex on X i , then Pi (x) is convex and x i ∈ Pi (x) for all x ∈ X. If pi is continuous, then Pi has an open graph. Thus we can replace preference correspondences {Pi }i∈Λ in Theorem 2 by cost functions { pi }i∈Λ and have the following: Theorem 5. Let X = i∈Λ X i and {Λ, (X i )i∈Λ , ( pi )i∈Λ } be a generalized n-person game. Suppose the following conditions are satisfied for each i ∈ Λ: 1. X i is a nonempty convex subset of a Hausdorff topological vector space E i ; 2. the function pi is lower semicontinuous on each compact subset of X and, for each fixed yi ∈ X i , the function pi (yi , .) is upper semicontinuous on each compact subset of X −i ; 3. the function pi (., x −i ) is quasiconvex on X i for each x; and 4. suppose there exists a compact set D of X such that, for each finite subset S of X, there is a compact convex set K = i∈I K i containing S and satisfying, for each x ∈ K /D, inf inf ( pi (z i , x −i ) − pi (x)) < 0.
i∈Λ z∈K
Then the game admits a pure-strategy Nash equilibrium. References [1] A. Borglin, H. Keiding, Existence of equilibrium actions of equilibrium: A note on the ‘new’ existence theorems, J. Math. Econom. 3 (1976) 313–316. [2] S.Y. Chang, Maximal elements in noncompact spaces with application to equilibria, Proc. Amer. Math. Soc 132 (2) (2003) 535–541. [3] X.P. Ding, K.K. Tan, On equilibria of noncompact generalized games, J. Math. Anal. Appl. 177 (1993) 226–238. [4] J.-C. Hou, Existence of equilibria for generalized games without para-compactness, Nonlinear Anal. 56 (2004) 625–632. [5] W.K. Kim, G.X.-Z. Yuan, Existence of equilibria for generalized games and generalized social systems with coordination, Nonlinear Anal. 45 (2001) 169–188.
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[6] D. Gale, A. Mass-Colell, On the role of complete, transitive preferences in equilibrium theory, in: G. Schw¨odiauer (Ed.), Equilibrium and Disequlibrium in Economic Theory, Reidel, Dordrecht, 1978, pp. 7–14, 243–251. [7] S. Toussaint, On the existence of equlibria in economies with infinitely many commodities and without ordered preferences, J. Econom. Theory 33 (1984) 98–115. [8] N.C. Yannelis, N.D. Prabhakar, Existence of maximal elements and equilibria in linear topological spaces, J. Math. Econom. 12 (1983) 233–245. [9] G.X.-Z. Yuan, The existence of equilibria for noncompact generalized games, Appl. Math. Lett. 13 (2000) 57–63.