Abstract
Benefitting from the popular uses of internet technologies, two-sided market has been playing an increasing prominent role in modern times. Users and developers can interact with each other through two-sided platforms. The two-sided market structure has been investigated profoundly. Through building a dynamics two-sided market model with bounded rational, stability conditions of the two-sided market competition system are presented. With the help of bifurcation diagram, Lyapunov exponent, and strange attractor, the stability of the two-sided market competition model is simulated. At last, we use the time-delayed feedback control (TDFC) method to control the chaos. Our main results are as follows: (1) when the adjustment speed of two-sided increases, the system becomes bifurcation, and chaos state happens finally. When the system is stable, the consumer fee is positive while developer fee is negative. (2) When the user externality increases, the stable area of the system increases, and the difference in user externality leads the whole system more stable. When the system is stable, the developer fee decreases. (3) The stable area becomes larger when developer externality increases; when the system is stable, the user fee becomes lower and developer fee becomes higher when developer externality increases. (4) The TDFC method is presented for controlling the chaos; we find that the system becomes more stable under the TDFC method.
1. Introduction
With the rapid development of internet technologies, platforms are becoming a common form in real-world business. There are different examples of two-sided markets in the today market [1, 2], such as newspaper, ride-sharing, and TV channels. Different agents have the cross-group relationship through the platforms, and two-sided market structure makes the competition between platforms more complex. Internet platform decreases the cost of communication between different agents [3]; however, there are lots of problems emerges as the different characters of platforms, for example, the develop of ride-sharing platform brings lots of challenge for government [4]. The reason of the chaos state is because the agents are bounded rational and the network externality between agents.
This paper related to the two-sided market theory. The earlier two-sided market articles are mostly about the horizontal competition, such as [5, 6]; two-sided markets compete for their network externality; when other sides have more agents, the consumer would has more utility. Affeldt et al. [7] extend the upward pricing pressure from the one-sided market to two-sided markets. Lam [8] shows switching costs have different effects in a dynamic two-sided model from the traditional market. The two-sided platform uses price decision as a competition strategy. Caillaud and Jullien [9] analyze the pricing strategy in the two-sided market. Choi [10] analyzes the pricing strategy in multihoming. Belleflamme and Toulemonde [11] study intragroup externality in the two-sided market.
The second branch of literature is dynamics competition. The second branch of literature is dynamics competition. A nonlinear behavior was widely studied in different fields, such as Bao et al. [12, 13], and Li et al. [14] investigate hyperchaos in memristor. The economy complexity behavior was discovered by many researchers. Onozaki et al. [15] analyze a dynamic of a cobweb market. Du et al. [16] find that the control of chaos improve performance of system. Dubiel-Teleszynski [17] studies nonlinear dynamics in a duopoly game with diseconomies of scale. Elsadany [18] investigates bounded rationality in dymamics of a Cournot duopoly game. Fanti and Gori [19] study quantity competition in the duopoly game. Jayanthi and Sinha [20] investigate the chaotic characteristics of innovation in high technology manufacturing. Guo et al. [21] show the information intermediary and the end users have the chaos behavior. Chaos theory has a close relationship with the industrial organization [12–23].
The two-sided market, such as TV channels, has a complex behavior in competition [24]. In order to deal with the complexity, there are several mechanisms related to the control of chaotic systems, such as feed-forward control, OGY method, and time-delayed feedback control (TDFC) [25]. TDFC becomes increasingly popular for its advantages that the feedback does not need fine understanding of the system [26]. Sukono et al. [27] and Vaidyanathan et al. [28] investigate the bifurcation and control in the financial risk system.
The rest of the paper is organized as follows: Section 2 describes the two-sided market competition model. Section 3 presents the fixed points of the system and stability of the fixed points. Section 4 shows the simulation of the two-sided market model and gives the time-delayed feedback control of system. Section 5 draws the conclusions.
2. The Model
Our two-sided market is shown in Figure 1, the two two-sided platforms make a fee decision about the user and developer, the developer is independent to each other, and the user and developer interacted with each other through two-sided platforms.
In our model, our two-sided market competition model is similar to Armstrong [5], but our model have new features. Our model is different from Armstrong’s model in that our model in the developer side is independent; in the real world, there are most common that the two-sided platform competing with each other in one side, such as different newspapers competes only in readers, but in the writer side, they have no competition.
There are two two-sided platforms who connect users and developers. Suppose that users are heterogeneous, whose preference distribute between a unit interval, two-sided platforms maximize their profits through charging fees on developers and users. Setis a user externality parameter which is from developers to users, as Hagiu and Halaburda [29], we set , is the fee that the two-sided platform charged on users, and is the developers quantity who join platform .
Users whose position in join platform 1 utility function is
In platform 2, the users’ utility function is denoted:
Set is a developer externality parameter which is from users to developers, for more consumer leads to more developers, as Hagiu and Halaburda [29], we assume . is the fee that the two-sided platform charged on developers, and is the users’ quantity who join platform .
The number of developers who joins platform 1 is
The number of developers who joins platform 2 is
The externality between two groups is asymmetric. For simplicity, we assume marginal costs is 0. Then, platform 1’s profit function is
Platform 2’s profit function is
Let , then the users quantity is
Inserting equation (7) into equations (3) and (4), we get the number of users and developers who join the platform 1 using
The number of users and developers who join the platform 1 is
The platform makes its price decision to maximize its profit; inserting equation (8) in equation (5), the platform 1’s profit function can be written as
Similarly, inserting equation (9) into equation (6), the platform 2’s profit function can be written as
Taking the first-order derivative of (10) with respect to , we obtain the following FOC:
Similarly, we take the first-order derivative of (11) with respect to , we obtain the following FOC:
From equation (12), platform 1 makes developer fee decision
From equation (13), platform 2 makes developer fee decision
Substitute equation (14) into platform 1 profit function (10), then we get platform 1’s profit function
Substitute equation (15) into platform 2 profit function (11), then we get platform 2’s profit function
Differentiating equation (16) with respect to , we get
Differentiating equation (17) with respect to , we get
When the former equations (18) and (19) become equilibrium, then from (10) and (11), we can get
Insert equation (20) into equation (18), then we can get platform 1’s profit maximum condition:
Insert equation (20) into equation (19), then we can get platform 2’s profit maximum condition
Equations (21) and (22) are the condition that platform 1 and platform 2’s maximize their profits.
We assume the platforms use bounded rational expectation, and each platform adopts the myopic adjustment [18]; then, the competition system can be written as
Following, we analyze the equilibrium of system (23).
3. Equilibrium Analysis
Now, we turn to the two-sided market competition equilibrium, when , ; then, we combined equations (21)–(23); then, we can conclude that there are four fixed points in system (23): , , , and . When and , it is obvious that four points are positive.
The point Jacobian matrix is
It is easy to see that the root of the determinant, , , is a repelling point, and is unstable [18].
Jacobian matrix is where , so .
, so , then the point is a saddle point [18].
Similarly, is a saddle point.
Jacobian matrix is where , .
It is easy to conclude that , .
The characteristic equation of Jacobian matrix is
The trace of Jacobian matrix is .
The determinant is .
So, the characteristic equation’s two roots are real [17, 19].
Following Jury rule [18, 19], the stability conditions for system (23) are
The (a) condition is
We can conclude that .
The (b) condition is
For and , so .
The (b) condition is satisfied.
The (c) condition is
We can conclude that .
The stable area can be illustrated in Figure 2. The dark area is the stable area, and the white area is the unstable area. We can see that when is small in an extent, system (23) is stable; when is large, the system would become bifurcation and finally chaos.
4. Simulation
Numerical examples such as bifurcation diagram, maximum Lyapunov exponents, and strange attractor can help us illustrate dynamics of system (23). In this section, our purpose is to show the adjustment speed, buyers’ externality parameter, and sellers’ externality parameter effects on stability of system (23).
4.1. Adjustment Speed Effect
We first fix the parameters set , the initial state is , after take iteration 700 times, and the bifurcation diagram is illustrated in Figure 3. We find that when , the Nash equilibrium point, is stable, when larger than 1.511, system (23) becomes bifurcation, and after , the chaos state happens. We can see that when system (23) is stable, the two platform’s equilibrium are exactly symmetric, and . The users’ fee is larger than developers’ fee, and the developers’ fee is lower than zero; this is a common phenomenon in the two-sided market pricing strategy.
As is shown in Figure 4, the maximum Lyapunov exponent (LE) changes corresponding to Figure 3, as [13], we calculate LE with Wolf’s Jacobian algorithm. The LE returns to zero when , which means system (23) becomes bifurcation; when , the LE becomes larger than zero, meaning system (23) enters the chaos state. And Figure 5 shows the strange attractor when . We can see that when system (23) is chaotic state, the and are in a certain interval.
4.2. Effects on Stable
The externality parameter that developers bring to users affects the stability of system (23); we fixed parameter set .
Figure 6 shows the stable area with different externality parameters . Compare the stable area in different values of , . We can see that when is larger, the stable area of system (23) increases. The externality in consumer side can be seen as the difference between two platforms; the difference in externality leads the whole system more stable.
Figure 7 shows fee bifurcation with the change of user externality parameter . When externality parameter increases , and change from the chaos state to thr two-period bifurcation, and after , system (23) becomes stable, and in the stable state, and decrease, and below zero, and platform’s profit mainly from the fee that consumer pays to the platform. This strategy is also known as the divide and conquer strategy [25].
Figure 8 illustrates the maximum Lyapunov exponents with change of ; when the , the LE is larger than zero, and system (23) is in chaos state; when , the system becomes bifurcation; when , the system becomes stable.
4.3. Effects on Stable
We first fix system (23) parameter set . Figure 9 shows stable area changers with respect to when other parameters are given. The stable area becomes larger when increasing; this means system (23) when is larger.
Figures 10 and 11 show the bifurcation diagram and maximum Lyapunov exponent for changes when other parameters are given. We can see that when , system (23) is in chaos state; after , system (23) becomes bifurcation. After , the system becomes stable and user fee becomes lower while developer fee becomes higher.
From Figure 11, we can see that when , the system is in a chaotic state, when , the system is in bifurcation, while , the system becomes stable.
From simulation, we can infer that the externality parameters in the developer side and user side make the system more chaotic; this means when the externality parameters are large enough, the system becomes out of control.
4.4. Time-Delayed Feedback Control (TDFC)
From first simulation in Section 5.1, given when are fixed, when large enough, system (23) become chaos state.
We use the TDFC to alleviate the chaos state. The TDFC method becomes a popular method to alleviate chaos; as [30], we propose the delay feedback control parameter , we assume a TDFC method , we insert this mechanism into system (23), and then, we can get
So, we get the TDFC system
Then, we fixed the parameter set , From Figure 5, we know the system is in a chaotic state. Figure 12 shows in this chaotic state; then, we let and ; we can see that when , system (23) goes into 2-period bifurcation; when , system (23) goes into a stable state.
Figure 13 shows that when , the original system is chaos; when becomes large, until , system (23) becomes change to bifurcation; and when system (23) becomes stable in the NE point. The simulation shows that the TDFC can alleviate the stability of the two-sided chaos system.
5. Conclusion and Limitations
5.1. Discussion and Conclusions
Two-sided platforms are used by different agents to interact with each other; however, the two-sided markets challenges arise as a result of the externality in different sides. We analyze the complexity in two-sided market competition.
We build a two-sided platform competition game model and then investigate the dynamics of competition between two platforms when two platforms are bounded rational; we find that the stability of system (23) is influenced by adjustment speed and users’ and developers’ externality parameters. When the adjustment speed of two-sided markets is larger, the systems becomes more instable. Both users’ and developers’ externality parameters make the system more stable and enhance the fee in its own side, while lowering the other side price. We then introduce the TDFC mechanism and find that TDFC alleviates the chaos state.
5.2. Theoretical Contributions
Our research results have several contributions. Our research focus on the chaotic phenomenon in the two-sided market. Compared with other previous studies, we combined the two-sided market theory and nonlinear dynamics. We assume that platform is bounded rational, and they can only achieve limit information; this is more in line with reality.
5.3. Managerial Implications
Our research has three strategic implications for platform decision. The platform might presents complexity behaviors. From a strategy perspective, our results imply that platforms have incentives to affect the consumer heterogeneous to alleviate chaotic state; however, this may reduce platform profit. Platform makes their decisions compare its profit target and management control.
5.4. Limitations and Further Research
There are several limitations to this research. For the theoretical study, the relevant assumptions were too strict, such as externality parameters which we assume are small; in fact, platforms have an extensive relationship with other sides; for simplifying research, we assume the platform only compete in the consumer side; if the platform competes in two sides, the conclusion would be more general. And the multihoming and single homing are general in platform; we think this should take more attention.
Data Availability
The (data type) data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was supported by the Natural Science Foundation Project of Guizhou Provincial Education Department (Qian Jiao He KY zi [2019] 207).