On Bertrand duopoly game with differentiated goods

Applied Mathematics and Computation 251 (2015) 169–179

Contents lists available at ScienceDirect

Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

On Bertrand duopoly game with differentiated goods E. Ahmed a, A.A. Elsadany b,c,⇑, Tonu Puu d a

Department of Mathematics, Faculty of Science, Mansoura University, Egypt Department of Basic Science, Faculty of Computers and Informatics, Ismailia, Suez Canal University, Egypt Department of Mathematics, Shanghai University, Shanghai 200444, China d Umea University, Faculty of Social Sciences, Centre for Regional Science (CERUM), 90187 Umea, Sweden b c

a r t i c l e

i n f o

Keywords: Bertrand game CES utility function Nash equilibrium point Bifurcation Chaos

a b s t r a c t The paper investigates a dynamic Bertrand duopoly with differentiated goods in which boundedly rational firms apply a gradient adjustment mechanism to update their price in each period. The demand functions are derived from an underlying CES utility function. We investigate numerically the dynamical properties of the model. We consider two specific parameterizations for the CES function and study the Nash equilibrium and its local stability in the models. The general finding is that the Nash equilibrium becomes unstable as the speed of adjustment increases. The Nash equilibrium loses stability through a period-doubling bifurcation and the system eventually becomes chaotic either through a series of period-doubling bifurcations or after a Neimark–Sacker bifurcation. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction Duopoly game is one of the oldest branches of mathematical economics see Augustin Cournot 1838 [1]. Two sellers were assumed to sell a homogenous commodity in the same market. As market price was assumed dependent on the total quantity sold by both competitors, the case was more complicated than monopoly. The optimal decision of each competitor became dependent, not only on repercussions on the demand side, but also on the expected retaliation of the other competitor on any of its own moves. Bertrand [2] pointed out that if the commodity really was conceived as completely homogenous by the consumers, then any competitor could use price instead of quantity for action, and take the entire market by just undercutting the other’s price by whatever tiny fraction. As elaborated by Edgeworth [3], and finally summarized in Chamberlin’s theory of monopolistic competition, the commodity could be considered as slightly heterogenous; the sellers would be offering close substitutes, but the consumers would have slight preferences for one or the other, so that they might be willing to pay a little more for the preferred supplier, but just a little. In this context, we should not forget Harold Hotelling’s ingenious solation 1929 [4] where the commodity with Cournot was assumed to be homogenous, but where the whole problem was put in a spatial context with local monopoly areas protected through transportation costs, but competition at the area boundaries. So, we still differentiate between homogenous Cournot oligopoly, where the competitors set quantities sold, and heterogenous Bertrand oligopoly, where they decide prices. Notably, in more recent literature, much more has been written about the Cournot case. This seems to be due to the fact that it presents more possibilities to formulate interesting dynamic

⇑ Corresponding author at: Department of Basic Science, Faculty of Computers and Informatics, Ismailia, Suez Canal University, Egypt. E-mail addresses: [email protected] (E. Ahmed), [email protected] (A.A. Elsadany), [email protected] (T. Puu). http://dx.doi.org/10.1016/j.amc.2014.11.051 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.

170

E. Ahmed et al. / Applied Mathematics and Computation 251 (2015) 169–179

models. A first textbook approach in all economics has been to assume linear demand functions; market price as a decreasing linear function of total demand, or the reverse. This case has its problems; due to economic facts, negative prices or demand/ supply quantities make no sense, so such linear functions must be stated as piecewise linear. Due to Joan Robinson 1933 [5], aggregate market demand composed from such, becomes a train of linear segments, which results in marginal cost consisting of discontinuous line segments, and multiple local optima, even for a single monopolist. In 1939 Tord Palander [6] extended this to Cournot duopoly, and discovered some complex dynamics, with multiple fixed points and coexistent oscillations. So, linear demand is not as simple as it appears, at least not if one wants to address global analysis. There are numerous studies, most addressing local issues, of Cournot duopoly, for instance the much cited 1959 Theocharis article [7]. Caplin and Nalebuff [8] presented a new approach to the theory of imperfect competition and apply it to study price competition among differentiated products. In addition, they have been shown both the existence and uniqueness of an equilibrium solution given any number of competing products. As for the Bertrand case, its dynamics under bounded rationality was studied in Zhang et al. [9]. Ahmed et al. [10] investigated the stability of a multi-team Bertrand game. Stability and instability in oligopoly has been discussed by Furth [11]. The stability of best reply and gradient systems with applications to imperfectly competitive models has been investigated in [12]. Bischi and Naimzada [13] has studied the dynamical properties of bounded rationality duopoly game. Agiza and Elsadany [14,15] has been introduced duopoly games with heterogeneous players, and in particular they analyzed the nonlinear dynamics emerging in these kinds of model. Nonlinear oligopolies have been surveyed in [16]. Dubiel-Teleszynski [17] studied a heterogeneous duopoly game with adjusting players and diseconomies of scale. Learning cycles in Bertrand competition with differentiated commodities and competing learning rules has been studied by Anufriev et al. [18]. Recently, behavioral rationality and heterogeneous expectations in complex economic systems has been surveyed by Hommes [19]. The stability of gradient learning in oligopoly was analyzed by [13– 15]. They prove that the steady state loses stability as the speed of adjustment increases. These findings are in line with our results. Agiza et al. [20] considers a similar setup as our model but in Cournot competition. They are investigated the existence of Nash equilibrium point and its local stability of the game and the route to chaos. Zhang [21] built a Bertrand repeated game model with linear demand function based on heterogeneous expectations, and investigated its system complexity. Puu [22] introduces bounded rationality in the model in a different way so-called Puu’s incomplete information. It has main advantage that it is realistic since a firm does not need to know the form of the profit function to get an estimate of the quantity produced in the next time step. Recently, Ahmed et al. [23] have reported models based on Puu’s techniques are numerically unstable when approaching to the equilibrium point. Sun et al. [24] studied the complex dynamics of a nonlinear Chinese cold rolled steel market model based on Bertrand game. Xu et al. [25] analyze the dynamic model of a Bertrand game with delay in insurance market and analyzed the existence and stability of the Nash equilibrium point of the game. Zhang [26] considered an nonlinear Bertrand game of insurance market in which one of the two players in the market made decision only with bounded rationality without delay, and the other player made the delayed decision with one period and two periods. Fanti et al. [27] analyzed the dynamics of a Bertrand duopoly with differentiated products. The results showed that an increase in either the degree of substitutability or complementarity between products of different varieties was a source of complexity in a competition game. Zhao and Zhang [28] studied a Bertrand duopoly game with heterogeneous players participating in carbon emission trading and analyzed the asymptotic stability of the equilibrium points of the game. Other studies on the dynamics of Bertrand models with more firms and other modifications have been studied [29–34]. Other works have also explored evolutionary game theory [35] both in continuum and on networks [36,37]. Imitating the most successful player is studied in [38–40]. Xia et al. [41] analyzed the effects of delayed recovery and nonuniform transmission on the spreading of diseases in complex networks. Optimal interdependence between networks for the evolution of cooperation is introduced in [42]. Yuan and Xia [43] have introduced the role of investment heterogeneity in the cooperation on spatial public goods game. They deduced that the cooperative groups will receive the more investment. On the other hand, the cooperator who gives this group will also obtain the higher payoffs. This duplicate mechanism to promote the cooperation encourages more players to adopt the cooperation strategy and enhance the collective cooperation. However, most above studies about Bertrand game are investigated the game with linear demand function [9,10,21,27,28]. In real life, the relationship between price and demand of products is very complicated and is not just a simple linear relationship. So the nonlinear demand function can more accurately reflect the relationship between price and demand. To avoid the discontinuity problems inherent with linear demand, one of the present authors [44] in 1991 suggested a different and more smooth kind of demand function where total market demand and price are reciprocal. It has become quite popular, and goes under the name of ‘‘isoelastic’’ demand function.1 Economists have vary few shapes of demand functions that are both derivable from utility functions, and that result in derivable reaction functions. Unfortunately, the reaction function cannot be solved for in our case either, which is the reason for the assumed adjustment process. So it seems interesting to see what it can do in dynamics. In particular, Bertrand game based on nonlinear demand functions and its dynamical behavior needs to be further studied. This paper aims to set up a Bertrand duopoly game with differentiated goods and that is characterized by standard assumptions such as a nonlinear price function and linear cost function. We apply dynamic methods to investigate the dynamic behaviors of this nonlinear duopoly game.

1 Concerning terminology, economists have since ages measured derivatives of functions as logarithmic derivatives, called ‘‘elasticities’’. For the function y ¼ f ðxÞthe elasticity is d log y=d log x ¼ dy=dx x=y. The idea seems to have been to obtain a dimensionless measure. If the elasticity is constant along the whole range of a function, one says that it is iso-elastic.


171

The rest of the paper is organized as follows: In Section 2 we present the market structure of Bertrand duopoly game with nonlinear demand function and differentiated goods. In Section 3 we study a dynamical analysis for a ¼ 12. Dynamical analysis of the game for a ¼ 13 will be investigated in Section 4. Section 5 summarizes our results. 2. Market structure We said that for the Bertrand case we need the commodities to be close substitutes, so the Cobb-Douglas simply does not work. One needs a utility maximization where the demand functions come out so that the commodities are substitutes. A so pffiffiffiffiffi pffiffiffiffiffi called CES function will serve, for instance U ¼ q1 þ q2 ; subject to the same budget constraint as above. The result of the constrained utility maximization then is q1 ¼ p2 =ðp1 ðp1 þ p2 ÞÞ and q2 ¼ p1 =ðp2 ðp1 þ p2 ÞÞ.2 Replacing the square roots with other fractional exponents one can give much sharper substitutively which one needs for Bertrand game, but the computations become more awkward. In the sequel we derive the demand function for substitutes, as suitable for the Bertrand problem from the following utility function:

Uðq1 ; q2 Þ ¼ qa1 þ qa2 ;

0 0 > p ¼ p1 þ k1 > 2 2 > < 1 p1 þ p1 p2 > p1 p22 þ 2c2 p1 p2 þ c2 p21 > 0 > > p2 ¼ p2 þ k2 2 2 : p1 þ p1 p2

ð10Þ

where 0 denotes the unit-time advancement operator, that is, if the right-hand side variables are prices of period t, then the left-hand ones represent prices of period (t + 1). The fixed points of the game 10 are obtained as nonnegative solution of the nonlinear algebraic system 2 p2 1 p2 þ 2c 1 p1 p2 þ c 1 p2 ¼ 0 2 p1 p2 2 þ 2c 2 p1 p2 þ c 2 p1 ¼ 0

The solution of above algebraic system is Nash equilibrium point. Because, the former algebraic system represents that both marginal profits are zero in the fixed point and this means that both firms are operating on their reaction curves. The Nash equilibrium is characterized by

rffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 1þ 2 c1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi p p2 c2 ¼ c2 1 þ 1 c2

p1 c1 ¼ c1

See appendix B. 3.1. Bifurcation and chaos for a ¼ 12 However, it is difficult to study this system analytically because no expression of the Nash equilibrium point is available. So,we will use some numerical simulations to show the complicated behavior of the model (stability, period doubling bifurcation and chaos). We consider several interesting numerical examples in this section. If k2 ¼ 1; c1 ¼ 0:1 and c2 ¼ 0:15, the system 10 undergoes a flip bifurcation at k1 ¼ 0:385 (see Fig. 1). The Nash equilibrium loses stability via a period-doubling (flip) bifurcation. A 2-cycle appears and it attracts all the orbits previously attracted by the fixed point. See [19] for a good overview of dynamical concepts such as bifurcations and chaos and for their applications in economics and finance. For values of k1 lower than 0:385 Nash equilibrium is locally stable for each admissible value of k1 , while if k1 > 0:385 the Nash equilibrium is unstable and an increase in the value of k1 causes an advancement in the route to chaos. As k1 increase even more, the erratic fluctuations of the equilibrium price appear. The 2-cycle loses stability and stable invariant circles are created. As k1 takes higher values, there is a possibility of periodic attractors. Fig.1 illustrates the complex behavior of the price, changing from stable equilibrium to chaotic trajectories, through period-doubling and Neimark–Sacker bifurcations. The bifurcation diagram of the of price p1 with respect to c1 parameter is presented in Fig. 2. If we consider an increase in c1 , then the steady state gains stability via a backward flip bifurcation. If we, however, consider a decrease in c1 , then the steady state loses stability through a flip bifurcation. The period-1 behavior undergoes a period doubling bifurcation, followed by a Neimark–Sacker bifurcation occurring on the period-2 orbit.


173

Fig. 1. Bifurcation diagram of model 10 with respect to k1 .

Fig. 2. Bifurcation diagram of model 10 with respect to c1 .

Fig. 3. Chaotic attractor of Bertrand game 10.

Fig. 3 illustrates the chaotic attractor for the model 10 for the values ðk1 ; k2 Þ ¼ ð1:01; 1Þ. To demonstrate the sensitivity on initial conditions of the system 10, we compute two orbits with initial points ðp10 ; p20 Þ and ðp10 þ 0:001; p20 Þ at the parameters values k1 ¼ 1:01; k2 ¼ 1. The corresponding result is shown in Fig. 4. From this figure, it is clear that the result is indistinguishable at the beginning, but after a number of iterations, the difference between them builds up rapidly. Then, the time series of prices are exponentially sensitive to initial conditions.

174


Fig. 4. Sensitive dependence on initial conditions.

4. Dynamical analysis for a ¼ 13 In this section, we consider that a ¼ 13, then c ¼ 12. So Eqs. (3) become 1

p2 1 q1 ¼ 2 1 ; p1 p2 þ p12 1 2

1

p2 1 q2 ¼ 1 1 p2 p2 þ p12 1 2

ð11Þ

By the same method which used to have system 10, the Bertrand duopoly game is given by

8 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi ð1:5c1 p1 p2 þc1 p2 0:5p1 p1 p2 Þ 0 > > pffiffiffiffi pffiffiffiffi 2 < p1 ¼ p1 þ k1 ðp1 p1 þp1 p2 Þ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 1:5c p > ð 2 1 p2 þc 2 p1 0:5p2 p1 p2 Þ 0 > : y ¼ y þ k2 pffiffiffiffi pffiffiffiffi 2 ðp2 p1 þp2 p2 Þ

ð12Þ

where 0 denotes the unit-time advancement operator. 4.1. Nash equilibrium stability In order to analyze the relationship between the stationary state of the dynamical system 12 and the Nash equilibrium, we must seek the equilibrium points as the solution of the following algebraic system:

pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi ð1:5c1 p1 p2 þ c1 p2 0:5p1 p1 p2 Þ ¼ 0 pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi ð1:5c2 p1 p2 þ c2 p1 0:5p2 p1 p2 Þ ¼ 0

ð13Þ

It is difficult to solve the formula 13. So the values of parameters are set as c1 ¼ 0:1 and c2 ¼ 0:2. We only consider the stability of Nash equilibrium point p1 ¼ 0: 556 16; p2 ¼ 0: 912 31. The Jacobian matrix at ðp1 ; p2 Þ is given by

Jðp1 ; p2 Þ ¼

1 0:48965179k1

0:1165287387k1

0:06147515166k2

1 0:1853846250k2

ð14Þ

Characteristic polynomial of 14 is:

k2 Trk þ Det ¼ 0; where

Tr ¼ 2 0:48965179k1 0:185384625k2 ; Det ¼ 1 0:48965179k1 0:185384625k2 þ 0:0 836k1 k2 : The local stability of Nash equilibrium can be gained according to Jury conditions [45]

1 þ Tr þ Det > 0; 1 Tr þ Det > 0;

ð15Þ

jDet j < 1: The condition 15 gives the necessary and sufficient conditions of stable region of the Nash equilibrium point. The stability region of adjustment speeds (k1 ; k2 ) for the Nash equilibrium ð0: 556 16; 0: 912 31Þ described by Fig. 5. Economic meaning of


175

Fig. 5. Stability region.

the stable region is that whatever initial price is chosen by the two players in local stable region, they will eventually achieve the Nash equilibrium price after finite games. It is important to notice that the two players may accelerate the price adjustment speed in order to increase their profit. Price adjustment parameter does not change Nash equilibrium point. Once one player adjusts price too fast and pushes k1 and k2 out of the stable region, the game tends to become unstable and falls into chaos. It seems that, the expression 13 of the Nash equilibrium point is not available. So, we will use some numerical simulations to show the complicated behavior of the model 12. 4.2. Numerical simulations This section is devoted to a numerical analysis of the effects of changes in the parameters’ values on the stability of the Nash equilibrium of the Bertrand duopoly game. We expect that the parameters ki , representing the speed of reaction of the boundedly rational firms, plays a destabilizing role. That is, starting from a set of parameters for which the Nash equilibrium is locally stable, an increase in the speed of adjustment should cause a loss of stability. In fact, the Nash equilibrium may lose stability depending on the value of ki ; i ¼ 1; 2. The Nash equilibrium loses stability via a period-doubling (flip) bifurcation (Fig. 6). It occurs when moving the value of k1 parameter, one of the eigenvalues of the Jacobian matrix calculated at the Nash equilibrium, becomes lower than 1, while the other is still lower than 1 in absolute value. A 2-cycle appears and it attracts all the orbits previously attracted by the fixed point. The bifurcation diagram at

Fig. 6. Bifurcation diagram of system 12 with respect to the parameter k1 .

176


Fig. 7. The Largest Lyapunov exponent with respect to k1 corresponding with Fig. 6.

Fig. 8. Chaotic attractor of Bertrand game 12.

Fig. 9. Bifurcation diagram of system 12 with respect to c1 .

Fig. 6 shows that this is what happens by increasing the value of the speed of reaction parameter k1 , keeping fixed other parameters k2 ¼ 5; c1 ¼ 0:1 and c2 ¼ 0:2. For values of k1 lower than 4:1 the Nash equilibrium is locally stable for each admissible value of k1 , while if k1 > 4:1 the Nash equilibrium is unstable and an increase in the value of k1 causes an advancement in the route to chaos.


177

The largest Lyapunov exponent corresponding to Fig. 6 are calculated and plotted in Fig. 7. In the range 0 < k1 < 4:1 the Lyapunov exponents are negative, corresponding to a stable cycles of the system. When 4:1 < k1 < 8, most Lyapunov exponents are non-negative, and few are negative. This means that there exist stable fixed points or periodic windows in the chaotic band. It is apparent that the behavior of the model 12 is very complicated, including period-doubling and chaotic behavior. This is a typical route to chaos, as it is confirmed by the largest Lyapunov exponent, in Fig. 7. A chaotic attractor in the two-dimensional phase plane for the parameters constellation ðk1 ; k2 ; c1 ; c2 Þ ¼ ð5; 8; 0:1; 0:2Þ is shown in Fig. 8, which exhibit a fractal structure. From Fig. 6, we can see for high values of the speed of adjustment the steady state loses stability. In fact, it is possible to have a situation in which for low values of the marginal cost the Nash equilibrium is unstable and the orbits converge to a chaotic attractor. A bifurcation diagram with respect to the marginal cost of the first player c1 , while other parameters are fixed as follows k1 ¼ 5; k2 ¼ 8 and c2 ¼ 0:2 is shown in Fig. 9. For intermediate values of c1 the Nash equilibrium is locally stable but decreasing again the marginal cost it loses stability via a backward flip bifurcation. Also, as higher values of c1 the Nash equilibrium becomes locally stable again (see the bifurcation diagram in Fig. 9). 5. Conclusion The utility function of the form Uðq1 ; q2 Þ ¼ qa1 þ qa2 ; 0 < a < 1 result in demand functions for commodities that are close substitutes. They therefore provide a suitable underpinning for Bertrand duopoly with heterogenous goods. Here a dynamical Bertrand game, with demand functions derived from the above utility function, is studied. We also analyze equilibrium points and bifurcation behavior in two special cases when a ¼ 12 and a ¼ 13. By investigating the local stability and bifurcations of the Nash equilibrium we have found that the dynamics may become periodic, quasiperiodic or chaotic. As in all cases of dynamics, a periodic outcome may be learned by the acting agents and become a basis for strategy. A quasiperiodic time series still makes such things possible as useful approximation. The empirical importance of chaos, i.e., unpredictability is that it makes any kind of adaptation to it impossible. Appendix A In this appendix we describe the mathematical foundations that lead to the demand functions represented by Eqs. (3) in the main text. In order to have explicit demand functions for the goods, a specific demand utility function should be assumed. We assume the following utility function:

Uðq1 ; q2 Þ ¼ qa1 þ qa2 ; 0 < a 6 1

ðA:1Þ

which is maximized subject to the budget constraint:

p1 q1 þ p2 q2 ¼ 1: The Lagrangian is

Lðq1 ; q2 ; kÞ ¼ qa1 þ qa2 þ kð1 p1 q1 p2 q2 Þ; The first-order conditions are

@L ¼ aq1a1 kp1 ¼ 0; @q1 @L ¼ aq2a1 kp2 ¼ 0; @q2 @L ¼ 1 p1 q1 p2 q2 ¼ 0: @k Solving the first two equations for qa1 and qa2 , we have

q1 ¼ q2 ¼

k

a

k

a

p1 p2

1

a1

; 1

a1

Thus 1 a1 q1 p1 ¼ ; q2 p2

:

ðA:2Þ

178


Using the budget constraint to find

q1 ¼

1 a1 p1 1 p1 q1 ; p2 p2

Then a

q1 ¼

a p1 2 1

a p1 1

ð1 p1 q1 Þ:

Solve this for q1 . This gives us demand function a

q1 ¼

a p1 2 a

a

a 1a p1 p1 1 þ p2

:

ðA:3Þ

By using a similar method, we can find a

q2 ¼

a p1 1 a

a

a 1a p2 p1 1 þ p2

:

ðA:4Þ

By using standard maximization of utility function (A.1) with respect to the budget constraint. We deduced the corresponding demand functions (A.3) and (A.4) as given Eqs. (3) in the main text. Appendix B The fixed points of the game 10 are obtained as the solution of the algebraic system 2 p2 1 p2 þ 2c 1 p1 p2 þ c 1 p2 ¼ 0 2 p1 p2 2 þ 2c 2 p1 p2 þ c 2 p1 ¼ 0

ðB:1Þ

Thus, unique fixed point of system (B.1) is called Nash equilibrium of Bertrand game 10. The system (B.1) does not depend on price adjustment parameters k1 and k2 . By simple computation, we found that 2 p2 1 p2 2c1 p1 p2 ¼ c 1 p2 2 p1 p2 2 2c2 p1 p2 ¼ c 2 p1

Then 2 2 p2 1 2c 1 p1 þ c 1 ¼ c 1 þ c 1 p2 2 2 p2 2 2c 2 p2 þ c 2 ¼ c 2 þ c 2 p1

Solve first equation for p1 and second equation for p2 . Thus, the Nash equilibrium is characterized by

rffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 1þ 2 c1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi p1 p2 c2 ¼ c2 1 þ c2

p1 c1 ¼ c1

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

A. Cournot, Récherches sur les principes mathématiques de la théorie des richesses, Dunod, Paris, 1838. J. Bertrand, Théorie mathématique de la richesse sociale, J. des Savants 48 (1883) 499–508. F.Y. Edgeworth, La teoria pura del monopolio, Giornale dgli Economisti 15 (1897) 13–31. H. Hotelling, Stability in competition, Econ. J. 39 (1929) 41–57. J. Robinson, The Economics of Imperfect Competition, MacMillan, London, 1969. T.F. Palander, Konkurrens och marknadsjämvikt vid duopol och oligopol, Ekonomisk Tidskrift 41 (1939) 124–145. R.D. Theocharis, On the stability of the Cournot solution of the oligopoly problem, Rev. Econ. Stud. 27 (1959) 133–134. A. Caplin, B. Nalebuff, Aggregation and imperfect competition: on the existence of equilibrium, Econometrica 59 (1991) 25–59. J.X. Zhang, Q.L. Da, Y.H. Wang, The dynamics of Bertrand model with bounded rationality, Chaos, Solitons Fractals 39 (2009) 2048–2055. E. Ahmed, M.F. Elettreby, A.S. Hegazi, On Puu’s incomplete information formulation for the standard and multi-team bertrand game, Chaos, Solitons Fractals 30 (2006) 1180–1184. D. Furth, Stability and instability in oligopoly, J. Econ. Theory 40 (1986) 197–228. L. Corchon, A. Mas-Colell, On the stability of best reply and gradient systems with applications to imperfectly competitive models, Econ. Lett. 51 (1996) 59–65. G.I. Bischi, A. Naimzada, Global analysis of a dynamic duopoly game with bounded rationality, in: Advances in Dynamic Games and Applications, Springer, 2000, pp. 361–385. H.N. Agiza, A.A. Elsadany, Nonlinear dynamics in the Cournot duopoly game with heterogeneous players, Physica A 320 (2003) 512–524.


179

[15] H.N. Agiza, A.A. Elsadany, Chaotic dynamics in nonlinear duopoly game with heterogeneous players, Appl. Math. Comput. 149 (2004) 843–860. [16] G.I. Bischi, C. Chiarella, M. Kopel, F. Szidarovszky, Nonlinear Oligopolies: Stability and Bifurcations, Springer, 2009. [17] T. Dubiel-Teleszynski, Nonlinear dynamics in a heterogeneous duopoly game with adjusting players and diseconomies of scale, Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 296–308. [18] M. Anufriev, D. Kopányi, J. Tuinstra, Learning cycles in Bertrand competition with differentiated commodities and competing learning rules, J. Econ. Dyn. Control 37 (2013) 2562–2581. [19] C. Hommes, Behavioral Rationality and Heterogeneous Expectations in Complex Economic Systems, Cambridge University Press, 2013. [20] H.N. Agiza, A.A. Elsadany, M.M. El-Dessoky, On a new Cournot duopoly game, J. Chaos 2013 (2013) 5. Article ID 487803. [21] J. Zhang, G. Wang, Complex Dynamics of Bertrand duopoly games with bounded rationality, World Acad. Sci. Eng. Technol. 79 (2013) 106–110. [22] T. Puu, The chaotic monopolist, Chaos, Solitons Fractals 5 (1995) 35–39. [23] E. Ahmed, M.F. Elettreby, A.S. Hegazi, On Puu’s incomplete information formulation for the standard and multi-team Bertrand game, Chaos, Solitons Fractals 30 (2006) 1180–1184. [24] Z. Sun, J. Ma, Complexity of triopoly price game in Chinese cold rolled steel market, Nonlinear Dyn. 67 (2012) 2001–2008. [25] W. Xu, J. Ma, Study on the dynamic model of a duopoly game with delay in insurance market, WSEAS Trans. Math. 11 (2012) 615–624. [26] J.L. Zhang, Research on delayed complexity based on nonlinear price game of insurance market, WSEAS Trans. Math. 10 (2011) 368–376. [27] L. Fanti, L. Gori, C. Mammana, E. Michetti, The dynamics of a Bertrand duopoly with differentiated products: synchronization, intermittency and global dynamics, Chaos, Solitons Fractals 52 (2013) 73–86. [28] L. Zhao, J. Zhang, Analysis of a duopoly game with heterogeneous players participating in carbon emission trading, Nonlinear Anal. Model. Control 19 (2014) 118–131. [29] N. Singh, X. Vives, Price and quantity competition in a differentiated duopoly, RAND J. Econ. 15 (1984) 546–548. [30] C.H. Tremblay, V.J. Tremblay, The Cournot–Bertrand model and the degree of product differentiation, Econ. Lett. 111 (2011) 233–235. [31] C. Tremblay, M. Tremblay, V. Tremblay, A general Cournot–Bertrand model with homogenous goods, Theor. Econ. Lett. 1 (2011) 38–40. [32] J. Ding, Q. Mei, H. Yao, Dynamics and adaptive control of a duopoly advertising model based on heterogeneous expectations, Nonlinear Dyn. 67 (2012) 129–138. [33] B. Xin, T. Chen, On a master–slave Bertrand game model, Econ. Model. 28 (2011) 1864–1870. [34] J. Ma, K. Wu, Complex system and influence of delayed decision on the stability of a triopoly price game model, Nonlinear Dyn. 73 (2013) 1741–1751. [35] J. Hofbauer, K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998. [36] M.A. Nowak, Evolutionary Dynamics: Exploring the Equations of Life, Harvard University Press, Cambridge, MA, 2006. [37] G. Szabo, G. Fath, Evolutionary games on graphs, Phys. Rep. 446 (2007) 97. [38] Z. Wang, A. Szolnoki, M. Perc, If players are sparse social dilemmas are too: importance of percolation for evolution of cooperation, Sci. Rep. 2 (2012) 369. [39] Z. Wang, A. Szolnoki, M. Perc, Evolution of public cooperation on interdependent networks: the impact of biased utility functions, Europhys. Lett. 97 (2012) 48001. [40] S. Meloni, A. Buscarino, L. Fortuna, M. Frasca, J. Gomez-Gardenes, et al, Effects of mobility in a population of prisoners dilemma players, Phys. Rev. E 79 (2009) 067101. [41] C. Xia, Z. Wang, J. Sanz, S. Meloni, Y. Moreno, Effects of delayed recovery and nonuniform transmission on the spreading of diseases in complex networks, Physica A 392 (2013) 1577–1585. [42] Z. Wang, A. Szolnoki, M. Perc, Optimal interdependence between networks for the evolution of cooperation, Sci. Rep. 3 (2013) 2470. [43] W. Yuan, C. Xia, Role of investment heterogeneity in the cooperation on spatial public goods game, PLoS One 9 (3) (2014) e91012. [44] T. Puu, Chaos in duopoly pricing, Chaos, Solitons Fractals 1 (1991) 573–581. [45] S. Elaydi, An Introduction to Difference Equations, Springer-Verlag, New York, 2005.