# Approximate Solutions for Nash Differential Games

Authors: A. A. Megahed; A. A. Hemeda; H. F. A. Madkour
Journal: Journal of Mathematics
 (2023)

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## Abstract

In this paper, we are concerned with an open-loop Nash differential game. The necessary conditions for an open-loop Nash equilibrium solution are obtained, also the existence for the solution of the dynamical system of the differential game is studied. Picard method is used to find an approximate solution, and the uniform convergence is proved. Finally, we constructed figures for the analysis of the differential game. These results can be applied between economic and financial firms as well as industrial firms.


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## Body

## 1. Introduction

Differential equations have a great importance in our life and many applications in physics and engineering fields [1]. Differential equations help us in describing all the phenomena in which there are rates of change and provide a description of the way this change works, for example, population growth, chemical reactions, launching rockets into space, the spread of diseases, and climate changes. The differential game is a direct application of a differential equation and game theory. The game theory is an important field in mathematics [2]. It has applications in almost all fields of social science, as well as in logic, system science, and computer science. It has an important role in Economics. John Forbes Nash is one of the mathematicians who made fundamental contributions to the game theory. Nash proposed a solution of a noncooperative game including two or more players in which every player is assumed to know the equilibrium strategies of the players. No one can change the strategy or move without the other players knowing that the players have the same strategies. In the game theory, differential games are a group of problems related to the modeling and analysis of conflict (competition) expressed as a dynamical system [3]. It means here a state variable evolves over time according to a differential equation. A state variable is one of the set of variables that are used to describe the mathematical “state” of a dynamical system. The state of a system describes enough about the system to determine its future behaviour in the absence of any external forces affecting the system. There are many applications of differential game in our life and it is an important case ([4–8]).In [9], Hemeda introduced an integral iterative method (IIM) as a modification for PM to solve nonlinear integrodifferential and systems of nonlinear integrodifferential equations.In [10], Joseph presented a duopolistic market problem in which two firms sell the same product competitively in a certain time. Each firm has its own market share and the strategy here is one for the two firms, that is, the advertising efforts. Kristina solved the differential game by using the maximum principle with a general inequality constraints theorem. She used a numerical method for finding the solutions. In this work, we used another method and a theorem for finding the necessary conditions for an open-loop Nash equilibrium differential game. We found the approximate solutions using the Picard method, constructed graphs for the solutions, and making comparisons to the two firms simultaneously. We illustrated those comparisons with each graph.The rest of the paper is organized as the following: In Section2, the dynamical system of the problem, its payoff functionals, and the necessary conditions for an open-loop Nash equilibrium are presented. The aim of Section 3 is to find an approximate solution using the Picard method. Section 4 is preserved for the discussion of the results. Finally, Section 5 contains a conclusion.


## 2. Problem Formulation

In this section, first we explain the dynamical system of the problem, the payoff functionals, and the open-loop Nash equilibrium.


### 2.1. The Dynamical System

We consider two firms, Firm 1 and Firm 2. The two firms sell the same product in the market competitively [10], where xt is the market share of Firm 1 at time t, 1−xt is the market share of Firm 2 at time t. u1t and u2t are the controls which defined as follows: u1t is the advertising efforts of Firm 1 at time t, and u2t is the advertising efforts of Firm 2 at time t.In Firm 1, its number of customers increases by its advertising efforts. The advertising efforts of Firm 2 take away the customers of Firm 1. Therefore, the system dynamics can be expressed as the following:(1)dxdt=u1t1−xt−u2txt,x0=x0,t∈0,T,with constraints given by(2)0≤xt≤1.As we are concerned with a differential game with two players, then we have the following definition.Definition 1.
(2-players differential game). In the differential game of two players on the time intervalt0,tf, we have the following:(1)
A set of playersN=1,2(2)
For each playeri∈N, there is a vector of controls uit∈Ui∈Rni, where Ui is the set of admissible control values for player i(3)
A vector of state variablesx∈X⊂Rn, where X is the set of admissible states(4)
A strategy setΨi, where the strategy ψi∈Ψi is a decision rule that defines the control uit∈Ui as a function of the information available at time t


### 2.2. Payoff Functionals and Open-Loop Nash Equilibrium

We consider the state equation which describes the state of the game and the payoff functionals as the following:(3)dxdt=fxt,u1t,u2t,t,(4)Jiu1t,u2t=∫t0tfIixt,u1t,u2t,tdt;i=1,2.Since the information structure is open-loop, then the equilibrium strategy of playeri will be ui∗t,t∈t0,tf,i=1,2.For obtaining these strategies, we have Hamiltonian’s function for each playeri.Hiλi,x,u1,u2,t=Iixt,u1t,u2t,t+λiTfx,u1,u2,t;t∈t0,tf,i=1,2, where λi is the costate vector for player i.Definition 2.
In the 2-players differential game given by Definition1 of a duration t0,tf, we say that player i,s information structure is open-loop; if at time t, the only information available to player i is the initial state of the game x0; hence, his strategy set can be written as Ψit=x0,t∈t0,tf.Definition 3.
IfJ1u1t,u2t and J2u1t,u2t are cost functionals for players 1,2, respectively, then an ordered pair control u1∗,u2∗ is a Nash equilibrium strategy if for each i=1,2, we have(5)Jiu1∗,u2∗≤Jiu1,u2∗.
For simplicity, the Nash equilibrium concept means that if one player tries to change his strategy from his own side, he cannot improve his own optimization criterion.
Now, we can define the payoff functionals of the two firms of our problem (1) and (2) as the following:(6)J1=∫0Te−r1tϕ1xt−c1u1tdt,J2=∫0Te−r2tϕ21−xt−c2u2tdt,such that t0,tf=0,T, ri is the interest rate of Firm i, ϕi is the fractional revenue potential of Firm i, and cis is the advertising cost function of the two firms.
Assuming thatcis=kis2/2; where ki is a positive constant and(7)fxt,u1t,u2t,t=u1t1−xt−u2txt,I1xt,u1t,u2t,t=ϕ1xt−c1u1t,I2xt,u1t,u2t,t=ϕ21−xt−c2u2t.
The Hamiltonian function of player 1 is(8)H1λ1,x,u1,u2,t=ϕ1xt−k1u122+λ1u1t1−xt−u2txt.
The Hamiltonian function of player 2 is(9)H2λ2,x,u1,u2,t=ϕ21−xt−k2u222+λ2u1t1−xt−u2txt.Theorem 1.
Letfxt,u1t,u2t,t and Iixt,u1t,u2t,t, i=1,2 be continuously differentiable on Rn i.e., fxt,u1t,u2t,t:Rn×Rs×0,T⟶R,f∈C1,s=Σj=1Nsj,i≠j, Iixt,u1t,u2t,t:Rn×Rs×0,T⟶R,Ii∈C1,i=1,2. If ui∗t,t0≤t≤tf is an open-loop Nash-equilibrium solution and x∗t,t0≤t≤tf be the corresponding state trajectory, then there exists 2 costate vectors λi:t0,tf⟶Rn, and 2-Hamiltonian functions Hiλi,x,u1,u2,t=Iixt,u1t,u2t,t+λiTfx,u1,u2,t such that the following conditions are satisfied:(10)dx∗dt=fx∗t,u1∗t,u2∗t,t,x∗0=x0,dλitdt=−∂Hiλi,x∗,u1∗,u2∗,t∂x,i=1,2,∂H1λ1,x∗,u1∗,u2∗,t∂u1=0,∂H2λ2,x∗,u1∗,u2∗,t∂u2=0,with boundary conditions,(11)x∗0=x0,λitf=0,i=1,2,where ui∗,i=1,2, is an open-loop Nash equilibrium strategy, and x∗ is the corresponding equilibrium state trajectory.The proof of the thereom is proved in [11].


## 2.1. The Dynamical System

We consider two firms, Firm 1 and Firm 2. The two firms sell the same product in the market competitively [10], where xt is the market share of Firm 1 at time t, 1−xt is the market share of Firm 2 at time t. u1t and u2t are the controls which defined as follows: u1t is the advertising efforts of Firm 1 at time t, and u2t is the advertising efforts of Firm 2 at time t.In Firm 1, its number of customers increases by its advertising efforts. The advertising efforts of Firm 2 take away the customers of Firm 1. Therefore, the system dynamics can be expressed as the following:(1)dxdt=u1t1−xt−u2txt,x0=x0,t∈0,T,with constraints given by(2)0≤xt≤1.As we are concerned with a differential game with two players, then we have the following definition.Definition 1.
(2-players differential game). In the differential game of two players on the time intervalt0,tf, we have the following:(1)
A set of playersN=1,2(2)
For each playeri∈N, there is a vector of controls uit∈Ui∈Rni, where Ui is the set of admissible control values for player i(3)
A vector of state variablesx∈X⊂Rn, where X is the set of admissible states(4)
A strategy setΨi, where the strategy ψi∈Ψi is a decision rule that defines the control uit∈Ui as a function of the information available at time t


## 2.2. Payoff Functionals and Open-Loop Nash Equilibrium

We consider the state equation which describes the state of the game and the payoff functionals as the following:(3)dxdt=fxt,u1t,u2t,t,(4)Jiu1t,u2t=∫t0tfIixt,u1t,u2t,tdt;i=1,2.Since the information structure is open-loop, then the equilibrium strategy of playeri will be ui∗t,t∈t0,tf,i=1,2.For obtaining these strategies, we have Hamiltonian’s function for each playeri.Hiλi,x,u1,u2,t=Iixt,u1t,u2t,t+λiTfx,u1,u2,t;t∈t0,tf,i=1,2, where λi is the costate vector for player i.Definition 2.
In the 2-players differential game given by Definition1 of a duration t0,tf, we say that player i,s information structure is open-loop; if at time t, the only information available to player i is the initial state of the game x0; hence, his strategy set can be written as Ψit=x0,t∈t0,tf.Definition 3.
IfJ1u1t,u2t and J2u1t,u2t are cost functionals for players 1,2, respectively, then an ordered pair control u1∗,u2∗ is a Nash equilibrium strategy if for each i=1,2, we have(5)Jiu1∗,u2∗≤Jiu1,u2∗.
For simplicity, the Nash equilibrium concept means that if one player tries to change his strategy from his own side, he cannot improve his own optimization criterion.
Now, we can define the payoff functionals of the two firms of our problem (1) and (2) as the following:(6)J1=∫0Te−r1tϕ1xt−c1u1tdt,J2=∫0Te−r2tϕ21−xt−c2u2tdt,such that t0,tf=0,T, ri is the interest rate of Firm i, ϕi is the fractional revenue potential of Firm i, and cis is the advertising cost function of the two firms.
Assuming thatcis=kis2/2; where ki is a positive constant and(7)fxt,u1t,u2t,t=u1t1−xt−u2txt,I1xt,u1t,u2t,t=ϕ1xt−c1u1t,I2xt,u1t,u2t,t=ϕ21−xt−c2u2t.
The Hamiltonian function of player 1 is(8)H1λ1,x,u1,u2,t=ϕ1xt−k1u122+λ1u1t1−xt−u2txt.
The Hamiltonian function of player 2 is(9)H2λ2,x,u1,u2,t=ϕ21−xt−k2u222+λ2u1t1−xt−u2txt.Theorem 1.
Letfxt,u1t,u2t,t and Iixt,u1t,u2t,t, i=1,2 be continuously differentiable on Rn i.e., fxt,u1t,u2t,t:Rn×Rs×0,T⟶R,f∈C1,s=Σj=1Nsj,i≠j, Iixt,u1t,u2t,t:Rn×Rs×0,T⟶R,Ii∈C1,i=1,2. If ui∗t,t0≤t≤tf is an open-loop Nash-equilibrium solution and x∗t,t0≤t≤tf be the corresponding state trajectory, then there exists 2 costate vectors λi:t0,tf⟶Rn, and 2-Hamiltonian functions Hiλi,x,u1,u2,t=Iixt,u1t,u2t,t+λiTfx,u1,u2,t such that the following conditions are satisfied:(10)dx∗dt=fx∗t,u1∗t,u2∗t,t,x∗0=x0,dλitdt=−∂Hiλi,x∗,u1∗,u2∗,t∂x,i=1,2,∂H1λ1,x∗,u1∗,u2∗,t∂u1=0,∂H2λ2,x∗,u1∗,u2∗,t∂u2=0,with boundary conditions,(11)x∗0=x0,λitf=0,i=1,2,where ui∗,i=1,2, is an open-loop Nash equilibrium strategy, and x∗ is the corresponding equilibrium state trajectory.The proof of the thereom is proved in [11].


## 3. The Approximate Solution by Using the Picard Method

The purpose of this section is to find the existence and convergence of the solution for the problem.


### 3.1. Existence and Convergence of the Solution

Now, we study the existence of the solution for the problem (10)–(11) (see [12, 13]) and we apply the Picard method for finding an approximate solution and studying the convergence of this solution.Consider the following system after reducing the necessary conditions of an open-loop Nash differential game as the following:(12)dxdt=f1x,λ1,λ2,t,dλ1dt=f2x,λ1,λ2,t+ϕ1,dλ2dt=f3x,λ1,λ2,t−ϕ2,x0=x0,λ1T=0,λ2T=0.Assume these assumptions for our problem.(1)
fixt,λ1t,λ2t,t:Rn×Rn×Rn×0,T⟶R are continuous and there are positive constants Mi s.t fi≤Mi,i=1,2,3.(2)
fi satisfies Lipschitz condition with Lipschitz constants Li,0<Li<1,i=1,2,3 such that(13)f1x,λ1,λ2,t−f1y,λ1,λ2,t≤L1x−y,f2x,λ1,λ2,t−f2x,p1,λ2,t≤L2λ1−p1,f3x,λ1,λ2,t−f3x,λ1,p2,t≤L3λ2−p2.We prove the existence of the solution of system (12), then we have to integrate the differential equation in (12), we get(14)xt=x0+∫0tf1x,λ1,λ2,tdt,(15)λ1t=ϕ1t−T+∫Ttf2x,λ1,λ2,tdt,(16)λ2t=−ϕ1t−T+∫Ttf3x,λ1,λ2,tdt.By differentiating the integral equations (14)–(16) with respect to t, we have(17)dxdt=f1x,λ1,λ2,t,dλ1dt=f2x,λ1,λ2,t+ϕ1,dλ2dt=f3x,λ1,λ2,t−ϕ2.Substitutingt=0 in (14), t=T in (15) and (16), we obtain(18)x0=x0+∫00f1x,λ1,λ2dt,t=x0,λ1T=ϕ1T−T+∫TTf2x,λ1,λ2,tdt=0,λ2T=−ϕ1T−T+∫TTf3x,λ1,λ2dt,t=0.Hence, the existence is proved from the equivalence between the system (12) and the integral equations ((14)–(16)). Therefore, there is a solution for our system.Now, we apply the Picard method to find the solutions of the integral equations (14)–(16). The solution is constructed by the sequences,(19)xnt=x0+∫0tf1xn−1,λ1n−1,λ2n−1,tdt,x0=x0,λ1tn=ϕ1t−T+∫Ttf2xn−1,λ1n−1,λ2n−1,tdt,λ10=0,λ2tn=−ϕ2t−T+∫Ttf3xn−1,λ1n−1,λ2n−1,tdt,λ20=0,n=1,2,3,….xnt,λ1tn,λ2tn can be written as the following:(20)xnt=x0+∑j=1nxj−xj−1,λ1tn=ϕ1t−T+∑j=1nλ1j−λ1j−1,λ2tn=−ϕ2t−T+∑j=1nλ2j−λ2j−1.Ifxn,λ1n, and λ2n are convergent, then the infinite series ∑xj−xj−1,∑λ1j−λ1j−1 and ∑λ2j−λ2j−1 are convergent, and the solution will be x,λ1, and λ2, where(21)xt=limn⟶∞xn,λ1t=limn⟶∞λ1n,λ2t=limn⟶∞λ2n.If the three series are converged, then the three sequencesxn,λ1n and λ2n will be converged, respectively, to xt,λ1t and λ2t. For proving the uniform convergence of xn,λ1n and λ2n, we have to consider the three associated series,(22)∑n=1∞xn−xn−1,∑n=1∞λ1n−λ1n−1,∑n=1∞λ2n−λ2n−1.Forn=1, we get(23)x1−x0=∫0tf1x0,λ10,λ20,tdt,x1−x0=∫0tf1x0,λ10,λ20,tdt,x1−x0≤M1T,λ11−λ10=ϕ1t−T+∫Ttf2x0,λ10,λ20,tdt,λ11−λ10≤ϕ1+M2t−T,λ11−λ10≤ϕ1+M2T,λ21−λ20=−ϕ2t−T+∫Ttf3x0,λ10,λ20,tdt,λ21−λ20≤ϕ2+M3t−T,λ21−λ20≤ϕ2+M3T.Now, we shall get an estimation forxn−xn−1, λ1n−λ1n−1, and λ2n−λ2n−1(24)xn−xn−1=∫0tf1xn−1,λ1n−1,λ2n−1,tdt−∫0tf1xn−2,λ1n−2,λ2n−2,tdt,xn−xn−1≤L1Txn−1−xn−2,λ1n−λ1n−1=∫Ttf2xn−1,λ1n−1,λ2n−1,tdt−∫Ttf2xn−2,λ1n−2,λ2n−2,tdt,λ1n−λ1n−1≤L2Tλ1n−1−λ1n−2,λ2n−λ2n−1=∫Ttf3xn−1,λ1n−1,λ2n−1,tdt−∫Ttf3xn−2,λ1n−2,λ2n−2,tdt,λ2n−λ2n−1≤L3Tλ2n−1−λ2n−2.By using the first estimation in (24) and putting n=2, we get(25)x2−x1≤L1Tx1−x0,x2−x1≤L1M1T2.Atn=3, we have(26)x3−x2≤L1Tx2−x1,x3−x2≤L12M1T3,and so on(27)xn−xn−1≤L1n−1M1Tn.By using the second estimation in (24), we have at n=2(28)λ12−λ11≤L2Tλ11−λ10,λ12−λ11≤L2ϕ1+M2T2.Atn=3(29)λ13−λ12≤L22ϕ1+M2T3,and so on(30)λ1n−λ1n−1≤L2n−1ϕ1+M2Tn.By using the third estimation in (24), we get at n=2(31)λ22−λ21≤L3Tλ21−λ20,λ22−λ21≤L3ϕ2+M3T2.Atn=3,(32)λ23−λ22≤L32ϕ2+M3T3,and so on(33)λ2n−λ2n−1≤L3n−1ϕ2+M3Tn.SinceLi<1 such that i=1,2,3 and T<1, then ∑n=1∞xn−xn−1,∑n=1∞λ1n−λ1n−1 and ∑n=1∞λ2n−λ2n−1 are uniformly convergent, and thus the sequences xnt, λ1tn and λ2tn are uniformly convergent.Now, we can apply the Picard method to obtain an approximate solution for our problem.From Theorem1, we get the system of the problem as the following:(34)dxdt=u11−x−u2x,dλ1dt=−λ1u1−λ1u2+ϕ1,λ2dt=−λ2u1−λ2u2−ϕ2,u1=λ1k11−x,u2=−λ2k2x,x0=x0,λ1T=0,λ2T=0.After reducing these equations, we obtain(35)dxdt=λ1k11−x2+λ2k2x2,dλ1dt=−λ12k11−x+λ1λ2k2x+ϕ1,dλ2dt=−λ1λ2k11−x+−λ22k2x−ϕ2,x0=x0,λ1T=0,λ2T=0.By integrating the differential equations in (35), we have the following system:(36)xt=x0+∫0tλ1k1−2λ1k1x+λ1k1+λ2k2x2dt,(37)λ1t=∫Tt−λ12k1+λ12k1x+λ1λ2k2x+ϕ1dt,(38)λ2t=∫Tt−λ1λ2k1+λ1λ2k1x+λ22k2x−ϕ2dt.By applying the Picard method to equations (36)–(38), we have(39)xnt=x0+∫0tλ1n−1k1−2λ1n−1k1xn−1+λ1n−1k1+λ2n−1k2xn−12dt,λ1tn=∫Tt−λ1n−12k1+λ1n−12k1xn−1+λ1n−1λ2n−1k2xn−1+ϕ1dt,λ2tn=∫Tt−λ1n−1λ2n−1k1+λ1n−1λ2n−1k1xn−1+λ2n−12k2xn−1−ϕ2dt,x0=x0,λ10=0,λ20=0.Forn=1, we have(40)x1t=x0+∫0tλ10k1−2λ10k1x0+λ10k1+λ20k2x02dt,λ1t1=∫Tt−λ102k1+λ102k1x0+λ10λ20k2x0+ϕ1dt,λ2t1=∫Tt−λ10λ10k1+λ10λ20k1x0+λ202k2x0−ϕ2dt,x0=x0,λ10=0,λ20=0.Therefore, the first approximation forx,λ1 and λ2 is the following:(41)x1t=x0,λ1t1=∫Ttϕ1dt=ϕ1t−T,λ2t1=∫Tt−ϕ2dt=ϕ2t−T.Hence, the first approximation ofu1,u2,c1, and c2 is the following:(42)u11=ϕ1t−Tk11−x0,u21=−ϕ2t−Tk21−x0,c11=k12ϕ1t−Tk11−x02,c21=k22−ϕ2t−Tk21−x02.Forn=2, we have(43)x2t=x0+∫0tλ11k1−2λ11k1x1+λ11k1+λ21k2x12dt,λ1t2=∫Tt−λ112k1+λ112k1x1+λ11λ21k2x1+ϕ1dt,λ2t2=∫Tt−λ11λ21k1+λ11λ21k1x1+λ212k2x1−ϕ2dt.Therefore, the second approximation forx,λ1 and λ2 is the following:(44)x2t=x0−t2ϕ12k1+tTϕ1k1+t2x02ϕ12k1−tTx02ϕ1k1−t2x02ϕ22k2+tTx02ϕ2k2,λ1t2=ϕ11/3t−T3k2−1+x0ϕ1+k1t−Tk2−1/3t−T3x0ϕ2k1k2,λ2t2=ϕ2−1/3t−T3k2−1+x0ϕ1+k1−tk2+Tk2+1/3t−T3x0ϕ2k1k2.Hence, the second approximation ofu1,u2,c1, and c2 is the following:(45)u12=ϕ11/3t−T3k2−1+x0ϕ1+k1t−Tk2−1/3t−T3x0ϕ2k12k21−x0−t2ϕ12k1+tTϕ1k1+t2x02ϕ12k1−tTx02ϕ1k1−t2x02ϕ22k2+tTx02ϕ2k2,u22=ϕ2−1/3t−T3k2−1+x0ϕ1+k1−tk2+Tk2+1/3t−T3x0ϕ2k12k2x0−t2ϕ12k1+tTϕ1k1+t2x02ϕ12k1−tTx02ϕ1k1−t2x02ϕ22k2+tTx02ϕ2k2,c12=ϕ11/3t−T3k2−1+x0ϕ1+k1t−Tk2−1/3t−T3x0ϕ2k12k21−x0−t2ϕ12k1+tTϕ1k1+t2x02ϕ12k1−tTx02ϕ1k1−t2x02ϕ22k2+tTx02ϕ2k2,c22=k22−ϕ2−1/3t−T3k2−1+x0ϕ1+k1tk2+Tk2+1/3t−T3x0ϕ2k1k22x0−t2ϕ12k1+tTϕ1k1+t2x02ϕ12k1−tTx02ϕ1k1−t2x02ϕ22k2+tTx02ϕ2k22.By puttingx0=0,ϕ1=0.1,ϕ2=0.3,k1=0.25,k2=0.5, and T=0.5, then we have a comparison between the obtained approximate solutions of the two firms in our problem in the following figures:Figure1 indicates the approximated optimal solutions of the state for our problem. In our example x and 1−x represented, respectively, the market share of Firm 1 and Firm 2 at time t, and from this figure, we found that the market share of Firm 1 is increasing and the market share of Firm 2 is decreasing with the time, and notice that Firm 2 has a greater market share than Firm 1 on the interval t∈0,0.5.Figure 1
Market share of firm 1 and firm 2 at timet.In Figure2, we show that the approximated optimal controls to the problem. In the problem, the controls u1 and u2 represented, respectively, the advertising efforts of Firm 1 and Firm 2 at time t, and from this figure, we noticed that the advertising efforts of Firm 1 is decreasing and the advertising efforts of Firm 2 increases until it reaches a certain time and then begins to decrease again. We found that Firm 1 has a greater advertising effort than Firm 2 on the interval t∈0,0.5.Figure 2
The advertising efforts of firm 1 and firm 2 at the timet.In Figure3, we explore the approximated solutions of the advertising cost function to the problem. In the problem, c1 and c2 represented respectively the advertising cost function of Firm 1 and Firm 2 at time t, and from this figure we noticed that the advertising cost function of Firm 1 is decreasing and the advertising cost function of Firm 2 is marginally increasing. We found that Firm 1 has a greater advertising cost function than Firm 2 on the interval t∈0,0.5.Figure 3
The advertising cost function of firm 1 and firm 2 at the timet.In Figure4, we present the approximated optimal solutions of the costate variables to the problem. The costate here represents the marginal value of the market share. In other words, the adjoint for this problem is the rate of change in the payoff for small changes in the market share. From this figure we see that in Firm 1, λ1 increases while λ2 decreases in Firm 2 on the time interval t∈0,0.5.Figure 4
The costate variables of firm 1 and firm 2 and the timet.


## 3.1. Existence and Convergence of the Solution

Now, we study the existence of the solution for the problem (10)–(11) (see [12, 13]) and we apply the Picard method for finding an approximate solution and studying the convergence of this solution.Consider the following system after reducing the necessary conditions of an open-loop Nash differential game as the following:(12)dxdt=f1x,λ1,λ2,t,dλ1dt=f2x,λ1,λ2,t+ϕ1,dλ2dt=f3x,λ1,λ2,t−ϕ2,x0=x0,λ1T=0,λ2T=0.Assume these assumptions for our problem.(1)
fixt,λ1t,λ2t,t:Rn×Rn×Rn×0,T⟶R are continuous and there are positive constants Mi s.t fi≤Mi,i=1,2,3.(2)
fi satisfies Lipschitz condition with Lipschitz constants Li,0<Li<1,i=1,2,3 such that(13)f1x,λ1,λ2,t−f1y,λ1,λ2,t≤L1x−y,f2x,λ1,λ2,t−f2x,p1,λ2,t≤L2λ1−p1,f3x,λ1,λ2,t−f3x,λ1,p2,t≤L3λ2−p2.We prove the existence of the solution of system (12), then we have to integrate the differential equation in (12), we get(14)xt=x0+∫0tf1x,λ1,λ2,tdt,(15)λ1t=ϕ1t−T+∫Ttf2x,λ1,λ2,tdt,(16)λ2t=−ϕ1t−T+∫Ttf3x,λ1,λ2,tdt.By differentiating the integral equations (14)–(16) with respect to t, we have(17)dxdt=f1x,λ1,λ2,t,dλ1dt=f2x,λ1,λ2,t+ϕ1,dλ2dt=f3x,λ1,λ2,t−ϕ2.Substitutingt=0 in (14), t=T in (15) and (16), we obtain(18)x0=x0+∫00f1x,λ1,λ2dt,t=x0,λ1T=ϕ1T−T+∫TTf2x,λ1,λ2,tdt=0,λ2T=−ϕ1T−T+∫TTf3x,λ1,λ2dt,t=0.Hence, the existence is proved from the equivalence between the system (12) and the integral equations ((14)–(16)). Therefore, there is a solution for our system.Now, we apply the Picard method to find the solutions of the integral equations (14)–(16). The solution is constructed by the sequences,(19)xnt=x0+∫0tf1xn−1,λ1n−1,λ2n−1,tdt,x0=x0,λ1tn=ϕ1t−T+∫Ttf2xn−1,λ1n−1,λ2n−1,tdt,λ10=0,λ2tn=−ϕ2t−T+∫Ttf3xn−1,λ1n−1,λ2n−1,tdt,λ20=0,n=1,2,3,….xnt,λ1tn,λ2tn can be written as the following:(20)xnt=x0+∑j=1nxj−xj−1,λ1tn=ϕ1t−T+∑j=1nλ1j−λ1j−1,λ2tn=−ϕ2t−T+∑j=1nλ2j−λ2j−1.Ifxn,λ1n, and λ2n are convergent, then the infinite series ∑xj−xj−1,∑λ1j−λ1j−1 and ∑λ2j−λ2j−1 are convergent, and the solution will be x,λ1, and λ2, where(21)xt=limn⟶∞xn,λ1t=limn⟶∞λ1n,λ2t=limn⟶∞λ2n.If the three series are converged, then the three sequencesxn,λ1n and λ2n will be converged, respectively, to xt,λ1t and λ2t. For proving the uniform convergence of xn,λ1n and λ2n, we have to consider the three associated series,(22)∑n=1∞xn−xn−1,∑n=1∞λ1n−λ1n−1,∑n=1∞λ2n−λ2n−1.Forn=1, we get(23)x1−x0=∫0tf1x0,λ10,λ20,tdt,x1−x0=∫0tf1x0,λ10,λ20,tdt,x1−x0≤M1T,λ11−λ10=ϕ1t−T+∫Ttf2x0,λ10,λ20,tdt,λ11−λ10≤ϕ1+M2t−T,λ11−λ10≤ϕ1+M2T,λ21−λ20=−ϕ2t−T+∫Ttf3x0,λ10,λ20,tdt,λ21−λ20≤ϕ2+M3t−T,λ21−λ20≤ϕ2+M3T.Now, we shall get an estimation forxn−xn−1, λ1n−λ1n−1, and λ2n−λ2n−1(24)xn−xn−1=∫0tf1xn−1,λ1n−1,λ2n−1,tdt−∫0tf1xn−2,λ1n−2,λ2n−2,tdt,xn−xn−1≤L1Txn−1−xn−2,λ1n−λ1n−1=∫Ttf2xn−1,λ1n−1,λ2n−1,tdt−∫Ttf2xn−2,λ1n−2,λ2n−2,tdt,λ1n−λ1n−1≤L2Tλ1n−1−λ1n−2,λ2n−λ2n−1=∫Ttf3xn−1,λ1n−1,λ2n−1,tdt−∫Ttf3xn−2,λ1n−2,λ2n−2,tdt,λ2n−λ2n−1≤L3Tλ2n−1−λ2n−2.By using the first estimation in (24) and putting n=2, we get(25)x2−x1≤L1Tx1−x0,x2−x1≤L1M1T2.Atn=3, we have(26)x3−x2≤L1Tx2−x1,x3−x2≤L12M1T3,and so on(27)xn−xn−1≤L1n−1M1Tn.By using the second estimation in (24), we have at n=2(28)λ12−λ11≤L2Tλ11−λ10,λ12−λ11≤L2ϕ1+M2T2.Atn=3(29)λ13−λ12≤L22ϕ1+M2T3,and so on(30)λ1n−λ1n−1≤L2n−1ϕ1+M2Tn.By using the third estimation in (24), we get at n=2(31)λ22−λ21≤L3Tλ21−λ20,λ22−λ21≤L3ϕ2+M3T2.Atn=3,(32)λ23−λ22≤L32ϕ2+M3T3,and so on(33)λ2n−λ2n−1≤L3n−1ϕ2+M3Tn.SinceLi<1 such that i=1,2,3 and T<1, then ∑n=1∞xn−xn−1,∑n=1∞λ1n−λ1n−1 and ∑n=1∞λ2n−λ2n−1 are uniformly convergent, and thus the sequences xnt, λ1tn and λ2tn are uniformly convergent.Now, we can apply the Picard method to obtain an approximate solution for our problem.From Theorem1, we get the system of the problem as the following:(34)dxdt=u11−x−u2x,dλ1dt=−λ1u1−λ1u2+ϕ1,λ2dt=−λ2u1−λ2u2−ϕ2,u1=λ1k11−x,u2=−λ2k2x,x0=x0,λ1T=0,λ2T=0.After reducing these equations, we obtain(35)dxdt=λ1k11−x2+λ2k2x2,dλ1dt=−λ12k11−x+λ1λ2k2x+ϕ1,dλ2dt=−λ1λ2k11−x+−λ22k2x−ϕ2,x0=x0,λ1T=0,λ2T=0.By integrating the differential equations in (35), we have the following system:(36)xt=x0+∫0tλ1k1−2λ1k1x+λ1k1+λ2k2x2dt,(37)λ1t=∫Tt−λ12k1+λ12k1x+λ1λ2k2x+ϕ1dt,(38)λ2t=∫Tt−λ1λ2k1+λ1λ2k1x+λ22k2x−ϕ2dt.By applying the Picard method to equations (36)–(38), we have(39)xnt=x0+∫0tλ1n−1k1−2λ1n−1k1xn−1+λ1n−1k1+λ2n−1k2xn−12dt,λ1tn=∫Tt−λ1n−12k1+λ1n−12k1xn−1+λ1n−1λ2n−1k2xn−1+ϕ1dt,λ2tn=∫Tt−λ1n−1λ2n−1k1+λ1n−1λ2n−1k1xn−1+λ2n−12k2xn−1−ϕ2dt,x0=x0,λ10=0,λ20=0.Forn=1, we have(40)x1t=x0+∫0tλ10k1−2λ10k1x0+λ10k1+λ20k2x02dt,λ1t1=∫Tt−λ102k1+λ102k1x0+λ10λ20k2x0+ϕ1dt,λ2t1=∫Tt−λ10λ10k1+λ10λ20k1x0+λ202k2x0−ϕ2dt,x0=x0,λ10=0,λ20=0.Therefore, the first approximation forx,λ1 and λ2 is the following:(41)x1t=x0,λ1t1=∫Ttϕ1dt=ϕ1t−T,λ2t1=∫Tt−ϕ2dt=ϕ2t−T.Hence, the first approximation ofu1,u2,c1, and c2 is the following:(42)u11=ϕ1t−Tk11−x0,u21=−ϕ2t−Tk21−x0,c11=k12ϕ1t−Tk11−x02,c21=k22−ϕ2t−Tk21−x02.Forn=2, we have(43)x2t=x0+∫0tλ11k1−2λ11k1x1+λ11k1+λ21k2x12dt,λ1t2=∫Tt−λ112k1+λ112k1x1+λ11λ21k2x1+ϕ1dt,λ2t2=∫Tt−λ11λ21k1+λ11λ21k1x1+λ212k2x1−ϕ2dt.Therefore, the second approximation forx,λ1 and λ2 is the following:(44)x2t=x0−t2ϕ12k1+tTϕ1k1+t2x02ϕ12k1−tTx02ϕ1k1−t2x02ϕ22k2+tTx02ϕ2k2,λ1t2=ϕ11/3t−T3k2−1+x0ϕ1+k1t−Tk2−1/3t−T3x0ϕ2k1k2,λ2t2=ϕ2−1/3t−T3k2−1+x0ϕ1+k1−tk2+Tk2+1/3t−T3x0ϕ2k1k2.Hence, the second approximation ofu1,u2,c1, and c2 is the following:(45)u12=ϕ11/3t−T3k2−1+x0ϕ1+k1t−Tk2−1/3t−T3x0ϕ2k12k21−x0−t2ϕ12k1+tTϕ1k1+t2x02ϕ12k1−tTx02ϕ1k1−t2x02ϕ22k2+tTx02ϕ2k2,u22=ϕ2−1/3t−T3k2−1+x0ϕ1+k1−tk2+Tk2+1/3t−T3x0ϕ2k12k2x0−t2ϕ12k1+tTϕ1k1+t2x02ϕ12k1−tTx02ϕ1k1−t2x02ϕ22k2+tTx02ϕ2k2,c12=ϕ11/3t−T3k2−1+x0ϕ1+k1t−Tk2−1/3t−T3x0ϕ2k12k21−x0−t2ϕ12k1+tTϕ1k1+t2x02ϕ12k1−tTx02ϕ1k1−t2x02ϕ22k2+tTx02ϕ2k2,c22=k22−ϕ2−1/3t−T3k2−1+x0ϕ1+k1tk2+Tk2+1/3t−T3x0ϕ2k1k22x0−t2ϕ12k1+tTϕ1k1+t2x02ϕ12k1−tTx02ϕ1k1−t2x02ϕ22k2+tTx02ϕ2k22.By puttingx0=0,ϕ1=0.1,ϕ2=0.3,k1=0.25,k2=0.5, and T=0.5, then we have a comparison between the obtained approximate solutions of the two firms in our problem in the following figures:Figure1 indicates the approximated optimal solutions of the state for our problem. In our example x and 1−x represented, respectively, the market share of Firm 1 and Firm 2 at time t, and from this figure, we found that the market share of Firm 1 is increasing and the market share of Firm 2 is decreasing with the time, and notice that Firm 2 has a greater market share than Firm 1 on the interval t∈0,0.5.Figure 1
Market share of firm 1 and firm 2 at timet.In Figure2, we show that the approximated optimal controls to the problem. In the problem, the controls u1 and u2 represented, respectively, the advertising efforts of Firm 1 and Firm 2 at time t, and from this figure, we noticed that the advertising efforts of Firm 1 is decreasing and the advertising efforts of Firm 2 increases until it reaches a certain time and then begins to decrease again. We found that Firm 1 has a greater advertising effort than Firm 2 on the interval t∈0,0.5.Figure 2
The advertising efforts of firm 1 and firm 2 at the timet.In Figure3, we explore the approximated solutions of the advertising cost function to the problem. In the problem, c1 and c2 represented respectively the advertising cost function of Firm 1 and Firm 2 at time t, and from this figure we noticed that the advertising cost function of Firm 1 is decreasing and the advertising cost function of Firm 2 is marginally increasing. We found that Firm 1 has a greater advertising cost function than Firm 2 on the interval t∈0,0.5.Figure 3
The advertising cost function of firm 1 and firm 2 at the timet.In Figure4, we present the approximated optimal solutions of the costate variables to the problem. The costate here represents the marginal value of the market share. In other words, the adjoint for this problem is the rate of change in the payoff for small changes in the market share. From this figure we see that in Firm 1, λ1 increases while λ2 decreases in Firm 2 on the time interval t∈0,0.5.Figure 4
The costate variables of firm 1 and firm 2 and the timet.


## 4. Discussion

In this study, we discussed the competing between two firms in the market.(1)
Firm 2 is existing in the market and there is no competitor to it, and its market share is1−xt. Firm 1 entered the market to compete this firm, and its market share is xt.(2)
Firm 1 started to make an advertising campaign by getting a loan. It earned profits and increased its sales.(3)
Firm 2 had no advertisements, while the number of customers of the new firm increased and so, its market share increased; therefore, the new firm controlled the market.(4)
Firm 2 started to lose and its sales decreased. It got a loan for doing advertisements because of its loss.(5)
The cost of Firm 1 was too much and after increasing its sales, the cost started to decrease and also the advertisements started to increase from the negative value. This means that debts decrease with time. After a specific time, the advertisements of the old firm started to increase and this increase is almost slight, so the cost seems constant.(6)
Finally, we used an iterative method (Picard method) for finding the solution and proved the convergence of the solution. On the other hand, Kristina used a numerical algorithm for the solution (Forward-Backward sweep algorithm), but she did not determine the solution [10].


## 5. Conclusion

In this paper, we concerned with an open-loop Nash differential game. We proved that there is a solution for the problem, we studied the existence for the solution of the dynamical system of the differential game, and we used the Picard method for finding an approximate solution. Also, we studied the convergence for the Picard method to make sure that the approximate solution is uniformly convergent. Finally, we added the figures to compare between the behavior of the two firms in the problem with respect to market share, the advertising efforts, the advertising cost function, and the costate variables. We proved the convergence atT<1, so we chose the interval [0, 0.5]. If we reach to T=1, the solution will be divergent.



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*Source: 7374882-2023-01-17.xml*