# Generic Stability of the Weakly Pareto-Nash Equilibrium with Strategy Transformational Barriers

Authors: Luping Liu; Wensheng Jia; Li Zhou
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The object of this paper is to establish a new model with strategy transformational barriers for a class of generalized multileader multifollower multiple objective games (GMLMFMOG) and further deduce some new results of the weakly Pareto-Nash equilibrium (WPNE) with strategy transformational barriers for the GMLMFMOG. First, we investigate the existence of the WPNE with strategy transformational barriers for the GMLMFMOG by using the Kakutani-Fan-Glicksberg fixed point theory. Next, we study the generic stability of the GMLMFMOG with strategy transformational barriers in Hausdorff space. Finally, we obtain that the majority of the WPNE with strategy transformational barriers for the GMLMFMOG are essential on the meaning of Baire’s category. In addition, we demonstrate that there is at least an essential component for the GMLMFMOG with strategy transformational barriers.


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## Body

## 1. Introduction

Barriers, such as market competition [1], the Lévy risk process [2, 3], the optimal dividend problem [4], and the marketing ethics of medical schemes [5], are common in the field of economics. Transformational barriers, an important aspect of barriers, represent many factors that make the behaviour of shift strategy more difficult or costly for consumers. Furthermore, the payoff function with the strategy transformational barriers may be an abstract partial order rather than a numerical order. Game theory is an important tool for studying the interactions among the decision-making behaviours of players in many fields, such as economics, political science, psychology, and biology. Glicksberg [6] and Mas-Colell [7] provided a maximum element method to analyze the decision-making behaviours of players with the strategy transformational barriers. Therefore, the payoff function with strategy transformational barriers was introduced into game model to further study the decision-making behaviour of players based on the fact that there is a cost for players to change their strategies in practical life.Fort [8] first presented the essential fixed point in 1950. Wu and Jiang [9] first provided the concept of essential equilibrium for a finite game through using fixed point theory for continuous mapping. Afterwards, Yu and Luo and Yu [10, 11] extended previous work to the general n-person noncooperative game, generalized game, or other games by using entirely different approaches. Recently, Scalzo [12, 13] and Carbonell-Nicolau and Carbonell-Nicolau and Wohl [14, 15] provided some extensions about discontinuous payoffs and further studied the essential stability of discontinuous games. Yang and Zhang [16] proved some existence and essential stability results of cooperative equilibrium for population games. We can also refer to [17–20] for more details on the essential stability. Hence, the essential stability has become one of the important topics in nonlinear analysis and game theory.The weakly Pareto-Nash equilibrium (WPNE) of the multiple objective game was proposed by Shapley and Rigby [21]. Pang and Fukushima [22] studied the existence of a type of multileader multifollower multiobjective game by using quasivariational inequalities. Sherali [23] obtained the existence and uniqueness results of a WPNE regarding the multileader multifollower game. Kulkarni and Shanbhag [24] considered multileader multifollower game with shared-constraint approach to obtain local Nash equilibrium (NE), Nash B-stationary point, and Nash strong-stationary point. Yu and Wang [25] verified some existence theorems for 2-leader multifollower game in locally convex topological space. Yang and Ju [26] obtained some consequences on existence and stability of solution for multileader multifollower game. Jia et al. [27] provided the existence and stability of a WPNE for the generalized multileader multifollower multiple objective game (GMLMFMOG). Inspired by the above research work, this paper establishes a new generalized multiobjective multileader multifollower model with strategy transformational barriers by considering the influence of strategy transformational barriers and analyzes the strategy selection of the players. The leaders consider multiple objectives when selecting their strategies. The followers also consider multiple objectives when selecting their strategies with complete knowledge and make optimal responses to the leaders’ strategies. The goals of all players are to maximize their own incomes. Furthermore, the existence of the WPNE with strategy transformational barriers of a GMLMFMOG is proved, and the generic stability of the GMLMFMOG with strategy transformational barriers is obtained. We prove that the solution set of the GMLMFMOG with the strategy transformational barriers is essential and that there is at least one essential component of the WPNE with the strategy transformational barriers under the meaning of the Baire’s category.This paper is outlined as follows. We present necessary preliminaries and the GMLMFMOG model with strategy transformational barriers in Section 2. In Section 3, we provide the existence of the WPNE with the strategy transformational barriers of the GMLMFMOG. In Section 4, we investigate some generic stability results of the GMLMFMOG with strategy transformational barriers. In Section 5, we show that the majority of WPNE with strategy transformational barriers of the GMLMFMOG are essential, and then there is at least an essential component. Finally, some brief and concise conclusions are given.


## 2. Preliminaries and Model

### 2.1. Preliminaries

In this paragraph, we introduce some substantial definitions, lemmas, and game models.Definition 1 (see [28]).
Suppose thatA is not empty subset of Hausdorff topological vector space (HTVS) F, L⊂F is not empty convex cone, and a vector-valued correspondence is denoted by S:A⟶F. We define S is L-usc (resp. L-lsc) at a0∈A if, for each open neighbourhood V of the 0 element in F, there exists an open neighbourhood Oa0 of a0 such that Sa∈Sa0+V−L (resp. Sa∈Sa0+V+L), ∀a∈Oa0. Furthermore, we say S is L-usc (resp. L-lsc) on A, if S is L-usc (resp. L-lsc) for all a∈A. We call S is L-continuous on A, if S is L-usc and L-lsc on A. S is closed if GraphS=a,f∈A×Ff∈Sa is closed on A×F.Definition 2 (see [29]).
LetA and B be two HTVSs, L⊂B be a closed convex pointed cone, intL≠∅, D⊂A be not empty convex subset, and S:D⟶B be a vector-valued correspondence. If, ∀a1,a2∈D and θ∈0,1, Sθa1+1−θa2−θSa1+1−θSa2∉−intL holds, then S is L- concave, and −S is L- convex. If, ∀a1,a2∈D, b∈B, and θ∈0,1, Sa1∉b−intL, Sa2∉b−intL such that Sθa1+1−θa2∉b−intL, then S is L- quasiconcave-like, and −S is L- quasiconvex-like.Remark 3.
ForB=−∞,+∞, L=0,+∞, if S is L- quasiconcave-like, then S is obviously quasiconcave. However, D=0,1, B=−∞,+∞×−∞,+∞, L=0,+∞×0,+∞, and f=f1,f2=a,−a, g=g1,g2=a2,a2. We know that f is L- concave but not L- quasiconcave-like, and g is L- quasiconcave-like but not L- concave. Thus, L- quasiconcave-like and L- concave do not include each other.Definition 4 (maximal element theorem, see [30]).
LetA be not empty compact convex subset (NECCS) of HTVS F and S:A⟶2A with the following conditions, where 2A denotes all nonempty subsets of A:
(1)
∀a∈A, a∉convSa, where convSa denotes the convex hull of Sa(2)
∀b∈A, S−1b=a∈Ab∈Sa is open in A
Then, there isa∗=a1∗,a2∗,⋯,an∗∈A such that Sa∗=∅.


### 2.2. Model

A model of the GMLMFMOG with strategy transformational barrier is denoted by a tupleℕ,M,A,B,V,P, where
(i)
ℕ=1,⋯,n and M=1,⋯,m indicate the index set of leaders and followers, respectively(ii)
∀i∈ℕ, ∀j∈M, Ai, and Bj denote the strategy set of the ith leader and the jth follower, separately. The leaders’ strategy represents a=ai,a−i∈A, where A=∏i∈ℕAi, A−i=∏l∈ℕ\iAl. Meanwhile, the strategy of the followers denotes b=bj,b−j∈B, where B=∏j∈MBj, B−j=∏k∈M\jBk(iii)
∀i∈ℕ, Ui=B, U=∏i∈ℕUi, and U−i=∏k∈ℕ\iUk. Let Yi=χ1i,⋯,χli:Ai×A−i×Ui⟶R+l be the payoff function of the ith leader. Let Ψj=ψ1j,⋯,ψkj:A×Bj×B−j⟶Rk, ∀j∈M be a payoff function of the jth follower and Gj:A×B−j⟶2Bj be a constraint correspondence of the jth follower(iv)
LetVi:Ai×Ai⟶R+l be the strategy transformational barrier function of the leader i. ∀i∈ℕ, there exists ai∈Ai such that(1)Viai,ai′>0,∀ai′∈Ai,where Viai,a′i denotes the strategy transformational barriers of the leader i changing from strategy ai to strategy a′i. In particular, Viai,a′i=0 denotes that the ith leader has no transformational strategy
(v)
The followers are a generalized constraint multiobjective with the strategy parametric game (PGCMOG) after fixing the strategya∈A of the leaders. Let P:Ai×A−i⟶2B be the solution mapping of the WPNE with strategy transformational barriers for the PGCMOG. Particularly, ∀b∗∈Pai,a−i such that there is bj∗∈Gja,b−j∗,∀j∈M, and we have Ψja,bj,b−j∗−Ψja,bj∗,b−j∗∉intR+k, ∀bj∈Gja,b−j∗. Furthermore, if there is ui∗∈Ui such that ui∗∈Pai∗,a−i∗,∀i∈ℕ, satisfying(2)Yia′i,a−i∗,ui−Yia−i∗,a−i∗,ui∗−Via−i∗,a′i∉intR+l,∀a′i,ui∈Ai×Pa′i,a−i∗,then a∗=a−i∗,a−i∗∈A is called a WPNE with strategy transformational barriers of the GMLMFMOG, where intR+l=a1,a2,⋯,al∈Rl:ai>0,i=1,⋯,l, Via−i∗,a′i denotes the leader i’s cost changing from strategy a−i∗ to strategy a′iLeti, a−i, and u−i be elements in ℕ, A−i, and U−i, respectively. By Definition 4, then we have the best response of the ith leader with strategy transformation barriers to the other players, i.e.,
(3)Bia−i,u−i=ai∈Ai,ui∈Pai,a−iYiai′,a−i,vi−Yiai,a−i,ui−Viai,ai′∉intR+l,∀a′i,vi∈Ai×Pa′i,a−i,where Bi is independent of u−i∈Ui.Fixinga−i∈A−i, we know that the player’s set-valued mapping Bi provides the order relation “≥a−i” as follows:
(4)wi,ui≥a−iai,ui⇔wi,ui∈Bia−i,u−i.In general, the order relation is not transitive, and we give a sufficient condition for the transitivity of the order relation “≥a−i” with the following propositions.Proposition 5.
Letℕ,M,A,B,V,P be a GMLMFMOG with strategy transformational barriers, if, for any wi,ai,zi∈Ai, and
(5)Vizi,wi+Viwi,ai≤Vizi,aii.e.,Vihasnegativesubadditivity.
Then, the order relation “≥a−i” has transitivity.Proof.
Settingwi,ai,zi which are three elements in Ai and u−i,vi∈U−i such that zi,ui≥a−iwi,ui≥a−iai,ui holds, we obtain
(6)Yiwi,a−i,vi−Yizi,a−i,ui−Vizi,wi∉intR+l,Yiai,a−i,vi−Yiwi,a−i,ui−Viwi,ai∉intR+l,
byzi,ui∈Bia−i,u−i, wi,ui∈Bia−i,u−i, and the definition of best response mapping Bia−i,u−i. Then, we attain
(7)Yiai,a−i,vi−Yizi,a−i,ui−Vizi,wi−Viwi,ai−Yiwi,a−i,ui+Yiwi,a−i,vi∉intR+l⇒Yiai,a−i,vi−Yizi,a−i,ui−Vizi,wi+Viwi,ai−Yiwi,a−i,ui−Yiwi,a−i,vi∉intR+l⇒Yiai,a−i,vi−Yizi,a−i,ui−Vizi,ai∉intR+l.
SinceBi is not dependent on u−i∈Ui, we can see that Yiwi,a−i,ui−Yiwi,a−i,vi is equal to zero element of R+l. Therefore, zi,ui∈Bia−i,u−i⇔zi,ui≥a−iai,ui; then, the order relation “≥a−i” has transitivity.Example 1.
Considering the Hotelling model [31], the influence of the strategy transformational barrier function can be added. Assume that consumers are evenly distributed on a street and that businessmen (ℕ=1,2) choose their shop location on the street. Suppose that the street can be abstracted to a line segment with a length of 1, namely, [0, 1]. Meanwhile, c∈0,1 and d∈0,1 represent the positions of the two businessmen. The strategy set of the businessmen is 0,1, and the payoff functions f1,f2:0,1⟶R are expressed as
(8)f1=c+d2,c<d,1−c+d2,c>d12,c=d,,f2=c+d2,d<c,1−c+d2,d>c,12,d=c.It is a well-known fact thatc,d=1/2,1/2 is the unique NE point of the Hotelling game [31], which can better explain the phenomenon of shop centralization. However, it is worth noting that shops may not be concentrated in the centre because of the influence of relocation costs and other factors. In reality, the distribution of shop locations corresponds to a WPNE with strategy transformational barriers, which means a state of equilibrium under weaker conditions.Suppose that the strategy transformational barrier functions areVc1,c2 and Vd1,d2, respectively. If Vc1,c2 and Vd1,d2 are
(9)Vc1,c2=α1c1−c2+β1,∀c1,c2∈0,1,Vd1,d2=α2d1−d2+β2,∀d1,d2∈0,1.Setting businessmen 1 takingd=1/2, c1=1/4, c2=3/4, and c3=3/8, we have
(10)f1c1,d=f114,12=38,f1c2,d=f134,12=38,f1c3,d=f138,12=716.Ifα1=1/3, β1=0, then
(11)f1c1,d=38≥f1c2,d−Vc1,c2=38−13c1−c2=524,i.e.,c1≥bc2,f1c2,d=38≥f1c3,d−Vc2,c3=716−13c2−c3=516,i.e.,c2≥bc3,but
(12)f1c1,d=38≤f1c3,d−Vc1,c3=716−13c1−c3=716−124=1948.Furthermore,(13)Vc1,c2=13c1−c2=13×14−34=16,Vc2,c3=13c2−c3=13×34−38=18,Vc1,c3=13c3−c1=13×38−14=124.SinceVc1,c2+Vc2,c3≰Vc1,c3, “c1≥dc3” has no negative subadditivity. Then, “c1≥dc3” does not hold; thus, the order relation “≥” is not satisfied to transitive.Remark 6.
When the strategy transformational barrier function does not have negative subadditivity, the order relationship “≥a−i” does not have transitivity. Furthermore, the game with a strategy transformational barrier function may not have a numerical payoff function since the strategy transformational barrier function often possesses subadditivity rather than negative subadditivity.


## 2.1. Preliminaries

In this paragraph, we introduce some substantial definitions, lemmas, and game models.Definition 1 (see [28]).
Suppose thatA is not empty subset of Hausdorff topological vector space (HTVS) F, L⊂F is not empty convex cone, and a vector-valued correspondence is denoted by S:A⟶F. We define S is L-usc (resp. L-lsc) at a0∈A if, for each open neighbourhood V of the 0 element in F, there exists an open neighbourhood Oa0 of a0 such that Sa∈Sa0+V−L (resp. Sa∈Sa0+V+L), ∀a∈Oa0. Furthermore, we say S is L-usc (resp. L-lsc) on A, if S is L-usc (resp. L-lsc) for all a∈A. We call S is L-continuous on A, if S is L-usc and L-lsc on A. S is closed if GraphS=a,f∈A×Ff∈Sa is closed on A×F.Definition 2 (see [29]).
LetA and B be two HTVSs, L⊂B be a closed convex pointed cone, intL≠∅, D⊂A be not empty convex subset, and S:D⟶B be a vector-valued correspondence. If, ∀a1,a2∈D and θ∈0,1, Sθa1+1−θa2−θSa1+1−θSa2∉−intL holds, then S is L- concave, and −S is L- convex. If, ∀a1,a2∈D, b∈B, and θ∈0,1, Sa1∉b−intL, Sa2∉b−intL such that Sθa1+1−θa2∉b−intL, then S is L- quasiconcave-like, and −S is L- quasiconvex-like.Remark 3.
ForB=−∞,+∞, L=0,+∞, if S is L- quasiconcave-like, then S is obviously quasiconcave. However, D=0,1, B=−∞,+∞×−∞,+∞, L=0,+∞×0,+∞, and f=f1,f2=a,−a, g=g1,g2=a2,a2. We know that f is L- concave but not L- quasiconcave-like, and g is L- quasiconcave-like but not L- concave. Thus, L- quasiconcave-like and L- concave do not include each other.Definition 4 (maximal element theorem, see [30]).
LetA be not empty compact convex subset (NECCS) of HTVS F and S:A⟶2A with the following conditions, where 2A denotes all nonempty subsets of A:
(1)
∀a∈A, a∉convSa, where convSa denotes the convex hull of Sa(2)
∀b∈A, S−1b=a∈Ab∈Sa is open in A
Then, there isa∗=a1∗,a2∗,⋯,an∗∈A such that Sa∗=∅.


## 2.2. Model

A model of the GMLMFMOG with strategy transformational barrier is denoted by a tupleℕ,M,A,B,V,P, where
(i)
ℕ=1,⋯,n and M=1,⋯,m indicate the index set of leaders and followers, respectively(ii)
∀i∈ℕ, ∀j∈M, Ai, and Bj denote the strategy set of the ith leader and the jth follower, separately. The leaders’ strategy represents a=ai,a−i∈A, where A=∏i∈ℕAi, A−i=∏l∈ℕ\iAl. Meanwhile, the strategy of the followers denotes b=bj,b−j∈B, where B=∏j∈MBj, B−j=∏k∈M\jBk(iii)
∀i∈ℕ, Ui=B, U=∏i∈ℕUi, and U−i=∏k∈ℕ\iUk. Let Yi=χ1i,⋯,χli:Ai×A−i×Ui⟶R+l be the payoff function of the ith leader. Let Ψj=ψ1j,⋯,ψkj:A×Bj×B−j⟶Rk, ∀j∈M be a payoff function of the jth follower and Gj:A×B−j⟶2Bj be a constraint correspondence of the jth follower(iv)
LetVi:Ai×Ai⟶R+l be the strategy transformational barrier function of the leader i. ∀i∈ℕ, there exists ai∈Ai such that(1)Viai,ai′>0,∀ai′∈Ai,where Viai,a′i denotes the strategy transformational barriers of the leader i changing from strategy ai to strategy a′i. In particular, Viai,a′i=0 denotes that the ith leader has no transformational strategy
(v)
The followers are a generalized constraint multiobjective with the strategy parametric game (PGCMOG) after fixing the strategya∈A of the leaders. Let P:Ai×A−i⟶2B be the solution mapping of the WPNE with strategy transformational barriers for the PGCMOG. Particularly, ∀b∗∈Pai,a−i such that there is bj∗∈Gja,b−j∗,∀j∈M, and we have Ψja,bj,b−j∗−Ψja,bj∗,b−j∗∉intR+k, ∀bj∈Gja,b−j∗. Furthermore, if there is ui∗∈Ui such that ui∗∈Pai∗,a−i∗,∀i∈ℕ, satisfying(2)Yia′i,a−i∗,ui−Yia−i∗,a−i∗,ui∗−Via−i∗,a′i∉intR+l,∀a′i,ui∈Ai×Pa′i,a−i∗,then a∗=a−i∗,a−i∗∈A is called a WPNE with strategy transformational barriers of the GMLMFMOG, where intR+l=a1,a2,⋯,al∈Rl:ai>0,i=1,⋯,l, Via−i∗,a′i denotes the leader i’s cost changing from strategy a−i∗ to strategy a′iLeti, a−i, and u−i be elements in ℕ, A−i, and U−i, respectively. By Definition 4, then we have the best response of the ith leader with strategy transformation barriers to the other players, i.e.,
(3)Bia−i,u−i=ai∈Ai,ui∈Pai,a−iYiai′,a−i,vi−Yiai,a−i,ui−Viai,ai′∉intR+l,∀a′i,vi∈Ai×Pa′i,a−i,where Bi is independent of u−i∈Ui.Fixinga−i∈A−i, we know that the player’s set-valued mapping Bi provides the order relation “≥a−i” as follows:
(4)wi,ui≥a−iai,ui⇔wi,ui∈Bia−i,u−i.In general, the order relation is not transitive, and we give a sufficient condition for the transitivity of the order relation “≥a−i” with the following propositions.Proposition 5.
Letℕ,M,A,B,V,P be a GMLMFMOG with strategy transformational barriers, if, for any wi,ai,zi∈Ai, and
(5)Vizi,wi+Viwi,ai≤Vizi,aii.e.,Vihasnegativesubadditivity.
Then, the order relation “≥a−i” has transitivity.Proof.
Settingwi,ai,zi which are three elements in Ai and u−i,vi∈U−i such that zi,ui≥a−iwi,ui≥a−iai,ui holds, we obtain
(6)Yiwi,a−i,vi−Yizi,a−i,ui−Vizi,wi∉intR+l,Yiai,a−i,vi−Yiwi,a−i,ui−Viwi,ai∉intR+l,
byzi,ui∈Bia−i,u−i, wi,ui∈Bia−i,u−i, and the definition of best response mapping Bia−i,u−i. Then, we attain
(7)Yiai,a−i,vi−Yizi,a−i,ui−Vizi,wi−Viwi,ai−Yiwi,a−i,ui+Yiwi,a−i,vi∉intR+l⇒Yiai,a−i,vi−Yizi,a−i,ui−Vizi,wi+Viwi,ai−Yiwi,a−i,ui−Yiwi,a−i,vi∉intR+l⇒Yiai,a−i,vi−Yizi,a−i,ui−Vizi,ai∉intR+l.
SinceBi is not dependent on u−i∈Ui, we can see that Yiwi,a−i,ui−Yiwi,a−i,vi is equal to zero element of R+l. Therefore, zi,ui∈Bia−i,u−i⇔zi,ui≥a−iai,ui; then, the order relation “≥a−i” has transitivity.Example 1.
Considering the Hotelling model [31], the influence of the strategy transformational barrier function can be added. Assume that consumers are evenly distributed on a street and that businessmen (ℕ=1,2) choose their shop location on the street. Suppose that the street can be abstracted to a line segment with a length of 1, namely, [0, 1]. Meanwhile, c∈0,1 and d∈0,1 represent the positions of the two businessmen. The strategy set of the businessmen is 0,1, and the payoff functions f1,f2:0,1⟶R are expressed as
(8)f1=c+d2,c<d,1−c+d2,c>d12,c=d,,f2=c+d2,d<c,1−c+d2,d>c,12,d=c.It is a well-known fact thatc,d=1/2,1/2 is the unique NE point of the Hotelling game [31], which can better explain the phenomenon of shop centralization. However, it is worth noting that shops may not be concentrated in the centre because of the influence of relocation costs and other factors. In reality, the distribution of shop locations corresponds to a WPNE with strategy transformational barriers, which means a state of equilibrium under weaker conditions.Suppose that the strategy transformational barrier functions areVc1,c2 and Vd1,d2, respectively. If Vc1,c2 and Vd1,d2 are
(9)Vc1,c2=α1c1−c2+β1,∀c1,c2∈0,1,Vd1,d2=α2d1−d2+β2,∀d1,d2∈0,1.Setting businessmen 1 takingd=1/2, c1=1/4, c2=3/4, and c3=3/8, we have
(10)f1c1,d=f114,12=38,f1c2,d=f134,12=38,f1c3,d=f138,12=716.Ifα1=1/3, β1=0, then
(11)f1c1,d=38≥f1c2,d−Vc1,c2=38−13c1−c2=524,i.e.,c1≥bc2,f1c2,d=38≥f1c3,d−Vc2,c3=716−13c2−c3=516,i.e.,c2≥bc3,but
(12)f1c1,d=38≤f1c3,d−Vc1,c3=716−13c1−c3=716−124=1948.Furthermore,(13)Vc1,c2=13c1−c2=13×14−34=16,Vc2,c3=13c2−c3=13×34−38=18,Vc1,c3=13c3−c1=13×38−14=124.SinceVc1,c2+Vc2,c3≰Vc1,c3, “c1≥dc3” has no negative subadditivity. Then, “c1≥dc3” does not hold; thus, the order relation “≥” is not satisfied to transitive.Remark 6.
When the strategy transformational barrier function does not have negative subadditivity, the order relationship “≥a−i” does not have transitivity. Furthermore, the game with a strategy transformational barrier function may not have a numerical payoff function since the strategy transformational barrier function often possesses subadditivity rather than negative subadditivity.


## 3. Existence

In this paragraph, the existence of the WPNE with the strategy transformational barriers of the GMLMFMOG is demonstrated.Lemma 7 (Kakutani-Fan-Glicksberg, see [6]).
Assume thatA is a NECCS of locally convex Hausdorff space F, S:A⟶2A is a set-valued mapping, ∀a∈A, Sa is a nonempty, convex, compact set, and Sa is usc on A. Then, there exists a∗∈A such that a∗∈Sa∗.Lemma 8 (see [17]).
Assume thatA is a nonempty subset of Hausdorff space F and Y:A⟶R+l is a vector value correspondence, where Y=χ1,⋯,χl. In that case, Y is R+l-continuous if χi∀i=1,⋯,l is continuous.Lemma 9 (see [28]).
Suppose thatA and B are two Hausdorff spaces and B is compact. If a set-valued correspondence S:A⟶2B is closed, then S is usc.Lemma 10 (see [29]).
Assume thatA and B are two NECCSs of locally convex Hausdorff space F and H, respectively. Y:A×B⟶R+l is continuous correspondence; W:B⟶2A is a continuous set-valued correspondence on B, ∀b∈B, Wb is not empty and compact subset of A, as well as Wb=a∈Wb:Ya′,b−Ya,b∉intR+l,∀a′∈Wb. Then, we obtain that Wb is a compact, nonempty set as well as W:B⟶2A is usc on B.Theorem 11 (Fort theorem, see [8]).
Suppose thatA and B are Hausdorff and metric spaces, respectively. Given a set-valued mapping S:A⟶2B is usc on A with nonempty compact value (briefly, usco), then there is a residual subset Q in A such that S is lsc on Q.Remark 12 (see [29]).
IfA is Baire space, then the residual set in A is dense.Theorem 13.
Suppose thatAii∈ℕ and Bjj∈M are two NECCSs of locally convex Hausdorff space Fi and Hj, respectively. If ℕ,M,A,B,V,P satisfies the following conditions.
(1)
∀i∈ℕ, Yi=χ1i,⋯,χli:Ai×A−i×Ui⟶R+l is R+l-continuous(2)
∀i∈ℕ, Vi:Ai×Ai⟶R+l is R+l-continuous, ∀a′i∈Ai, ai⟶Vai,ai′ is convex(3)
∀i∈ℕ, ∀a−i∈A−i, ai,ui⟶Yiai,a−i,ui is R+l- quasiconcave-like(4)
∀i∈ℕ, P:Ai×A−i⟶2B is continuous, and ∀a=ai,a−i∈A, Pai,a−i is a nonempty and compact subset of B(5)
∀a−i∈A−i, the set-valued correspondence ai⟶Pai,a−i is convex (i.e., ∀θ∈0,1, ai1,ai2∈Ai, Pθai1+1−θai2,a−i⊂θPai1,a−i+1−θPai2,a−i)
Then, the GMLMFMOG with strategy transformational barriers contains at least a pointa−i∗,a−i∗,ui∗∈Ai×A−i×Ui such that ui∗∈Pa−i∗,a−i∗,∀i∈ℕ, satisfying
(14)Yiai,a−i∗,ui−Yia−i∗,a−i∗,ui∗−Via−i∗,ai∉intR+l,∀ai,ui∈Ai×Pai,a−i∗.Proof.
∀i∈ℕ, the set-valued correspondence Ti:A−i×U−i⟶2Ai×Ui is defined, ∀a−i∈A−i, u−i∈U−i, we have
(15)Tia−i,u−i=ai∈Ai,ui∈Pai,a−iYiai′,a−i,vi−Yiai,a−i,ui−Viai,ai′∉intR+l,∀ai′,vi∈Ai×Pai′,a−i,where Ti is independent of ui∈U−i.
By Lemma7, we only need to prove that the set-valued mapping Ti is usc mapping with nonempty convex compact value.
(1)
Tia−i,u−i≠∅. Because Ai is compact and P is a continuous correspondence with compact value, Pwi,a−i:wi∈Ai,∀i∈ℕ is compact. Y is R+l-continuous from Lemma 8; then, ∀i=1,⋯,l, Yi is R+l-continuous and Vi is also R+l-continuous. Thus, Tia−i,u−i≠∅ from Lemma 7(2)
Tia−i,u−i is convex. ∀ai1,ui1∈Tia−i,u−i,ai2,ui2∈Tia−i,u−i, i.e., ai1∈Ai, ui1∈Pai1,a−i, ai2∈Ai, ui2∈Pai2,a−i, and ∀i∈ℕ, we obtain(16)Yiai′,a−i,νi−Yiai1,a−i,ui1−Viai1,ai′∉intR+l,Yiai′,a−i,νi−Yiai2,a−i,ui2−Viai2,ai′∉intR+l,∀ai′,νi∈Ai×Pai′,a−i,i.e., we have
(17)Yiai1,a−i,ui1∉Yiai′,a−i,νi−Viai1,ai′−intR+l,Yiai2,a−i,ui2∉Yiai′,a−i,νi−Viai2,ai′−intR+l,∀ai′,νi∈Ai×Pai′,a−i
SinceAi is convex, θai1+1−θai2∈Ai, ∀θ∈0,1, and ∀a−i∈A−i by Theorem 13 (5), we have θai1+1−θai2∈Pθai1+1−θai2,a−i⊂θPai1,a−i+1−θPai2,a−i.
Since∀a−i∈A−i, ai,ui⟶Yiai,a−i,ui is R+l- quasiconcave-like, and ∀a′i∈Aiai⟶Vai,a′i is convex, we obtain
(18)Yiθai1+1−θai2,a−i,θui1+1−θui2∉Yiai′,a−i,νi−Viθai1+1−θai2,ai′−intR+l,i.e.,
(19)Yiai′,a−i,νi−Yiθai1+1−θai2,a−i,θui1+1−θui2−Vθai1+1−θai2,ai′∉intR+l.
Thus,θai1+1−θai2,a−i,θui1+1−θui2∈Tia−i,u−i, Tia−i,u−i is convex.
(3)
Tia−i,u−i is a usc mapping. According to Lemma 9, we just need to verify that GraphTi is closed. Thus, we next demonstrate that the set-valued correspondence Ca−i=ai′,νi∈Ai×Ui:ai′∈Ai,νi∈Pai′,νi is continuous
Suppose thata−iα:α∈K is any net on Ai, and a−iα⟶a−i, ∀a′iα,νiα∈Ca−iα, a′iα,νiα⟶ai′,νi∈Ai×Ui. Because P is a usc mapping with compact value and a′iα⟶a′i, νiα∈Pa′α,a−i, νiα⟶νi from Theorem 16.17 in [32], we attain νi∈Pai′,a−i. Therefore, ai′,νi∈Ca−i, C is closed. Since Ai×Ui is compact from Lemma 9, C is usc on A−i.
Meanwhile, assume thata−iα:α∈K is any net on Ai, a−iα⟶a−i, ∀ai′,νi∈Ca−i, then a′i∈Ai, νi∈Pa′i,νi. For any α∈K, we set a′iα=a′i, since P is continuous, from Theorem 16.19 in [32] if there is some νiα∈Pa′iα,a−iα=Pai′,a−iα, νiα⟶νi, ai′,νiα∈Ca−iα, and ai′,νiα⟶ai′,νi hold. Thus, C is lsc on A−i.
Hence, we have proved thatC is continuous with compact values. Tia−i,u−i is compact and Ti is a usc mapping from Lemma 10. On the basis of the above proof, we know that Ti is a usco correspondence.
A set-valued correspondenceS:A×U⟶2A×U is defined, and ∀a,u∈A,U contains Sa,u=T1a−1,u−1×⋯×Tna−n,u−n⊂A×U.
BecauseA×U is a NECCS of locally convex Hausdorff space, S is a usco mapping and Lemma 7, if there is a∗,u∗∈A,U, then a∗,u∗∈Sa∗,u∗ holds. We obtain a−i∗,a−i∗,ui∗∈Tia−i∗,u−i∗,∀i∈ℕ. Consequently, there is a−i∗,a−i∗,ui∗∈Ai×A−i×Ui such that ∀i∈ℕ, ui∗∈Pa−i∗,a−i∗, Yiai′,a−i∗,ui−Yia−i∗,a−i∗,ui∗−Via−i∗,ai′∉intR+l,∀ai′,ui∈Ai×Pai′,a−i∗. This concludes the proof.Remark 14.
In this paper, the WPNE with strategy transformational barriers are more broadly concepts than the WPNE in literature [27] in practical life, which means that the player needs to consider the impact of other some factors, such as the cost of changing strategies. In particular, if the leaders have no transformational strategy barriers, then the WPNE can be considered as the WPNE with strategy transformational barriers.


## 4. Generic Stability

In this paragraph, we prove the generic stability of the WPNE with the strategy transformational barriers of the GMLMFMOG.LetAii∈ℕ and Bjj∈M be two NECCSs of Banach space F and H, respectively, and Ω=ϕ=Y1,⋯,Yn,V1,⋯,Vn,Pforanyi∈ℕ,Yi,Vi and P satisfy all conditions provided in Theorem 13.Forϕ1=Y11,⋯,Yn1,V11,⋯,Vn1,P1 and ϕ2=Y12,⋯,Yn2,V12,⋯,Vn2,P2∈Ω, the distance on Ω is defined as follows:
(20)ϖϕ1,ϕ2=supai,ui∈Ai×Ui∑i=1nYi1ai,a−i,ui−Yi2ai,a−i,ui+supai,ai′∈Ai×Ai∑i=1nVi1ai,ai′−Vi2ai,ai′+supai,a−i∈Ai×A−iHP1ai,a−i,P2ai,a−i,where HP1ai,a−i,P2ai,a−i is the Hausdorff distance between P1ai,a−i and P2ai,a−i on A.Theorem 15.
Ω,ϖ is a complete metric space.Proof.
It is easy to see thatΩ,ϖ serves as a metric space. Then, we just need to check that Ω,ϖ is complete.
Settingϕα=Y1α,⋯,Ynα,V1α,⋯,Vnα,Pα∈Ω, Y1α,⋯,Ynα,V1α,⋯,Vnα,Pα⟶Y1,⋯,Yn,V1,⋯,Vn,P, we need to prove ϕ=Y1,⋯,Yn,V1,⋯,Vn,P∈Ω.
(1)
Letϕα=Y1α,⋯,Ynα,V1α,⋯,Vnα,Pα be any Cauchy sequence in Ω. ∀ε>0, there is a positive whole number Nε such that ϖϕα,ϕα~<ε,∀α,α~≥Nε. On the one hand, ∀i∈ℕ, ε>0 and α~>0, when α~>α, supai,ui∈Ai×UiYiαai,a−i,ui−Yiai,a−i,ui<ε/3, thus supa′i,ui∈Ai×UiYiα~ai′,a−i,ui−Yiai′,a−i,ui<ε/3. We know that Yiα~ is R+l-continuous by means of Theorem 13 (1); then, there is δ>0, ∀ai,a′i∈A; when ai−a′i<δ, we obtain Yiα~ai,a−i,ui−Yiα~ai′,a−i,ui<ε/3. Similarly,
(21)Yiai,a−i,ui−Yiai′,a−i,ui=Yiai,a−i,ui−Yiα~ai,a−i,ui+Yiα~ai,a−i,ui−Yiα~ai′,a−i,ui+Yiα~ai′,a−i,ui−Yiai′,a−i,ui≤Yiai,a−i,ui−Yiα~ai,a−i,ui+Yiα~ai,a−i,ui−Yiα~ai′,a−i,ui+Yiα~ai′,a−i,ui−Yiai′,a−i,ui<ε3+ε3+ε3=ε
Thus,Yi is R+l-continuous on A, ∀ai,a′i∈Ai×Ai and Vi is also R+l-continuous by proving the same method on A. Meanwhile, ∀i∈ℕ and ε>0, there is a positive integer Nε and ∀α,α~≥Nε, we obtain
(22)supai,a−i∈Ai×A−iHPαai,a−i,Pα~ai,a−i<ε.
Then,∀i∈ℕ, there is P:Ai×A−i⟶2B such that limα~⟶∞Pα~ai,a−i=Pai,a−i, and ∀α≥Nε, we have
(23)supai,a−i∈Ai×A−iHPαai,a−i,Pai,a−i≤ε.
Since the set-valued correspondencePα is continuous on A, it is easy to know that P is continuous on A.
(2)
Sinceai,ui⟶Yiαai,a−i,ui is R+l- quasiconcave-like, ai⟶Vαai,a′i is convex, fixing ai1,ai2∈Ai and ui1,ui2,νi∈U−i, if ∀θ∈0,1, θui1+1−θui2∈Ui holds, then ∀i∈ℕ, θui1+1−θui2∈Pθai1+1−θai2,a−i, we have(24)Yiαθai1+1−θai2,a−i,θui1+1−θui2∉Yiαai′,a−i,νi−Viαθai1+1−θai2,ai′−intR+l
SinceYiαa,u⟶Yia,u,Viαa,a′⟶Via,a′α⟶∞, ∀a∈A, ∀u∈U and the strategy space is closed, we conclude that
(25)Yiθai1+1−θai2,a−i,θui1+1−θui2∉Yiai′,a−i,νi−Viθai1+1−θai2,ai′−intR+l.
This indicates that∀a−i∈A−i, ai,ui⟶Yiai,a−i,ui is R+l- quasiconcave-likeand ai⟶Vai,a′i is convex
(3)
Since∀a−i∈A−i,ai⟶Pαai,a−i is convex, ∀ai1,ai2∈Ai, θ∈0,1, and ε>0, we have
(26)Pαθai1+1−θai2,a−i+ε⊂θPαai1,a−i+1−θPαai2,a−i+ε
Whenα is sufficiently large number, we have
(27)Pθai1+1−θai2,a−i⊂Pαθai1+1−θai2,a−i+ε,θPαai1,a−i+1−θPαai2,a−i+ε⊂θPai1,a−i+1−θPa−i2,a−i+2ε.
Thus,(28)Pθai1+1−θai2,a−i⊂θPai1,a−i+1−θPai2,a−i+2ε.
We takeε⟶0 because ε is arbitrary, and we can obtain Pθai1+1−θai2,a−i⊂θPai1,a−i+1−θPai2,a−i. Hence, ∀a−i∈A−i,ai⟶Pai,a−i is convex on A. In conclusion, ϕ=Y1,⋯,Yn,V1,⋯,Vn,P∈Ω, and Ω,ϖ is a complete metric space.∀ϕ∈Ω, we define Γ:Ω⟶2A1×U1×⋯×An×Un, where Γϕ=a−i∗,a−i∗,ui∗∈Ai×A−i×Ui:∀i∈ℕ,ui∗∈Pa−i∗,a−i∗,Yia′i,a−i∗,ui−Yia−i∗,a−i∗,ui∗−Via−i∗,a′i∉intR+l,∀a′i,ui∈Ai×Pa′i,a−i∗. By Theorem 13, there is a−i∗,a−i∗,ui∗∈Ai×A−i×Ui such that Γϕ≠∅. Then, Γ is also called an equilibrium mapping.Next, we denote to verify the generic stability result of the WPNE with the strategy transformational barriers of the GMLMFMOG.Lemma 16.
An equilibrium mappingΓ:Ω⟶2A1×U1×⋯×An×Un is a usco correspondence.Proof.
By means of the compactness ofA and Lemma 9, we need to demonstrate that the Γ is closed. In other words, if ∀ϕβ=Y1β,⋯,Ynβ,V1β,⋯,Vnβ,Pβ∈Ω, ϕβ⟶ϕ=Y1,⋯,Yn,V1,⋯,Vn,P, ∀a1β,u1β,⋯,anβ,unβ∈Γϕβ, a1β,u1β,⋯,anβ,unβ⟶a1∗,u1∗,⋯,an∗,un∗, then we only need to prove
(29)a1∗,u1∗,⋯,an∗,un∗∈Γϕ.(1)
SinceAi is compact, we assume that aiβ⟶a−i∗∈Ai, P is continuous, Paiβ,a−iβ⟶Pa−i∗,a−i∗, uiβ∈Pβaiβ,a−iβ. Let d be the distance on Ui; since ϕβ⟶ϕ, Pβ⟶P, and uiβ⟶ui∗, we have dui∗,Pa−i∗,a−i∗≤dui∗,uiβ+duiβ,Pβaiβ,a−iβ+HPβaiβ,a−iβ,Paiβ,a−iβ+HPaiβ,a−iβ,Pa−i∗,a−i∗⟶0. Thus, ui∗∈Pa−i∗,a−i∗,∀i∈ℕ(2)
We verify that∀i∈ℕ, ui∗∈Pa−i∗,a−i∗, and we have(30)Yiai′,a−i∗,ui−Yia−i∗,a−i∗,ui∗−Via−i∗,ai′∉intR+l,∀ai′,ui∈Ai×Pai′,a−i∗
By contradiction, suppose that formula (30) is not true, then there is some i∈ℕ such that ai′,ui∈Ai×Pai′,a−i∗, Yiai′,a−i∗,ui−Yia−i∗,a−i∗,ui∗−Via−i∗,ai′∈intR+l. Therefore, there exists some open neighbourhood V of the 0 element of R+l satisfying
(31)Yiai′,a−i∗,ui−Yia−i∗,a−i∗,ui∗−Via−i∗,ai′+V⊂intR+l.
BecauseYiβ⟶Yi, there is a positive integer β1 such that ∀β≥β1,
(32)Yiβai,a−iβ,νi−Yiβaiβ,a−iβ,uiβ−Viβaiβ,ai−Yiai,a−iβ,νi−Yiaiβ,a−iβ,uiβ−Viβaiβ,ai∈12V.
Furthermore, sinceai′,ui∈Ai×Pai′,a−i∗, Yiai′,a−i∗,ui−Yia−i∗,a−i∗,ui∗−Va−i∗,ai′ is lsc at ai,a−i,ui with aiβ,a−iβ⟶ai∗,a−i∗, there is a positive integer β2 and β2≥β1 such that ∀β≥β2,
(33)Yiai,a−iβ,ui−Yiaiβ,a−iβ,uiβ−Viaiβ,ai∈Yiai,a−i∗,ui−Yiai∗,a−i∗,ui∗−Viai∗,ai+12V+R+l.
Then,∀β≥β2, and we can obtain that
(34)Yiβai,a−iβ,ui−Yiβaiβ,a−iβ,uiβ−Viβaiβ,ai=Yiβai,a−iβ,ui−Yiβaiβ,a−iβ,uiβ−Viβaiβ,ai−Yiai,a−iβ,ui−Yiaiβ,a−iβ,uiβ−Viaiβ,ai+Yiai,a−iβ,ui−Yiaiβ,a−iβ,uiβ−Viaiβ,ai∈12V+Yiai,a−i∗,ui−Yiai∗,a−i∗,ui∗−Viai∗,ai+12V+R+l=Yiai,a−i∗,ui−Yiai∗,a−i∗,ui∗−Viai∗,ai+V+R+l⊂intR+l+R+l⊂intR+l.
It is a contradiction witha1β,u1β,⋯,anβ,unβ∈Γϕβ. Thus, we can obtain a1∗,u1∗,⋯,an∗,un∗∈Γϕ; i.e., Γ is a closed correspondence and Γ is a usco correspondence on Ω by means of Lemma 9.
Next, we define a set-valued mapT:A1×U1×⋯×An×Un⟶A1×⋯×An, wherein Ta1,u1,⋯,an,un=a1,⋯,an∈A1×⋯×An, ∀a1,u1,⋯,an,un∈A1×U1×⋯×An×Un. It is obvious that T is continuous on A1×U1×⋯×An×Un.
Finally, we define a set-valued mappingF=TΓ:Ω⟶2A1×⋯×An, where ∀ϕ∈Ω, Fϕ=TΓϕ represents the set of WPNE with strategy transformational barriers for the GMLMFMOG. According to Theorem 13, Γϕ≠∅, then Fϕ=TΓϕ≠∅.Lemma 17.
A set-valued mappingF=TΓ:Ω⟶2A1×⋯×An is a usco correspondence.Proof.
According to Lemma16, Γ:Ω⟶2A1×U1×⋯×An×Un is usc on Ω, and Γϕ is compact ∀ϕ∈Ω. Since T is continuous on A1×U1×⋯×An×Un, it is obvious to check that F=TΓ:Ω⟶2A1×⋯×An is also a usco correspondence on Ω.Definition 18.
(1)
An equilibrium pointa∈A of the game ϕ∈Ω is referred to essential if for every Oa of a, there is one Oϕ of ϕ such that ∀ϕ′∈Oϕ, and there exists at least an equilibrium point a′ of ϕ′ with a′∈Oa. If all equilibria points of the game ϕ∈Ω are essential, then the game ϕ is an essential game(2)
A setm~ϕ of the game ϕ∈Ω is referred to essential set if for each open set O of A is associated with m~ϕ⊂O, and there is an ε>0 satisfying ∀ϕ′∈Ω, ϖϕ,ϕ′<ε, and Fϕ′∩O=∅. Given that m~ϕ is one minimal element in total essential sets of Fϕ which are ordered by inclusion relations, then m~ϕ is a minimal essential set(3)
∀ϕ∈Ω, Fϕ is composed of the union of the pairing of disjoint connected subsets [33], i.e.,(35)Fϕ=⋃κ∈KCκϕ,wherein K signifies one index set. Given a component Cκϕ of Fϕ is essential, then Cκϕ is one essential setTheorem 19.
∀ϕ∈Ω, there is a dense Q in Ω such that Ω is essential.Proof.
Ω,ϖ is complete by using Theorem 15, and F:Ω⟶2A1×⋯×An is a usco correspondence by means of Lemma 17. By Theorem 11 and Remark 12, F serves as lsc on one dense Q of Ω such that Ω is essential.Remark 20.
By Theorem19, we proved that most of ϕ∈Ω have a stable solution set in the dense Q of Ω on the meaning of Baire’s category.


## 5. Essential Component

In this paragraph, we derive the essential component results of the WPNE with the strategy transformation barrier solution sets of the GMLMFMOG.Theorem 21.
Fϕ encompasses at least one minimal essential set ∀ϕ∈Ω, where F:Ω⟶2A1×⋯×An.Proof.
Forϕ∈Ω, F:Ω⟶2A1×⋯×An is usco mapping by Lemma 17, and then, Fϕ is one essential set of itself. Suppose that E is the collections of all essential sets of Fϕ, which is defined by the set inclusion order relation, we obtain E≠∅. Assume that any total order subset be eγϕ:γ∈K on A, where K denote the index set. Let eϕ=⋂γ∈Keγϕ, then eϕ serves as compact. If eϕ=∅, then Fϕ=Fϕ\eϕ=⋃γ∈KFϕ\eγϕ. Note that Fϕ\eγϕ is one open set as well as Fϕ is compact, then there are e1ϕ,e2ϕ,⋯,enϕ such that Fϕ=⋃i=1nFϕ\eiϕ by using the open covering theorem. It is obvious that ⋂i=1neiϕ=∅ from Fϕ=⋃i=1nFϕ\eiϕ=Fϕ\⋂i=1neiϕ. It means that ⋂i=1neiϕ=∅ is in contradiction with ⋂i=1neiϕ≠∅. Thus, eϕ≠∅. Given any open set O with eϕ⊂O, if ∀γ∈K, there exists aγ∈eγϕ⊂Fϕ with aγ∉O; then, we can assume that aγ⟶a∈Fϕ. Because ∀γ∈K,eγϕ is compact and eγϕγ∈K is totally order set, then aγ1∈eγϕ when γ1>γ and a∈eγϕ,∀γ∈K. Hence, a∈⋂γ∈Keγϕ=eϕ⊂O, which contradicts with aγ⟶a and aγ∉O,∀γ∈K. Therefore, there exists aγ0∈K such that eγ0ϕ⊂O. Since eγ0ϕ is an essential set of Fϕ, ∀ε>0, there is δ>0 such that ϕ1∈Ω with ϖϕ,ϕ1<δ, Fa′⋂O≠∅ with a−a′<ε, ∀a′∈Fϕ1. Thus, eϕ is essential, and there must be a lower bound of eγϕ:γ∈K in E. According to Zorn’s lemma, there is one minimal element m~ϕ in E such that Fϕ includes at least one minimal essential set m~ϕ.Theorem 22.
∀ϕ∈Ω, each minimal essential set of Fϕ is connected.Proof.
Letm~ϕ be a minimum essential set of Fϕ. By contradiction, we assume that m~ϕ is disconnected. There are two not empty closed sets c~1ϕ and c~2ϕ with m~ϕ=c~1ϕ∩c~2ϕ, as well as two disjoint open sets O1 and O2 with O1∩O2=∅ such that c~1ϕ⊂O1 and c~2ϕ⊂O2.
Sincem~ϕ is the minimum essential set, c~1ϕ and c~2ϕ are not essential set for Fϕ. Therefore, there exist two open sets, namely, D1 and D2, with c~1ϕ⊂D1 and c~2ϕ⊂D2 such that ∀δ>0, ϕ1,ϕ2∈Ω; we obtain ϖϕ,ϕ1<δ and ϖϕ,ϕ2<δ, but Fϕ∩D1=∅, Fϕ⋂D2=∅. Suppose that U1=O1∩D1 and U2=O2∩D2 are open sets and that c~1ϕ⊂U1 and c~2ϕ⊂U2. Beacuse c~1ϕ and c~2ϕ are compact, there are two open sets, namely, Z1 and Z2, such that c~1ϕ⊂Z1⊂Z¯1⊂U1 and c~2ϕ⊂Z2⊂Z¯2⊂U2. Since m~ϕ is one essential set of Fϕ and m~ϕ⊂Z1∪Z2, there is δ′>0 such that ∀ϕ¯∈Ω with ϖϕ,ϕ¯<δ′, and
(36)Fϕ¯∩Z1∪Z2≠∅.
SinceZ1⊂O1 and Z2⊂O2, there exist ψ1∈Ω and ψ2∈Ω such that ϖϕ,ψ1<δ′/2 and ϖϕ,ψ2<δ′/2 with Fψ1∩Z1=∅ and Fψ2∩Z2=∅.
We define a GMLMFMOG with strategy transformational barrierψ=Ai3,Yi3,Vi3,P3i∈ℕ by a linear combination function between ψ1=Ai1,Yi1,Vi1,P1i∈ℕ and ψ1=Ai2,Yi2,Vi2,P2i∈ℕ as follows:
(37)Yi3a,u=vaYi1a,u+uaYi2a,u,Vi3ai,ai′=vaVi1ai,ai′+uaVi2ai,ai′,P3ai,a−i=vaHP1ai,a−i,P2ai,a−i+uaHP2ai,a−i,P2ai,a−i,where
(38)va=ℏa,Z¯2ℏa,Z¯1+ℏa,Z¯2,ua=ℏa,Z¯1ℏa,Z¯1+ℏa,Z¯2,and ℏ represents the distance function on A. Note that va and ua are continuous and nonnegative; furthermore, va+ua=1,∀a∈A.
We can check thatψ=Ai3,Yi3,Vi3,P3i∈ℕ∈Ω. Noting that
(39)ϖϕ,ψ=supa,ui∈A×Ui∑i=1nYia,ui−Yi3a,ui+supai,ai′∈Ai×Ai∑i=1nViai,ai′−Vi3ai,ai′+supai,a−i∈Ai×A−iHPai,a−i,P3ai,a−i,=supa,ui∈A×Ui∑i=1nvaYia,ui+uaYia,ui−vaYi1a,ui−uaYi2a,ui+supai,ai′∈Ai×Ai∑i=1nvaViai,ai′+uaViai,ai′−vaVi1ai,ai′−uaVi2ai,ai′+supai,a−i∈Ai×A−ivaHPai,a−i,P1ai,a−i+uaHPai,a−i,P2ai,a−i≤ϖw,ψ1+ϖw,ψ2<δ′2+δ′2=δ′,we obtain Fψ∩Z1∪Z2≠∅ since ϖϕ,ψ<δ′. Next, we assume that Fψ∩Z1≠∅; then there exists a′∈Fψ∩Z1. By a′∈Z1, we attain wa′=1, ua′=0, Yi3a,ui=Yi1a,ui, Vi3ai,a′i=Vi1ai,a′i, and P3ai,a−i=P1ai,a−i. Then, we obtain a′∈Fψ, which implies
(40)Yi3ai′,a−i∗,ui−Yi3a−i∗,a−i∗,ui∗−Via−i∗,ai′∉intR+l,∀ai′,ui∈Ai×Pai′,a−i∗.
Thus,a′∈Fψ1. This contradicts the fact that Fψ1∩Z1=∅. Then, m~ϕ is connected.Theorem 23.
∀ϕ∈Ω, if there exists Fϕ=a (single point set), then ϕ is essential.Theorem 24.
∀ϕ∈Ω, there is at least an essential connected component of Fϕ.Proof.
According to Theorems21 and 22, Fϕ encompasses at least a minimum essential set m~ϕ and m~ϕ is connected. Aiming at a component Cκϕ of Fϕ as well as m~ϕ⊂Cκϕ, we obtain that Cκϕ is one essential connected component of Fϕ by Definition 18 (3).


## 6. Summaries

In this paper, we have investigated a new generalized multileader multifollower multiple objective game (GMLMFMOG) model with strategy transformational barriers and obtained some new stability results of the WPNE with the strategy transformational barriers for the GMLMFMOG. Furthermore, we have proved the existence of the WPNE with the strategy transformational barriers of the GMLMFMOG and studied its generic stability. In fact, we have obtained that most of the WPNE with the strategy transformational barriers of the GMLMFMOG serve as essential on the meaning of Baire’s category. In addition, we have demonstrated that there is at least an essential connected component of the GMLMFMOG with the strategy transformational barriers. These results extend the corresponding results obtained in reference [27] by introducing strategy transformational barrier function into the decision-making behaviour of players.



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*Source: 1689732-2022-10-06.xml*