aa r X i v : . [ m a t h . O C ] A p r Noname manuscript No. (will be inserted by the editor)
Existence of the equilibrium in choiceMonica Patriche
University of Bucharest e-mail: [email protected]
The date of receipt and acceptance will be inserted by the editor
Abstract
In this paper, we prove the existence of the equilibrium in choicefor games in choice form. These games have recently been introduced by A.Stefanescu, M. Ferrara and M. V. Stefanescu. Our results link the recentresearch to the older approaches, regarding games in normal form or quali-tative games.
Key words game in choice form , equilibrium in choice , selection theorem,fixed point theorem
It has been widely agreed upon the fact that Nash’s concept of equilibriumreflects the possibility of challenging the choices of the unilateral acts of theplayers involved in noncooperative games. Having the aim of gaining themost they can do, the players are in the situation of making choices in aprocess described, mathematically, by the notion of ”game” (this was pro-posed by Nash, and it was initially called ”game in normal form”). The orig-inal definitions of Nash [4],[5] have been extended, but the derived notionsof qualitative game or abstract economy and their corresponding conceptsof equilibrium reflect the initial meaning of flexible elections in any givencontext to permit the players not to deviate, once they agreed on the bestsolution for them.The new extension of a game, called ”the game in choice form”, is dueto Stefanescu, Ferrara and Stefanescu [9]. The game in choice form is afamily of the sets of individual strategies and choice profiles. The authors
Correspondence to : University of Bucharest, Faculty of Mathematics and Com-puter Science, 14 Academiei Street, 010014 Bucharest, Romania Monica Patriche also defined the concept of equilibrium in choice for this type of game. Theirinterpretations evolved along with the problem of noncooperative solutionsof the voting operators. Firstly, Stefanescu and Ferrara proposed the conceptof Nash equilibrium in choice in [8] and they renamed it in [9], foundingconditions under which the equilibrium in choice exists. The new adopteddefinitions are coherent with the old underlying formalism. For instance,when the utility functions represent the players’ options, a choice profile canbe seen as the family of players’ graphs best reply mappings. In this case,the set of equilibria in choice coincides with the set of Nash equilibria. Thegenerality of the new concepts raise the interest of the formalist theorist ofgames, which explicitly can show their significance. The authors themselvesdeveloped these ideas in their work. They referred to the fact that theplayers’ preferences need not be explicitly represented, at the same timeconsidering the possibility of recuperating the known solutions as particularcases. The second problem raised is the one of the nonexistence of a bestreply. This interest is obviously at the core of our original research. In thispaper, we are looking for new conditions, in order to obtain the existenceof the equilibrium in choice. Our assumptions are different from the onesproposed when the new theory was framed. They concern the properties ofthe sets of choice profiles. We are now exploring a method of proof based oncontinuous selection and fixed point theorems for correspondences definedby using the upper sections of the sets which form the game.The rest of the paper is organised as follows: In Section 2, some no-tation, terminological convention, basic definition and results about corre-spondences and games in choice form are given. Section 3 contains existenceresults for equlibrium in choice.
Throughout this paper, we shall use the following notations and definitions:Let A be a subset of a vector space X , 2 A denotes the family of allsubsets of A and co A denotes the convex hull of A . If A is a subset of atopological space X, cl A denotes the closure of A in X . If T , G : X → Y are correspondences, then co G and G ∩ T : X → Y are correspondencesdefined by (co G )( x ) :=co G ( x ) and ( G ∩ T )( x ) := G ( x ) ∩ T ( x ) , for each x ∈ X , respectively.Given a correspondence T : X → Y , for each x ∈ X, the set T ( x ) iscalled the upper section of T at x. For each y ∈ Y, the set T − ( y ) := { x ∈ X : y ∈ T ( x ) } is called the lower section of T at y . The correspondence T − : Y → X , defined by T − ( y ) = { x ∈ X : y ∈ T ( x ) } for y ∈ Y , is calledthe (lower) inverse of T. Let X , Y be topological spaces. The correspondence T : X → Y iscalled lower semicontinuous if for each x ∈ X and for each open set V in Y with T ( x ) ∩ V = ∅ , there exists an open neighborhood U of x in X so that T ( y ) ∩ V = ∅ for each y ∈ U .xistence of the equilibrium in choice The following lemma will be useful to our study of existence of equilib-rium in choice.
Lemma 1 (see Yuan [15]). Let X and Y be two topological spaces and let W be an open (resp. closed) subset of X. Suppose T : X → Y , T : X → Y are upper semicontinuous (resp. lower semicontinuous) correspondencessuch that T ( x ) ⊂ T ( x ) for all x ∈ W. Then the correspondence T : X → Y defined by T ( x ) = (cid:26) T ( x ), if x / ∈ W , T ( x ), if x ∈ W is also upper semicontinuous (resp. lower semicontinuous). Further, we present the main models of noncooperative games we willdeal with in this paper. The corresponding notions of equilibrium are alsorecalled.Let ( X i ) i ∈ N be the family of the individual sets of strategies and let X = Q i ∈ I X i .The normal form of an n -person game is ( X i , r i ) i ∈ N , where, for each i ∈ N , X i is a nonempty set (the set of individual strategies of player i ) and r i is the preference relation on X of player i . The individual preferences r i are often represented by utility functions , i.e. for each i ∈ N there exists areal valued function u i : X → R (called the utility function of i ), such that: xr i y ⇔ u i ( x ) ≥ u i ( y ) , ∀ x, y ∈ X. Then the normal form of n-person game is ( X i , u i ) i ∈ N .We denote x − i = ( x , ..., x i − , x i +1 , ..., x n ), X − i = Q i = j X i and ( x − i , X i ) = { ( x − i , x i ) : x i ∈ X i } . The
Nash equilibrium for the game ( X i , u i ) i ∈ N is a point x ∗ ∈ X whichsatisfies for each i ∈ N : u i ( x ∗ ) ≥ u i ( x ∗− i , x i ) for each x i ∈ X i . For each i ∈ N, the player’s i’s best reply mapping is the correspondence B i : X − i → X i , defined by B i ( x − i ) = { x i ∈ X i : u i ( x − i , x i ) ≥ u i ( x − i , y i )for each y i ∈ X i } . Then, x ∗ ∈ X is a Nash equilibrium if only if x ∗ ∈ T i ∈ N Gr( B i ) . The element x ∗ ∈ X is called weak Nash equilibrium (Stefanescu, Fer-rara, Stefanescu [9]): u i ( x ∗ ) ≥ u i ( x ∗− i , x i ) for each x i ∈ X i and i ∈ N such that B i ( x ∗− i ) = ∅ . A qualitative game Γ = ( X i , P i ) i ∈ N is defined as a family of n orderedtriplets ( X i , P i ) , where for each i ∈ N : P i : X → X i is a preferencecorrespondence. An equilibrium for Γ is a point x ∗ ∈ X which satisfies foreach i ∈ N : P i ( x ∗ ) = ∅ . A weak equilibrium (Stefanescu, Ferrara, Stefanescu[9]) of Γ = ( X i , P i ) i ∈ N is a point x ∗ ∈ X which satisfies P i ( x ∗ ) = ∅ for each i ∈ N such that { x i ∈ X i : P i ( x ∗− i , x i ) = ∅} 6 = ∅ . A choice profile (Stefanescu, Ferrara, Stefanescu [9]) is any collection C := ( C i ) i ∈ N of nonempty subsets of X. A game in choice form is a doublefamily (( X i ) i ∈ N , ( C i ) i ∈ N ), where C = ( C i ) i ∈ N is a choice profile.We denote C ( x − i ) the upper section through x − i of the set C i , i.e., C ( x − i ) = { y i ∈ X i : ( x − i , y i ) ∈ C i } . Monica Patriche
We will make the following assumption:(A) assume that for each x ∈ X, there exists i ∈ N such that C i ( x − i ) = ∅ . The game strategy x ∗ ∈ X is an equilibrium in choice (denoted EC) (A.Stefanescu, M. Ferrara, V. Stefanescu [9]) if ∀ i ∈ N, ( x ∗− i , X i ) ∩ C i = ∅ ⇒ x ∗ ∈ C i , equivalently, x ∗ i ∈ C i ( x ∗− i ) , for every i ∈ N for which C i ( x ∗− i ) = ∅ . The strategy x ∗ ∈ X is a strong equilibrium in choice (denoted SEC) if x ∗ i ∈ T i ∈ N C i . This section provides a summary of different theorems concerning the exis-tence of equilibria in choice for games in choice form. In order to underlinethe novelty and the importance of our work, we also must discuss here theadditional benefit of obtaining corollaries which state, under new conditions,the existence of the weak Nash equilibria for games in normal form, or theexistence of the weak equilibria of qualitative games. To prove our point,we will use continuous selection theorems or fixed point theorems for thecorrespondences that we will form, considering upper sections of the setsdefining the game of the choice form. The advantage we have by doing this,deserves a great prominence in the assessment of the new assumptions whichcharacterize the new statements. These results differ very much from theones obtained by Stefanescu, Ferrara and Stefanescu in [9] and they link therecent research to the older approaches, regarding games in normal form orqualitative games. In order to suggest priorities to the reader, we keep therelationship between the main theorems and their consequences on particu-lar games. For a better understanding of the paper, we recall all propertiesof the correspondences which will be used and the tools of proofs. Our studygives a new perspective of unifying of different approaches and results on theequilibrium concepts and the existence of noncooperative theory of games.Finally, we note that we obtain the existence of the strong equilibrium inchoice for all situations considered in this section, if we suppose, in addition,that C i ( x − i ) = ∅ for each x − i ∈ X − i . Let X , Y be topological spaces. We recall that the correspondence T : X → Y has the local intersection property if x ∈ X with T ( x ) = ∅ impliesthe existence of an open neighborhood V ( x ) of x such that ∩ z ∈ V ( x ) T ( z ) = ∅ . To prove Theorem 1, we need the following lemma.
Lemma 2 (Wu, Shen, [13]). Let X be a nonempty paracompact subset of aHausdorff topological space E and Y be a nonempty subset of a Hausdorfftopological vector space. Let S , T : X → Y be correspondence which verify:a) for each x ∈ X, S ( x ) = ∅ and co S ( x ) ⊂ T ( x ) ; b) S has the local intersection property.Then, T has a continuous selection. xistence of the equilibrium in choice Our main result cites conditions which ensure the existence of equilibriain choice for a game in choice form in the lack of convexity of the uppersections of the sets C i . The framework for our general next theorem consistsof Hausdorff topological vector spaces. The proof is based on an argumentthat implies the above lemma. Theorem 1
Let (( X i ) i ∈ N , ( C i ) i ∈ N ) be a game in choice form. Assume that,for each i ∈ N, the following conditions are fulfilled:a) X i is a nonempty, convex and compact set in a Hausdorff topologicalvector space E i ; b) there exists a nonempty subset D i of C i such that W i = { x − i ∈ X − i : D i ( x − i ) = ∅} is closed and D i ( x − i ) = ∅ if only if C i ( x − i ) = ∅ ; c) if D i ( x − i ) = ∅ , there exists an open neighborhood V ( x − i ) of x − i such that ∩ z − i ∈ V ( x − i ) D i ( z − i ) = ∅ ; d) D i ( x − i ) is convex or empty for each x − i ∈ X − i . Then, the game admits equilibria in choice.Proof
For each i ∈ N, let us define the correspondences S i , T i : X − i → X i , by S i ( x − i ) = (cid:26) co( S { y − i : D i ( y − i ) = ∅} D i ( y − i )) if x − i / ∈ W i ; D i ( x − i ) if x − i ∈ W i and T i ( x − i ) = (cid:26) co( S { y − i : C i ( y − i ) = ∅} C i ( y − i )) if x − i / ∈ W i ; C i ( x − i ) if x − i ∈ W i . We call T i the correspondence of the upper sections of the sets C i . The correspondence S i has nonempty and convex values and S i ( x − i ) ⊂ T i ( x − i ) for each x − i ∈ X − i . Assumption c) implies that S i | ( X − i ∩ W i ) has the local intersection prop-erty. If x − i ∈ C W i , then, S i ( x − i ) =co( S { y − i : D i ( y − i ) = ∅} D i ( x − i )) = ∅ . Ac-cording to b), there exists an open neighborhood V i ( x − i ) of x − i such that D i ( z − i ) = ∅ for each z − i ∈ V i ( x − i ) . Then, S i ( z − i ) =co( S { y − i : D i ( y − i ) = ∅} D i ( y − i ))for each z − i ∈ V i ( x − i ) and, ∩ z − i ∈ V i ( x − i ) S i ( z − i ) =co( S { y − i : D i ( y − i ) = ∅} D i ( y − i )) = ∅ . It follows that S i : X − i → X i has the local intersection property.The Wu-Shen Lemma implies that T i has a continuous selection f i : X − i → X i . Let f : X → X be defined by f ( x ) = Q i ∈ N f i ( x − i ) for each x ∈ X. Thefunction f is continuous, and, according to the Brouwer fixed point Theo-rem, there exists x ∗ ∈ X such that f ( x ∗ ) = x ∗ . Hence, x ∗ ∈ Q i ∈ N T i ( x ∗− i )and obviously, x ∗ i ∈ T i ( x ∗− i ) for each i ∈ N. Suppose that ( x ∗− i , X i ) ∩ C i = ∅ , for some i ∈ N. Then, C i ( x ∗− i ) = ∅ and x ∗ i ∈ C i ( x ∗− i ) , which implies x ∗ = ( x ∗− i , x ∗ i ) ∈ C i . As a corollary, we obtain sufficient conditions for a game in normal formto admit weak Nash equilibria. The main assumption is new in literatureand it refers to the existence of a best reply for each player i to the commonstrategy of the other players, which lies in open intervals of the product Monica Patriche space X − i . The meaning is that each player i can remain stable in thechoice of his own best strategy in the situation that the decisions of theopponents can vary slightly in any manner profitable to themselves. Corollary 1
Let (( X i ) i ∈ N , ( u i ) i ∈ N ) be a game in normal form. Assumethat, for each i ∈ N, the following conditions are fulfilled:a) X i is a nonempty, convex and compact set in a Hausdorff topologicalvector space E i ; b) the set { x ∈ X : u i ( x ) ≥ u i ( x − i , y i ) for each y i ∈ X i } is nonempty; c) W i = { x − i ∈ X − i : B i ( x − i ) = ∅} is closed ; d) if B i ( x − i ) = ∅ , there exists an open neighborhood V ( x − i ) of x − i sothat ∩ z − i ∈ V ( x − i ) B i ( z − i ) = ∅ ; e) B i ( x − i ) is convex or empty for each x − i ∈ X − i . Then, the game admits weak Nash equilibria.
The following corollary concerns the existence of the weak equilibria forqualitative games.
Corollary 2
Let (( X i ) i ∈ N , ( P i ) i ∈ N ) be a qualitative game. Assume that, foreach i ∈ N, the following conditions are fulfilled:a) X i is a nonempty, convex and compact set in a Hausdorff topologicalvector space E i ; b) the set { x ∈ X : P i ( x ) = ∅} is nonempty and W i = { x − i ∈ X − i : ∃ x i ∈ X i such that P i ( x − i , x i ) = ∅} is closed ; c) for each x − i ∈ X − i , if there exists x i ∈ X i such that P i ( x ) = ∅ , then, there exist an open neighborhood V ( x − i ) of x − i and z i ∈ X i suchthat P i ( z − i , z i ) = ∅ for each z − i ∈ V i ( x − i ); d) { x i ∈ X i : P i ( x − i , x i ) = ∅} is convex or empty for each x − i ∈ X − i . Then, the game admits weak equilibria.
A new approach to the existence of equilibria in choice implies the con-cept of weakly convex graph for a correspondence, proposed by X. Ding andY He in [2]. Firstly, we recall the definition.
Definition 1 (see [2]). Let X be a nonempty convex subset of a topologicalvector space E and Y be a nonempty subset of E . The correspondence T : X −→ Y is said to have weakly convex graph (in short, it is a WCGcorrespondence) if for each n ∈ N and for each finite set { x , x , ..., x n } ⊂ X , there exists y i ∈ T ( x i ) , ( i = 1 , , ..., n ) , such that co( { ( x , y ) , ( x , y ) , ..., ( x n , y n ) } ) ⊂ Gr( T ) . We note that either the graph Gr( T ) is convex, or T { T ( x ) : x ∈ X } 6 = ∅ ,then T has a weakly convex graph.It is obvious that a WCG correspondence may have nonempty valuesand may not be convex-valued.xistence of the equilibrium in choice Example 1 T : [0 , → [0 , , T ( x ) = (cid:26) [0 , ] ∪ [ ,
2] if x = 1 , [0 , − x ] if x ∈ [0 , { } is aWCG correspondence (since ∩{ T ( x ) : x ∈ [0 , } = { } 6 = ∅ ), but T (1) isnot convex and Gr( T ) is not convex either.The theorem below is a continuous selection theorem for correspondenceshaving weakly convex graph. Lemma 3 (Patriche, [6]). Let Y be a nonempty subset of a topological vec-tor space E and K be a ( n − - dimensional simplex in a topological vectorspace F . Let T : K → Y be a WCG correspondence. Then, T has a con-tinuous selection on K . The following lemma guarantees the existence of a fixed point for a prod-uct of lower semi-continuous correspondences. It will be useful for provingour second result on equilibria in choice.
Lemma 4 (Wu, [12]) Let I be an index set. For each i ∈ I, let X i bea nonempty convex subset of a Hausdorff locally convex topological vectorspace E i , D i a nonempty compact metrizable subset of X i and S i , T i : X → D i two correspondences with the following conditions: (1) for each x ∈ X , clco S i ( x ) ⊂ T i ( x ) and S i ( x ) = ∅ .(2) S i is lower semi-continuous.Then, there exists x ∗ = Q i ∈ I x ∗ i ∈ D = Q i ∈ I D i such that x ∗ i ∈ T i ( x ∗ ) foreach i ∈ I. The assumption that each correspondence of the upper sections of thesets C i has a selection which is an WCG correspondence also assures theexistence of the equilibria in choice. The following theorem presents preciselythis.
Theorem 2
Let (( X i ) i ∈ N , ( C i ) i ∈ N ) be a game in choice form. Assume that,for each i ∈ N, the following conditions are fulfilled:a) X i is a nonempty convex compact metrizable set in a locally convexspace E i and C i is nonempty;b) for each x i ∈ X i , W i = { x − i ∈ X − i : C i ( x − i ) = ∅} is a ( n i − -dimensional simplex in X ;c) there exists a WCG correspondence S i : W i → X i such that S i ( x − i ) ⊂ C i ( x − i ) for each x − i ∈ W i ; d) C i ( x − i ) is convex or empty for each x − i ∈ X − i . Then, the game admits equilibria in choice.Proof.
Let be i ∈ I. From the assumption (c) and the selection Lemma3, it follows that there exists a continuous function f i : W i → X i so thatfor each x − i ∈ W i , f i ( x − i ) ∈ S i ( x − i ) ⊂ C i ( x − i ) . Let us define the correspondence T i : X − i → X i , by Monica Patriche T i ( x − i ) := (cid:26) { f i ( x − i ) } if x − i ∈ W i ;co( S { y − i : C i ( y − i ) = ∅} C i ( y − i )) if x − i / ∈ W i . We appeal to Lemma 1 to conclude that T i is lower semicontinuous on X . Tychonoff’s Theorem implies that X is compact.According to Wu’s fixed-point Theorem, applied for the correspondences S i = T i and T i : X → X i , there exists x ∗ ∈ X such that for each i ∈ I , x ∗ i ∈ T i ( x ∗ ). Suppose that ( x ∗− i , X i ) ∩ C i = ∅ , for some i ∈ N. Then, C i ( x ∗− i ) = ∅ and x ∗ i ∈ C i ( x ∗− i ) , which implies x ∗ = ( x ∗− i , x ∗ i ) ∈ C i . Remark 1
We can obtain two corollaries to the above theorem, if we replaceassumption c) with:c’) there exists a correspondence S i : W i → X i such that S i has aconvex graph and S i ( x − i ) ⊂ C i ( x − i ) for each x − i ∈ W i ;orc”) there exists a correspondence S i : W i → X i with closed values suchthat S i has the property that for any finite set { x − i , x − i , ...x m − i } ⊂ X, T mj =1 S i ( x j − i ) = ∅ and S i ( x − i ) ⊂ C i ( x − i ) for each x − i ∈ W i ; Proof . In the first case, since a correspondence with a convex graph is aWCG one, it follows that S i verifies Assumption c) from Theorem 2, thenwe can apply this theorem.In the second case, X is a compact space and for each i ∈ I the closed sets S i ( x − i ) , x − i ∈ X − i have the finite intersection property, then T { S i ( x − i ) : x − i ∈ X − i } 6 = ∅ . It follows that S i is a WCG correspondence and theconclusion comes from Theorem 2.As in the first case, we obtain the following corollaries concerning theexistence of the weak Nash equilibria for games in normal form and, respec-tively, of the weak equilibria for qualitative games. Corollary 3
Let (( X i ) i ∈ N , ( u i ) i ∈ N ) be a game in normal form. Assumethat, for each i ∈ N, the following conditions are fulfilled:a) X i is a nonempty, convex and compact set in a Hausdorff topologicalvector space E i ;b) the set { x ∈ X : u i ( x ) ≥ u i ( x − i , y i ) for each y i ∈ X i } is nonempty ; c) for each x i ∈ X i , W i = { x − i ∈ X − i : B i ( x − i ) = ∅} is a ( n i − -dimensional simplex in X ; d) there exists a WCG correspondence S i : W i → X i such that S i ( x − i ) ⊂ B i ( x − i ) for each x − i ∈ W i ; e) B i ( x − i ) is convex or empty for each x − i ∈ X − i . Then, the game admits weak Nash equilibria.
Corollary 4
Let (( X i ) i ∈ N , ( P i ) i ∈ N ) be a qualitative game. Assume that, foreach i ∈ N, the following conditions are fulfilled: xistence of the equilibrium in choice a) X i is a nonempty, convex and compact set in a Hausdorff topologicalvector space E i ;b) the set { x ∈ X : P i ( x ) = ∅} is nonempty;c) for each x i ∈ X i , W i = { x − i ∈ X − i : ∃ x i ∈ X i such that P i ( x − i , x i ) = ∅} is a ( n i − -dimensional simplex in X ; d) there exists a WCG correspondence S i : W i → X i such that S i ( x − i ) ⊂{ x i ∈ X i such that P i ( x − i , x i ) = ∅} for each x − i ∈ W i ; e) { x i ∈ X i : P i ( x − i , x i ) = ∅} is convex or empty for each x − i ∈ X − i . Then, the game admits weak equilibria.
Now, we present a continuous selection lemma on Banach spaces whichwas proposed by Yuan [15].
Lemma 5 (see [15]). Let X be a paracompact space, Y be a Banach spaceand T : X → Y be a lower semicontinuous correspondences with closedconvex values. Let S : X → Y be a correspondence whose graph is openin X × Y such that T ( x ) ∩ S ( x ) = ∅ for each x ∈ X. Then, there exists acontinuous function f : X → Y such that f ( x ) ∈ co ( T ( x ) ∩ S ( x )) for each x ∈ X . The previous lemma leads us to the enunciation of Theorem 3, whichgives new conditions under which the equilibria in choice exist.
Theorem 3
Let (( X i ) i ∈ N , ( C i ) i ∈ N ) be a game in choice form. Assume that,for each i ∈ N, the following conditions are fulfilled:a) X i is a nonempty, convex and compact set in a Banach space E i ; b) C i is nonempty and open;c) the set W i = { x − i ∈ X − i : C i ( x − i ) = ∅} is nonempty and open ; d) there exists a lower semicontinuous correspondence S i : W i → X i with closed convex values such that C i ( x − i ) ∩ S i ( x − i ) = ∅ for each x − i ∈ X − i ;e) C i ( x − i ) is convex or empty for each x − i ∈ X − i . Then, the game admits equilibria in choice.Proof
Let be i ∈ I. From the assumption d) and the above lemma, it followsthat there exists a continuous function f i : W i → X i such that f i ( x − i ) ∈ C i ( x − i ) ∩ S i ( x − i ) for each x − i ∈ W i .Let us define the correspondence T i : X − i → X i , by T i ( x − i ) := (cid:26) { f i ( x − i ) } if x − i ∈ W i ;co( S { y − i : C i ( y − i ) = ∅} C i ( y − i )) if x − i / ∈ W i . We appeal to Lemma 1 to conclude that T i is upper semicontinuous on X . According to Kakutani’s fixed-point Theorem, there exists x ∗ ∈ X suchthat for each i ∈ I , x ∗ i ∈ T i ( x ∗− i ). Suppose that ( x ∗− i , X i ) ∩ C i = ∅ , forsome i ∈ N. Then, C i ( x ∗− i ) = ∅ and x ∗ i = f i ( x ∗− i ) ∈ C i ( x ∗− i ) , which implies x ∗ = ( x ∗− i , x ∗ i ) ∈ C i . As corollaries of Theorem 3, we obtain the following results which assumethe lower semicontinuity of the involved correspondences. We note thatthese results are different from the old ones obtained in literature so far.
Corollary 5
Let (( X i ) i ∈ N , ( u i ) i ∈ N ) be a game in normal form. Assumethat, for each i ∈ N, the following conditions are fulfilled:a) X i is a nonempty, convex and compact set in a Banach space E i ;b) the set { x ∈ X : u i ( x ) ≥ u i ( x − i , y i ) for each y i ∈ X i } is nonemptyand open ; c) the set W i = { x − i ∈ X − i : B i ( x − i ) = ∅} is nonempty and open ; d) there exists a lower semicontinuous correspondence S i : X − i → X i with closed convex values such that S i ( x − i ) ∩ B i ( x − i ) = ∅ for each x − i ∈ X − i ;e) B i ( x − i ) is convex or empty for each x − i ∈ X − i . Then, the game admits weak Nash equilibria.
Corollary 6
Let (( X i ) i ∈ N , ( P i ) i ∈ N ) be a qualitative game. Assume that, foreach i ∈ N, the following conditions are fulfilled:a) X i is a nonempty, convex and compact set in a Banach space E i ;b) the set { x ∈ X : P i ( x ) = ∅} is nonempty and open;c) the set W i = { x − i ∈ X − i : ∃ x i ∈ X i such that P i ( x − i , x i ) = ∅} isnonempty and open ; d) there exists a lower semicontinuous correspondence S i : W i → X i with closed convex values such that S i ( x − i ) ∩{ x i ∈ X i : P i ( x − i , x i ) = ∅} 6 = ∅ for each x − i ∈ X − i ;e) { x i ∈ X i : P i ( x − i , x i ) = ∅} is convex or empty for each x − i ∈ X − i . Then, the game admits weak equilibria.
Further, we will prove the existence of the equilibrium in choice undernew conditions. The proof we will provide explicitly relies on lemmas con-cerning the fixed points for the correspondences we will construct based onthe upper sections of the sets ( C i ) i ∈ N . First, we recall the following definition.If X is a nonempty set and Y is a topological space, the correspondence T : X → Y is said to be transfer open-valued [11] if for any ( x, y ) ∈ X × Y with y ∈ T ( x ) , there exists an x ′ ∈ X such that y ∈ int T ( x ′ ) . Further, we present the following useful statement about the transferopen-valued correspondences (Proposition 1 in [3]).
Lemma 6
Let Y be a nonempty set, X be a topological space and S : X → Y be a correspondence. The following assertions are equivalent:a) S − : Y → X is transfer open-valued and S has nonempty values;b) X = S y ∈ Y int S − ( y ) . In this context, Ansari and Yao proved in [1] a fixed point result.xistence of the equilibrium in choice Lemma 7 (Ansari, Yao [1]). Let X be a compact convex subset of a Haus-dorff topological vector space. Let S : X → X be a correspondence withnonempty convex values. If X = ∪{ int X S − ( y ) : y ∈ X } (or, S − : Y → X is transfer open-valued) , then, S has fixed points. We will apply the previous lemma in order to prove the existence of theequilibria in choice for games in choice form.
Theorem 4
Let (( X i ) i ∈ N , ( C i ) i ∈ N ) be a game in choice form. Assume that,for each i ∈ N, the following conditions are fulfilled:a) X i is a nonempty, convex and compact set in a Hausdorff topologicalvector space E i and C i is nonempty;b) X = S y ∈ X { int X T i ∈ N ( C W i ∪ { x − i ∈ X − i : x − i ∈ C i ( y i ) } ) } , where W i = { x − i ∈ X − i : C i ( x − i ) = ∅} ; c) C i ( x − i ) is convex or empty for each x − i ∈ X − i . Then, the game admits equilibria in choice.Proof
For each i ∈ N, let us define the correspondence T i : X − i → X i , by T i ( x − i ) = (cid:26) co( S { y − i : C i ( y − i ) = ∅} C i ( y − i )) if x i ; / ∈ W i C i ( x − i ) if x − i ∈ W i . The correspondence S i has nonempty and convex values . Let T : X → X be defined by T ( x ) = Q i ∈ N T i ( x − i ) for each x ∈ X. The correspondence S also has nonempty and convex values . If y ∈ X, then T − ( y ) = T i ∈ N { x ∈ X : y i ∈ T i ( x − i ) } = T i ∈ N ( C W i ∪{ x − i ∈ X − i : x − i ∈ C i ( y i ) } ) .X = S y ∈ X int X T − ( y ), according to assumption b).We can apply the Ansari and Yao Lemma and we obtain that thereexists x ∗ ∈ X such that x ∗ ∈ T ( x ∗ ) . Obviously, x ∗ i ∈ T i ( x ∗− i ) for each i ∈ N. Suppose that ( x ∗− i , X i ) ∩ C i = ∅ , for some i ∈ N. Then, C i ( x ∗− i ) = ∅ and x ∗ i ∈ C i ( x ∗− i ) , which implies x ∗ = ( x ∗− i , x ∗ i ) ∈ C i . Remark 2
According to Lemma 7, we can replace condition b) in Theorem4 with b’) if x i ∈ C i ( y − i ) , then, there exists z i ∈ X i such that y − i ∈ int X − i { x − i ∈ X − i : z i ∈ C i ( x − i ) } and the set W i = { x − i ∈ X − i : C i ( x − i ) = ∅} is closed. A new result involving the equilibria in choice will naturally follow di-rectly from Theorem 4.
Theorem 5
Let (( X i ) i ∈ N , ( C i ) i ∈ N ) be a game in choice form. Assume that,for each i ∈ N, the following conditions are fulfilled:a) X i is a nonempty, convex and compact set in a Hausdorff topologicalvector space E i and C i is nonempty;b) for each x i ∈ X i , { x − i ∈ X − i : x i ∈ C i ( x − i ) } ∪ C W i is open, where W i = { x − i ∈ X − i : C i ( x − i ) = ∅} ; c) C i ( x − i ) is convex or empty for each x − i ∈ X − i . Then, the game admits equilibria in choice.
Another proof of the above theorem appeals to Yannelis and Prabhakar’continuous selection lemma [14] applied for the correspondences T i : X − i → X i defined by T i ( x − i ) = (cid:26) co( S { y − i : C i ( y − i ) = ∅} C i ( y − i )) if x − i / ∈ W i ; C i ( x − i ) if x − i ∈ W i . for each i ∈ N. We present below the lemma.
Lemma 8 (Yannelis and Prabhakar, [14]). Let X be a paracompact Haus-dorff topological space and Y be a Hausdorff topological vector space. Let T : X → Y be a correspondence with nonempty convex values and for each y ∈ Y , T − ( y ) is open in X . Then, T has a continuous selection that is,there exists a continuous function f : X → Y so that f ( x ) ∈ T ( x ) for each x ∈ X . For the proof, we note that for each i ∈ N, the correspondence T i hasnonempty and convex values and if x i ∈ X i , then T − i ( x i ) = C W i ∪ { x − i ∈ X − i : x i ∈ C i ( x − i ) } is an open set, according to assumption b).We can apply the Yannelis and Prabhakar Lemma and we obtain thatthere exists f i : X − i → X i , a continuous selection of T i . Let f : X → X be defined by f ( x ) = Q i ∈ N f i ( x − i ) for each x ∈ X. The function f iscontinuous, and, according to the Brouwer fixed point Theorem, there exists x ∗ ∈ X such that f ( x ∗ ) = x ∗ . Hence, x ∗ ∈ Q i ∈ N T i ( x ∗− i ) and obviously, x ∗ i ∈ T i ( x ∗− i ) for each i ∈ N. Suppose that ( x ∗− i , X i ) ∩ C i = ∅ , for some i ∈ N. Then, C i ( x ∗− i ) = ∅ and x ∗ i ∈ C i ( x ∗− i ) , which implies x ∗ = ( x ∗− i , x ∗ i ) ∈ C i . Remark 3
If for each x i ∈ X i , { x − i ∈ X − i : x i ∈ C i ( x − i ) } is open, and theset W i = { x − i ∈ X − i : C i ( x − i ) = ∅} is closed, then the above Theoremholds.Corollary 7 is mainly obtained by verifying an assumption concerningthe union of all lower sections of the best reply correspondences for a gamein normal form . Corollary 7
Let (( X i ) i ∈ N , ( u i ) i ∈ N ) be a game in normal form. Assumethat, for each i ∈ N, the following conditions are fulfilled:a) X i is a nonempty, convex and compact set in a Hausdorff topologicalvector space E i ;b) the set { x ∈ X : u i ( x ) ≥ u i ( x − i , y i ) for each y i ∈ X i } is nonempty ; c) X = S y ∈ X { int X T i ∈ N ( C W i ∪ B − i ( y i )) } , where W i = { x − i ∈ X − i : B i ( x − i ) = ∅} ; d) B i ( x − i ) is convex or empty for each x − i ∈ X − i . Then, the game admits weak Nash equilibria. xistence of the equilibrium in choice Remark 4
In Corollary 7, condition c) can also be replaces withc’) the best reply correspondence B i : X − i → X i is transfer open valuedand the set W i = { x − i ∈ X − i : B i ( x − i ) = ∅} is closed . A new statement can be deduced explicitly from Corollary 7.
Corollary 8
Let (( X i ) i ∈ N , ( u i ) i ∈ N ) be a game in normal form. Assumethat, for each i ∈ N, the following conditions are fulfilled:a) X i is a nonempty, convex and compact set in a Hausdorff topologicalvector space E i ;b) the set { x ∈ X : u i ( x ) ≥ u i ( x − i , y i ) for each y i ∈ X i } is nonempty ; c) B − i ( x i ) ∪ { x − i ∈ X − i : B i ( x − i ) = ∅} is open for each x i ∈ X i ; d) B i ( x − i ) is convex or empty for each x − i ∈ X − i . Then, the game admits weak Nash equilibria.
The following results refer to the existence of weak equilibria for thequalitative games. They are consequences of Theorem 5.
Corollary 9
Let (( X i ) i ∈ N , ( P i ) i ∈ N ) be a qualitative game. Assume that, foreach i ∈ N, the following conditions are fulfilled:a) X i is a nonempty, convex and compact set in a Hausdorff topologicalvector space E i ;b) the set { x ∈ X : P i ( x ) = ∅} is nonempty;c) X = S y ∈ X { int X T i ∈ N ( C W i ∪ { x − i ∈ X − i : P i ( x − i , y i ) = ∅} ) } , where W i = { x − i ∈ X − i : ∃ x i ∈ X i such that P i ( x − i , x i ) = ∅} ; d) { x i ∈ X i : P i ( x − i , x i ) = ∅} is convex or empty for each x − i ∈ X − i . Then, the game admits weak equilibria.Remark 5
In Corollary 9, condition c) can also be replaced withc”) if P i ( y − i , x i ) = ∅ , then, there exists z i ∈ X i such that y − i ∈ int Y − i { x − i ∈ X − i : P i ( x − i , z i ) = ∅} and the set W i = { x − i ∈ X − i : ∃ x i ∈ X i such that P i ( x − i , x i ) = ∅} is open. Corollary 10
Let (( X i ) i ∈ N , ( P i ) i ∈ N ) be a qualitative game. Assume that,for each i ∈ N, the following conditions are fulfilled:a) X i is a nonempty, convex and compact set in a Hausdorff topologicalvector space E i ;b) the set { x ∈ X : P i ( x ) = ∅} is nonempty;c) { x − i ∈ X − i : ∄ x i ∈ X i such that P i ( x − i , x i ) = ∅} ∪ { x − i ∈ X − i : P i ( x − i , x i ) = ∅} is open for each x i ∈ X i ; d) { x i ∈ X i : P i ( x − i , x i ) = ∅} is convex or empty for each x − i ∈ X − i . Then, the game admits weak equilibria.
A very great importance in the fixed point theory has Tarafdar’s fixedpoint Theorem, which we present below.
Lemma 9 (Tarafdar, [10]). Let { X i } i ∈ I be a family of nonempty compactconvex sets, each in a topological vector space E i , where I is an index set.Let X = Q i ∈ I X i . For each i ∈ I, let S i : X → X i be a correspondence suchthat a) for each x ∈ X, S i ( x ) is a nonempty, convex subset of X i ;b) for each x i ∈ X i , S − i ( x i ) contains a relatively open subset O x i of X such that ∪ x i ∈ X i O x i = X ( O x i may be empty for some x i ) . Then, there exists a point x ∈ X such that x ∈ S ( x ) = Q i ∈ I S i ( x ) , thatis, x i ∈ S i ( x ) for each i ∈ I, where x i is the projection of x onto X i foreach i ∈ I. By using Lemma 9, we establish Theorem 6, which is a slightly differentversion of Theorem 4.
Theorem 6
Let (( X i ) i ∈ N , ( C i ) i ∈ N ) be a game in choice form. Assume that,for each i ∈ N, the following conditions are fulfilled:a) X i is a nonempty, convex and compact set in a Hausdorff topologicalvector space E i and C i is nonempty;b) for each x i ∈ X i , { x − i ∈ X − i : x i ∈ C i ( x − i ) } contains a relativelyopen subset O x i of X − i , such that ∪ x i ∈ X i O x i = X − i ( O x i may be empty for some x i ); c) C i ( x − i ) is convex or empty for each x − i ∈ X − i . Then, the game admits equilibria in choice.Proof
For each i ∈ N, let us define the correspondence T i : X − i → X i , by T i ( x − i ) = (cid:26) co( S { y − i : C i ( y − i ) = ∅} C i ( y − i )) if x − i / ∈ W i ; C i ( x − i ) if x − i ∈ W i , , where W i = { x − i ∈ X − i : C i ( x − i ) = ∅} . The correspondence T i has nonempty and convex values . If x i ∈ X i , then T − i ( x i ) = C W i ∪ { x − i ∈ X − i : x i ∈ C i ( x − i ) } . According to assumption b), for each x i ∈ X i , T − i ( x i ) contains a rela-tively open subset O x i of X such that ∪ x i ∈ X i O x i = X − i ( O x i may be empty for some x i ) . Then, X = ∪ x i ∈ X i ( O x i × X i ) . For each i ∈ N, let us define the correspondence S i : X → X i , by S i ( x ) = T i ( x − i ) . If x i ∈ X i , then S − i ( x i ) = T − i ( x i ) × X i . We denote U x i = O x i × X i and we obtain that for each x i ∈ X i , S − i ( x i ) containsa relatively open subset U x i of X such that ∪ x i ∈ X i U x i = X ( U x i may beempty for some x i ) . We can apply the previous lemma and we obtain that there exists x ∗ ∈ X such that x ∗ i ∈ S i ( x ∗ ) = T i ( x ∗− i ) for each i ∈ N. Suppose that ( x ∗− i , X i ) ∩ C i = ∅ , for some i ∈ N. Then, C i ( x ∗− i ) = ∅ and x ∗ i ∈ C i ( x ∗− i ) , which implies x ∗ = ( x ∗− i , x ∗ i ) ∈ C i . Now, we get the following corollaries from the previous result.xistence of the equilibrium in choice Corollary 11
Let (( X i ) i ∈ N , ( u i ) i ∈ N ) be a game in normal form. Assumethat, for each i ∈ N, the following conditions are fulfilled:a) X i is a nonempty, convex and compact set in a Hausdorff topologicalvector space E i ;b) the set { x ∈ X : u i ( x ) ≥ u i ( x − i , y i ) for each y i ∈ X i } is nonempty ; c) for each x i ∈ X i , B − i ( x i ) contains a relatively open subset O x i of X such that ∪ x i ∈ X i O x i = X − i ( O x i may be empty for some x i ); d) B i ( x − i ) = { x i ∈ X i : u i ( x ) ≥ u i ( x − i , y i ) for each y i ∈ X i } is convexor empty for each x − i ∈ X − i . Then, the game admits weak Nash equilibria.
Corollary 12
Let (( X i ) i ∈ N , ( P i ) i ∈ N ) be a qualitative game. Assume that,for each i ∈ N, the following conditions are fulfilled:a) X i is a nonempty, convex and compact set in a Hausdorff topologicalvector space E i ;b) the set { x ∈ X : P i ( x ) = ∅} is nonempty;c) for each x i ∈ X i , { x − i ∈ X − i : P i ( x − i , x i ) = ∅} contains a relativelyopen subset O x i of X such that ∪ x i ∈ X i O x i = X − i ( O x i may be empty for some x i ); d) { x i ∈ X i : P i ( x − i , x i ) = ∅} is convex or empty for each x − i ∈ X − i . Then, the game admits weak equilibria.Remark 6
In a particular case, we can weaken condition b) of Theorem 6by condition b’):b’) for each x i ∈ X i , { x − i ∈ X − i : x i ∈ C i ( x − i ) } = O x i is an opensubset of X such that ∪ x i ∈ X i O x i = X − i ( O x i may be empty for some x i ) . According to Lemma7, this condition is equivalent with the fact that the correspondence T − i : X i → X − i is transfer open-valued and T i has nonempty values, where T i : X − i → X i is defined by T i ( x − i ) = C i ( x − i ) for each x − i ∈ X − i . In this case, we obtain the following theorem concerning the existenceof the strong equilibrium in choice.
Theorem 7
Let (( X i ) i ∈ N , ( C i ) i ∈ N ) be a game in choice form. Assume that,for each i ∈ N, the following conditions are fulfilled:a) X i is a nonempty, convex and compact set in a Hausdorff topologicalvector space E i and C i is nonempty;b) for each x i ∈ X i , { x − i ∈ X − i : x i ∈ C i ( x − i ) } = O x i is an open subsetof X such that ∪ x i ∈ X i O x i = X − i ( O x i may be empty for some x i ); c) C i ( x − i ) is convex or empty for each x − i ∈ X − i . Then, the game admits strong equilibria in choice.
Our study is a new perspective unifying different approaches and resultson the equilibrium concepts and the existence of noncooperative theory ofgames. We have proposed to the reader a synthesis of theorems and conse-quences which state, under new conditions, the existence of the equilibriumfor games in choice form, in normal form and also for qualitative games.Our approach differs essentially from the one of Stefanescu, Ferrara andStefanescu, who proposed the new concept of game in choice form and thecorresponding equilibrium in choice (2012). A further research may consistof the integration of all research instruments and perspectives. The advan-tage of using these new ideas is that they are more systematic and cancover more general situations. This paper reflects the integrity of this kindof thinking and can reopen the problem of the equilibrium existence undernew conditions.
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