开环非合作微分对策及其在经济中的应用
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摘要
本文主要利用最大值原理和动态规划方法探讨微分对策在经济上的相关分析和应用.全文分为四章,主要内容如下.
在第一章绪论中,我们介绍了本文的研究背景及需要的研究工具,并详细给出了微分对策的基本概念和解的分类.
在第二章,我们研究了随机开环二人零和线性二次型微分对策的鞍点存在性.利用Peng最大值原理及倒向随机微分方程与原方程的关系,证明了开环鞍点的存在性等价于对策上,下值的存在性,而不需要对策上,下值相等这一条件.这一结果与其他学者用Hilbert空间上对策问题分析方法给出的结论一致.
在第三章,我们讨论了反恐措施与经济发展综合考虑的微分对策模型.政府的反恐能力应与其经济实力息息相关,而前人在研究时没有考虑到这一情况.本章利用Pontryagin最大值原理和带跳扩散随机最大值原理分别讨论了确定型与随机型微分对策问题,给出了相应的开环鞍点,并分析了数学结果在现实中实际意义.
在第四章,我们考虑了n-人Bertrand竞争寡头垄断模型.由于前人未考虑到实际需求与理想化需求量的差别,本章通过建立一个动态的实际需求函数来刻画一个快消品的市场环境.证明了n-人Bertrand竞争的稳定点就是微分对策的鞍点,并通过对鞍点的分析,探讨了数学结果在经济生活中的意义.
Game Theory was established in 1944 by Von Nenmann and Morgerstern in "The Theory of Games and Economic Behavior". And Differential game is one kind problem of Game Theory, which state and control function relating time. Its theory was estab-lished in 1965 by R.Isaacs in "Differential Games". Differential game is a mathematical method that studies how to establish the most advantageous strategies of players in the conflict situation.
Game theory is the most appropriate method to establish economic behavior. So we discuss differential game relating the economic applications in this paper.
The main content of this paper is following. We study one kind of two-player zero-sum linear quadratic stochastic differential game problem on a finite horizon. By the theory of Peng's stochastic maximum principle, no equivalent of the lower and upper values, we can prove the existence of the open-loop saddle point if and only if the lower and upper values exist. And we discuss the differential game model about countert-crror measures and economic development. We still consider a differential oligopoly game under n-Bertrand competition. Open-loop saddle points of above problems are given and the meaning of economic life about the mathematical results also is discussed.
As the foundation of differential game theory, the existence of the saddle point is the concern problem by scholars. Linear quadratic differential game has the important position in economic model. Zhang and Delfour had proved existence of saddle point of determinate open-loop two-player zero-sum linear quadratic differential game if and only if the lower and upper values exist until 2007. And Mou and Yong got same results thorough study of general two-player zero-sum linear quadratic games in Hilbert spaces.
In this paper, we can prove the same results in stochastic differential game. Let x be a solution of the following stochastic differential equation For any choice of controls u, v we have the following payoff function We assume thatΩbe a bounded smooth domain in Rn. Let (Ω, F, P) be a probability space with filtration Ft. Let W(·) be an Rn-valued standard Wiener process. F is an n×n matrix and that A(t), B1(t),B2(t), C1(t), C2(t), D(t) and Q(t) are matrix func-tions of appropriate order that are measurable and bounded a.e. in [0,T]. Moreover, F and Q(t) are symmetrical.
Since the payoff function is quadratic, it is infinitely differentiable. We can prove
dCx0(u, v; u, v)=E(Fx(T)·y(T)+(Qx,y)+(u,u)-(v,v)),
where x is the solution of (0.0.9) and y is the solution of
We define the Hamiltonian by The adjoint equation of (0.0.9) (0.0.10) is whereσ=Dx+C1u+C2v∈[0,T]×Rn×Rm×Rk andσ=(σ1,σ2,…,σd). Moreover (p(·),K(·))∈L2(0,T;Rn)×(L2(0,T; Rn))d.
Proposition 0.1 According to adjoint equation (0.0.11), we can rewrite expression (0.0.10) in the following form.
Similarly, the second order bidirectional derivative of payoff function is the follow-ing form where In particular, then for all x0, u, v, u and v d2Cxo(u,v;u,v;u,v)= C0(u,v). Namely, the second order bidirectional derivative of payoff function is independent of x0 and (u,v).
Theorem 0.2 If V(x0)≠(?) and U(x0)≠(?), then the saddle point of payoff Co(u,v) exists and it is (0,0).
Theorem 0.3 There exists a solution (x,p,K) of the adjoint system If then
The main results in this paper is the following theorem.
Theorem 0.4 Consider the stochastic differential game(0.0.9),(0.0.10).The follow-ing statements are equivalent.
(ⅰ)There exists an open loop saddle point of Cx0(u,v).
(ⅱ)The value of the game exists.
(ⅲ)Both the lower value and the upper value of the game exist.
We prove the above Theorem by the following theorems.
Theorem 0.5 The following statements are equivalent.
(ⅰ)There exist u*∈L2(0,T;Rm)and v*∈L2(0,T；Rk)such that
(ⅱ)The open loop lower value v-(x0)of the game exists.
(ⅲ)There exists a solution(x,p,K)of the adjoint system(0.0.12)such that B2*p+C2*K∈
V(x0),the solution pairs(u*,v*)is(0.0.13),and the open loop lower value are given by(0.0.14).
Theorem 0.6 The following statements are equivalent.
(ⅰ)There exist u*∈L2(0,T;Rm)and v*∈L2(0,T;Rk)such that
(ⅱ)The open loop upper value v+(x0)of the game exists.
(ⅲ)There exists a solution(x,p,K)of the adjoint system(0.0.12)such that-B1*p- C1*K∈U(x0), the solution pairs(u*,v*) is (0.0.13), and the open loop lower value is given by (0.0.14).
Then, according to the literature [77], [37], [38], [48] only considering counterterror measures, we consider how to allocate the resources of governments such that benefit economic development and defense terrorist organizations simultaneously. We give the saddle points of determinate and stochastic differential games. And we analyze the relationship of the national economic development and counterterror measures.
We consider a determinate differential game. There are two state functions x(t),y(t), where x(t) describes the size of RTO, which include personnel, technology, capital, and weapons etc. of terrorist organization. And y(t) describes resources of government that is economic income. Government and terrorist organization are two players in this dif-ferential game. The control function of the player 1 (government) u(t),0≤u(t)≤1, is the devoted proportion for counterterror, and 1-u(t) is the part one for economic development. v(t)≥0 is the control function of the player 2 (terrorist organization), which is intensity of attacks. The dynamic system can be written as where 0 0 on [0,∞). b>0 is the coefficient of the player 2's resources growth.α>0 is the coefficient of the player l's income growth. Unit loss of the player 2's income is denoted as f＞0.
Let the objective function of the two players be where l, c, k, p are positive constants. Player 1 hopes to use control function u(t) so that increase objective function J(u,v) as far as possible. On the contrary, player 2 hopes to use control function v(t) so that decrease objective function J(u, v) as far as possible. p is the discount rate. We let p> b, p>αin this paper.
Proposition 0.7 There are two saddle points of the differential game (0.0.15)(0.0.16), which(u1(t),vi(t))=(0,0) and (u2(t), v2(t)) is given by and where
The above proposition can be interpreted as follow. This differential game admits two saddle points, one of them is (0,0). It means that the relation between government and the terrorist organization is peaceful, i.e. the terrorist organizations do not orga-nize any attack at all. Even there is no terrorist organization. In this situation, the government need not worry about any attack. All the income can be spent on economic reproduction without any counterterror spending. The government and the terrorist organizations both can obtain the optimal benefit in this case. But in fact, this case is almost impossible. When the terrorist organizations attack, the optimal strategy of the terrorist organizations is keeping certain intensity of attacks. It is expressed that the optimal strategy is a constant in mathematic model, i.e. v2. At this time, no matter what attacks happen, whether economic grow or decay, the optimal strategy of the government is spending certain funds for counterterror all the time. This phe-nomenon is expressed as a constant u2y. So, the differential game reaches equilibrium. We find that ifαis large enough, then the income of the government is positive growth along with the increase of coefficient a. If a is increasing, the government is attacked more frequently. It can be interpreted that terrorist organizations often choose the government of which economic development is faster as the attack object.
It is easy to occur emergencies in a conflict, such as injuring civilian by countert-error etc. A stochastic differential game is used to described these problems. Nt=Nt-(?)λ(s)ds is a compensated Poisson process, where Nt is Poisson rando measures with Levy measuresλ(t) on a filtered probability space (Ω, F,{Ft}t≥0, P) satisfying the usual conditions.
For each u(t) and v(t), the objective function of the both players is where E is the expectation under the probability P.
Proposition 0.8 We assume the saddle point is (u,v), and corresponding state func-tion is (x,y), then
In addition, We compare the saddle points of determinate game and numerical solution of saddle points of stochastic game in graph.
In final, we consider a differential oligopoly game under Bertrand competition. In this paper, a FMCG market environment is described by a dynamic practical demand function, which determines the steady point of the n-player Bertrand is the saddle point of deferential game. The outcome of the optimal prices and demand quantity are different, depending on the situation of the manufacturers. It is different from other literature,in which the solutions are in the same way.Via the analysis of optimal price and demand quantity,we get to know the key point of a manufacturer to get a bigger profit is to control its cost and occupy market rapidly.
Each player chooses his control variable pi(t)over time,from the present to infinity, i.e.,t∈[0,∞),in order to maximize the value of the profit.The problem of firm i can be written as following
Proposition 0.9 The steady state of the differential game is a saddle point.
Proposition 0.10 The open-loop Nash equilibrium,i.e.the optimal market price and demand quantity are where
Simple computations on the steady price and demand quantity lead to the follow-ing results.
(1)((?)pi*)/((?)fi)>0 and ((?)qi*)/((?)fi)<0.A higher cost means a high price for customers and lower market share for manufacturer i.Therefore,reduction of the cost effectively may im-prove the competitive power of products.
(2)((?)pi*)/((?)Ai)＞0 and ((?)pi*)/((?)Ai)>0.A higher initial market size for manufacturer i means it can make a bigger profit by increasing price and output. That's why every manufacturer attempts to occupy the market by promoting new products as soon as possible. Once the products accepted by consumers, no matter how many similar products come out following, the earlier one may get bigger profit, because it has already occupied a bigger market share. In one word, the competition of the manufacturers is the competition of market share.
在第一章绪论中,我们介绍了本文的研究背景及需要的研究工具,并详细给出了微分对策的基本概念和解的分类.
在第二章,我们研究了随机开环二人零和线性二次型微分对策的鞍点存在性.利用Peng最大值原理及倒向随机微分方程与原方程的关系,证明了开环鞍点的存在性等价于对策上,下值的存在性,而不需要对策上,下值相等这一条件.这一结果与其他学者用Hilbert空间上对策问题分析方法给出的结论一致.
在第三章,我们讨论了反恐措施与经济发展综合考虑的微分对策模型.政府的反恐能力应与其经济实力息息相关,而前人在研究时没有考虑到这一情况.本章利用Pontryagin最大值原理和带跳扩散随机最大值原理分别讨论了确定型与随机型微分对策问题,给出了相应的开环鞍点,并分析了数学结果在现实中实际意义.
在第四章,我们考虑了n-人Bertrand竞争寡头垄断模型.由于前人未考虑到实际需求与理想化需求量的差别,本章通过建立一个动态的实际需求函数来刻画一个快消品的市场环境.证明了n-人Bertrand竞争的稳定点就是微分对策的鞍点,并通过对鞍点的分析,探讨了数学结果在经济生活中的意义.
Game Theory was established in 1944 by Von Nenmann and Morgerstern in "The Theory of Games and Economic Behavior". And Differential game is one kind problem of Game Theory, which state and control function relating time. Its theory was estab-lished in 1965 by R.Isaacs in "Differential Games". Differential game is a mathematical method that studies how to establish the most advantageous strategies of players in the conflict situation.
Game theory is the most appropriate method to establish economic behavior. So we discuss differential game relating the economic applications in this paper.
The main content of this paper is following. We study one kind of two-player zero-sum linear quadratic stochastic differential game problem on a finite horizon. By the theory of Peng's stochastic maximum principle, no equivalent of the lower and upper values, we can prove the existence of the open-loop saddle point if and only if the lower and upper values exist. And we discuss the differential game model about countert-crror measures and economic development. We still consider a differential oligopoly game under n-Bertrand competition. Open-loop saddle points of above problems are given and the meaning of economic life about the mathematical results also is discussed.
As the foundation of differential game theory, the existence of the saddle point is the concern problem by scholars. Linear quadratic differential game has the important position in economic model. Zhang and Delfour had proved existence of saddle point of determinate open-loop two-player zero-sum linear quadratic differential game if and only if the lower and upper values exist until 2007. And Mou and Yong got same results thorough study of general two-player zero-sum linear quadratic games in Hilbert spaces.
In this paper, we can prove the same results in stochastic differential game. Let x be a solution of the following stochastic differential equation For any choice of controls u, v we have the following payoff function We assume thatΩbe a bounded smooth domain in Rn. Let (Ω, F, P) be a probability space with filtration Ft. Let W(·) be an Rn-valued standard Wiener process. F is an n×n matrix and that A(t), B1(t),B2(t), C1(t), C2(t), D(t) and Q(t) are matrix func-tions of appropriate order that are measurable and bounded a.e. in [0,T]. Moreover, F and Q(t) are symmetrical.
Since the payoff function is quadratic, it is infinitely differentiable. We can prove
dCx0(u, v; u, v)=E(Fx(T)·y(T)+(Qx,y)+(u,u)-(v,v)),
where x is the solution of (0.0.9) and y is the solution of
We define the Hamiltonian by The adjoint equation of (0.0.9) (0.0.10) is whereσ=Dx+C1u+C2v∈[0,T]×Rn×Rm×Rk andσ=(σ1,σ2,…,σd). Moreover (p(·),K(·))∈L2(0,T;Rn)×(L2(0,T; Rn))d.
Proposition 0.1 According to adjoint equation (0.0.11), we can rewrite expression (0.0.10) in the following form.
Similarly, the second order bidirectional derivative of payoff function is the follow-ing form where In particular, then for all x0, u, v, u and v d2Cxo(u,v;u,v;u,v)= C0(u,v). Namely, the second order bidirectional derivative of payoff function is independent of x0 and (u,v).
Theorem 0.2 If V(x0)≠(?) and U(x0)≠(?), then the saddle point of payoff Co(u,v) exists and it is (0,0).
Theorem 0.3 There exists a solution (x,p,K) of the adjoint system If then
The main results in this paper is the following theorem.
Theorem 0.4 Consider the stochastic differential game(0.0.9),(0.0.10).The follow-ing statements are equivalent.
(ⅰ)There exists an open loop saddle point of Cx0(u,v).
(ⅱ)The value of the game exists.
(ⅲ)Both the lower value and the upper value of the game exist.
We prove the above Theorem by the following theorems.
Theorem 0.5 The following statements are equivalent.
(ⅰ)There exist u*∈L2(0,T;Rm)and v*∈L2(0,T；Rk)such that
(ⅱ)The open loop lower value v-(x0)of the game exists.
(ⅲ)There exists a solution(x,p,K)of the adjoint system(0.0.12)such that B2*p+C2*K∈
V(x0),the solution pairs(u*,v*)is(0.0.13),and the open loop lower value are given by(0.0.14).
Theorem 0.6 The following statements are equivalent.
(ⅰ)There exist u*∈L2(0,T;Rm)and v*∈L2(0,T;Rk)such that
(ⅱ)The open loop upper value v+(x0)of the game exists.
(ⅲ)There exists a solution(x,p,K)of the adjoint system(0.0.12)such that-B1*p- C1*K∈U(x0), the solution pairs(u*,v*) is (0.0.13), and the open loop lower value is given by (0.0.14).
Then, according to the literature [77], [37], [38], [48] only considering counterterror measures, we consider how to allocate the resources of governments such that benefit economic development and defense terrorist organizations simultaneously. We give the saddle points of determinate and stochastic differential games. And we analyze the relationship of the national economic development and counterterror measures.
We consider a determinate differential game. There are two state functions x(t),y(t), where x(t) describes the size of RTO, which include personnel, technology, capital, and weapons etc. of terrorist organization. And y(t) describes resources of government that is economic income. Government and terrorist organization are two players in this dif-ferential game. The control function of the player 1 (government) u(t),0≤u(t)≤1, is the devoted proportion for counterterror, and 1-u(t) is the part one for economic development. v(t)≥0 is the control function of the player 2 (terrorist organization), which is intensity of attacks. The dynamic system can be written as where 0
Let the objective function of the two players be where l, c, k, p are positive constants. Player 1 hopes to use control function u(t) so that increase objective function J(u,v) as far as possible. On the contrary, player 2 hopes to use control function v(t) so that decrease objective function J(u, v) as far as possible. p is the discount rate. We let p> b, p>αin this paper.
Proposition 0.7 There are two saddle points of the differential game (0.0.15)(0.0.16), which(u1(t),vi(t))=(0,0) and (u2(t), v2(t)) is given by and where
The above proposition can be interpreted as follow. This differential game admits two saddle points, one of them is (0,0). It means that the relation between government and the terrorist organization is peaceful, i.e. the terrorist organizations do not orga-nize any attack at all. Even there is no terrorist organization. In this situation, the government need not worry about any attack. All the income can be spent on economic reproduction without any counterterror spending. The government and the terrorist organizations both can obtain the optimal benefit in this case. But in fact, this case is almost impossible. When the terrorist organizations attack, the optimal strategy of the terrorist organizations is keeping certain intensity of attacks. It is expressed that the optimal strategy is a constant in mathematic model, i.e. v2. At this time, no matter what attacks happen, whether economic grow or decay, the optimal strategy of the government is spending certain funds for counterterror all the time. This phe-nomenon is expressed as a constant u2y. So, the differential game reaches equilibrium. We find that ifαis large enough, then the income of the government is positive growth along with the increase of coefficient a. If a is increasing, the government is attacked more frequently. It can be interpreted that terrorist organizations often choose the government of which economic development is faster as the attack object.
It is easy to occur emergencies in a conflict, such as injuring civilian by countert-error etc. A stochastic differential game is used to described these problems. Nt=Nt-(?)λ(s)ds is a compensated Poisson process, where Nt is Poisson rando measures with Levy measuresλ(t) on a filtered probability space (Ω, F,{Ft}t≥0, P) satisfying the usual conditions.
For each u(t) and v(t), the objective function of the both players is where E is the expectation under the probability P.
Proposition 0.8 We assume the saddle point is (u,v), and corresponding state func-tion is (x,y), then
In addition, We compare the saddle points of determinate game and numerical solution of saddle points of stochastic game in graph.
In final, we consider a differential oligopoly game under Bertrand competition. In this paper, a FMCG market environment is described by a dynamic practical demand function, which determines the steady point of the n-player Bertrand is the saddle point of deferential game. The outcome of the optimal prices and demand quantity are different, depending on the situation of the manufacturers. It is different from other literature,in which the solutions are in the same way.Via the analysis of optimal price and demand quantity,we get to know the key point of a manufacturer to get a bigger profit is to control its cost and occupy market rapidly.
Each player chooses his control variable pi(t)over time,from the present to infinity, i.e.,t∈[0,∞),in order to maximize the value of the profit.The problem of firm i can be written as following
Proposition 0.9 The steady state of the differential game is a saddle point.
Proposition 0.10 The open-loop Nash equilibrium,i.e.the optimal market price and demand quantity are where
Simple computations on the steady price and demand quantity lead to the follow-ing results.
(1)((?)pi*)/((?)fi)>0 and ((?)qi*)/((?)fi)<0.A higher cost means a high price for customers and lower market share for manufacturer i.Therefore,reduction of the cost effectively may im-prove the competitive power of products.
(2)((?)pi*)/((?)Ai)＞0 and ((?)pi*)/((?)Ai)>0.A higher initial market size for manufacturer i means it can make a bigger profit by increasing price and output. That's why every manufacturer attempts to occupy the market by promoting new products as soon as possible. Once the products accepted by consumers, no matter how many similar products come out following, the earlier one may get bigger profit, because it has already occupied a bigger market share. In one word, the competition of the manufacturers is the competition of market share.
引文
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