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Ordering stability of Nash equilibria for a class of differential games

  • Keke Jia , Shihuang Hong EMAIL logo and Jieqing Yue
Published/Copyright: November 21, 2023
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Open Mathematics
From the journal Open Mathematics Volume 21 Issue 1

Abstract

This study is concerned with the stability of Nash equilibria for a class of n -person noncooperative differential games. More precisely, due to a preorder induced by a convex cone on a real linear normed space, we define a new concept called ordering stability of equilibria against the perturbation of the right-hand side functions of state equations for the differential game. Moreover, using the set-valued analysis theory, we present the sufficient conditions of the ordering stability for such differential games.

Keywords: differential games; order relation; stability; Nash equilibrium; set-valued mapping
MSC 2010: 49L20; 49N70; 49N90; 91A23

1 Introduction

Differential games provide a natural extension of the standard model of control theory to the case where two or more individuals are present, and each one of them seeks to minimize his own payoff. Differential games well model the optimal decision-making problems in economics and ecology, which include agents interacting with the system and so are complex in nature. The theory of differential games was first developed by Isaacs [1]. Since then, it has gained wide attention in academia and a large number of relevant literature has been published (see for example, [2–10] and references therein).

In the case of n -person games, one usually considers a system whose state x in a finite dimensional space evolves according to some ordinary differential equation (state equation). Therefore, Nash equilibria of differential games rely on the right-hand side function f , which determines the state equation. Some natural questions are what are the impacts on the Nash equilibrium if the right-hand side function of state equations changes slightly? How will the set of Nash equilibria change? These problems are all concerned with the stability of Nash equilibria for differential games with respect to the right-hand side functions. To this end, Wu and Jiang [11] put forward the concept of essential equilibria for n -person noncooperative games, which is widely applied to study the stability of Nash equilibria in various games including differential games with respect to linear and nonlinear state functions (see, for instance, [9,12–28]). For instance, by means of considering some proper conditions and using the set-valued analysis theory, Yu and Peng [9] studied the stability of equilibria against the perturbation of the right-hand side functions for noncooperative differential games and further characterized that the differential games whose equilibria are all stable form a dense residual set, and so every differential game can be approximated arbitrarily by a sequence of essential differential games. For other types of stability for differential games, we mention that in [10] the author established the characterization of the infinitesimal forms of u - and v -stability properties for the deterministic mean field-type differential game expressed in terms of directional derivatives.

In this article, we are concerned with the stability of Nash equilibria for n -person noncooperative differential games. We first equip with a preorder induced by a convex cone on a real linear normed space. For the order relation and its properties on the power set 2 Y of Y , we refer to [29,30]. Due to the preorder, we define a new concept called ordering stability of equilibria against the perturbation of the right-hand side functions of state equations for the differential game, which is a more strict concept in the sense of “stability” relative to the existing literature. Moreover, using the set-valued analysis theory, we present the sufficient conditions of the ordering stability for such a differential game. Our results reveal that the Nash equilibria of the differential games are all stable under certain conditions relative to the literature [9]. In particular, our results essentially extend and improve the corresponding results in [9] and other existing works. Finally, some examples are given to illustrate the applicability of our results.

This article is organized as follows. In Section 2, we recall the general notation used in this study, including the corresponding concept of the n -person noncooperative differential game, introducing the preorder and certain properties of the set-valued functions. The main result of this article is presented in Section 3. The ordering stability for the Nash equilibria and the sufficient conditions of ordering stability to the differential games are defined. Finally, several examples are presented to illustrate the applicability of our results.

2 Preliminaries

We always denote { 1 , 2 , 3 , … } by N * , the m -dimensional real space by R m and the set of all real numbers by R if m = 1 . Furthermore, let C ( A , B ) stand for the set consisting of all continuous functions from a set A into a set B . For fixed t 0 , T ∈ R with t 0 < T , in the model control theory [31], the state of a system is described by a function x : [ t 0 , T ] → R m and the control is described by a vector-valued function u : [ t 0 , T ] → R n for m , n ∈ N * . The state evolves over time and is affected by the control function. The evolution of the system is determined by the following ordinary differential equation:

(1) d x d t = f ( t , x ( t ) , u ( t ) ) , t ∈ [ t 0 , T ] , x ( t 0 ) = x 0 ,

where the right-hand side function f : ( [ t 0 , T ] , R m , R n ) → R m is a vector-valued function and x 0 ∈ R m .

The Bolza optimal control problem is to find a vector-valued control function u * : [ t 0 , T ] → R n that minimizes the payoff:

(2) J f ( u ) ≔ h ( x ( T ) ) + ∫ t 0 T g ( t , x ( t ) , u ( t ) ) d t ,

where h : R m → R is a terminal payoff and g : ( [ t 0 , T ] , R m , R n ) → R is a running cost, i.e.,

(3) h ( x * ( T ) ) + ∫ t 0 T g ( t , x * ( t ) , u * ( t ) ) d t = min u ∈ U J f ( u ) ≔ h ( x ( T ) ) + ∫ t 0 T g ( t , x ( t ) , u ( t ) ) d t ,

where U is a set of admissible control functions and x * is a solution of (1) under the optimal control u * .

This study deals with a two-player noncooperative differential game, denoted by Γ f , by applying the control theory in which the control system is the following differential equation:

(4) d x d t = f ( t , x ( t ) , u ( t ) , v ( t ) ) , t ∈ [ t 0 , T ] , x ( t 0 ) = x 0 ,

where vector-valued functions u : [ t 0 , T ] → U , v : [ t 0 , T ] → V are the control functions of the two players and U = ( U , ‖ ⋅ ‖ u ) , V = ( V , ‖ ⋅ ‖ v ) stand for the normed linear spaces endowed with the norm ‖ ⋅ ‖ u , ‖ ⋅ ‖ v , respectively. The goal of the ith player is to minimize its payoff:

(5) J i f ( u , v ) ≔ h i ( x ( T ) ) + ∫ t 0 T g i ( t , x ( t ) , u ( t ) , v ( t ) ) d t ,

where x : [ t 0 , T ] → R m is determined by equation (4) and h i and g i are a terminal payoff and a running cost of the ith player, respectively. More precisely, we consider the stability of Nash equilibria for Γ f . For the sake of convenience, we need the following hypotheses and background materials. Let

U = { u ∣ u ∈ C ( [ t 0 , T ] , U ) } , V = { v ∣ v ∈ C ( [ t 0 , T ] , V ) }

be endowed the norm:

‖ u ‖ = max t 0 ≤ t ≤ T ‖ u ( t ) ‖ u and ‖ v ‖ = max t 0 ≤ t ≤ T ‖ v ( t ) ‖ v ,

for any u ∈ U and v ∈ V . In addition, denote U = U × V . For each ( u 1 , v 1 ) , ( u 2 , v 2 ) ∈ U , we define the distance as

d ( ( u 1 , v 1 ) , ( u 2 , v 2 ) ) ≔ ‖ u 1 − u 2 ‖ + ‖ v 1 − v 2 ‖ .

The following notations will also be used.

A ≔ { ( t , x , u , v ) ∈ R 1 + m × U × V : t ∈ [ t 0 , T ] , x ∈ R m , u ∈ U , v ∈ V } , D ≔ { f ∣ f = ( f 1 , f 2 , … , f m ) : A → R m and f i ∈ C ( A , R ) for i = 1 , 2 , … , m } .

We define the metric on D as, for each f 1 , f 2 ∈ D ,

ρ ( f 1 , f 2 ) ≔ sup ( t , x , u , v ) ∈ A ∥ f 1 ( t , x , u , v ) − f 2 ( t , x , u , v ) ∥ .

Let ( X , ‖ ⋅ ‖ ) be a normed vector space, 2 X represents the power set of X (i.e., the family of all nonempty subsets of X ); B ( θ , r ) represents a ball in X with the radius of r centered around the null element θ , and B X = B ( θ , 1 ) denotes the unit closed ball in X . In this study, unless specifically stated, by C , we always mean that C ⊂ X is a solid cone of X , i.e., θ ∈ C ≠ X , C is closed convex, C ∩ ( − C ) = { θ } , C + C ⊂ C , λ C ⊂ C for every λ ≥ 0 , and the interior of C denoted by int C in X is nonempty. Moreover, assume that ≤ c is a partial order relation induced by C on X as follows: for any x , y ∈ X ,

x ≤ c y ⇔ y − x ∈ C .

An element X 1 ∈ 2 X is called C -closed if X 1 + C is closed. Next, we introduce a relation.

Definition 2.1

[30] For any X , Y ∈ 2 X , the lower set less order relation ≤ l on 2 X is defined by:

X ≤ l Y ⇔ Y ⊂ X + C .

Moreover, the strict lower set less order relation < l on 2 X is defined via

X < l Y ⇔ ∃ η ∈ C \ { θ } , X + η ≤ l Y .

Remark 2.1

It is clear that the relation ≤ l is a preorder on 2 X , i.e., ≤ l is reflexive and transitive. In addition, for subsets X , Y of X , X ≤ l Y if and only if for any y ∈ Y , there exists x ∈ X such that x ≤ c y . < l is nonreflexive, namely, X ≮ l X for X ∈ 2 X (see, for instance, [32]).

From now on, we will adopt the following notation:

J ˆ i f ( v ) = J i f ( U , v ) = { J i f ( u , v ) : u ∈ U } , J ˜ i f ( u ) = J i f ( u , V ) = { J i f ( u , v ) : v ∈ V } ,

and J i f ( u , v ) sometimes also means a single point set { J i f ( u , v ) } for i = 1 , 2 .

Definition 2.2

A pair ( u * , v * ) ∈ U is said to be

  1. A Nash equilibrium of Γ f if J 1 f ( U , v * ) ≤ l J 1 f ( u * , v * ) and J 2 f ( u * , V ) ≤ l J 2 f ( u * , v * ) imply J 1 f ( u * , v * ) ≤ l J 1 f ( U , v * ) and J 2 f ( u * , v * ) ≤ l J 2 f ( u * , V ) ;

  2. A weakly Nash equilibrium of Γ f if J 1 f ( U , v * ) ≰ l J 1 f ( u * , v * ) and J 2 f ( u * , V ) ≰ l J 2 f ( u * , v * ) .

It is worth noting that for some f ∈ D , a Nash equilibrium may not exist. To this end, we consider the subset of D as follows:

G = { f ∈ D : there exists at least one Nash equilibrium of Γ f related to function f } .

Definition 2.3

[33] Y 1 ⊂ Y is a nonempty subset. A set-valued mapping Φ : Y 1 → 2 X is said to be locally order-Lipschitz continuous (LOLC) at y 0 ∈ Y 1 if there exist a constant L > 0 and a neighborhood U ( y 0 ) of y 0 such that

Φ ( y 2 ) + L ‖ y 1 − y 2 ‖ B X ≤ l Φ ( y 1 ) , ∀ y 1 , y 2 ∈ U ( y 0 ) ∩ Y 1 .

If Φ is LOLC at every y 0 ∈ Y 1 , then we say that Φ is LOLC on Y 1 .

Furthermore, the set-valued mapping Φ : Y 1 → 2 X is said to be C -closed at y 0 ∈ Y 1 if Φ ( y 0 ) + C is closed and we usually say that Φ has C -closed values at y 0 , in special, C R -closed values at y 0 if C is a solid cone of R .

Definition 2.4

A set-valued mapping Φ : Y → 2 X is said to be

  1. Upper semicontinuous (USC) at y ∈ Y if for each open set W in X with Φ ( y ) ⊂ W , there exists a neighborhood U ( y ) of y in Y such that

    Φ ( y ¯ ) ⊂ W , ∀ y ¯ ∈ U ( y ) ;

  2. Lower semicontinuous (LSC) at y ∈ Y if for each open set W in X with Φ ( y ) ∩ W ≠ ∅ , there exists a neighborhood U ( y ) of y in Y such that

    Φ ( y ¯ ) ∩ W ≠ ∅ , ∀ y ¯ ∈ U ( y ) .

Φ is said to be USC (LSC) on Y if it is USC (LSC) at every y ∈ Y . Φ is called continuous on Y if it is both USC and LSC on Y .

Remark 2.2

Regarding the details of the set-valued mapping, we refer to [34, 35]. If Φ ( y ) is compact, then Φ is USC at y ∈ Y iff for any sequence { y n } ⊂ Y , y n → y and for any z n ∈ Φ ( y n ) , there are z ∈ Φ ( y ) and subsequence { z n k } ⊂ { z n } such that z n k → z .

Φ is LSC at y ∈ Y iff for any sequence { y n } ⊂ Y with y n → y and z ∈ Φ ( y ) , there exists z n ∈ Φ ( y n ) for n = 1 , 2 , … such that z n → z .

Definition 2.5

Let V be a nonempty convex set. A set-valued mapping Φ : V → 2 R is said to be (strictly) quasiconvex on V if for any u 1 , u 2 ∈ V with u 1 ≠ u 2 and for any λ ∈ ( 0 , 1 ) :

Φ ( u 2 ) ≤ l Φ ( u 1 ) ⇒ Φ ( λ u 2 + ( 1 − λ ) u 1 ) ≤ l ( < l ) Φ ( u 1 ) .

Lemma 2.1

(Ky Fan’s Lemma [36]) Let X be a normed vector space and X 1 ⊂ X . If the set-valued mapping Φ : X 1 → X satisfies the following conditions:

  1. c o { x 1 , x 2 , … , x n } ⊂ ∪ i = 1 n Φ ( x i ) , where c o A stands for the convex hull of the subset A of X ;

  2. Φ ( x ) is closed for every x ∈ X 1 ;

  3. There exists x 0 ∈ X 1 such that Φ ( x 0 ) is compact.

Then, ∩ x ∈ X 1 Φ ( x ) ≠ ∅ .

3 Main results

In this section, we always assume that G is nonempty. For each f ∈ G , S ( f ) represents the set of all Nash equilibria of Γ f related to f in (4); from the definition of G , we have S ( f ) ≠ ∅ . Furthermore, S : G → U is a set-valued mapping. Next, we will study some stability of S ( f ) . For this purpose, let us give the definition of the ordering stability.

Definition 3.1

Let f ∈ G . S ( f ) is called ordering stable if it satisfies the following:

  1. Left-stability, i.e., for any ε > 0 , there exists δ > 0 such that S ( f ) ⊂ S ( f ′ ) + B ( θ , ε ) ∩ C for any f ′ ∈ G with ρ ( f ′ , f ) < δ ;

  2. right-stability, i.e., for any ε > 0 , there exists δ > 0 such that S ( f ′ ) ⊂ S ( f ) + B ( θ , ε ) ∩ C for any f ′ ∈ G with ρ ( f ′ , f ) < δ .

Γ f is called ordering stable iff each nonempty S ( f ) is ordering stable.

Before stating and proving our main results, we need the following notation. A lower-level mapping Ψ f : U → 2 U is defined by:

(6) Ψ f ( u ¯ , v ¯ ) = { ( u , v ) ∈ S ( f ) : J ˆ i f ( v ) ≤ l J ˆ i f ( v ¯ ) and J ˜ i f ( u ) ≤ l J ˜ i f ( u ¯ ) for i = 1 , 2 } ,

for all ( u ¯ , v ¯ ) ∈ U and given f ∈ G . It is easy to see that Ψ f ( u ¯ , v ¯ ) ≠ ∅ for all ( u ¯ , v ¯ ) ∈ U . In addition, the set-valued mapping Ψ : U × G → 2 U is given by:

Ψ ( ( u , v ) , f ) = Ψ f ( u , v ) .

Furthermore, for given f ∈ G and a positive constant δ , define the following set-valued mappings from G into 2 G :

Λ δ , f l ( f ′ ) = { g ∈ G : ρ ( f , g ) ≤ ρ ( f , f ′ ) < δ , S ( g ) ≤ l S ( f ) , S ( f ′ ) ≤ l S ( f ) } , for all f ′ ∈ G , Λ δ , f r ( f ′ ) = { g ∈ G : ρ ( f , g ) ≤ ρ ( f , f ′ ) < δ , S ( f ) ≤ l S ( g ) , S ( f ) ≤ l S ( f ′ ) } , for all f ′ ∈ G .

It is obvious that f ∈ Λ δ , f l ( f ′ ) ( Λ δ , f r ( f ′ ) ) , i.e., Λ δ , f l ( f ′ )   ( Λ δ , f r ( f ′ ) ) ≠ ∅ , for all f ′ ∈ G .

Lemma 3.1

For any given f ∈ G , let the set-valued mappings J ˆ i f : V → 2 R and J ˜ i f : U → 2 R be strictly quasiconvex on S ( f ) for i = 1 , 2 . If S ( f ) is convex, then for each ( u ¯ , v ¯ ) ∈ S ( f ) , we have

Ψ f ( u ¯ , v ¯ ) = { ( u ¯ , v ¯ ) } .

Proof

For ( u ¯ , v ¯ ) ∈ S ( f ) , we assert that J ˆ i f ( v ) ≮ l J ˆ i f ( v ¯ ) and J ˜ i f ( u ) ≮ l J ˜ i f ( u ¯ ) for all ( u , v ) ∈ S ( f ) and i = 1 , 2 . Suppose that the assertion is not true. Then, say, for i = 1 , there exists ( u 0 , v 0 ) ∈ S ( f ) such that J ˆ 1 f ( v 0 ) < l J ˆ 1 f ( v ¯ ) , i.e., J ˆ 1 f ( v 0 ) ≤ l J ˆ 1 f ( v ¯ ) and J ˆ 1 f ( v 0 ) ≠ J ˆ 1 f ( v ¯ ) . On the other hand, from ( u ¯ , v ¯ ) ∈ S ( f ) , we have J ˆ 1 f ( v ¯ ) ≤ l J ˆ 1 f ( v 0 ) . This implies that J ˆ 1 f ( v ¯ ) < l J ˆ 1 f ( v ¯ ) , which is a contradiction.

Let ( u , v ) ∈ Ψ f ( u ¯ , v ¯ ) with ( u , v ) ≠ ( u ¯ , v ¯ ) . Clearly, ( u ¯ , v ¯ ) ∈ Ψ f ( u ¯ , v ¯ ) and

J ˆ i f ( v ) ≤ l J ˆ i f ( v ¯ ) , J ˜ i f ( u ) ≤ l J ˜ i f ( u ¯ ) , i = 1 , 2 .

Note that J ˆ i f and J ˜ i f are strictly quasiconvex on S ( f ) for i = 1 , 2 , we have

J ˆ i f ( t v + ( 1 − t ) v ¯ ) < l J ˆ i f ( v ¯ ) and J ˜ i f ( t u + ( 1 − t ) u ¯ ) < l J ˜ i f ( u ¯ ) ,

for i = 1 , 2 and t ∈ ( 0 , 1 ) . This contradicts our assertion since S ( f ) is convex. Consequently, Ψ f ( u ¯ , v ¯ ) = { ( u ¯ , v ¯ ) } .□

Lemma 3.2

Let sequences { ( u ¯ n , v ¯ n ) } , { ( u ˜ n , v ˜ n ) } ⊂ U , and { f n } ⊂ G with ( u ¯ n , v ¯ n ) → ( u ¯ , v ¯ ) , ( u ˜ n , v ˜ n ) → ( u ˜ , v ˜ ) , and f n → f ∈ G for n → ∞ . Suppose that the following conditions are satisfied:

  1. The set-valued mapping J ˆ i f : V → 2 R and J ˜ i f : U → 2 R for i = 1 , 2 are LOLC at v ¯ , v ˜ ∈ V and u ¯ , u ˜ ∈ U , respectively;

  2. J ˆ i f and J ˜ i f for i = 1 , 2 have C R -closed values at v ¯ ∈ V and u ¯ ∈ U , respectively.

Then, J ˆ i and J ˜ i are order preserving, namely, if J ˆ i f n ( v ¯ n ) ≤ l J ˆ i f n ( v ˜ n ) and J ˜ i f n ( u ¯ n ) ≤ l J ˜ i f n ( u ˜ n ) ( i = 1 , 2 ) for n sufficiently large, then

J ˆ i f ( v ¯ ) ≤ l J ˆ i f ( v ˜ ) a n d J ˜ i f ( u ¯ ) ≤ l J ˜ i f ( u ˜ ) , i = 1 , 2 .

Proof

From the property of LOLC, there exist constants L j > 0 ( j = 1 , 2 , 3 , 4 ) such that

(7) J ˆ i f ( v ¯ ) + L 1 ‖ v ¯ n − v ¯ ‖ B R ≤ l J ˆ i f n ( v ¯ n ) , J ˆ i f n ( v ˜ n ) + L 2 ‖ v ˜ n − v ˜ ‖ B R ≤ l J ˆ i f ( v ˜ ) ;

(8) J ˜ i f ( u ¯ ) + L 3 ‖ u ¯ n − u ¯ ‖ B R ≤ l J ˜ i f n ( u ¯ n ) , J ˜ i f n ( u ˜ n ) + L 4 ‖ u ˜ n − u ˜ ‖ B R ≤ l J ˜ i f ( u ˜ ) ,

for n sufficiently large and i = 1 , 2 , where B R is the unit closed ball in R . Since there is J ˆ i f n ( v ¯ n ) ≤ l J ˆ i f n ( v ˜ n ) , J ˜ i f n ( u ¯ n ) ≤ l J ˜ i f n ( u ˜ n ) ( i = 1 , 2 ) when n is large enough, combining (7) and (8), we can obtain

(9) J ˆ i f ( v ¯ ) + L 1 ‖ v ¯ n − v ¯ ‖ B R + L 2 ‖ v ˜ n − v ˜ ‖ B R ≤ l J ˆ i f ( v ˜ ) ,

(10) J ˜ i f ( u ¯ ) + L 3 ‖ u ¯ n − u ¯ ‖ B R + L 4 ‖ u ˜ n − u ˜ ‖ B R ≤ l J ˜ i f ( u ˜ ) ,

for n sufficiently large and i = 1 , 2 . Let α ∈ int C R , then there is a constant λ > 0 such that B R ⊂ − λ α + C R . In view of (9) and (10), we obtain

(11) J ˆ i f ( v ¯ ) − λ α L 1 ‖ v ¯ n − v ¯ ‖ − λ α L 2 ‖ v ˜ n − v ˜ ‖ ≤ l J ˆ i f ( v ˜ ) ,

(12) J ˜ i f ( u ¯ ) − λ α L 3 ‖ u ¯ n − u ¯ ‖ − λ α L 4 ‖ u ˜ n − u ˜ ‖ ≤ l J ˜ i f ( u ˜ ) ,

for n sufficiently large and i = 1 , 2 . According to the C R -closed valued property of J ˆ i f and J ˜ i f at ( u ¯ , v ¯ ) ∈ U for i = 1 , 2 when n → ∞ , together with (11) and (12), we have

J ˆ i f ( v ¯ ) ≤ l J ˆ i f ( v ˜ ) and J ˜ i f ( u ¯ ) ≤ l J ˜ i f ( u ˜ ) , i = 1 , 2 .

Thus, we conclude the proof.□

Lemma 3.3

Let G ′ ⊂ G be a subset such that for any f ∈ G ′ , one possesses that

  1. S is USC on G ′ and S ( f ) is compact for each f ∈ G ′ ;

  2. The set-valued mappings J ˆ i f : V → 2 R and J ˜ i f : U → 2 R are LOLC with C R -closed values for i = 1 , 2 .

Then Ψ is USC on S ( f ) × G ′ for each f ∈ G ′ .

Proof

Assuming that there exists f ∈ G ′ such that Ψ is not USC on S ( f ) × { f } , namely, there exists ( u ¯ , v ¯ ) ∈ S ( f ) such that Ψ f is not USC at ( u ¯ , v ¯ ) . Therefore, there is an open set W in U with Ψ f ( u ¯ , v ¯ ) ⊂ W , the sequence { f n } ⊂ G , f n → f , and sequence { ( u n , v n ) } ∈ S ( f n ) with ( u n , v n ) → ( u ¯ , v ¯ ) such that

(13) Ψ f n ( u n , v n ) ⊄ W , for all n ∈ N * .

This shows that there exists a sequence { ( u ¯ n , v ¯ n ) } ⊂ Ψ f n ( u n , v n ) such that

( u ¯ n , v ¯ n ) ∉ W , for all n ∈ N * .

Since S is USC and S ( f ) is compact with f ∈ G ′ , ( u ¯ n , v ¯ n ) ∈ S ( f n ) , by virtue of Remark 2.2, there exist ( u ˜ , v ˜ ) ∈ S ( f ) and subsequence { ( u ¯ n k , v ¯ n k ) } ⊂ { ( u ¯ n , v ¯ n ) } such that ( u ¯ n k , v ¯ n k ) → ( u ˜ , v ˜ ) . By means of the definition of Ψ f n k ( u n k , v n k ) and ( u ¯ n k , v ¯ n k ) ∈ Ψ f n k ( u n k , v n k ) , we have

J ˆ i f n k ( v ¯ n k ) ≤ l J ˆ i f n k ( v n k ) , J ˜ i f n k ( u ¯ n k ) ≤ l J ˜ i f n k ( u n k ) , i = 1 , 2 ,

for any k ∈ N * . In the light Condition (ii), together with Lemma 3.2, we obtain

J ˆ i f ( v ˜ ) ≤ l J ˆ i f ( v ¯ ) and J ˜ i f ( u ˜ ) ≤ l J ˜ i f ( u ¯ ) , i = 1 , 2 .

Thus, ( u ˜ , v ˜ ) ∈ Ψ f ( u ¯ , v ¯ ) ⊂ W . This implies that { ( u ¯ n k , v ¯ n k ) } ⊂ W for large enough k , which contradicts (13). Consequently, Ψ is USC as desired.□

The following theorem is a main result of this study.

Theorem 3.1

If the following hypotheses hold,

  1. U is compact and S : G → U is convex;

  2. For any given f ∈ G , the set-valued mappings J ˆ i f and J ˜ i f satisfy (ii) in Lemma 3.3 and are strictly quasiconvex on S ( f ) for i = 1 , 2 ;

  3. For any f ∈ G , there exists a constant δ > 0 such that c o { f 1 , f 2 , … , f n } ⊂ ∪ i = 1 n Λ δ , f l ( f i ) for any finite subset { f 1 , f 2 , … , f n } of G ;

  4. There exist a compact set G ⊂ G and f 0 ′ ∈ G such that S ( f 0 ′ ) ≰ l S ( f ) for every f ∈ G \ G .

Then, differential game Γ f is left-stable.

Proof

This proof will be divided into three steps.

Step 1. We check G = G ′ , where G ′ is given as in Lemma 3.3. To this end, we first verify that S has compact values on G ; it suffices to verify that S has closed values since U is compact. For any f ∈ G , let the sequence { ( u n , v n ) } ⊂ S ( f ) such that { ( u n , v n ) } → ( u 0 , v 0 ) ∈ U . We claim that ( u 0 , v 0 ) ∈ S ( f ) . From the fact that J ˆ i , J ˜ i ( i = 1 , 2 ) are LOLC at ( u 0 , v 0 ) , there exist constants L 1 , L 2 > 0 such that

(14) J ˆ i f ( v 0 ) + L 1 ‖ v n − v 0 ‖ B R ≤ l J ˆ i f ( v n ) , J ˜ i f ( u 0 ) + L 2 ‖ u n − u 0 ‖ B R ≤ l J ˜ i f ( u n ) ,

for large enough n , where B R is the unit closed ball in R . Let α ∈ int C R . Then, there is a constant λ > 0 such that B R ⊂ − λ α + C R . This, combining (14), implies that

(15) J ˆ i f ( v 0 ) − λ α L 1 ‖ v n − v 0 ‖ ≤ l J ˆ i f ( v n ) , J ˜ i f ( u 0 ) − λ α L 2 ‖ u n − u 0 ‖ ≤ l J ˜ i f ( u n ) .

{ ( u n , v n ) } ⊂ S ( f ) guarantees that J ˆ i f ( v n ) ≤ l J ˆ i f ( v ) and J ˜ i f ( u n ) ≤ l J ˜ i f ( u ) for any ( u , v ) ∈ U . This, combining with (15), implies that

(16) J ˆ i f ( v 0 ) − λ α L 1 ‖ v n − v 0 ‖ ≤ l J ˆ i f ( v ) , J ˜ i f ( u 0 ) − λ α L 2 ‖ u n − u 0 ‖ ≤ l J ˜ i f ( u ) .

Let n → ∞ in (16) and note that J ˆ i f and J ˜ i f ( i = 1 , 2 ) have C R -closed values on U ; we have J ˆ i f ( v 0 ) ≤ l J ˆ i f ( v ) and J ˜ i f ( u 0 ) ≤ l J ˜ i f ( u ) . This guarantees that ( u 0 , v 0 ) ∈ S ( f ) , and hence, S ( f ) is closed. Furthermore, the arbitrariness of f implies that S has closed values.

We next verify that S is USC at f for any f ∈ G . Suppose that this is false. There exists an element f ∈ G and an open set W in U with S ( f ) ⊂ W such that we can find some sequence { f n } ⊂ G with f n → f and

S ( f n ) ⊄ W , for all n ∈ N * .

This yields that there exists a sequence { ( u n , v n ) } ⊂ S ( f n ) such that

(17) ( u n , v n ) ∉ W , for all n ∈ N * .

From the definition of S ( G ) , it follows that

(18) J ˆ i f n ( v n ) ≤ l J ˆ i f n ( v ) , J ˜ i f n ( u n ) ≤ l J ˜ i f n ( u ) , ( u , v ) ∈ U ,

for any n ∈ N * . Note that U is compact, there exist a sequence { ( u n , v n ) } ⊂ U and a point ( u 0 , v 0 ) ∈ U such that ( u n , v n ) → ( u 0 , v 0 ) for n → ∞ . By Lemma 3.2, we obtain J ˆ i f ( v 0 ) ≤ l J ˆ i f ( v ) and J ˜ i f ( u 0 ) ≤ l J ˜ i f ( u ) . This infers ( u 0 , v 0 ) ∈ S ( f ) ⊂ W , and hence, { ( u n , v n ) } ⊂ W for sufficiently large n . This contradicts (17). Thus, S : G → U is USC on G .

To sum up, Hypotheses (i) and (ii) in Lemma 3.3 are satisfied on G , and hence, G = G ′ .

Step 2. Check that S is LSC on G . In fact, if there exists f ∈ G such that S is not LSC at f , then there exists an open set W 1 ⊂ U such that S ( f ) ∩ W 1 ≠ ∅ . Thus, we are able to choose a point ( u ′ , v ′ ) ∈ S ( f ) ∩ W 1 and a neighborhood W of zero element θ in U such that ( u ′ , v ′ ) + W ⊂ W 1 , for which we possess a sequence { f n } ⊂ G with f n → f such that

(19) [ ( u ′ , v ′ ) + W ] ∩ S ( f n ) = ∅ , for all n ∈ N * .

Let us choose ( u n , v n ) ∈ ( u ′ , v ′ ) + W with ( u n , v n ) → ( u ′ , v ′ ) as n → ∞ . In order to achieve a contradiction, we consider the lower-level mapping Ψ f defined in (6). From Condition (ii) and Lemma 3.1, it follows that

(20) Ψ f ( u ′ , v ′ ) = ( u ′ , v ′ ) ∈ ( u ′ , v ′ ) + W .

Now, Lemma 3.3 guarantees that Ψ is USC at ( ( u ′ , v ′ ) , f ) . Therefore,

(21) Ψ f n ( u n , v n ) ⊂ ( u ′ , v ′ ) + W ,

for n sufficiently large.

By virtue of the definition of Ψ f , we have Ψ f n ( u n , v n ) ⊂ S ( f n ) . Consequently, Ψ f n ( u n , v n ) ⊂ [ ( u ′ , v ′ ) + W ] ∩ S ( f n ) . This is a contradiction.

Step 3. Finally, we verify that S ( f ) is left-stable for each f ∈ G . This is equivalent to proving that for any ε > 0 , there exists δ > 0 such that for any f ′ ∈ G with ρ ( f ′ , f ) < δ , we possess

(22) S ( f ) − S ( f ′ ) ⊂ B ( θ , ε ) and S ( f ′ ) ≤ l S ( f ) ,

where A − B = { z : ∀ x ∈ A , ∃ y ∈ B such that z = x − y } .

To check the first formula in (22), we suppose that this is false. Then, there exists ε > 0 such that for any δ > 0 , we can find some f δ ∈ G satisfying

(23) sup ( u , v ) ∈ S ( f ) ‖ ( u , v ) − ( u ′ , v ′ ) ‖ ≥ ε ,

for ρ ( f δ , f ) < δ and any ( u ′ , v ′ ) ∈ S ( f δ ) . In special, take δ = 1 n and f 1 n ≕ f n for n ∈ N * , we have ρ ( f n , f ) < 1 n , and hence, the sequence { f n } converges to f . Note that S is LSC by Step 2, and from Remark 2.2, for any ( u , v ) ∈ S ( f ) , we can take ( u n , v n ) ∈ S ( f n ) such that ( u n , v n ) → ( u , v ) as n → ∞ . Thus, there exists a positive integer n 0 such that d ( ( u , v ) , ( u n , v n ) ) < ε 2 < ε for every n > n 0 . This contradicts (23), and hence, the desired result holds.

To check the second formula in (22), it is necessary to prove that Λ δ , f l ( f ′ ) satisfies all conditions of Lemma 2.1. We first verify that for any fixed f ∈ G and δ > 0 given as in (iii), Λ δ , f l ( f ′ ) is closed for every f ′ ∈ G . Let the sequence { f n } ⊂ Λ δ , f l ( f ′ ) with f n → f ˜ ∈ G . In what follows, we verify f ˜ ∈ Λ δ , f l ( f ′ ) . It suffices to verify S ( f ) ⊂ S ( f ˜ ) + C . By virtue of the definition of Λ δ , f l ( f ′ ) , we have S ( f n ) ≤ l S ( f ) and ρ ( f , f n ) ≤ ρ ( f , f ′ ) < δ . This means that ρ ( f ˜ , f ) ≤ ρ ( f , f ′ ) < δ and S ( f ) ⊂ S ( f n ) + C . Furthermore, this implies that for each ( u , v ) ∈ S ( f ) , there exists ( u n , v n ) ∈ S ( f n ) such that ( u n , v n ) − ( u , v ) ≤ c θ . We observe that S is USC and apply Remark 2.2; there exist ( u 0 , v 0 ) ∈ S ( f ˜ ) and a subsequence { ( u n k , v n k ) } ⊂ { ( u n , v n ) } such that ( u n k , v n k ) → ( u 0 , v 0 ) for k → ∞ . In the light of the isotonicity of linear normed spaces, we obtain ( u 0 , v 0 ) − ( u , v ) ≤ c θ . This infers ( u , v ) ∈ S ( f ˜ ) + C . The arbitrariness of ( u , v ) guarantees that S ( f ) ⊂ S ( f ˜ ) + C , which implies that Λ δ , f l ( f ′ ) is closed.

We are in a position to show that there exists f 0 ′ ∈ G such that Λ δ , f l ( f 0 ′ ) is compact. From Hypothesis (iv), it follows that Λ δ , f l ( f 0 ′ ) ⊂ G . It is known that Λ δ , f l ( f 0 ′ ) is closed, and hence, Λ δ , f l ( f 0 ′ ) is compact.

Finally, by virtue of Lemma 2.1, we obtain ∩ f ′ ∈ G Λ δ , f l ( f ′ ) ≠ ∅ for any fixed f ∈ G and δ > 0 given as in (iii). Let g ∈ ∩ f ′ ∈ G Λ δ , f l ( f ′ ) . Then, for any f ′ ∈ G with ρ ( f ′ , f ) < δ , we have g ∈ Λ δ , f l ( f ′ ) . According to the definition of Λ δ , f l , we have S ( f ′ ) ≤ l S ( f ) , i.e., S ( f ) ⊂ S ( f ′ ) + C .

Consequently, (22) holds, and hence, S ( f ) is left-stable. This proof is complete.□

Theorem 3.2

Suppose that all conditions of Theorem 3.1 are satisfied except (iii) and (iv) being changed to

  1. For any f ∈ G , there exists a constant δ > 0 such that c o { f 1 , f 2 , … , f n } ⊂ ∪ i = 1 n Λ δ , f r ( f i ) for any finite subset { f 1 , f 2 , … , f n } of G ;

  2. There exist a compact set G ⊂ G and f 0 ′ ∈ G such that S ( f ) ≰ l S ( f 0 ′ ) for every f ∈ G \ G .

Then, differential game Γ f is right-stable.

Proof

Comparing the proof process of Theorem 3.1, the rest of the proof is the same, as long as Step 3 is modified as follows. To verify that S ( f ) is right-stable for each f ∈ G , it is equivalent to proving that for any ε > 0 , there exists δ > 0 such that

(24) S ( f ′ ) − S ( f ) ⊂ B ( θ , ε ) and S ( f ) ≤ l S ( f ′ ) ,

for any f ′ ∈ G with ρ ( f ′ , f ) < δ .

The first formula in (24) is similar to that in (22) for the proof, and therefore, we omit it.

To check the second formula in (24), it is necessary to prove that Λ δ , f r ( f ′ ) satisfies all conditions of Lemma 2.1. First, for any fixed f ∈ G and δ > 0 given as in (iii’), we verify that Λ δ , f r ( f ′ ) is closed for every f ′ ∈ G . To this end, let us take the sequence { f n } ⊂ Λ δ , f r ( f ′ ) with f n → f ˜ ∈ G . We prove f ˜ ∈ Λ δ , f r ( f ′ ) . By virtue of the definition of Λ δ , f r ( f ′ ) , we have S ( f ) ≤ l S ( f n ) and ρ ( f , f n ) ≤ ρ ( f , f ′ ) < δ . This implies that ρ ( f ˜ , f ) ≤ ρ ( f , f ′ ) < δ and S ( f n ) ⊂ S ( f ) + C . Therefore, we just prove S ( f ) ≤ l S ( f ˜ ) , i.e., S ( f ˜ ) ⊂ S ( f ) + C . Indeed, for any ( u 0 , v 0 ) ∈ S ( f ˜ ) , since S is LSC at f ˜ , by Remark 2.2, there exists ( u n , v n ) ∈ S ( f n ) such that ( u n , v n ) → ( u 0 , v 0 ) for n → ∞ . This, combined with ( u n , v n ) ∈ S ( f ) + C and the fact that S ( f ) is a closed set, implies ( u 0 , v 0 ) ∈ S ( f ) + C . Now, the arbitrariness of ( u 0 , v 0 ) implies S ( f ˜ ) ⊂ S ( f ) + C . Hence, f ˜ ∈ Λ δ , f r ( f ′ ) , which implies that Λ δ , f r ( f ′ ) is closed. Moreover, from Hypothesis (iv’), it follows that Λ δ , f r ( f 0 ′ ) ⊂ G . Hence, Λ δ , f r ( f 0 ′ ) is compact since it is closed.

Finally, by virtue of Lemma 2.1, we obtain ∩ f ′ ∈ G Λ δ , f r ( f ′ ) ≠ ∅ for any fixed f ∈ G and δ > 0 given as in (iii’). Let g ∈ ∩ f ′ ∈ G Λ δ , f r ( f ′ ) . Then, for any f ′ ∈ G with ρ ( f ′ , f ) < δ , we have g ∈ Λ δ , f r ( f ′ ) . According to the definition of Λ δ , f r , we have S ( f ) ≤ l S ( f ′ ) , i.e., S ( f ′ ) ⊂ S ( f ) + C .

Consequently, (24) holds, and hence, S ( f ) is right-stable. This proof is complete.□

Definition 3.2

[9] Let f ∈ G and ( u , v ) ∈ S ( f ) . If for any ε > 0 , there exists δ > 0 such that for all f ′ ∈ G with ρ ( f ′ , f ) < δ , there exists ( u ′ , v ′ ) ∈ S ( f ′ ) satisfying ‖ ( u , v ) − ( u ′ , v ′ ) ‖ < ε , then ( u , v ) ∈ S ( f ) is called an essential Nash equilibrium. The differential game Γ f is called essential if each ( u , v ) ∈ S ( f ) is an essential Nash equilibrium.

Remark 3.1

The differential game Γ f with f ∈ G is essential if it is left-stable.

Proof

From the left-stability, it follows that for any ε > 0 , there exists δ > 0 such that S ( f ) ⊂ S ( f ′ ) + B ( θ , ε ) ∩ C for any f ′ ∈ G with ρ ( f ′ , f ) < δ . Therefore, for any given ( u , v ) ∈ S ( f ) , there exists ( u ′ , v ′ ) ∈ S ( f ′ ) such that ‖ ( u , v ) − ( u ′ , v ′ ) ‖ < ε . This implies that S ( f ) with f ∈ G is essential.□

We remark that while Remark 3.1 is valid, the converse does not hold. See the example in the following.

Example 3.1

Let Γ f be a two-person differential game with the admissible control sets:

U = V = u k : u k ( t ) = k t , t ∈ [ 0 , 1 ] , k ∈ − 3 π 4 , 3 π 4 .

If the right-hand side function of the differential equation (4) and the payoff function are as follows:

f ∈ G ≔ { f : f ( t , x ( t ) , u ( t ) , v ( t ) ) = cos ( u ( t ) + v ( t ) ) } , J i f ( u ( t ) , v ( t ) ) = f ( 1 , x ( 1 ) , u ( 1 ) , v ( 1 ) ) , i = 1 , 2 , t ∈ [ 0 , 1 ] , ( u , v ) ∈ U × V ,

then Γ f is essential but not left-stable.

Proof

For t ∈ [ 0 , 1 ] and k , k ′ ∈ − 3 π 4 , 3 π 4 , one has

f ( t , x ( t ) , u ( t ) , v ( t ) ) = cos ( k + k ′ ) t , u = u k , v = u k ′ and

J i f ( u ( t ) , v ( t ) ) = cos ( k + k ′ ) , u = u k , v = u k ′ , i = 1 , 2 .

Thus, the set of Nash equilibria of Γ f associated with f is

S ( f ) = ( u k , u k ′ ) : k + k ′ = − π or k + k ′ = π for any k , k ′ ∈ − 3 π 4 , 3 π 4 .

For any t ∈ [ 0 , 1 ] , n ∈ N * , we take

f n ( t , x ( t ) , u ( t ) , v ( t ) ) = cos k t + k ′ t − 1 n , u = u k , v = u k ′ .

Then,

J i f n ( u ( t ) , v ( t ) ) = cos k + k ′ − 1 n , u = u k , v = u k ′ , i = 1 , 2 ,

and ρ ( f n , f ) → 0 when n goes to infinite. Hence, the set of Nash equilibria of Γ f n associated with f n is

S ( f n ) = ( u k n , u k n ′ ) : k n + k n ′ = − π + 1 n or k n + k n ′ = π + 1 n for any k n , k n ′ ∈ − 3 π 4 , 3 π 4 .

For any ( u k , u k ′ ) ∈ S ( f ) , we take ( u k n , u k n ′ ) ∈ S ( f n ) with ( u k n , u k n ′ ) ≔ ( u k , u k ′ + 1 n ) or ( u k n , u k n ′ ) ≔ ( u k + 1 n , u k ′ ) . Since

‖ ( u k , u k ′ ) − ( u k n , u k n ′ ) ‖ = ‖ u k − u k n ‖ + ‖ u k ′ − u k n ′ ‖ = 1 n → 0 , as n → ∞ ,

we derive that every ( u k , u k ′ ) ∈ S ( f ) is an essential Nash equilibrium of Γ f . By Definition 3.2, Γ f is essential.

On the other hand, let k = − 3 π 4 and k ′ = − π 4 . Then, f ( t , x , u , v ) = cos ( − π t ) and ( u − 3 π 4 , u − π 4 ) ∈ S ( f ) . For ε 0 = 1 and any δ > 0 , take ( u k n , u k n ′ ) ∈ S ( f n ) for n ∈ N * with n > 1 δ . Then, ρ ( f , f n ) ≤ 1 n < δ and k n + k n ′ = − π + 1 n or k n + k n ′ = π + 1 n . This implies that at least one of u k ≤ c u k n and u k ′ ≤ c u k n ′ is true. In other words, we have ( u k n , u k n ′ ) ≰ c ( u k , u k ′ ) for any ( u k n , u k n ′ ) ∈ S ( f n ) . This guarantees that ( u k , u k ′ ) ∉ S ( f n ) + C , i.e., S ( f ) ⊄ S ( f n ) + C for n > 1 δ . Thus, S ( f ) ⊄ S ( f n ) + B ( θ , ε 0 ) ∩ C for n > 1 δ . By virtue of Definition 3.1, Γ f is not left-stable.□

Remark 3.2

If Nash equilibrium set S ( f ) with f ∈ G is ordering stable, then it is stable, i.e., for any ε > 0 , there exists δ > 0 such that H ( S ( f ) , S ( f ′ ) ) < ε for any f ′ ∈ G with ρ ( f , f ′ ) < δ , where H is the Hausdorff distance due to distance d on U .

Proof

According to the definition of Hausdorff distance, we have

H ( S ( f ) , S ( f ′ ) ) = max { sup ( u , v ) ∈ S ( f ) inf ( u ′ , v ′ ) ∈ S ( f ′ ) d ( ( u , v ) , ( u ′ , v ′ ) ) , sup ( u ′ , v ′ ) ∈ S ( f ′ ) inf ( u , v ) ∈ S ( f ) d ( ( u , v ) , ( u ′ , v ′ ) ) } .

We consider the following two cases:

Case 1. sup ( u , v ) ∈ S ( f ) inf ( u ′ , v ′ ) ∈ S ( f ′ ) d ( ( u , v ) , ( u ′ , v ′ ) ) < ε and

Case 2. sup ( u ′ , v ′ ) ∈ S ( f ′ ) inf ( u , v ) ∈ S ( f ) d ( ( u , v ) , ( u ′ , v ′ ) ) < ε .

Suppose that Case 1 is false. There exists ε 0 > 0 such that for any δ > 0 , we can find f ′ ∈ G with ρ ( f , f ′ ) < δ satisfying

(25) sup ( u , v ) ∈ S ( f ) inf ( u ′ , v ′ ) ∈ S ( f ′ ) d ( ( u , v ) , ( u ′ , v ′ ) ) ≥ ε 0 .

Note that S ( f ) is left-stable, so for any ε > 0 with ε < ε 0 2 , there exists δ > 0 such that S ( f ) ⊂ S ( f ′ ) + B ( θ , ε ) ∩ C for any f ′ ∈ G with ρ ( f ′ , f ) < δ . This yields that S ( f ) ⊂ S ( f ′ ) + B ( θ , ε ) . Thus, for any ( u , v ) ∈ S ( f ) , there exists ( u ′ , v ′ ) ∈ S ( f ′ ) such that d ( ( u , v ) , ( u ′ , v ′ ) ) ≤ ε , i.e., for any ( u , v ) ∈ S ( f ) , one has inf ( u ′ , v ′ ) ∈ S ( f ′ ) d ( ( u , v ) , ( u ′ , v ′ ) ) ≤ ε . From the arbitrariness of ( u , v ) , we obtain

sup ( u , v ) ∈ S ( f ) inf ( u ′ , v ′ ) ∈ S ( f ′ ) d ( ( u , v ) , ( u ′ , v ′ ) ) ≤ ε < ε 0 .

This contradicts (25).

We can similarly prove Case 2 using the right-stability of S ( f ) . Hence, the Nash equilibrium set S ( f ) with f ∈ G is stable.□

4 Examples

As an application of our main results, we first consider a game problem of interest competition for power generation companies [37]. In the electricity market environment, the income of each power generation company depends not only on its bidding but also on the influence of bidding from other companies, which constitutes a game problem of interest competition.

Example 4.1

Suppose that two power generation companies are competing whose bidding strategy sets denoted by:

U 1 = { u 1 : u 1 ( t ) = u 1 k ( t ) = k t , t ∈ [ 0 , 1 ] , k ∈ [ 0 , 1 ] } , U 2 = { u 2 : u 2 ( t ) = u 2 k ( t ) = k t 2 , t ∈ [ 0 , 1 ] , k ∈ [ 1 , 2 ] } .

and the corresponding earning function is as follows:

π i f ( t ) = q i ( t ) x ( t ) − c i ( q i ( t ) ) , i = 1 , 2 .

Here, f is the right-hand side function in the differential equation (4), x : R + → R + is a market clearing price function, q i : R + → R + is the generated energy of power generation company i , and c i : R + → R + for i = 1 , 2 is the actual power generation cost function of power generation company whose functional expression is as follows:

c i ( q i ( t ) ) = α i + β i q i ( t ) + 0.5 γ i q i 2 ( t ) , i = 1 , 2 ,

where the positive constants α i , β i , and γ i are the no-load operating cost, the intercept of the marginal cost curve, and the slope of the marginal cost curve, respectively.

Conclusion. The differential game Γ f whose control system is (4) under the control functions u i ∈ U i ( i = 1 , 2 ) is ordering stable.

Proof

Since the bidding of the generator should not be lower than the marginal cost price, using the marginal cost function β i + γ i q i ( t ) ( i = 1 , 2 ) , we can obtain the expression of bidding b i of the generator i :

b i ( t ) = u i ( t ) + γ i q i ( t ) , i = 1 , 2 ,

with u i ( t ) > β i . We have x ( t ) = b i ( t ) at the balance of market supply and demand, i.e.,

q i ( t ) = x ( t ) − u i ( t ) γ i , i = 1 , 2 .

Therefore, the earning function of the power generation company i can be expressed as:

π i f ( t ) = [ x ( t ) − u i ( t ) ] [ x ( t ) + u i ( t ) − 2 β i ] 2 γ i − α i , i = 1 , 2 .

The generator needs to seek the optimal bidding strategy through continuous checking and continuous adjustment of the bidding strategy, so we establish a differential equation about the bidding checking of the generator. We replace (4) with the following differential equation to describe the relationship between the market clearing price x ( t ) and the generated energy q i ( t ) in the market:

d x ( t ) d t = μ [ ξ − η ( q 1 ( t ) + q 2 ( t ) ) − x ( t ) ] , x ( t 0 ) = x 0 ,

where μ > 0 is the regulating parameter and ξ and η > 0 are the demand parameters. Note that the right-hand side function is

f ( t , x ( t ) , u 1 ( t ) , u 2 ( t ) ) = μ [ ξ − η ( q 1 ( t ) + q 2 ( t ) ) − x ( t ) ] .

Under continuous dynamic changes, generators need to constantly adjust their bidding strategies to maximize their discounted earnings ∫ 0 1 e − ρ t π i f ( t ) d t , i.e., minimize payoff function J i . Therefore, we consider the following optimization problem under the control system (4):

min J i f ( u 1 ( t ) , u 2 ( t ) ) = ∫ 0 1 e − ρ t α i − [ x ( t ) − u i ( t ) ] [ x ( t ) + u i ( t ) − 2 β i ] 2 γ i d t , i = 1 , 2 , s.t. d x ( t ) d t = μ [ ξ − η ( q 1 ( t ) + q 2 ( t ) ) − x ( t ) ] .

For the convenience of calculation, we set α i = β i = ξ = η = μ = ρ = 1 , γ i = 2 for i = 1 , 2 and only consider the case when Q is the maximum available supply of the power generation companies. In view of assigning values to the parameters and via solving the aforementioned differential equation, we have

x ( t ) = 1 − Q + e − t , f ( t , x ( t ) , u 1 ( t ) , u 2 ( t ) ) = − e − t , t ∈ [ 0 , 1 ] .

For any f ∈ G , k ∈ [ 0 , 1 ] , k ′ ∈ [ 1 , 2 ] , this guarantees

J ˆ 1 f ( u 2 ) = J 1 f ( u 1 , u 2 ) = ∫ 0 1 e − ρ t α 1 − [ x ( t ) − u 1 ( t ) ] [ x ( t ) + u 1 ( t ) − 2 β 1 ] 2 γ 1 d t : ∀ u 1 ( t ) ∈ U 1 = J 1 f ( u 1 , u 2 ) = 1 12 e 3 − Q 4 e 2 + − 5 k 2 + 4 k + Q 2 − 5 4 e + Q − Q 2 4 + k 2 − k 2 + 7 6 : ∀ k ∈ [ 0 , 1 ] , J ˆ 2 f ( u 2 ) = J 2 f ( u 1 , u 2 ) = ∫ 0 1 e − ρ t α 2 − [ x ( t ) − u 2 ( t ) ] [ x ( t ) + u 2 ( t ) − 2 β 2 ] 2 γ 2 d t : ∀ u 1 ( t ) ∈ U 1 = J 2 f ( u 1 , u 2 ) = 1 12 e 3 − Q 4 e 2 + − 65 k ′ 2 + 10 k ′ + Q 2 − 5 4 e + Q − Q 2 4 + 6 k ′ 2 − k ′ + 7 6 : ∀ k ∈ [ 0 , 1 ] , J ˜ 1 f ( u 1 ) = J 1 f ( u 1 , u 2 ) = ∫ 0 1 e − ρ t α 1 − [ x ( t ) − u 1 ( t ) ] [ x ( t ) + u 1 ( t ) − 2 β 1 ] 2 γ 1 d t : ∀ u 2 ( t ) ∈ U 2 = J 1 f ( u 1 , u 2 ) = 1 12 e 3 − Q 4 e 2 + − 5 k 2 + 4 k + Q 2 − 5 4 e + Q − Q 2 4 + k 2 − k 2 + 7 6 : ∀ k ′ ∈ [ 1 , 2 ] , J ˜ 2 f ( u 1 ) = J 2 f ( u 1 , u 2 ) = ∫ 0 1 e − ρ t α 2 − [ x ( t ) − u 2 ( t ) ] [ x ( t ) + u 2 ( t ) − 2 β 2 ] 2 γ 2 d t : ∀ u 2 ( t ) ∈ U 2 = J 2 f ( u 1 , u 2 ) = 1 12 e 3 − Q 4 e 2 + − 65 k ′ 2 + 10 k ′ + Q 2 − 5 4 e + Q − Q 2 4 + 6 k ′ 2 − k ′ + 7 6 : ∀ k ′ ∈ [ 1 , 2 ] .

Clearly, U × V is compact. This deduces that the Nash equilibrium set S ( f ) = { ( u 1 1 , u 2 1 ) } and S ( f ) is convex. It is easy to obtain that J ˆ i f and J ˜ i f for i = 1 , 2 satisfy the property of LOLC with C -closed values. Moreover, they are strictly quasiconvex on S ( f ) for i = 1 , 2 .

Obviously,

G = { f : f ( t , x ( t ) , u 1 ( t ) , u 2 ( t ) ) = − e − t , t ∈ [ 0 , 1 ] } .

Let { f 1 , f 2 , … , f n } ⊂ G and { λ 1 , λ 2 , … , λ n } ⊂ [ 0 , 1 ] be such that ∑ i = 1 n λ i = 1 . Put f ˜ = ∑ i = 1 n λ i f i ∈ G . Take δ = 1 , for any f ∈ G , one has ρ ( f ˜ , f ) ≤ ρ ( f i , f ) < 1 . From S ( f ) = { ( u 1 1 , u 2 1 ) } for any f ∈ G , it follows that S ( f ˜ ) ≤ l S ( f ) and S ( f i ) ≤ l S ( f ) for any i = 1 , 2 , … , n . This implies that f ˜ = c o { f 1 , f 2 , … , f n } ⊂ ∪ i = 1 n Λ δ , f l ( f i ) . Finally, for any compact set G ⊂ G and any f k ∈ G , by means of the converse-negative proposition of (iv), there exists f ∈ G \ G such that S ( f k ) ≤ l S ( f ) .

Consequently, all conditions of Theorem 3.1 are satisfied, and hence, this differential game is left-stable. By virtue of the similar method, we will derive the differential game is right-stable. In addition, it is also stable and essential.□

The following examples further illustrate that our main results are available.

Example 4.2

Consider a two-person differential game Γ f with the admissible control sets:

U = u k : u k ( t ) = sin k t , t ∈ [ 0 , 1 ] , k ∈ π 8 , π 4 , V = { v k : v k ( t ) = k t + k , t ∈ [ 0 , 1 ] , k ∈ [ 0 , 1 ] } .

Moreover, assume that the right-hand side function of differential game (4) and the payoff function are as follows:

f ( t , x ( t ) , u ( t ) , v ( t ) ) = u 2 ( t ) + v 2 ( t ) ,

J i f ( u ( t ) , v ( t ) ) = − x ( 1 ) , i = 1 , 2 , t ∈ [ 0 , 1 ] , ( u , v ) ∈ U × V .

Then, Γ f is ordering stable.

Proof

For any t ∈ [ 0 , 1 ] , k ∈ π 8 , π 4 , k ′ ∈ [ 0 , 1 ] , if u = u k , v = v k ′ , we have

f ( t , x ( t ) , u ( t ) , v ( t ) ) = 1 − cos 2 k t 2 + k ′ 2 ( t 2 + 2 t + 1 ) .

By solving the differential equation (4), we have

x ( t ) = t 2 − sin 2 k t 4 k + k ′ 2 1 3 t 3 + t 2 + t .

In addition,

J 1 f ( u ( t ) , v ( t ) ) = J 2 f ( u ( t ) , v ( t ) ) = − 1 2 + sin 2 k 4 k − 7 3 k ′ 2 .

Thus, the set of Nash equilibria of Γ f associated with f is S ( f ) = { ( u π 4 , v 1 ) } . Let

f n ( t , x ( t ) , u ( t ) , v ( t ) ) = u 2 ( t ) + v 2 ( t ) + 1 n , t ∈ [ 0 , 1 ] , u ∈ U , v ∈ V , n ∈ N * .

This infers that ρ ( f n , f ) = 1 n → 0 when n goes to infinity. Then, for i = 1 , 2 , in the light of the similar methods as the aforementioned methods, we obtain

f n ( t , x ( t ) , u ( t ) , v ( t ) ) = 1 − cos 2 k t 2 + k ′ 2 ( t 2 + 2 t + 1 ) + 1 n ,

x ( t ) = t 2 − sin 2 k t 4 k + k ′ 2 ( 1 3 t 3 + t 2 + t ) + 1 n t ,

J i f n ( u ( t ) , v ( t ) ) = − 1 2 + sin 2 k 4 k − 7 3 k ′ 2 − 1 n .

Similarly, the set of Nash equilibria of Γ f n associated with f n is S ( f n ) = { ( u π 4 , v 1 ) } . Trivially, the conditions of Theorems 3.1 and 3.2 are satisfied, and hence, Γ f is ordering stable.□

Example 4.3

Let Γ f be an n -person differential game with the admissible control set, the right-hand side function, and the payoff function, respectively, as follows:

U i = { u k : u k ( t ) = k t + k , t ∈ [ 0 , 1 ] , k = 1 , 2 , … , m } , i ∈ I , f ( t , x ( t ) , u ( t ) ) = ∑ i = 1 n u i 2 ( t ) , u ( t ) = ( u 1 ( t ) , … , u n ( t ) ) ∈ ∏ i = 1 n U i and J i f ( u ( t ) ) = x 3 ( 1 ) , t ∈ [ 0 , 1 ] , i ∈ I ,

where the set of persons is denoted as I = { 1 , 2 , … , n } . Then, Γ f is ordering stable.

Proof

Clearly, ∏ i = 1 n U i is compact and

f ( t , x ( t ) , u ( t ) ) = ∑ i = 1 n ( k i t + k i ) 2 , x ( t ) = ∑ i = 1 n k i 2 1 3 t 3 + t 2 + t , J i f ( u ( t ) ) = 343 27 ∑ i = 1 n k i 2 3 ,

for t ∈ [ 0 , 1 ] , where u i = u k i with k i ∈ { 1 , 2 , … , m } . Thus, the set of Nash equilibria of Γ f associated with f is S ( f ) = { ( u 1 , u 1 , … , u 1 ) } being convex. We take

G = f l : f l ( t , x ( t ) , u ( t ) ) = ∑ i = 1 n u i 2 ( t ) + 1 l = ∑ i = 1 n ( k i t + k i ) 2 + 1 l , t ∈ [ 0 , 1 ] , u i ∈ U i , l ∈ N * .

So, ρ ( f l , f ) = 1 l → 0 when l → ∞ . Note that

x l ( t ) = ∑ i = 1 n k i 2 1 3 t 3 + t 2 + t + 1 l t and J i f l ( u ( t ) ) = 7 3 ∑ i = 1 n k i 2 + 1 l 3 ,

we obtain that S ( f l ) = { ( u 1 , u 1 , … , u 1 ) } is convex. Obviously, for any f l ∈ G , J ˆ i f l and J ˜ i f l satisfy the property of LOLC with C -closed values. Moreover, J ˆ i f l and J ˜ i f l are strictly quasiconvex on S ( f l ) for i ∈ I . It is easy to see that Γ f is ordering stable using a similar method to Example 4.1.□

Acknowledgement

The author thanks the referees for their valuable comments and constructive suggestions that helped improving this manuscript.

  1. Funding information: This study was supported by the National Natural Science Foundation of China (71771068).

  2. Conflict of interest: The author states no conflicts of interest.

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Received: 2022-10-27
Revised: 2023-08-04
Accepted: 2023-09-22
Published Online: 2023-11-21

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2023-0132
Keywords for this article
differential games; order relation; stability; Nash equilibrium; set-valued mapping
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  85. Existence of solutions for semilinear retarded equations with non-instantaneous impulses, non-local conditions, and infinite delay
  86. On the quadratic residues and their distribution properties
  87. On average theta functions of certain quadratic forms as sums of Eisenstein series
  88. Connected component of positive solutions for one-dimensional p-Laplacian problem with a singular weight
  89. Some identities of degenerate harmonic and degenerate hyperharmonic numbers arising from umbral calculus
  90. Mean ergodic theorems for a sequence of nonexpansive mappings in complete CAT(0) spaces and its applications
  91. On some spaces via topological ideals
  92. Linear maps preserving equivalence or asymptotic equivalence on Banach space
  93. Well-posedness and stability analysis for Timoshenko beam system with Coleman-Gurtin's and Gurtin-Pipkin's thermal laws
  94. On a class of stochastic differential equations driven by the generalized stochastic mixed variational inequalities
  95. Entire solutions of two certain Fermat-type ordinary differential equations
  96. Generalized Lie n-derivations on arbitrary triangular algebras
  97. Markov decision processes approximation with coupled dynamics via Markov deterministic control systems
  98. Notes on pseudodifferential operators commutators and Lipschitz functions
  99. On Graham partitions twisted by the Legendre symbol
  100. Strong limit of processes constructed from a renewal process
  101. Construction of analytical solutions to systems of two stochastic differential equations
  102. Two-distance vertex-distinguishing index of sparse graphs
  103. Regularity and abundance on semigroups of partial transformations with invariant set
  104. Liouville theorems for Kirchhoff-type parabolic equations and system on the Heisenberg group
  105. Spin(8,C)-Higgs pairs over a compact Riemann surface
  106. Properties of locally semi-compact Ir-topological groups
  107. Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2
  108. Ordering stability of Nash equilibria for a class of differential games
  109. A new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order
  110. About a dubious proof of a correct result about closed Newton Cotes error formulas
  111. Ricci ϕ-invariance on almost cosymplectic three-manifolds
  112. Schur-power convexity of integral mean for convex functions on the coordinates
  113. A characterization of a ∼ admissible congruence on a weakly type B semigroup
  114. On Bohr's inequality for special subclasses of stable starlike harmonic mappings
  115. Properties of meromorphic solutions of first-order differential-difference equations
  116. A double-phase eigenvalue problem with large exponents
  117. On the number of perfect matchings in random polygonal chains
  118. Evolutoids and pedaloids of frontals on timelike surfaces
  119. A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine
  120. The 𝔪-WG° inverse in the Minkowski space
  121. Stability result for Lord Shulman swelling porous thermo-elastic soils with distributed delay term
  122. Approximate solvability method for nonlocal impulsive evolution equation
  123. Construction of a functional by a given second-order Ito stochastic equation
  124. Global well-posedness of initial-boundary value problem of fifth-order KdV equation posed on finite interval
  125. On pomonoid of partial transformations of a poset
  126. New fractional integral inequalities via Euler's beta function
  127. An efficient Legendre-Galerkin approximation for the fourth-order equation with singular potential and SSP boundary condition
  128. Eigenfunctions in Finsler Gaussian solitons
  129. On a blow-up criterion for solution of 3D fractional Navier-Stokes-Coriolis equations in Lei-Lin-Gevrey spaces
  130. Some estimates for commutators of sharp maximal function on the p-adic Lebesgue spaces
  131. A preconditioned iterative method for coupled fractional partial differential equation in European option pricing
  132. A digital Jordan surface theorem with respect to a graph connectedness
  133. A quasi-boundary value regularization method for the spherically symmetric backward heat conduction problem
  134. The structure fault tolerance of burnt pancake networks
  135. Average value of the divisor class numbers of real cubic function fields
  136. Uniqueness of exponential polynomials
  137. An application of Hayashi's inequality in numerical integration
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