Dynamics of Hotelling triopoly model with bounded rationality

Applied Mathematics and Computation 373 (2020) 125027

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

Dynamics of Hotelling triopoly model with bounded rationality Yu Peng a,b,∗, Qian Lu a,b, Xue Wu a, Yueru Zhao a, Yue Xiao c a

School of Science, Southwest University of Science and Technology, Mianyang 621010, China Institute of Modeling and Algorithm, Southwest University of Science and Technology, Mianyang 621010, China c School of Management, University of Science and Technology of China, Hefei 230026, China b

a r t i c l e

i n f o

Article history: Received 8 May 2018 Revised 18 December 2019 Accepted 23 December 2019

Keywords: Hotelling triopoly model Bounded rationality Bifurcation Chaos control

a b s t r a c t It is devoted to introduce and research a triopoly Hotelling model with a triangular market in this paper. Based on bounded rationality, a three-dimensional discrete dynamical system is formulated. The complex dynamics which occur in the process of chaotic evolution have been investigated. The solution of Nash equilibrium gives rise to complicated phenomena as some parameters of the model are varied. Some numerical simulations are used to portray such chaotic behavior. Furthermore, proper control scheme is applied to keep the system from instability and chaos, which is also supported by some simulations. © 2019 Elsevier Inc. All rights reserved.

1. Introduction The spatial duopoly model was originally suggested by Hotelling, who assumed that the uniformly distributed continuum of consumers had to choose homogeneous product in a city represented by single line segment [1]. According to the literature on Hotelling model and its modifications, most works focused on the existence of noncooperative solutions. Ref. [2] investigated the situation of a “Principle of Maximal or Minimum Differentiation”, which researched the interaction between consumers’ preferences and firms’ optimal strategies and estimated whether the product diversity would be affected. However, under horizontal product differentiation, an established result is that a pure-strategy equilibrium in prices may not exist [3]. Some related subsequent studies are based on linear markets. Ref. [4] showed that no pure strategy pricing equilibrium existed when the firms were sufficiently close together (but not at the same location) within the Hotelling model. Ref. [5] investigated the existence of a unique pure-strategy price equilibrium by reformulating the spatial duopoly model with linear transportation costs as a differential game where product differentiation was the result of firms’ R&D investments. Nevertheless, the major part of literature concentrates on duopoly markets only. The problem that the number of firms would affect the equilibrium outcome is ignored widely. Some models considering a larger number of firms have been analyzed [6,7]. For the oligopoly model of n firms located in a circumference, it has shown that a symmetric equilibrium (where

∗

Corresponding author at: School of Science, Southwest University of Science and Technology, Mianyang 621010, China. E-mail addresses: [email protected] (Y. Peng), [email protected] (Q. Lu).

https://doi.org/10.1016/j.amc.2019.125027 0 096-30 03/© 2019 Elsevier Inc. All rights reserved.

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Y. Peng, Q. Lu and X. Wu et al. / Applied Mathematics and Computation 373 (2020) 125027

successive firms are equidistant and firms charge the same price) exists in the short-run price game [6]. Ref. [7] analyzed oligopolistic competition of more than two firms with products defined by their characteristics, substituting [0, 1] for the space of characteristics. As to the modeling of transport cost, the literature on Hotelling models and its modifications diverges. Some of the literature continued with the linear transportation cost model, whereas the rest parts mainly followed d’Aspremont et al. who introduced a quadratic transportation cost function motivated by the tractability problem of the linear cost model [4]. Although a linear market may form a triangle, a square, or even a polygon, some studies employ other market shape. A circular market has been considered in Ref. [6] where consumers can walk along the circle. The Hotelling model with a triangular market has been discussed in Ref. [8]. They developed a duopoly model in which consumers distributed uniformly along three sides of a triangular region and each consumer would walk across the region in the shortest way to the lowestprice firm to buy one unit of a product. It is noteworthy that people are only permitted to travel along the circle when shopping in Ref. [6]. And Ref. [8] assumed that consumers can walk across the open space. We put the dynamic pricing into a triopoly market in this paper and suppose a triangular open space in three-dimensional space, just as a city built by mountains. Humans distribute uniformly in the triangular region. There are three identical firms located at the vertexes of the triangle city. The city has a very developed subway system, and people can reach firms in the shortest way by subway easily. This is an extension of the Hotelling game which rarely modeled in the literature. The competition among firms may be different from that in traditional models. Nowadays, it is a hot topic in economics about establishing and studying nonlinear dynamic systems. The behaviors of an oligopoly game are much complicated because firms must consider not only the market demand, but also the strategies of the competitors. The general formula of the oligopoly model with bounded rationality has been investigated [9,10]. The dynamics of the game can lead to complex behaviors such as cycles and chaos. With a nonlinear demand function and bounded rationality, the complex dynamics of a duopoly game have been studied [11]. Ref. [12] investigated the dynamics of a Cournot duopoly game with differentiated goods. According to the strategies that firms try their best to maximize the expectations on output, the modification of the duopoly game has been discussed [13,14]. Ref. [15] examined the dynamical behaviors of Bowley’s model. Ref. [16] applied the Jury condition to discuss the stability of a modification of Puu’s model. There are some works on the dynamics of oligopoly models with more firms and other modifications. Refs. [17–19] formulated and studied complex dynamics on a duopoly Stackelberg game model. Refs.[20,21] analyzed the complex dynamics on Bertrand game with differentiated products. Ref. [22] established a nonlinear quadropoly game based on Cournot model with fully heterogeneous players. The analysis of a triopoly game or multi-player will be more difficult because the dynamics are represented by multidimensional dynamical system. The case of multi-player is interesting as it provides for much more variety than might be expected from merely increasing the number of players from two to three or more than three. Refs. [23–28] obtain some interesting analytical results under the assumption of homogeneous expectations among firms. The complex dynamics on a triopoly game with bounded rationality and radical form inverse demand function has been analyzed [23]. Ref. [24] investigated the complex dynamics of an R&D two-stage input competition triopoly Bertrand game model with bounded rational expectations. Ref. [25] studied the complex dynamic characteristics of a tripolistic Stackelberg game by the myopic mechanism (bounded rationality). Associated with the assumption of heterogeneous expectations, the system will become quite difficult to be analyzed by using theoretical tools with the increase of the number of players. Numerical experiments become more and more important in order to analyze the possible dynamic outcome of the model. Assumed that the inverse demand function is quadratic and the total cost function is cubic, Ref. [29] investigated the dynamic behaviors of fouroligopolist game model with different expectations. Refs. [28,30] extended the analysis to the case of three heterogeneous firms with nonlinear cost functions. They found the conditions on the parameters causing the loss of stability of the Nash equilibrium via a flip bifurcation. Recent trends in tackling economic competition have adopted complex networks. They are aptly used to describe such competition or social systems. Complex networks have been devoted to evolutionary games and in particular to the evolution of cooperation. The emergence and persistence of cooperation among selfish individuals is one of the fundamental and central problems in evolutionary game. Many 2 × 2 games (such as the prisoner’s dilemma game) and multi-player games (such as public goods game), as general metaphors, are often used in evolutionary games [31,32]. Ref. [33] presented a data envelopment analysis (DEA) efficient rule with emergence of high and stable cooperator frequency on two-dimensional regular lattices. Most of the previous works are based on the Cournot model and the modifications to discuss the complex dynamics. There is little literature dealing with Hotelling model and the modifications in studying dynamical behaviors. The present work aims to formulate a triopoly Hotelling game with bounded rationality and study the dynamical behaviors. Each firm adjusts its strategy according to the expected marginal profit and depends on local information on output to make decision. In this work, every firm tries to maximize its profit according to local information of its strategy. Under the above assumptions, it is possible to obtain interesting analytical results. This paper is organized as follows. In Section 2, the triopoly Hotelling game with a triangular market is briefly described with bounded rationality. The dynamics of a triopoly Hotelling game model with bounded rationality are analyzed. In Section 3, we present the numerical simulations to verify our theoretical results. In Section 4, we exert controlling method on the chaos which occur in the evolution process of this Hotelling model. Finally, some remarks are presented in Section 5.

Y. Peng, Q. Lu and X. Wu et al. / Applied Mathematics and Computation 373 (2020) 125027

3

1 C 0.8 0.6 M2

S3

0.4 0.2

P S2

M3 0 0

B

S1 M1

0.5 1A

0

0.2

0.4

0.6

0.8

1

Fig. 1. The triangular market and three subregions.

2. The model 2.1. The Hotelling triopoly game with a triangular market The original Hotelling model is assumed that consumers are distributed uniformly over a line segment, and travel, at a constant cost per unit, to a firm to buy one unit of a good. As is well known, this can be thought, alternatively, as a geographic space or as a space of product characteristics in which consumer preferences are defined. Each consumer inclines to choose the firm which charges lower and less travel costs there. The firms compete in a two-staged location-price game, in where simultaneously choose their location and afterwards set their prices in order to maximize their profits. We extend the original Hotelling model to triopoly with triangular market. Let  denote the market.

 = { ( x1 , x2 , x3 )| x1 + x2 + x3 = 1, xi ≥ 0, i = 1, 2, 3}. We know that  is a equilateral triangle. Point A(1, 0, 0), point B(0, 1, 0), and point C(0, 0, 1) are the vertices of the triangle. As shown in Fig. 1, let point P(x1 , x2 , x3 ) ∈ , let line segment PM1 , PM2 and PM3 be perpendicular to line segment AB, BC and CA respectively, i.e., PM1 ⊥AB, PM2 ⊥BC and PM3 ⊥CA. The region ABC is divided into three subregions, whose areas are denoted by S1 , S2 and S3 respectively for simplicity. √ √ √ By simple computation, we obtain S1 = 43 ((x2 + x3 )2 + 2x2 x3 ), S2 = 43 ((x1 + x3 )2 + 2x1 x3 ) and S3 = 43 ((x1 + x2 )2 + 2x1 x2 ). Since x1 + x2 + x3 = 1, therefore the area S of the region ABC is

√ √ 3 3 2 S = S1 + S2 + S3 = ( x1 + x2 + x3 ) = 2 2

We consider a Hotelling game with 3 firms, labelled by i = 1, 2, 3. Firm 1, firm 2, and firm 3 are located at point A, point B and point C respectively. These firms provide goods which are homogeneous in all characteristics other than the available location. Each firm is able and willing to supply the entire market with sufficient products. They take produce cost ci into account to determine the outputs quantity qi and fix unit price pi . The consumers are distributed uniformly with unit density in the region ABC and pick up exactly a single unit of the given commodity from the lowest-cost firm. The costs paid by a consumer include the product’s mill price and a transportation cost (with respect to distance at a transport rate r). If a generic consumer located at P(x1 , x2 , x3 ) in ABC, the distances to point A, point B and point C are 1 − x1 , 1 − x2 and 1 − x3 respectively. Pi = pi + r (1 − xi ) stands for the costs purchasing from firm i (i = 1, 2, 3 ). Consumers prefer the optimal choice on the basis of the rule: purchase from firm i satisfying Pi ≤ Pj , (i = j). To identify the location in the region , P(x1 , x2 , x3 ) is defined as the coordinate of consumer. The equilibrium conditions are P1 = P2 = P3 . The equations can be rewrite as

p1 + r ( 1 − x1 ) = p2 + r ( 1 − x2 ) = p3 + r ( 1 − x3 )

(1)

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Y. Peng, Q. Lu and X. Wu et al. / Applied Mathematics and Computation 373 (2020) 125027

 Since 3i=1 xi = 1, we can easily derive from (1) that the location P(x1 , x2 , x3 ) of the consumer who is indifferent among the three firms at generic price and location.

xi = where p¯ =

1 1 + ( pi − p¯ ), i = 1, 2, 3 3 r 1 3

3

i=1

(2)

pi . All firms have a non-empty market share if and only if P(x1 , x2 , x3 ) ∈ . Hence, the prices satisfy

−r ≤ δi ≤ 2r, i = 1, 2, 3

(3)

3

where δi = 3 pi − j=1 p j = 3( pi − p¯ ). We can easily obtain that δ1 + δ2 + δ3 = 0. As shown in Fig. 1, we suppose that point P(x1 , x2 , x3 ) is the indifferent point in region . The consumer located at the point on subregion Si pays Pi to obtain the product if the purchase is from firm i (i = 1, 2, 3). We know that Pi is the lowest-cost among the three costs. According to the decision rule, the consumer located on the subregion Si would choose firm i, i.e., firm 1 has consumers on subregion S1 , while firm 2 has consumers on subregion S2 and firm 3 has consumers on subregion S3 . Therefore, the total demand quantity offered by firm i is given by

qi =

Si 1 = ((x j + xk )2 + 2x j xk ), i = j = k, i, j, k = 1, 2, 3 S 2

(4)

Substituting (2) into (4), the demand functions are obtained as

⎧ 2 p1 δ1 − (2 p21 + p22 + p23 ) + 4 p2 p3 r − δ1 ⎪ ⎪ q1 = + ⎪ ⎪ 3r 6r 2 ⎪ ⎨ 2 2 p2 δ2 − ( p1 + 2 p22 + p23 ) + 4 p1 p3 r − δ2 q2 = + ⎪ 3r 6r 2 ⎪ ⎪ ⎪ 2 ⎪ ⎩q = r − δ3 + 2 p3 δ3 − ( p1 + p22 + 2 p23 ) + 4 p1 p2 3

3r

(5)

6r 2

In this work we assume that the firms use different production method and the cost function is proposed in the linear form

Ci (qi ) = ci qi , i = 1, 2, 3 With these assumptions, the single-period profit of the firm i is given by

i = ( pi − ci )qi , i = 1, 2, 3 The empirical estimate of marginal profit for the firm i at the point (p1 , p2 , p3 ) is given by

∂ i ∂ qi = qi + ( pi − ci ) , i = 1, 2, 3 ∂ pi ∂ pi

(6)

Substituting (5) into (6) to get

⎧ ∂ 1 2c1 + r − δ1 − 2 p1 δ12 − 2c1 δ1 + 2( p21 − p22 − p23 + p2 p3 ) ⎪ = + ⎪ ⎪ ∂ p1 3r 6r 2 ⎪ ⎪ ⎨ ∂ 2 2c2 + r − δ2 − 2 p2 δ22 − 2c2 δ2 + 2(−p21 + p22 − p23 + p1 p3 ) = + ∂ p2 3r 6r 2 ⎪ ⎪ ⎪ ⎪ 2 2 2 2 ⎪ δ − 2 c δ + 2 ( −p 3 3 ⎩ ∂ 3 = 2c3 + r − δ3 − 2 p3 + 3 1 − p2 + p3 + p1 p2 ) 2 ∂ p3 3r 6r

(7)

Based on Eq. (7), the firms’ reaction function relative to its competitor can be obtained by using the backward induction method, which is calculated for every possible given price by the other two firms. According to the profit maximization principle, we can obtain the Nash equilibrium solution. If the firm could predict the price choices of concurrent firms in advance, it would make optimal profits decisions, i.e., let the partial derivative of marginal profit respect to pi equal to 0. Even if it will be possible to obtain analytical form of the Nash equilibrium price, it would be quite difficult to deal with them because the structure of Eq. (7) is complex. We prefer to perform some numerical experiments. 2.2. Dynamics under Hotelling competition We consider a triopoly game in the sense that triopolists adopt the bounded rationality adjustment mechanism [9] to decide the price of each time period. The boundedly rational firm has no complete knowledge of the market. Every firm ∂ should try to use local information based on the marginal profit ∂ p i . At each time period t, a firm decides to increase i

(decrease) its price for the period t + 1 if it perceives positive (negative) marginal profit on the basis of information held at time t, according to the following dynamic adjustment mechanism

pi (t + 1 ) = pi (t ) + αi pi (t )i ( pi (t )), i = 1, 2, 3

(8)

Y. Peng, Q. Lu and X. Wu et al. / Applied Mathematics and Computation 373 (2020) 125027

5

∂  (t )

where i ( pi (t )) = ∂ p i(t ) is the marginal profit of firm i in period t, and α i is a positive constant which is called the speed i of adjustment of firm i [34]. By combining Eqs. (7) and (8), the dynamic Hotelling triopoly game with bounded rational players can be expressed as the following three-dimensional dynamic system.

⎧ ⎨ p1 (t + 1 ) = p1 (t )[1 + α1 1 ( p1 (t ))] p2 (t + 1 ) = p2 (t )[1 + α2 2 ( p2 (t ))] ⎩ p3 (t + 1 ) = p3 (t )[1 + α3 3 ( p3 (t ))]

where δi (t ) = 3 pi (t ) −

3 j=1

(9)

p j (t ), i = 1, 2, 3, and

⎧ 2c + r − δ1 (t ) − 2 p1 (t ) δ12 (t ) − 2c1 δ1 (t ) + 2[ p21 (t ) − p22 (t ) − p23 (t ) + p2 (t ) p3 (t )] ⎪ ⎪ 1 ( p1 (t )) = 1 + ⎪ ⎪ 3r 6r 2 ⎪ ⎨ 2 2 2c + r − δ2 (t ) − 2 p2 (t ) δ2 (t ) − 2c2 δ2 (t ) + 2[−p1 (t ) + p22 (t ) − p23 (t ) + p1 (t ) p3 (t )] 2 ( p2 (t )) = 2 + ⎪ 3r 6r 2 ⎪ ⎪ ⎪ 2 2 ⎪ ⎩ ( p (t )) = 2c3 + r − δ3 (t ) − 2 p3 (t ) + δ3 (t ) − 2c3 δ3 (t ) + 2[−p1 (t ) − p22 (t ) + p23 (t ) + p1 (t ) p2 (t )] 3 3 2 3r

6r

The analytical method and numerical simulation are adopted to investigate the stability and complex behaviors of system (9). It is very difficult to derive the analytic form of the fixed points for system (9) because the structure of system (9) is highly complex. Accordingly, we analyze the stability of Nash equilibrium point with a numerical example. Let r = 0.2, c1 = 0.15, c2 = 0.22 and c3 = 0.2. Let pi (t + 1 ) = pi (t ) (i = 1, 2, 3 ) in system (9), the fixed points of system (9) are obtained by combining (3) and pi > ci (i = 1, 2, 3 ). The system (9) has unique Nash equilibrium point:

E ∗ = ( p∗1 , p∗2 , p∗3 ) = (0.2663, 0.3056, 0.2933 ) The study of local stability for the fixed point E∗ depends on the eigenvalues of its Jacobian matrix which takes the form



J ( p1 , p2 , p3 ) = where

a11 a21 a31

4

a11 = 1 + α1 p∗1 − a12 = a13 = a21 = a22 = a23 = a31 = a32 = a33 =

a12 a22 a32

+

a13 a23 a33





2 (3 p∗1 − p∗2 − p∗3 − c1 ) = 1 − 1.58α1 3r 2

3r

1 α + 2 (−2 p∗1 − p∗2 + 2 p∗3 + c1 ) = 0.245α1 3r 3r

1 ∗ 1 α1 p1 + 2 (−2 p∗1 + 2 p∗2 − p∗3 + c1 ) = 0.2869α1 3r 3r

1 ∗ 1 α2 p2 + 2 (−p∗1 − 2 p∗2 + 2 p∗3 + c2 ) = 0.3593α2 3r 3r 4

2 1 + α2 p∗2 − + 2 (−p∗1 + 3 p∗2 − p∗3 − c2 ) = 1 − 1.3691α2 3r 3r 1

1 α2 p∗2 + 2 (2 p∗1 − 2 p∗2 − p∗3 + c2 ) = 0.1072α2 3r 3r

1 ∗ 1 α3 p3 + 2 (−p∗1 + 2 p∗2 − 2 p∗3 + c3 ) = 0.3649α3 3r 3r

1 ∗ 1 α3 p3 + 2 (2 p∗1 − p∗2 − 2 p∗3 + c3 ) = 0.0708α3 3r 3r 4

2 ∗ 1 + α3 p3 − + 2 (−p∗1 − p∗2 + 3 p∗3 − c3 ) = 1 − 1.3668α3 3r 3r ∗ 1 p1

1

Then, the characteristic equation is

λ3 + K2 λ2 + K1 λ + K0 = 0 where

K2 = −(a11 + a22 + a33 ) = 3 − 1.58α1 − 1.3691α2 − 1.3668α3 K1 = a22 a33 + a11 (a22 + a33 ) − a23 a32 − a12 a21 − a13 a31 = 3 − 3.16α1 − 2.7382α2 − 2.7336α3 + 2.0752α1 α2 + 2.0545α1 α3 + 1.8637α2 α3

(10)

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0.5 0.45 0.4 0.35 p2

0.3

p3 p1

0.25 0.2 0.15 0.1

0

0.2

0.4

0.6

0.8

v1

1

1.2

1.4

1.6

1.8

Fig. 2. The bifurcation diagram (a) and the corresponding largest Lyapunov exponents (b) for the system (9) with v1 ∈ (0, 1.8], for r = 0.2, c1 = 0.15, c2 = 0.22, c3 = 0.2, v2 = 0.75 and v3 = 0.65.

K0 = a11 a23 a32 + a12 a21 a33 + a13 a22 a31 − a11 a22 a33 − a12 a23 a31 − a13 a21 a32 = −1 + 1.58α1 + 1.3691α2 + 1.3668α3 − 2.0752α1 α2 − 1.8637α2 α3 − 2.0548α1 α3 + 2.6642α1 α2 α3 The necessary and sufficient conditions for the local stability of Nash equilibrium point E∗ are given as follows by using the equivalent conditions of Jury test [20]. (i) (ii) (iii) (iv)

1 + K2 + K1 + K0 > 0, 1 − K2 + K1 − K0 > 0, 1 − K1 + K0 K1 − K02 > 0, 1 + K1 − K0 K1 − K02 > 0.

These conditions define a region of stability in a bounded three-dimensional space with the adjustment speed (α 1 , α 2 , α 3 ). For the parameters (α 1 , α 2 , α 3 ) inside the stability region, the Nash equilibrium point E∗ is a stable node and it loses its stability. Finally, all the inputs are stable at Nash equilibrium point E∗ after a limited number of games for any given initial datum if the input speed adjustment parameters are in the stability domain. 3. Numerical simulation There are many practical examples of Hotelling model applications. Based on a modified analytical and a numerical Hotelling model, the long term effects of an increase in the domestic gas price in Russia on the European gas market had been assessed [35]. By using a fixed initial non-renewable resource reserve, Ref. [36] built a Hotelling model to develop a generic numerical optimisation model. Simulations were used to portray the non-renewable resource management regimes and the effects of different policy instruments deployed at different stages of the resource’s life cycle. Ref. [37] investigated the economic implications of additive manufacturing technology from the perspective of Hotelling model. The results showed that there is scope for the improvement of consumer welfare arising from local production by consumer producers. This section is devoted to a numerical analysis of the effects for changing in the parameters’ values on the stability of the Nash equilibrium. Numerical experiments are simulated to show the stability and period doubling bifurcation route to chaos for the three-dimensional system (9). In the simulation here, we take r = 0.2, c1 = 0.15, c2 = 0.22 and c3 = 0.2. If we vary one parameter, for example v1 , while fixing the rests, bifurcations and chaos would occur which are detected by using Lyapunov exponents. Fig. 2 presents the bifurcation diagrams with respect to the adjustment speed v1 . From Fig. 2(a) we see that a low adjustment speed can keep the system stable and its increase will make the equilibrium become unstable even chaotic. Fig. 2(a) shows that the system is always stable when v1 is located in a large interval. Bifurcation occurs only when v1 is numerically shown close about to 1.22. Then, the period-doubling bifurcations occur, that is, period-doubling, period four, period eight, and the chaotic behaviors appear when v1 > 1.61. The maximum Lyapunov exponent is plotted in Fig. 2(b) to show bifurcations and chaos, where positive values show the chaotic behaviors. At bifurcation point the maximum Lyapunov exponents is zero. Fig. 3 shows the bifurcation diagram with respect to r in system (9). We can see that the system dynamics is chaotic if r is small. As r increases, there exist period-halving bifurcations. The system (9) experiences chaos and period-halving bifurcation.

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Fig. 3. The bifurcation diagram of the system (9) with respect to r for the values of parameters c1 = 0.15, c2 = 0.22, c3 = 0.2, v1 = 0.65, v2 = 0.75 and v3 = 0.65.

Fig. 4. The strange attractor for the system (9) for the values of parameters r = 0.2, c1 = 0.15, c2 = 0.22, c3 = 0.2, v1 = 1.75, v2 = 0.75 and v3 = 0.65.

The strange attractor is another characteristic of chaos, and it reflects the inherent regularity of the complex phenomena in a chaotic state. Players can forecast the market output in a short term according to inherent regularity while the system is in chaos. Fig. 4 shows the change situation of strange attractor of the system (9) for r = 0.2, c1 = 0.15, c2 = 0.22, c3 = 0.2, v1 = 1.75, v2 = 0.75 and v3 = 0.65. From Fig. 4, we can see the bifurcation processes, where invariant curves take place after the equilibrium bifurcates properly and chaos occurs when v1 , v2 and v3 take their value big enough. The system trajectories are plotted in a three-dimensional phase plane of the variables p1 , p2 and p3 . Fig. 5 demonstrates the sensitivity of system (9) to initial conditions. For each state variable Fig. 5 plots two orbits initially from the slightly deviated points ( p11 (0 ), p12 (0 ), p13 (0 )) and ( p21 (0 ), p22 (0 ), p23 (0 )) = ( p11 (0 ) + 10−4 , p12 (0 ), p13 (0 )), respectively. The red curves (labeled by superscript 1) start from (0.2663, 0.3056, 0.2933) and the blue curves (labeled by 2) start form (0.2664, 0.3056, 0.2933). It suffices to show that although the two orbits of each variable are indistinguishable

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Fig. 5. Sensitive dependence of system (9) on initial conditions. The numerical simulations are done by setting v1 = 1.8, v2 = 0.75 and v3 = 0.65. The system orbits in the time periods [900, 1000] are plotted. The red curves are associated with the initial state p1 (0 ) = 0.2663, p2 (0 ) = 0.3056 and p3 (0 ) = 0.2933. The blue ones are initially associated with the initial state p1 (0 ) = 0.2664, p2 (0 ) = 0.3056 and p3 (0 ) = 0.2933. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

at the beginning, the difference between them is obviously built after a series of iterations. Thus, in chaotic states, system (9) is sensitive to the initial conditions.

4. Chaos control Based on the discussions above, we observe that the adjustment speed v1 , v2 and v3 tremendously influence the stability of system (9). If the parameters fail to locate in the stable region, the dynamic behaviors of the system will be much complicated, such as period-doubling bifurcation and chaos. The appearance of chaos in the economic system is not expected and even is harmful. It is very difficult to obtain a complete understanding of the structure of the market when the triopoly game has complex dynamic conditions. We hope that the system chaos can be avoided or controlled on some level such that the dynamic system could perform better. A wide variety of methods have been proposed for controlling chaos in oligopoly models. Feedback and parameter variation are two methods for the chaos control. Recently, a new control method has been put forward which is called as control strategy of the state variables feedback and parameter variation. This method had been considered in the four oligopolist model [29]. The same method had been used to control the chaos [17,18,38]. In this section, we apply the method to control

Y. Peng, Q. Lu and X. Wu et al. / Applied Mathematics and Computation 373 (2020) 125027

9

Fig. 6. The bifurcation diagram with respect to control parameter μ. The three bifurcation diagrams show that the system chaos has being gradually controlled with the parameter μ increasing and the system will be led to stability when μ is large enough.

Fig. 7. The time series of system (11) when control parameter μ = 0.28.

the chaotic behavior of system (9). We change three-dimensional discrete dynamic system (9) into the following format



p1 (t + 1 ) = p2 (t + 1 ) = p3 (t + 1 ) =

(1 − μ ) p1 (t )[1 + α1 1 ( p1 (t ))] + μ p1 (t ) (1 − μ ) p2 (t )[1 + α2 2 ( p2 (t ))] + μ p2 (t ) (1 − μ ) p3 (t )[1 + α3 3 ( p3 (t ))] + μ p3 (t )

(11)

where 0 < μ < 1 is the control parameter. System (9) will fall into instability region and chaos with the change of product modification speed for player 1, player 2 and player 3. The chaotic state of system (9) with the change of speed adjustment v1 for player 1, v2 for player 2 and v3 for

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player 3 are controlled. Fig. 6 is the bifurcation diagram of controlled system (11) with the change of control parameter μ after adding control to the chaotic state (c1 = 0.15, c2 = 0.22, c3 = 0.2, v1 = 1.7, v2 = 0.75 and v3 = 0.65). It can be seen from Fig. 6 that the period-doubling bifurcation disappears, and the controlled system (11) stabilizes at the Nash equilibrium point for 0 < μ < 0.265 as shown numerically. The system gradually gets out of chaos and becomes stable when the controlling parameter μ is properly large. Fig. 7 shows that the chaotic system is controlled at fixed point when μ = 0.28 which means the chaos controlling method (11) is successful.

5. Conclusion In this paper, a triopoly Hotelling model with a triangular market has been proposed. We investigated the nonlinear complex dynamics of a price competition triopoly game model with bounded rational expectation. The local stability conditions of the Nash equilibrium points have been studied. Basic properties of the game have been analyzed by means of bifurcation diagram, largest Lyapunov exponents, strange attractor and sensitive dependence on initial condition. The Nash equilibrium point is locally asymptotically stable for a relatively low adjustment speed. The result demonstrates that the stability of Nash equilibrium, as some parameters of the model are varied, gives rise to complex dynamics such as cycles and chaos. We used the state variables feedback and parameter variation to control chaos. The model is quickly arrived at the Nash equilibrium point when a suitable controlling parameter is chosen. Similar methods can be used to study the dynamics of the triopoly Hotelling game and some modifications.

Acknowledgment The authors thank the editor and the referees for their valuable comments and suggestions which improved the quality of this paper.

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