（節(jié)選）外文翻譯--具有不同類型參與人的古諾投資博弈動力學(xué)研究（英文）

1、Dynamics in a Cournot investment game with heterogeneous playersZhanwen Ding ?, Qiang Li, Shumin Jiang, Xuedi WangFaculty of Science, Jiangsu University, Zhenjiang 212013, PR Chinaa r t i c l e i n f oKeywords:Cournot ga

2、meInvestmentHeterogeneous expectationComplex dynamicsFeedback controla b s t r a c tIn this paper we concern the investment process in a duopoly game played by heteroge-neous players. A discrete and dynamic system is bui

3、lt for the case that a boundedly rationalplayer adjusts its investment decision by the locally marginal profit and a naïve playerchooses its strategy according to the opponent’s action in the previous period. By sta

4、bilityanalysis of the system, we show that the boundary equilibrium is unstable and obtain thestability conditions for the interior equilibrium. Numerical simulations are used to provideevidence for the influence of the

5、model parameters on the system stability and on the com-plicated behaviors in the system evolution. It is shown that the system with varying modelparameters may drive to chaos and the loss of stability may be caused by p

6、eriod doublingbifurcations or Neimark–Sacker bifurcations. It is also shown that the time-delayed feed-back control method can be used to keep the system from instability and chaos. All thenumerical simulations show that

7、 the capital depreciation rate has great influence on thesystem evolution: a smaller depreciation rate has a stronger stabilization effect on the sur-vival of the system and makes the system easier to control from chaos.

8、? 2015 Elsevier Inc. All rights reserved.1. IntroductionThe dynamical behaviors of oligopoly games are complex because every oligopolistic producer in each period must con-sider not only its own decision but also the rea

9、ctions of all other competitors. The earliest model giving a mathematical description of the competition in a duopolistic market was originally introduced by Cournot [1]. In the classic model, each participant uses a na&

10、#239;ve expectation to suppose that the opponents’ output keeps the same level as in the previous period and adopts an output strategy to maximize the expected profit. Many researchers have analyzed the system stability

11、and the complex phenomena in Cournot oligopoly games with this kind of expectation (e.g., [2–9]).In an early work by Bischiy and Naimzadaz [10], a kind of bounded rationality was assumed for the dynamical Cournotgame, wh

12、ere each producer does not have complete knowledge of the market and updates its production by the local profit maximization method. That is, a producer with bounded rationality increases its output if it perceives a pos

13、itive marginal profit and decreases its production if the perceived marginal profit is negative.In recent years, a great amount of work has been done on the dynamical Cournot games with homogeneous or hetero-geneous expe

14、ctations. Bounded rationality assumed in the marginal profit method is related to all producers in the models considering homogeneous expectation (e.g., [10–13]). The models with heterogeneous expectations (naïve, b

15、oundedly rational or adaptive) have been discussed in many other researches (e.g., [14–20]).http://dx.doi.org/10.1016/j.amc.2015.01.0600096-3003/? 2015 Elsevier Inc. All rights reserved.? Corresponding author.E-mail addr

16、ess: dgzw@ujs.edu.cn (Z. Ding).Applied Mathematics and Computation 256 (2015) 939–950Contents lists available at ScienceDirectApplied Mathematics and Computationjournal homepage: www.elsevier.com/locate/amcx2ðt 

17、4; 1Þ ¼ aB2 ? B2c2 ? 2bB2 2ð1 ? hÞK2ðt ? 1Þ ? bB1B2ðð1 ? hÞK1ðt ? 1Þ þ x1ðtÞÞ ? 12bB2 2: ð9ÞFrom all the Eqs. (4), (6), (7) and (9), we

18、 finally obtain a nonlinear dynamics with four variables x1; x2 , K1 and K2 :x1ðt þ 1Þ ¼ x1ðtÞ þ ax1ðtÞðaB1 ? B1c1 ? 2bB2 1ðð1 ? hÞK1ðt ? 1Þ 

19、4; x1ðtÞÞ?bB1B2ðð1 ? hÞK2ðt ? 1Þ þ x2ðtÞÞ ? 1Þx2ðt þ 1Þ ¼aB2?B2c2?2bB22ð1?hÞK2ðt?1Þ?bB1B2ðð1?hÞK1

20、40;t?1Þþx1ðtÞÞ?12bB22K1ðtÞ ¼ ð1 ? hÞK1ðt ? 1Þ þ x1ðtÞK2ðtÞ ¼ ð1 ? hÞK2ðt ? 1Þ þ x2ðtÞ:8 > &g

21、t; > > > > >> > > > > > :ð10ÞIf we denote Kiðt ? 1Þ by IiðtÞ and hence KiðtÞ by Iiðt þ 1Þ ði ¼ 1; 2Þ, then we can rew

22、rite system (10) as the following standard dynamics:x1ðt þ 1Þ ¼ x1ðtÞ þ ax1ðtÞðaB1 ? B1c1 ? 2bB2 1ðð1 ? hÞI1ðtÞ þ x1ðtÞÞ?bB1B

23、2ðð1 ? hÞI2ðtÞ þ x2ðtÞÞ ? 1Þx2ðt þ 1Þ ¼aB2?B2c2?2bB22ð1?hÞI2ðtÞ?bB1B2ðð1?hÞI1ðtÞþx1ðtÞ

24、2;?12bB22I1ðt þ 1Þ ¼ ð1 ? hÞI1ðtÞ þ x1ðtÞI2ðt þ 1Þ ¼ ð1 ? hÞI2ðtÞ þ x2ðtÞ:8 > > > > > > >

25、> > > > > > :ð11ÞThe nonlinear and discrete system (11) describes a duopoly game where a boundedly rational player and a naïve player make their decisions in a process of dynamical invest

26、ment. In the following section, the stability properties of this model will be discussed.3. The equilibrium points and stabilityIn order to study the qualitative behavior of system (11), we first find out its equilibrium

27、 points, which can be obtained bysetting xiðt þ 1Þ ¼ xiðtÞ and Iiðt þ 1Þ ¼ IiðtÞ in (11) so that the following algebraic system is satisfied:ax1ðtÞ

28、40;aB1 ? B1c1 ? 2bB2 1ðð1 ? hÞI1ðtÞ þ x1ðtÞÞ ? bB1B2ðð1 ? hÞI2ðtÞ þ x2ðtÞÞ ? 1Þ ¼ 0aB2?B2c2?2bB22ð1?hÞI2ð

29、tÞ?bB1B2ðð1?hÞI1ðtÞþx1ðtÞÞ?12bB22? x2ðtÞ ¼ 0x1ðtÞ ? hI1ðtÞ ¼ 0x2ðtÞ ? hI2ðtÞ ¼ 0:8 > > > > &g

30、t;> > > > :ð12ÞSolving the equation system (12), we obtain two equilibrium points:E ¼ 0; hðaB2 ? B2c2 ? 1Þ2bB2 2; 0; aB2 ? B2c2 ? 12bB2 2!;E? ¼ ðx?1; x?2; I?1; I?2Þ;whe

31、rex?1 ¼ hðB1 ? 2B2 þ B1B2ða ? 2c1 þ c2ÞÞ3bB2 1B2; ð13aÞx?2 ¼ hðB2 ? 2B1 þ B1B2ða þ c1 ? 2c2ÞÞ3bB1B22; ð13bÞI? 1 ¼ B1 ? 2B2 &#

32、254; B1B2ða ? 2c1 þ c2Þ3bB2 1B2; ð13cÞI? 2 ¼ B2 ? 2B1 þ B1B2ða þ c1 ? 2c2Þ3bB1B22: ð13dÞE is a boundary equilibrium point and E? is an interior one. In order to

33、 ensure their economic significance, we only con-sider the case that E and E? are nonnegative. Since a; b; c1; c2; B1; B2 and h are all positive parameters, E and E? will be non- negative provided that the following ineq