# Analysis of the Chaotic Dynamics Duopoly Game of ISPs Bounded Rational

Authors: Mohamed El Amrani; Hamid Garmani; Driss Ait Omar; Mohamed Baslam; Brahim Minaoui
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

We propose in this paper an investigation of the dynamical behaviors of a duopoly model with two Internet services providers (ISPs). The competition between them is assumed to take place in terms of their pricing decisions and their number of cached items. According to the ISPs’ rationality level, we consider a scenario where both ISPs are bounded rational. Each ISP in any period uses the marginal profit observed from the previous period to choose its strategies. We compute explicitly the steady states of the dynamic system induced by bounded rationality and establish a necessary and sufficient condition for the stability of its Nash equilibrium (NE). Numerical simulations show that if some parameters of the model are varied, the stability of the NE point is lost and the complex (periodic or chaotic) behavior occurs. The chaotic behavior of the system is stabilized on the NE point by applying control.


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## Body

## 1. Introduction

The coexistence of interests leads to conflict situations; the latter can be studied by game theory; interacting players choose their actions based on the actions of other players. Moreover, each player makes its decisions in order to maximize its utility function, which generally depends on the actions of all the players.In last years, game theory has been used to solve many problems in communication networks [1–7]. It has been used to propose new pricing strategies for Internet services [8, 9]. Many other issues relating to wireless networks have been modeled and analyzed using game theory, such as resource allocation [10, 11], power control [12], network routing [13], network caching [14], security [15], etc.Many researches have paid great attention to the dynamics of games (see [16–20]). Researchers tried to make the theory more realistic, by exhibiting various bounded rationality behaviors, by combining game theory with dynamic systems, and by introducing a delay in terms of bounded rationality. All these aspects gave birth to the bounded rational dynamic game. Expectations are important factors when formulating such games. There are several expectation techniques that have been studied in the literature for a player to update and adjust his strategies, namely: adaptive expectations, naive expectations, bounded rationality expectations, and best response expectations [21–23]. In [21], the authors investigated the dynamics of the Cournot duopoly game with heterogeneous players, in which case, one player adopts bounded rational expectation, and the other takes naive expectation. In [24], the authors studied the dynamics of the duopoly game model. The authors in [25] studied duopoly game with homogenous players in the electric power industry, where both players adjust their strategies according to bounded rational expectation. The authors in [16] assumed that one player adopts adaptive expectation and the other uses bounded rational expectation. In [21], the authors investigated the dynamics of a Cournot duopoly game in which bounded rational players adopt a gradient adjustment mechanism to update their outputs in each period. The authors in [26] considered bounded rationality in the telecommunication networks. The authors in [8, 27, 28] studied competition between service providers where end-users are bounded rational. In [29], the authors investigated a batch matching system with boundedly rational end-users. The authors in [30] developed a duopoly game with heterogeneous players participating in carbon emission trading and investigated the asymptotic stability of the equilibrium points of the game. In [31, 32], the authors investigated the long-term price competition in the multichannel supply chain and showed complex phenomena such as bifurcation and chaos. The authors in [33] developed the chaotic phenomenon in the closed-loop supply chain and performed effective chaotic control for the system. In [34] the authors studied a dynamic Stackelberg game model of the supply chain and analyzed the influences of market parameters on the stability of a dual-channel supply chain. The authors in [35] considered a dynamic epiphytic supply chain game model with two players and showed that the system had two routes to chaos: Neimark-Sacker bifurcation and flip bifurcation. In [36], the authors studied the local stability of the Nash equilibrium under quantity or price competition among firms with differentiated expectations. The dynamics of the game can lead to complex behaviors such as chaos. The authors in [37] showed complex dynamics such as bifurcation and chaos, in a duopoly remanufacturing game with bounded rationality. In [38], the authors considered a duopoly model of technological innovation with bounded rationality and studied the stability of the equilibrium points. They showed, through numerical simulation, a series of chaotic phenomena cause period-doubling bifurcation and strange attractors. The main idea in [39] is synchronization of fractional-order uncertain chaotic systems in the finite time. In [40], the authors design and implement a fast reaching finite-time synchronization technique for chaotic systems along with its application to medical image encryption. For the robust synchronization of uncertain delayed chaotic systems, in [41], the authors propose a disturbance observer-based Sliding Mode Control approach. In [42], the authors investigate a chaotic secure communication system between mobile equipment and a base transmitter station based on hybrid chaotic encryption method and finite-time chaos synchronization.Social media proliferation rapidly increases Internet traffic. This high increase in demand for content on the Internet and the need for new approaches that control the massive volume of information have motivated the development of a new approach called Information-Centric Network (ICN). This last is a new communication paradigm which aims to increase the efficiency of content delivery and content availability. In this new concept, the network infrastructure actively contributes to content caching and distribution and every ICN node can cache and serve the requested content. The authors in [43] studied a new business model where an ISP charges CP for caching service because CP benefits from more efficient content distribution. A pricing model is proposed in [44] to study the economic incentive for caching and sharing content in ICN, where ICN consists of access ICN, transit ICN, and CP. In [45] the authors considered a noncooperative game between CP and ISP in ICN, where CP and ISP are playing with their pricing strategies and fixing the caching policy. The authors in [46] analyzed the impact of caching cost in joint caching and pricing strategies in ICN with one CP, two access ICNs, and one transit ICN. In [47], the authors studied a noncooperative game between one CP and one ISP in ICN, where ISP cached content. It shows that caching investment is beneficial for ISP and CP. The authors in [48] modeled the caching game between CP and ISP as a cooperative game, where ISP cached a fraction of content.In the above literature on oligopoly games, most papers focused on games with a bounded rational player. Namely, players adopted different expectations (Naive expectations, adaptive expectations, and bounded rationality) to update their output; another branch of the literature studied competition among ISPs in ICN market where ISPs are rational and adopt best response expectation. However, we can hardly find a few papers that considered bounded rational ISP in an ICN market. On the other hand we followed the methods of [19]. The present paper moves toward this less explored direction.The main contributions of this work can, therefore, be summarized as follows:(i)
We study the competition between two ISPs’ bounded rational in the ICN market.(ii)
We present new features in the mathematical modeling that include the price to access the content in the cache, the number of items in the cache, and the ISPs revenues.(iii)
We model the interactions in prices, quality of service, and caching among ISPs in ICN, where each ISP is bounded rational. Each ISP acts under bounded rationality and adjusts its strategies according to its expected marginal profit.(iv)
Users’ behavior is modeled as a function of ISP strategies (i.e., network access price, the price for access to the content in the cache, quality of service (QoS), and the number of items in the cache).(v)
The existence and stability of the equilibrium solution of the game and the bifurcation of the system have been further discussed by the nonlinear dynamics theory.(vi)
We complement our analysis with numerical results that demonstrate that the variation of the system parameters causes a period-doubling bifurcation and chaotic behavior. Then an appropriate controlling method can be used to force the system back to the stability state.The rest of this paper is as follows: in Section2, we develop our model, and we provide a detailed description of the duopoly game with bounded rationality model. In Sections 3–5, we have presented two prices and the number of cached items games with bounded rationality and existence and stability of Nash equilibrium are studied. Section 6 presents a numerical study to validate our claims. In Section 7, we apply the control scheme on the proposed system to suppress the chaotic behavior that appeared. Finally, we conclude the paper in Section 8.


## 2. Model

We consider a hierarchical ICN market model with two ISPs and an arbitrary number of end-users who can switch from one ISP to another. Each ISP decides its network access pricepsj, price to access to the content in cache pcj, quality of service qsj, and the number of cached items kj in order to maximize their profits.


### 2.1. Demand Model

The demand ofISPj is a linear function with respect to the network access price psj, price to access to the content in cache pcj, quality of service qsj, and number of cached items kj (see [2, 8, 49]). This demand depends also on the price ps−j, price pc−j, quality of service qs−j, and number of cached items k−j where ps−j=ps1,ps2,…,psj−1,psj+1,…,psM. Then the demand function depends on ps=ps1,…,psM, pc=pc1,…,pcM, qs=pq1,…,pqM, and k=k1,…,kM. For ISPj, the demand Dj is decreasing w.r.t psj, pcj and increasing w.r.t psm, pcm, j≠m. Dj is increasing w.r.t qsj, kj and decreasing w.r.t qsm, km, j≠m. The demand function can be written as follows:(1)Djps,pc,qs,k=dj−αjjpsj+βjjqsj+σjjkj−γjjpcj+∑m=1,m≠jMαjmpsm−βjmqsm−σjmkm+γjmpcm.The parameterdj expresses the total potential demand of end-users. αjm, βjm, γjm, and σjm are positive constants representing, respectively, the responsiveness of the ISPj to price psm, QoS qsm, price pcm, and number of cached items km.


### 2.2. Utility

In the system model, the revenue of an ISP gained by providing services to end-users can be represented by a utility function. Each ISP preferentially uses the content cached to satisfy end-user requests; when the requested content does not exist in the cache, end-user requests will be forwarded to the content provider. According to those mentioned above, the utility function of an ISP is given as(2)Uj=psj−ptN−kjDj+psj+pcj−cjkjDj−vjN−kjBj,where cj is the caching cost. pt is the transmission price. The first term psj−ptDjN−kj is the revenue of network access. The second term psj+pcj−cjkjDj is the revenue of caching. vj is the cost of a unit of backhaul bandwidth. Bj is the backhaul bandwidth needed to serve the demand Dj. The third term vjN−kjBj is the bandwidth cost. The quality of service qsj defined as the expected delay is computed by the Kleinrock function (see [50, 51]):(3)qsj=1Delay=Bj−Dj,and it means that(4)Bj=qsj2+Dj.Then, the utility of theISPj given by(5)Uj=psj−ptN−kjDj+psj+pcj−cjkjDj−vjN−kjqsj2+Dj.


## 2.1. Demand Model

The demand ofISPj is a linear function with respect to the network access price psj, price to access to the content in cache pcj, quality of service qsj, and number of cached items kj (see [2, 8, 49]). This demand depends also on the price ps−j, price pc−j, quality of service qs−j, and number of cached items k−j where ps−j=ps1,ps2,…,psj−1,psj+1,…,psM. Then the demand function depends on ps=ps1,…,psM, pc=pc1,…,pcM, qs=pq1,…,pqM, and k=k1,…,kM. For ISPj, the demand Dj is decreasing w.r.t psj, pcj and increasing w.r.t psm, pcm, j≠m. Dj is increasing w.r.t qsj, kj and decreasing w.r.t qsm, km, j≠m. The demand function can be written as follows:(1)Djps,pc,qs,k=dj−αjjpsj+βjjqsj+σjjkj−γjjpcj+∑m=1,m≠jMαjmpsm−βjmqsm−σjmkm+γjmpcm.The parameterdj expresses the total potential demand of end-users. αjm, βjm, γjm, and σjm are positive constants representing, respectively, the responsiveness of the ISPj to price psm, QoS qsm, price pcm, and number of cached items km.


## 2.2. Utility

In the system model, the revenue of an ISP gained by providing services to end-users can be represented by a utility function. Each ISP preferentially uses the content cached to satisfy end-user requests; when the requested content does not exist in the cache, end-user requests will be forwarded to the content provider. According to those mentioned above, the utility function of an ISP is given as(2)Uj=psj−ptN−kjDj+psj+pcj−cjkjDj−vjN−kjBj,where cj is the caching cost. pt is the transmission price. The first term psj−ptDjN−kj is the revenue of network access. The second term psj+pcj−cjkjDj is the revenue of caching. vj is the cost of a unit of backhaul bandwidth. Bj is the backhaul bandwidth needed to serve the demand Dj. The third term vjN−kjBj is the bandwidth cost. The quality of service qsj defined as the expected delay is computed by the Kleinrock function (see [50, 51]):(3)qsj=1Delay=Bj−Dj,and it means that(4)Bj=qsj2+Dj.Then, the utility of theISPj given by(5)Uj=psj−ptN−kjDj+psj+pcj−cjkjDj−vjN−kjqsj2+Dj.


## 3. PricePs Game

We suppose both ISPs have bounded rationality and use the marginal profit method to update their strategy in the next period, as assumed in the existing work on the classical Cournot games for output competition [52–54]. It means that each ISP will increase its strategies in a period t+1 if the marginal profit in the current period t is positive; otherwise, the firm will decrease its investment. Then the investment adjustment mechanism of a player can be modeled as(6)ps1t+1=ps1t+θ1ps1t∂U1ps1t,ps2et+1∂ps1,ps2t+1=ps2t+θ2ps2t∂U2ps1et+1,ps2t∂ps2.As it was considered by Gibbons in [55], it makes sense to assume that, in the bounded rationality term, the expected price psjet+1 decided by ISPj is equal to its previous value psjt. However to anticipate psjet+1 , it may make more sense to assume that each ISP has a memory for storing prices decided in previous slot time before time t. So to anticipate the expected price of his competitor at t+1, ISPj uses previous prices of ISPm, i.e., psmt−1,psmt−2,…,psmt−T, j≠m with different weights; this point of view has been studied [56, 57] in different contexts. Generally, the expected price of ISPm becomes pmet+1=∑l=0TwlpmT−l,wl≥0 and ∑l=0Twl=1, the constants wl, l=0,1,…,T, are the weights given to previous prices, and T represents the size of memory, so (6) becomes(7)ps1t+1=ps1t+θ1ps1tl1+ps1tα11−2N+k1−1+α12N−k1+1∑f=0Tω2ps2t−f,ps2t+1=ps2t+θ2ps2tl2+ps2tα22−2N+k2−1+α21N−k2+1∑f=0Tω1ps1t−f,where l1=Nd1+β11Nqs1+σ11Nk1−γ11Npc1−β12Nqs2−σ12Nk2−Npt1+α11+Nv1α11+k1α11pt1−v1−pc1+c1, l2=Nd2+β22Nqs2+σ22Nk2−γ22Npc2−β21Nqs1−σ21Nk1−Npt2+α22+Nv2α22+k2α22pt2−v2−pc2+c2.For simplicity, we setT=1; in this case, the previous dynamic model, with one step T=1, is given by(8)ps1t+1=ps1t+θ1ps1tl1+ps1tα11−2N+k1−1+α12N−k1+1ω2ps2t+1−ω2ps2t−1,ps2t+1=ps2t+θ2ps2tl2+ps2tα22−2N+k2−1+α21N−k2+1ω1ps1t+1−ω1ps1t−1.To study the stability of dynamic system (8), we rewrite it as a fourth-dimensional system in the form(9)x1t+1=ps1t,x2t+1=ps2t,ps1t+1=ps1t+θ1ps1tl1+ps1tα11−2N+k1−1+α12N−k1+1ω2ps2t+1−ω2x2t,ps2t+1=ps2t+θ2ps2tl2+ps2tα22−2N+k2−1+α21N−k2+1ω1ps1t+1−ω1x1t.Equilibrium points of (9) can be calculated by setting psit+1=psit, i=1,2, in (9). This gives the following algebraic two equations:(10)θ1ps1tl1+ps1tα11−2N+k1−1+α12N−k1+1ps2t=0,θ2ps2tl2+ps2tα22−2N+k2−1+α21N−k2+1ps1t=0.Solving equations in (10), we obtain four equilibrium states of dynamics (9), which are listed as follows:(11)E1=0,0,0,0,E2=l1α11−2N+k1−1,0,l1α11−2N+k1−1,0,E3=0,l2α22−2N+k2−1,0,l2α22−2N+k2−1,E4=Nl2α12−k2l2α12−α22−2N+k2−1l1+l2α12α12α21NN−k1−k2+2+k1k2−k1−k2+1−α22α11−2N+k2−1−2N+k1−1Nl1α21−k1l1α21−α11−2N+k1−1l2+l1α21α12α21NN−k1−k2+2+k1k2−k1−k2+1−α22α11−2N+k1−1−2N+k1−1Nl2α12−k2l2α12−α22−2N+k2−1l1+l2α12α12α21NN−k1−k2+2+k1k2−k1−k2+1−α22α11−2N+k2−1−2N+k1−1Nl1α21−k1l1α21−α11−2N+k1−1l2+l1α21α12α21NN−k1−k2+2+k1k2−k1−k2+1−α22α11−2N+k1−1−2N+k1−1The equilibrium pointsE1, E2, and E3 are boundary equilibrium points and the equilibrium point E4 is a unique Nash equilibrium point. For the sake of the economic significance, all the equilibrium points E2, E3, and E4 of system (9) should be nonnegative. The equilibrium point of dynamical system (9) has economic meaning when l1<0 and l2<0.To analyze the local stability properties of the equilibrium points of system (9), the eigenvalues of the Jacobian matrix need to be calculated. The equilibrium point of a nonlinear system is stable if and only if the modules of all the eigenvalues of the Jacobian matrix evaluated at the equilibrium points are less than one.The Jacobian matrix of system (9) corresponding to the state variables r1,r2,ps1,ps2 is calculated as follows:(12)Jr1,r2,ps1,ps2=001000010p11−ω2θ1ps1θ1l1+2θ1ps1p2+θ1ω2ps2p1+θ11−ω2x2p1+1ω2p1θ1ps1p31−ω1θ2ps20ω1p3θ2ps2θ2l2+2θ2ps2p4+θ2ps1ω1p3+θ2x11−ω1p3+1,where p1=α12N−k1+1, p2=α11−2N+k1−1, p3=α21N−k2+1, and p4=α22−2N+k2−1.Theorem 1.
Steady statesE1, E2, and E3 are unstable equilibrium points of system (9).
The proof of the above theorem can be found in Appendix A.
To study the local stability of the Nash equilibriumE4 of (9), we estimate the Jacobian matrix J at E4, which is(13)JE4=001000010p11−ω2θ1ps1θ1l1+2θ1ps1p2+p1θ1ps2+1ω2p1θ1ps1p31−ω1θ2ps20ω1p3θ2ps2θ2l2+2θ2ps2p4+p3θ2ps1+1.
The characteristic polynomial ofJE4 can be described as(14)Pλ=λ4+a1λ3+b1λ2+c1λ+d1,where(i)
a1=2+θ1l1+θ2l2+2θ1p2+p3θ2ps1+p1θ1+2θ2p4(ii)
b1=1+θ2l2+θ1l1+θ1θ2l1l2+θ1θ2l1p3+2θ1θ2l1p2+p3θ2+2θ1p1ps1+2θ2p4+2θ1θ2l1p4+p1θ1+θ1θ2p1l2ps2+2θ1θ2p2p3ps12+2θ1θ2p1p4ps22+θ1θ24p2p4+p1p31−ω1ω2ps1ps2(iii)
c1=θ1θ2p1p2ω1+ω2−2ω1ω2(iv)
d1=−p1p3θ1θ21−ω1−ω2+ω1ω2ps1ps2
Following a standard stability analysis, a sufficient and necessary condition for the local stability of Nash equilibriumE4 is that the eigenvalues of the Jacobian matrix JE4 are inside the unit circle of the complex plane; this is true if and only if the following Jury’s conditions [58] hold.(1)
1+a1+b1+c1+d1>0(2)
1−a1+b1−c1+d1>0(3)
1−d11−d14−b11−d12+a1−c1c1−a1d1>0(4)
3+3d1>b1(5)
|d1|<1


## 4. PricePc Game

According to the above section, a bounded rational player modifies his price according to his marginal profit:∂Uipc/∂pci, i∈1,2. So, the dynamical system for decision of ISPs has this form:(15)pc1t+1=pc1t+ϑ1pc1t∂U1pc1t,pc2et+1∂pc1,pc2t+1=pc2t+ϑ2pc2t∂U2pc1et+1,pc2t∂pc2,where ϑj represents the speed of adjustment of price of ISPj, and pcjet+1 is the expected price of the opponent in the next time step t+1.As explained above, and if we consider thatT=1, the expected price pcjet+1 is given by(16)pcjet+1=ωjpcjt+1−ωjpcjt−1,where the positive constants ωj are the weights given to the previous prices.Thus, the dynamical model (13) becomes(17)pc1t+1=pc1t+ϑ1pc1tl3−2γ11k1pc1t+γ12k1ω2pc2t+1−ω2pc2t−1,pc2t+1=pc2t+ϑ2pc2tl4−2γ22k2pc2t+γ21k2ω1pc1t+1−ω1pc1t−1,where l3=−ps1−pt1γ11N−k1−k1γ11ps1−c1+vjγ11N−k1+d1k1−α11ps1k1+β11qs1k1σ11k12+α12ps2k1+β12qs2k1σ12k1k2, l4=−ps2−pt2γ22N−k2−k2γ22ps2−c2+v2γ22N−k2+d2k2−α22ps2k2+β22qs2k2σ22k22+α21ps1k2+β21qs1k2σ21k1k2To study the dynamic system (17), we write it as a fourth-dimensional system in the form:(18)y1t+1=pc1t,y2t+1=pc2t,pc1t+1=pc1t+ϑ1pc1tl3−2γ11k1pc1t+γ12k1ω2pc2t+1−ω2y2t,pc2t+1=pc2t+ϑ2pc2tl4−2γ22k2pc2t+γ21k2ω1pc1t+1−ω1y1t.The steady states of the dynamic system (18) are(19)E5=0,0,0,0,E6=l32γ11k1,0,l32γ11k1,0,E7=0,l42γ22k2,0,l42γ22k2,E8=k1l4γ12+2k2l3γ22k1k24γ11γ22−γ21γ12,k2l3γ21+2k1l4γ11k1k24γ11γ22−γ21γ12,k1l4γ12+2k2l3γ22k1k24γ11γ22−γ21γ12,k2l3γ21+2k1l4γ11k1k24γ11γ22−γ21γ12.Obviously,E5, E6, and E7 are boundary equilibrium points and E8 is the unique Nash equilibrium point. The study of local stability of a fixed point in the four-dimensional system (18) depends on the eigenvalues of its Jacobian matrix. The Jacobian matrix Jy1,y2,pc1,pc2 at any point y1,y2,pc1,pc2 in the dynamic system (18) takes the form(20)Jy1,y2,pc1,pc2=001000010ϑ1pc1γ12k11−ω21+ϑ1l3−4γ11ϑ1k1pc1+ϑ1γ12k1ω2pc2+1−ω2y2ϑ1γ12k1ω2pc1γ21k21−ω1ϑ2pc20ω1ϑ2k2pc2ϑ2l4−4ϑ2γ22k2pc2+ϑ2k2γ21pc1ω1+y11−ω1+1.Theorem 2.
The trivial equilibrium pointsE5, E6, and E7 of system (18) are unstable.
The proof of the above theorem can be found in Appendix B. To analyze the local stability atE8, we get the Jacobian matrix as the form(21)JE8=001000010ϑ1pc1γ12k11−ω21+ϑ1l3−4γ11ϑ1k1pc1+ϑ1γ12k1pc2ϑ1γ12k1ω2pc1γ21k21−ω1ϑ2pc20ω1γ21ϑ2k2pc2ϑ2l4−4ϑ2γ22k2pc2+ϑ2k2γ21pc1+1.
And by direct calculation, we obtain the characteristic polynomialPλ for JE8 as follows:(22)Pλ=λ4+a2λ3+b2λ2+c2λ+d2,where(1)
a2=2+υ1l3+υ2l4−4γ11υ1k1+υ2k2γ21pc1+−4υ2γ22k2+υ1γ12k1pc2(2)
b2=1+υ2l4+υ1l3+υ1υ2l3l4+υ2k2γ21+υ1υ2l3k2γ21−4υ1υ2γ11l4k1−4γ11υ1k1pc1+−4γ22υ2k2−4υ1υ2l3γ22k2+υ1υ2γ12k1l4+υ1γ12k1pc2)−4γ12γ21υ1υ2k1k2pc12−4υ1υ2γ12γ22k1k2pc22+16υ1υ2γ21γ12k1k2+υ1υ2k1k2γ21γ21−υ1υ2k1k2ω1ω2γ12γ21,(3)
c2=υ1υ2γ21γ12k1k2ω1+ω2−2ω1ω2pc1pc2(4)
d2=γ21γ12k1k2υ1υ21−ω1−ω2+ω1ω2
System (18) will be stable only if the regions match Jury conditions [58], that is,(1)
1+a2+b2+c2+d2>0(2)
1−a2+b2−c2+d2>0(3)
1−d21−d24−b21−d22+a2−c2c2−a2d2>0(4)
3+3d2>b2(5)
d2<1


## 5. Cache Game

According the above section, the adjustment mechanism of number of cached items over time of thejth ISPs is described by(23)kjt+1=kjt+ϑikjt∂Uj(kjt,kmet+1)∂kj,j≠m∈1,2,where ϑj, j=1,2, are positive constants which stand for the speed of adjusting to market variations.As explained above, and if we consider thatT=1, the expected price kjet+1 is given by(24)kjet+1=ωjkjt+1−ωjkjt−1.The positive constantsωj are the weights given to the previous number of cached items.Thus, the dynamical model with bounded rationality ISPs is defined as(25)k1t+1=k1t+η1k1t(l5+2e1σ11k1t−e1σ12∑f=0Tw2k2t−f,k2t+1=k2t+η2k2t(l6+2e2σ22k2t−e2σ21∑f=0Tw1k1t−f,where e1=pt1+pc1−c1+v1, e2=pt2+pc2−c2+v2, l5=pt+pc1−c1+v1d1−α11ps1+β11qs1−γ11pc1+α12ps2−β12qs2+γ12pc2+ps1−ptNσ11+ps1+pc1−c1σ11+v1qs12−v1Nσ11 and l6=pt+pc2−c2+v2d2−α22ps2+β22qs2−γ22pc2+α21ps1−β21qs1+γ21pc1+ps2−ptNσ22+ps2+pc2−c2σ22+v2qs22−v2Nσ22.To study the dynamic system (25), we write it as a fourth-dimensional system in the form(26)z1t+1=k1t,z1t+1=k2t,k1t+1=k1t+η1k1t(l5+2e1σ11k1t−e1σ12ω2k2t+1−ω2z2t,k2t+1=k2t+η2k2t(l6+2e2σ22k2t−e2σ21ω1k1t+1−ω1z1t.As time evolves,kjt+1 are approximately equal to kjt when the market structure is stable enough at time t. Set k1t+1=k1t and k2t+1=k2t; the dynamic system (26) turns to be(27)η1k1t(l5+2e1σ11k1t−e1σ12k2t=0,η2k2t(l6+2e2σ22k2t−e2σ21k1t=0.Solving equations in (27), we obtain four equilibrium states of dynamic system (26), which are listed as follows:(28)E9=0,0,0,0,E10=l52σ11e1,0,l32σ11e1,0,E11=0,l62σ22e2,0,l42σ22e2,E12=−e1l6σ12+2e2l5σ22e1e24σ11σ22−σ21σ12,−e2l5σ21+2e1l6σ11e1e24σ11σ22−σ21σ12,−e1l6σ12+2e2l5σ22e1e24σ11σ22−σ21σ12,−e2l5σ21+2e1l6σ11e1e24σ11σ22−σ21σ12.The research of local stability at the points concluded above replies on the Jacobian matrix of the dynamic system (26), which takes the unified form(29)Jz1,z2,k1,k2=001000010−η1k1σ12e11−ω21+η1l5+4σ11η1e1k1−η1σ12e1ω2k2+1−ω2z2−η1σ12e1ω2k1−σ21e21−ω1η2k20−ω1η2e2k2η2l6+4η2σ22e2k2−η2e2σ21k1ω1+z11−ω1+1.Theorem 3.
The trivial equilibrium pointsE9, E10, and E11 of system (26) are unstable.
The proof of the above theorem can be found in Appendix C. We study the local stability of Nash equilibriumE12. The Jacobian matrix (29) at E12 takes the form(30)JE12=001000010−η1k1σ12e11−ω21+η1l5+4σ11η1e1k1−η1σ12e1k2−η1σ12e1ω2k1−σ21e21−ω1η2k20−ω1η2σ21e2k2η2l6+4η2σ22e2k2−η2e2σ21k1+1.
The characteristic polynomial ofJE12 is(31)Pλ=λ4+a3λ3+b3λ2+c3λ+d3,where(i)
a3=2+η1l5+η2l6+4σ11η1e1−η2e2σ12k1+−η1σ12e1+4η2σ22e2k2(ii)
b3=1+η2l6+η1η2l5l6+η1l5+−η2c2σ22−η1η2l5e2σ21+4σ11η1η2l6e1+4σ11η1e1k1+4η2σ22e2+4η1η2l5σ22e2−η1η2l6σ12e1−η1σ12e1k2−4σ11σ21η1η2e1e2k12−4η1η2e1e2σ12σ21k22+η1η2e1e2σ12σ21+16σ11σ22η1η2e1e2l5−ω1ω2η1η2e1e2σ12σ21k1k2(iii)
c3=η1η2σ21σ12e1e2ω1+ω2−2ω1ω2(iv)
d3=−η1η2e1e2σ21σ121−ω1−ω2+ω1ω2
The local stability of Nash equilibrium is given by using Jury’s conditions [58] which are(1)
1+a3+b3+c3+d3>0(2)
1−a3+b3−c3+d3>0(3)
1−d31−d34−b31−d32+a3−c3c3−a3d3>0(4)
3+3d3>b3(5)
d3<1


## 6. Numerical Simulation

In this section, we will provide the numerical evidence for the complex dynamical behaviors of systems (9), (18), and (26) when losing stability and exhibit how the system evolves when the model parameters take different levels of values.In Figure1(a), the bifurcation scenario occurrs; if θ1 is small, then there exists a stable equilibrium point. As one can see, the Nash equilibrium point E4 is locally stable for small values of θ1. As θ1 increases, the Nash equilibrium becomes unstable, the system appears for a two-period bifurcation, and then the system goes into two cycles, after four times for each cycle and eight period-doubling bifurcations; then the final system enters into a state of chaos.Figure 1
(a) Bifurcation diagrams ofps1 with respect to θ1; (b) Te corresponding to the largest Lyapunov exponent for ps1; (c) convergence speed.
(a)(b)(c)Figure2(a) presents the bifurcation diagrams of price pc1 with respect to the adjustment speed ϑ1. From Figure 2(a) we see that a low adjustment speed ϑ1<0.04 can make the system stable. With ϑ1 increasing, a flip bifurcation for system (18) takes place at ϑ1=0.04 and 2-period bifurcate as ϑ1>0.04. As long as the parameter ϑ1 increases, the Nash equilibrium point E8 becomes unstable and the bifurcation scenario occurs and ultimately leads to unpredictable (chaotic) motions that are observed.Figure 2
(a) Bifurcation diagrams ofpc1 with respect to ϑ1; (b) Te corresponding to the largest Lyapunov exponent for pc1; (c) convergence speed.
(a)(b)(c)Figure3(a) shows the bifurcation diagram of the number of cached items k1 with change of η1. It is clear that the Nash equilibrium point E12 of system (26) is stable when η1<0.025. As η1 increases the Nash equilibrium of point E12 becomes unstable and complex dynamic behavior occurs, including two-period bifurcation and chaos. From the perspective of economics, the ISPs adjustment speed should be in a certain range; otherwise, the system moves to the period-doubling bifurcation point and then four times bifurcation point until chaos while growing, which means irregularity, unpredictability, sensitivity to initial values, and harm to the economy.Figure 3
(a) Bifurcation diagrams ofk1 with respect to η1; (b) Te corresponding to the largest Lyapunov exponent for k1; (c) convergence speed.
(a)(b)(c)Figure4(a) shows the bifurcation diagrams of price ps1 with respect to the number of cached items k1. With the rise of the number of cached items k1, the Nash equilibrium point E4 increases gradually and is locally stable for small values of the parameter k1. As k1 increases, the Nash equilibrium point E4 becomes unstable, and complex dynamic behavior occurs such as chaos.Figure 4
(a)Bifurcation diagrams ofps1 with respect to k1; (b) Te corresponding to the largest Lyapunov exponent for ps1; (c) convergence speed.
(a)(b)(c)Figure5(a) shows the bifurcation diagram of the number of cached items k1 with respect to the caching cost c. From Figure 5(a), we can see that the Nash equilibrium E12 is locally stable for small values of c. As c increases, the Nash equilibrium point E12 becomes unstable and complex dynamic behavior occurs, including chaos. Economic meanings of different states of this system are as follows: stabilization means that the market is regular and fixed; every ISP can get fixed profit with charging fixed prices and offering fixed number of cached items in every time period in this situation. Stable state is usually good for ISPs to make long-term strategies and can help them get away from the trouble of changing strategies frequently. 2-period is regular but fluctuant; ISPs can know the development of the market but have to change their strategies frequently in this situation. Chaos means that the market is irregular and fluctuant; it is difficult for ISPs to make long-term strategies in this situation as the system depends on initial values sensitivity.Figure 5
(a) Bifurcation diagrams ofk1 with respect to c; (b) Te corresponding to the largest Lyapunov exponent for k1; (c) convergence speed.
(a)(b)(c)


## 7. Chaos Control

From the above section, we can see that the dynamical behavior of systems (9), (18), and (26) may be chaotic. The chaotic motion is always irregular and unpredictable. In practical application, the chaotic motion is always not desired, so we need to avoid the occurrence of chaos. Therefore, all ISPs in the ICN market always hope that the market can operate stably. Over the past decade, a good deal of methods for controlling chaos has been put forward. Here, we employ the widely used control method, that is, time-delayed feedback control method to control the chaotic phenomenon (e.g., [54, 59]).Using the time-delayed feedback control method, we adjust system (9) by inserting the control action κjpsjt−psjt+1 in the right hands of the last two equations, where psjt is the time-delayed state variable and κj>0 is the controlling coefficient which can express the control over the price adjustment speed or the learning adaptability of market competitors. Then we get the controlled system as follows:(32)x1t+1=ps1t,x2t+1=ps2t,ps1t+1=ps1t+θ1ps1t(l1+ps1tα11−2N+k1−1+α12N−k1+1ω2ps2t+1−ω2x2t+κ1ps1t−ps1t+1,ps2t+1=ps2t+θ2ps2t(l2+ps2tα22−2N+k2−1+α21N−k2+1ω1ps1t+1−ω1x1t+κ2ps2t−ps2t+1.It can be rewritten as(33)x1t+1=ps1t,x2(t+1)=ps2(t),ps1t+1=ps1t+θ1ps1tκ1+1(l1+ps1tα11−2N+k1−1+α12N−k1+1ω2ps2t+1−ω2x2t,ps2t+1=ps2t+θ2ps2tκ2+1tl2+ps2tα22−2N+k2−1+α21N−k2+1ω1ps1t+1−ω1x1t.Whenκi=0, the controlled system degenerates into the initial system (9).Adding the control actionξj(pcjt−pcjt+1) to system (18) and simplifying them, we obtained the following form of the controlled dynamical system:(34)y1t+1=pc1t,y2t+1=pc2t,pc1t+1=pc1t+ϑ1pc1tξ1+1l3−2γ11k1pc1t+γ12k1(ω2pc2t+1−ω2y2t,pc2t+1=pc2t+ϑ2pc2tξ2+1l4−2γ22k2pc2t+γ21k2ω1pc1t+1−ω1y1t.Adding the control actionςj(kjt−kjt+1) to system (26) and simplifying them, we obtained the following form of the controlled dynamical system:(35)z1(t+1)=k1t,z1t+1=k2t,k1t+1=k1t+η1k1tς1+1(l5+2e1σ11k1t−e1σ12ω2k2t+1−ω2z2t,k2t+1=k2t+eta2k2tς2+1(l6+2e2σ22k2t−e2σ21ω1k1t+1−ω1z1t.Figure6 shows that, with the control coefficient κ1 increasing, the controlled systems (33) are gradually controlled from the chaotic state, 4-period bifurcation, and 2-period bifurcation to a stable state. When κ1>0.024 , the controlled system (33) stabilizes at the Nash equilibrium point.Figure 6
Bifurcation diagrams ofps1 with respect to κ1.Figure7 shows the bifurcation diagram of controlled system (34) with respect to control parameter ξ1. We can see that, with an increase in the control parameter, the system is controlled at 8-cycle, 4-cycle, 2-cycle, and the Nash equilibrium point. One can conclude that the controlled system (34) is being gradually controlled with an increase in the control parameter ξ1, and the system will be led to stability when ξ1 is large enough.Figure 7
Bifurcation diagrams ofpc1 with respect to ξ1.Figure8 shows that the period-doubling bifurcation disappears gradually and the controlled system (35) gets rid of chaos to be stable when the controlling parameter ς1 is appropriately significant. So, the above control method can control chaos in the game, and the market game can switch from chaotic trajectories to regular periodic orbits or equilibrium state.Figure 8
Bifurcation diagrams ofk1 with respect to ς1.


## 8. Conclusion

In this paper, we established the duopoly game model regarding the prices and the number of cached items competition in an open market, based on the theory of a particular economic model. The dynamic of the duopoly game model with bounded rationality has been analyzed. The equilibrium points have been obtained as functions of the system parameters. A stability analysis of those points has been investigated. Numerical simulations show that while varying the model parameters, complex dynamic behaviors would occur, such as chaos. In addition, the control scheme has been applied to force the system back to its stable state.As part of future work, we plan to add content popularity to our analysis of duopoly game in the ICN market with ISPs and CPs.



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*Source: 6632993-2022-03-16.xml*