# Service Mechanism for the Cloud–Edge Collaboration System Considering Quality of Experience in the Digital Economy Era: An Evolutionary Game Approach

^{*}

## Abstract

**:**

## 1. Introduction

- We establish an evolutionary game model for the collaborative service mechanism of cloud service providers and edge operators and theoretically study the existence conditions and evolution rules of evolutionary stable strategies (ESSs), which contributes to analyzing the behaviors of cloud service providers and edge operators when collaboratively handling user service requests;
- We perform numerical simulations to illustrate the evolution of the cloud–edge collaboration system and show quantitatively the impact of the initial conditions and the variation in decision parameters on the evolutionary results;
- Finally, we propose some specific measures to promote the stability of the cloud–edge collaboration system, based on a theoretical analysis and simulation results.

## 2. Literature Review

## 3. Model Description

#### 3.1. Basic Assumptions and Parameter Descriptions

**Hypothesis**

**1:**

**Hypothesis**

**2:**

**Hypothesis**

**3:**

**Hypothesis**

**4:**

#### 3.2. Construction of Revenue Matrix

## 4. Results

#### 4.1. Stability Analysis of the Evolution of One-Party Strategies

#### 4.1.1. Evolutionary Stability Analysis of Cloud Service Provider

- If $A>B>0$, the stable point ${y}^{*}<0$, and $\left(A-B\right)y+B>0$ for $\forall y\in \left(0,1\right)$. When $x=1$, ${F}^{\prime}\left(x\right)<0$. Moreover, this type of condition satisfies $B>0$, i.e., ${\pi}_{12}^{\left(1\right)}-{\pi}_{22}^{\left(1\right)}>0$. Thus, in this case, for ${S}_{1}$, its coprocessing gain is larger than the solo-processing gain, i.e., ${\pi}_{12}^{\left(1\right)}>{\pi}_{22}^{\left(1\right)}$; hence, ${S}_{1}$ chooses the coprocessing strategy no matter how ${S}_{2}$ chooses its strategy.
- If $B<A<0$, the stable point ${y}^{*}>1$, and $\left(A-B\right)y+B<0$ for $\forall y\in \left(0,1\right)$. When $x=0$, ${F}^{\prime}\left(x\right)<0$. Moreover, this type of condition satisfies $B<0$, i.e., ${\pi}_{12}^{\left(1\right)}-{\pi}_{22}^{\left(1\right)}<0$. Thus, in this case, for ${S}_{1}$, its coprocessing gain is smaller than the solo-processing gain, i.e., ${\pi}_{12}^{\left(1\right)}<{\pi}_{22}^{\left(1\right)}$; hence, ${S}_{1}$ chooses the solo-processing strategy no matter how ${S}_{2}$ chooses its strategy.
- If $B<0<A$, the stable point ${y}^{*}\in \left(0,1\right)$, and $\left(A-B\right)y+B<0$ if $y<{y}^{*}$. When $x=0$, ${F}^{\prime}\left(x\right)<0$. If $y>{y}^{*}$, $\left(A-B\right)y+B>0$, and when $x=1$, ${F}^{\prime}\left(x\right)<0$. Moreover, this type of condition satisfies $B<0$, that is, ${\pi}_{12}^{\left(1\right)}-{\pi}_{22}^{\left(1\right)}<0$. Then, in this case, for ${S}_{1}$, its coprocessing gain is less than the solo-processing gain, i.e., ${\pi}_{12}^{\left(1\right)}<{\pi}_{22}^{\left(1\right)}$; thus, whether ${S}_{1}$ chooses solo-processing or coprocessing is influenced by the strategy choice of ${S}_{2}$.
- From $A-B=\left(1+\alpha \right){l}_{1}>0$, we know there is no $A-B<0$.

#### 4.1.2. Evolutionary Stability Analysis of Edge Operator

- If $H>Q>0$, the stable point ${x}^{*}<0$, and $\left(H-Q\right)x+Q>0$ for $\forall x\in \left(0,1\right)$. When $y=1$, ${G}^{\prime}\left(y\right)<0$. This type of condition satisfies $Q>0$, i.e., ${\pi}_{21}^{\left(2\right)}-{\pi}_{22}^{\left(2\right)}>0$. Thus, in this case, for ${S}_{2}$, its coprocessing gain is larger than the solo-processing gain, i.e., ${\pi}_{21}^{\left(2\right)}>{\pi}_{22}^{\left(2\right)}$; thus, ${S}_{2}$ will choose co-processing no matter how ${S}_{1}$ chooses its strategy.
- If $Q<H<0$, the stable points ${x}^{*}>1$, and $\left(H-Q\right)x+Q<0$ for $\forall x\in \left(0,1\right)$. When $y=0$, ${G}^{\prime}\left(y\right)<0$. This type of condition satisfies $Q<0$, i.e., ${\pi}_{21}^{\left(2\right)}-{\pi}_{22}^{\left(2\right)}<0$. Then, in this case, the coprocessing gain of ${S}_{2}$ is less than its solo-processing gain, i.e., ${\pi}_{21}^{\left(2\right)}<{\pi}_{22}^{\left(2\right)}$; thus, ${S}_{2}$ chooses solo-processing no matter how ${S}_{1}$ chooses its strategy.
- If $Q<0<H$, the stable point ${x}^{*}\in \left(0,1\right)$. If $x<{x}^{*}$, then $\left(H-Q\right)x+Q<0$. When $y=0$, ${G}^{\prime}\left(y\right)<0$. If $x>{x}^{*}$, then $\left(H-Q\right)x+Q>0$. When $y=1$, ${G}^{\prime}\left(y\right)<0$. This type of condition satisfies $Q<0$, i.e., ${\pi}_{21}^{\left(2\right)}-{\pi}_{22}^{\left(2\right)}<0$, which means that the coprocessing gain of ${S}_{2}$ is less than its solo-processing gain, i.e., ${\pi}_{21}^{\left(2\right)}<{\pi}_{22}^{\left(2\right)}$; thus, whether ${S}_{2}$ chooses solo-processing or coprocessing is indeed influenced by the strategy choice of ${S}_{1}$;
- From $H-Q=\left(1+\beta \right){l}_{2}>0$, we know there is no $H-Q<0$.

#### 4.2. Analysis of the Evolutionary Stability of the Combination Strategies of Both Game Parties in the System

- Mutual influence relationship: From the game process of the ${S}_{1}$ and ${S}_{2}$ strategy selection, there are three different states:
- The strategy choices of the two parties do not affect each other, as in the case of condition 1;
- One party is affected; for example, ${S}_{2}$ is affected by the choice of ${S}_{1}$’s strategy selection in condition 3;
- The two parties affect each other; for example, ${S}_{1}$ and ${S}_{2}$ are affected by each other’s strategy choice in condition 9.

- Evolutionary results: As can be seen from Table 4, there are four evolutionary results, i.e., $\left(0,0\right),\left(1,0\right),\left(0,1\right)$, and $\left(1,1\right)$, in the evolutionary game of ${S}_{1}$ and ${S}_{2}$. The evolutionary results are $\left(1,1\right)$ in conditions 1, 3, and 7, indicating that in these cases, ${S}_{1}$ and ${S}_{2}$ choose to collaborate in handling various service requests from users. In other words, regardless of the initial state of the whole system, the two parties eventually reach a stable cloud–edge cooperative relationship after continuously learning and adjusting their strategies, and the common conditions in these three cases are $A>0$ and $H>0$ through a comparative analysis, as shown in Equation (13):$$\left\{\begin{array}{c}MR-{C}_{d}-{I}_{s}-\left(1-a\right)\left(C+N{C}_{r}\right)-{R}_{s}+{C}_{s}+{C}_{r}+L>0\\ \left(1-M\right)R-{C}_{m}-{I}_{e}-a\left(C+N{C}_{r}\right)-{R}_{e}+{C}_{e}+L>0\end{array}\right.,$$$$\left\{\begin{array}{c}{\pi}_{11}^{\left(1\right)}-{\pi}_{22}^{\left(1\right)}+L>0\\ {\pi}_{11}^{\left(1\right)}-{\pi}_{22}^{\left(1\right)}+L>0\end{array}\right..$$

- Is the evolutionary stable strategy unique? From condition 9 in Table 4, it can be seen that there are two evolutionary-stable strategies, namely, $\left(0,0\right)$ and $\left(1,1\right)$, for the cloud–edge collaboration system composed of ${S}_{1}$ and ${S}_{2}$, which mainly depend on the values of the cost and benefit in the evolutionary game matrix and the initial state of this system, i.e., the saddle point $\left({x}^{*},{y}^{*}\right)$.

#### 4.3. Factors Affecting Evolutionary Stability and Evolutionary Results

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

**Theorem**

**2.**

**Proof**

**of**

**Theorem**

**2.**

- With increasing $L,R,{C}_{e},{C}_{s},\alpha ,\beta $, the possibility of the system evolving to $\left(1,1\right)$ increases;
- With increasing ${I}_{e},{R}_{e},{I}_{s},{R}_{s},C,{l}_{1},{l}_{2}$, the possibility of the system evolving to $\left(0,0\right)$ increases;
- For $M$, when $A-B>H-Q$, the possibility of the system evolving to $\left(0,0\right)$ increases with an increment in $M$; when $A-B<H-Q$, the possibility of the system evolving to $\left(1,1\right)$ increases with an increment in $M$;
- For $a$, when $A-B<H-Q$, the possibility of the system evolving to $\left(0,0\right)$ increases with an increment in $a$. When $A-B>H-Q$, the possibility of the system evolving to $\left(1,1\right)$ increases with an increment in $a$.

## 5. Numerical Simulation Analysis

#### 5.1. Simulation Analysis

- The influence of the initial willingness of both parties on the evolution of the system

- Factors affecting evolutionary stability and evolutionary results

#### 5.2. Further Discussion

- The strategy choices of cloud service providers and edge operators promote each other. The improvement of one party’s willingness to cooperate in processing drives the improvement of the other party’s willingness to cooperate, thus promoting the cooperative stability of the whole cloud–edge system.
- For the cloud service providers and edge operators, the smaller the solo-processing benefit, the larger the solo-processing cost, the less the user loss, the less the cost of data transmission, and the higher the emphasis level on the QoE, thus the more favorable the evolutionary stability of collaborative processing. For the cloud–edge collaboration system, the initial willingness of both parties has a particular influence on the system evolution results. The larger the cooperation benefit, the lower the cooperation cost, and the stronger the stability of the system, the more favorable it is to achieve cooperation.
- The lost fee due to the service constraint agreement breach should be at least larger than the difference between the aggregated benefit when the parties choose to handle processing separately and the benefit when they cooperate to handle processing together, in order to establish a stable cooperative processing relationship between the parties, and to avoid possible speculation by both parties.
- In the cloud–edge collaborative processing, profit-sharing and cost-sharing should be dynamically adjusted in real time with the changes in the market environment, and different shares have different effects on the stability of the cloud–edge collaboration system. This is related to the importance both parties attach to the QoE and the loss of users who quit or complain.
- The higher the emphasis on user experience both parties put, the stronger the cooperation intention is, and this effect is obvious.

- Focusing on improving the cooperation willingness of cloud service providers or edge operators can achieve the effect of improving the cooperation willingness of both parties, so as to promote the harmony and stability of the whole system.
- Cloud service providers and edge operators, as the two major stakeholders of digital services, can reasonably use the policy dividends of the digital economy era and Internet technology to accelerate product development and constantly upgrade and transform to reduce the various costs of user services and improve economic returns, so as to further construct a more stable and mutually reinforcing cooperative relationship between them and jointly promote the high-quality development of the digital economy.
- The governments can supply a sound system to provide a legal basis and guarantee for the cost of service constraint agreement breach, enhance the binding force and enforcement of the agreement, provide credit guarantees for both parties to improve each other’s credit, increase the cooperation stickiness of both parties, integrate all forces together to maintain a stable cloud–edge collaboration system to serve the digital transformation of enterprises, drive the innovation and evolution of business models, and increase the value creation of enterprises.
- The benefit and cost distribution proportion of both parties in the cloud–edge collaboration system should be dynamically adjusted. Cloud service providers and edge operators influence each other in multiple dimensions. Therefore, in the changing market economy environment, both parties should adjust their benefit and cost distribution strategy in real time according to the actual cost and contribution, so that the allocation of benefits and costs can quickly respond to the market and satisfy both parties, thus improving the enthusiasm of cooperation and ensuring the long-term stability of the cloud–edge collaboration system.
- The greater the emphasis on QoE by cloud service providers and edge operators, the more it helps the establishment and stability of the cloud–edge collaboration system. Therefore, we should lower market entry barriers, improve competition in similar services, and create a favorable competitive environment in the future.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 5.**Impacts of data transmission cost ${I}_{s}$ of ${S}_{1}$ on the stability of the cloud–edge collaboration system.

**Figure 6.**Impacts of user complaint loss ${l}_{2}$ of ${S}_{2}$ on the stability of the cloud–edge collaboration system.

**Figure 7.**Impacts of the emphasis level parameters $\alpha $ and $\beta $ for the QoE on the stability of the cloud–edge collaboration system when ${l}_{1}>{l}_{2}$, α = β = 0.

**Figure 8.**Impacts of the emphasis level parameters $\alpha $ and $\beta $ for the QoE on the stability of the cloud–edge collaboration system when ${l}_{1}>{l}_{2}$, α = β = 0.05.

**Figure 9.**Impacts of the emphasis level parameters $\alpha $ and $\beta $ for the QoE on the stability of the cloud–edge collaboration system when ${l}_{1}>{l}_{2}$, α = β = 0.09.

**Figure 10.**Impacts of the emphasis level parameters $\alpha $ and $\beta $ for the QoE on the stability of the cloud–edge collaboration system when ${l}_{1}<{l}_{2}$, α = β = 0.

**Figure 11.**Impacts of the emphasis level parameters $\alpha $ and $\beta $ for the QoE on the stability of the cloud–edge collaboration system when ${l}_{1}<{l}_{2}$, α = β = 0.05.

**Figure 12.**Impacts of the emphasis level parameters $\alpha $ and $\beta $ for the QoE on the stability of the cloud–edge collaboration system when ${l}_{1}<{l}_{2}$, α = β = 0.09.

**Figure 13.**Impacts of the emphasis level parameter $\alpha $ of ${S}_{1}$ for the QoE on the stability of the cloud–edge collaboration system when ${l}_{1}<{l}_{2}$.

**Figure 14.**Impacts of the emphasis level parameter $\alpha $ of ${S}_{1}$ for the QoE on the stability of the cloud–edge collaboration system when ${l}_{1}>{l}_{2}$.

**Figure 15.**Impacts of the emphasis level parameter $\beta $ of ${S}_{2}$ for the QoE on the stability of the cloud–edge collaboration system when ${l}_{1}<{l}_{2}$.

**Figure 16.**Impacts of the emphasis level parameter $\beta $ of ${S}_{2}$ for the QoE on the stability of the cloud–edge collaboration system when ${l}_{1}>{l}_{2}$.

Parameters | Parameter Descriptions |
---|---|

$R$ | Benefits of cooperation between ${S}_{1}$ and ${S}_{2}$ |

${R}_{s}$ | The unique benefits of ${S}_{1}$ handling user service requests alone |

${R}_{e}$ | The unique benefits of ${S}_{2}$ handling user service requests alone |

${I}_{s}$ | Information transmission cost of ${S}_{1}$ |

${I}_{e}$ | Information transmission cost of ${S}_{2}$ |

$C$ | Total cost of collaborative processing services for ${S}_{1}$ and ${S}_{2}$ |

${C}_{s}$ | The cost of ${S}_{1}$ to process the user’s service requests alone |

${C}_{e}$ | The cost of ${S}_{2}$ to process the user’s service requests alone |

${l}_{1}$ | Losses of ${S}_{1}$ due to user complaints |

${l}_{2}$ | Losses of ${S}_{2}$ due to user complaints |

$L$ | Liquidated damages for breach of the cloud–edge collaborative constraint agreement |

$M$ | Revenue distribution coefficient of ${S}_{1}$ when revenue is shared |

$a$ | Cost allocation ratio of ${S}_{2}$ when cost is shared |

$\alpha $ | The emphasis level of ${S}_{1}$ for QoE |

$\beta $ | The emphasis level of ${S}_{2}$ for QoE |

Both Parties to the Game | $\mathbf{Edge}\mathbf{Operators}\left({\mathit{S}}_{2}\right)$ | ||
---|---|---|---|

$\mathbf{Coprocessing}\left(\mathit{y}\right)$ | $\mathbf{Solo}-\mathbf{Processing}(1-\mathit{y})$ | ||

Cloud Service Providers $\left({S}_{1}\right)$ | $\mathrm{Coprocessing}\left(x\right)$ | ${\pi}_{11}^{\left(1\right)}$$,{\pi}_{11}^{\left(2\right)}$ | ${\pi}_{12}^{\left(1\right)}$$,{\pi}_{12}^{\left(2\right)}$ |

$\mathrm{Solo}\mathrm{processing}(1-x$) | ${\pi}_{21}^{\left(1\right)}$$,{\pi}_{21}^{\left(2\right)}$ | ${\pi}_{22}^{\left(1\right)}$$,{\pi}_{22}^{\left(2\right)}$ |

Conditions | $\mathit{H}>0,\mathit{Q}0$ | $\mathit{H}<0,\mathit{Q}0$ | $\mathit{H}>0,\mathit{Q}0$ |
---|---|---|---|

$A>0,B0$ | $\left(1,1\right)$ | $\left(1,0\right)$ | $\left(1,1\right)$ |

$A<0,B0$ | $\left(0,1\right)$ | $\left(0,0\right)$ | $\left(0,0\right)$ |

$A>0,B0$ | $\left(1,1\right)$ | $\left(0,0\right)$ | $\left(0,0\right)\left(1,1\right)$ |

Combination of Conditions | ESS | Impacts | Evolutionary Results |
---|---|---|---|

Condition 1: $A>0,B0,H0,Q0$ | $\left(1,1\right)$ | No impact | Collaboration |

Condition 2: $A>0,B0,H0,Q0$ | $\left(1,0\right)$ | No impact | |

Condition 3: $A>0,B0,H0,Q0$ | $\left(1,1\right)$ | ${S}_{2}$ affected | Collaboration |

Condition 4: $A0,B0,H0,Q0$ | $\left(0,1\right)$ | No impact | |

Condition 5: $A<0,B0,H0.Q0$ | $\left(0,0\right)$ | No impact | |

Condition 6: $A<0,B0,H0,Q0$ | $\left(0,0\right)$ | ${S}_{2}$ affected | |

Condition 7: $A0,B0,H0,Q0$ | $\left(1,1\right)$ | ${S}_{1}$ affected | Collaboration |

Condition 8: $A>0,B0,H0,Q0$ | $\left(0,0\right)$ | ${S}_{1}$ affected | |

Condition 9: $A0,B0,H0,Q0$ | $\left(0,0\right)\left(1,1\right)$ | Interactions | Not necessarily |

Equilibrium Points | $\mathit{D}\mathit{e}\mathit{t}\mathit{J}$ | Symbol | $\mathit{T}\mathit{r}\mathit{J}$ | Symbol | Stability |
---|---|---|---|---|---|

${E}_{1}\left(0,0\right)$ | $BQ$ | $+$ | $B+Q$ | $-$ | ESS |

${E}_{2}\left(0,1\right)$ | $-AQ$ | $+$ | $A-Q$ | $+$ | Unstable |

${E}_{3}\left(1,0\right)$ | $-BH$ | $+$ | $-B+H$ | $+$ | Unstable |

${E}_{4}\left(1,1\right)$ | $AH$ | $+$ | $-A-H$ | $-$ | ESS |

$O\left({x}^{*},{y}^{*}\right)$ | $-$ | $0$ | Unknown | Saddle point |

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## Share and Cite

**MDPI and ACS Style**

Li, S.; Xu, M.; Liu, H.; Sun, W.
Service Mechanism for the Cloud–Edge Collaboration System Considering Quality of Experience in the Digital Economy Era: An Evolutionary Game Approach. *Systems* **2023**, *11*, 331.
https://doi.org/10.3390/systems11070331

**AMA Style**

Li S, Xu M, Liu H, Sun W.
Service Mechanism for the Cloud–Edge Collaboration System Considering Quality of Experience in the Digital Economy Era: An Evolutionary Game Approach. *Systems*. 2023; 11(7):331.
https://doi.org/10.3390/systems11070331

**Chicago/Turabian Style**

Li, Shiyong, Min Xu, Huan Liu, and Wei Sun.
2023. "Service Mechanism for the Cloud–Edge Collaboration System Considering Quality of Experience in the Digital Economy Era: An Evolutionary Game Approach" *Systems* 11, no. 7: 331.
https://doi.org/10.3390/systems11070331