# Duopoly Price Competition in Wireless Sensor Network-Based Service Provision

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## Abstract

**:**

## 1. Introduction

- We study price competition in an IoT market, where two SPs compete to provide WSNs-based services to a common of end users. As different types of end users generally have different requirements for the quality of services [25], we take end users’ different willingness-to-pay (WTP) for service quality into consideration.
- We model the relationship between the two SPs and end users in the IoT market as a two-stage Stackelberg game (SG), where the two SPs set the prices for their services in the first stage. Then, based on the qualities and prices of the offered services of the two SPs, the end users make decisions to subscribe or not to services from one of the two SPs in the second stage. We note that although in [15] the relationship between the network operator and the SPs, and the relationship between SPs and users are both modelled as a two-stage SG, the solution methods in each stage are different from our work.
- In SG, the two SPs set the prices for their services sequentially, while in noncooperative strategic game (NSG), the two SPs set the prices for their services simultaneously. Different from many of the existing works that only consider SG, in this paper, we consider two competition scenarios between the two SPs, i.e., a NSG and SG, respectively. A unique equilibrium is achieved in each of the two scenarios.
- Numerical results are performed to verify the theoretical analysis. Our numerical analysis show that both SPs can obtain more profits if they offer services with better qualities. SP1 can attract more users in the SG scenario while SP2 can attract more user in the NSG scenario, and both SPs get more profits in the SG scenario. We also present the analysis on cost factors to show how they impact the profits of the two SPs.

## 2. System Model

**Remark**

**1.**

## 3. Duopoly Competitive IoT Market

- It will join SP1 if ${U}_{k,1}({\theta}_{k},{p}_{1})>{U}_{k,2}({\theta}_{k},{p}_{2})$, and ${U}_{k,1}({\theta}_{k},{p}_{1})>0$, which requires ${\theta}_{k}>{\theta}^{*}$ and ${\theta}_{k}>{\theta}_{1}$;
- It will join SP2 if ${U}_{k,2}({\theta}_{k},{p}_{2})>{U}_{k,1}({\theta}_{k},{p}_{1})$, and ${U}_{k,2}({\theta}_{k},{p}_{2})>0$, which requires ${\theta}_{2}<{\theta}_{k}<{\theta}^{*}$;
- It will join neither of the two SPs if ${U}_{k,1}({\theta}_{k},{p}_{1})<0$, and ${U}_{k,2}({\theta}_{k},{p}_{2})<0$, which requires ${\theta}_{k}<{\theta}_{1}$ and ${\theta}_{k}<{\theta}_{2}$.

**Proposition**

**1.**

- (1)
- (2)
- If ${\theta}_{1}<{\theta}^{*}$, which leads to $\frac{{R}_{1}}{{p}_{2}}<\frac{{R}_{2}}{{p}_{1}}$, from which we get ${\theta}_{2}<{\theta}_{1}<{\theta}^{*}$. According to Equations (7) and (8), we have ${F}_{1}=F\left({\theta}^{*}\right)$ and ${F}_{2}=F\left({\theta}^{*}\right)-F\left({\theta}_{2}\right)$;

**Players**: SP1 and SP2 are the two players in the game;**Strategies**: SP1 and SP2 determine subscription prices ${p}_{1}$ and ${p}_{2}$, respectively;**Payoff**: The profits of SPs, which will be defined later by ${\pi}_{1}={p}_{1}{N}_{1}$ and ${\pi}_{2}={p}_{2}{N}_{2}$.

#### 3.1. Nash Equilibrium in the Duopoly IoT Market

#### 3.2. Noncooperative Strategic Game (NSG)

**Problem1:**

**Problem2:**

**Proposition**

**2.**

**Corollary**

**1.**

#### 3.3. Stackelberg Game (SG)

**Problem3:**

**Problem4:**

**Proposition**

**3.**

**Corollary**

**2.**

**Proposition**

**4.**

**Corollary**

**3.**

## 4. Simulation Results

#### 4.1. Parameter Setting

#### 4.2. Impact of Quality of Data Rate

#### 4.3. Impact of Cost Factor

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

IoT | Internet of Things |

WSNs | Wireless Sensor Networks |

SPs | service providers |

NSG | noncooperative strategic game |

SG | Stackelberg game |

WTP | willingness-to-pay |

## Appendix A. Users’ Joining Decision Policies

## Appendix B. Proof of Proposition 2

## Appendix C. Proof of Proposition 3

**Problem5:**

## Appendix D. Proof of Proposition 4

**Problem6:**

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**Figure 4.**The number of users that choose SP1 with varying ${R}_{1}$ in the two competition scenarios for ${R}_{1}>{R}_{2}$.

**Figure 5.**The number of users that choose SP2 with varying ${R}_{1}$ in the two competition scenarios for ${R}_{1}>{R}_{2}$.

**Figure 6.**The equilibrium price of SP1 with varying ${R}_{1}$ in the two competition scenarios for ${R}_{1}>{R}_{2}$.

**Figure 7.**The equilibrium price of SP2 with varying ${R}_{1}$ in the two competition scenarios for ${R}_{1}>{R}_{2}$.

**Figure 8.**Comparing the profits of the two SPs with varying ${R}_{1}$ in the NSG scenario for ${R}_{1}>{R}_{2}$.

**Figure 9.**Comparing the profits of the two SPs with varying ${R}_{1}$ in the SG scenario for ${R}_{1}>{R}_{2}$.

**Figure 10.**The number of users that choose SP1 with varying ${R}_{2}$ in the two competition scenarios for ${R}_{1}>{R}_{2}$.

**Figure 11.**The number of users that choose SP2 with varying ${R}_{2}$ in the two competition scenarios for ${R}_{1}>{R}_{2}$.

**Figure 12.**The equilibrium price of SP1 varies with ${R}_{2}$ increasing in the two competition scenarios for ${R}_{1}>{R}_{2}$.

**Figure 13.**The equilibrium price of SP2 varies with ${R}_{2}$ increasing in the two competition scenarios for ${R}_{1}>{R}_{2}$.

**Figure 14.**Comparing the profits of the two SPs with ${R}_{2}$ increasing in the NSG scenario for ${R}_{1}>{R}_{2}$.

**Figure 15.**Comparing the profits of the two SPs with ${R}_{2}$ increasing in the SG scenario for ${R}_{1}>{R}_{2}$.

**Figure 16.**The equilibrium prices of the two SPs with varying ${R}_{1}$ in the SG scenario for ${R}_{1}<{R}_{2}$.

**Figure 17.**The equilibrium prices of the two SPs with varying ${R}_{2}$ in the SG scenario for ${R}_{1}<{R}_{2}$.

**Figure 18.**The profits of the two SPs with varying ${R}_{1}$ in the SG scenario for ${R}_{1}<{R}_{2}$.

**Figure 19.**The profits of the two SPs with varying ${R}_{2}$ in the SG scenario for ${R}_{1}<{R}_{2}$.

**Figure 20.**The impact of $\alpha $ on the profits of the two SPs in the NSG scenario for ${R}_{1}>{R}_{2}$.

**Figure 21.**The impact of $\alpha $ on the profits of the two SPs in the SG scenario for ${R}_{1}>{R}_{2}$.

**Figure 22.**The impact of $\beta $ on the profits of the two SPs in the NSG scenario for ${R}_{1}>{R}_{2}$.

**Figure 23.**The impact of $\beta $ on the profits of the two SPs in the SG scenario for ${R}_{1}>{R}_{2}$.

**Figure 24.**The impact of $\alpha $ on the profits of the two SPs in the SG scenario for ${R}_{1}<{R}_{2}$.

**Figure 25.**The impact of $\beta $ on the profits of the two SPs in the SG scenario for ${R}_{1}<{R}_{2}$.

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**MDPI and ACS Style**

Li, X.; Zhao, L.; Zhou, Z.; Gu, B.; Chen, G.; Cheng, F.; Zhang, H.
Duopoly Price Competition in Wireless Sensor Network-Based Service Provision. *Sensors* **2018**, *18*, 4422.
https://doi.org/10.3390/s18124422

**AMA Style**

Li X, Zhao L, Zhou Z, Gu B, Chen G, Cheng F, Zhang H.
Duopoly Price Competition in Wireless Sensor Network-Based Service Provision. *Sensors*. 2018; 18(12):4422.
https://doi.org/10.3390/s18124422

**Chicago/Turabian Style**

Li, Xianwei, Liang Zhao, Zhenyu Zhou, Bo Gu, Guolong Chen, Fanyong Cheng, and Haiyang Zhang.
2018. "Duopoly Price Competition in Wireless Sensor Network-Based Service Provision" *Sensors* 18, no. 12: 4422.
https://doi.org/10.3390/s18124422