# Game Analysis of Low Carbonization for Urban Logistics Service Systems

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Notations and Mathematical Model

_{1}. It requires marginal profit of m

_{1}. The outsourcing price the 3PL charged for its logistics service to the e-business enterprise is represented by p

_{1}. For the e-business firm, the purchasing price of the unit goods is c

_{2}. Its required marginal profit is m

_{2}. Unit retail price is set at p

_{2}. The following equations could be established accordingly.

_{2}, i.e., the potential market demand would be lower than zero. Combining Equations (1)–(3), another important inequality D – dc

_{1}− dc

_{2}>0 could be obtained, which will often be used in the subsequent game procedures.

## 3. Game Scenarios

## 4. Analytical Solution of Nash Equilibria

_{1}, m

_{2}and l.

#### 4.1.Scenario 1:LS Game

_{2}are obtained as:

_{2}. By letting the first-order derivative equal to zero, it could be inferred as Equation (7).

_{2}in Equation (4) with the value obtained from Equation (7).

_{1}could be obtained as follows.

_{1},l) is $2\tau d-\frac{{a}^{2}}{4}$.

_{1}, l). Therefore, let the two first order partial derivatives equal to 0, then

_{1}and l are:

_{1}≥ 0, $8\tau d-{a}^{2}>0$ can easily hold, then

#### 4.2.Scenario 2:ES Game

_{2}, l) are obtained.

_{1}, l). By letting the two first-order partial derivatives equal to 0, we obtain the following equations.

_{1}> 0 and l > 0, another stricter constraint could be got $4d\tau -{a}^{2}>0$.

_{2}.

_{2}is concave. By letting its first-order derivative equal to 0, the optimal m

_{2}is obtained

#### 4.3.Scenario 3:NA Game

_{2}.

#### 4.4.Scenario 4:LCNA Game

_{1}as follows.

_{1}. Let its first-order partial derivative equal to 0 to get Equation (8). Next, as in Scenario 1, we solve the derivative of Equation (5) to m

_{2}to get Equation (7). Then by combining Equations (7) and (8) to get

_{2}, m

_{1}into Equation (4). Solve its partial derivatives to l as follows

_{1}and m

_{2}, the optimal solutions of them are

#### 4.5.Scenario 5:FSCC Game

_{1}+m

_{2}and then

## 5. Discussions and Several Propositions

^{LCNA}< l

^{LS}< l

^{ES}< l

^{FSCC}; l

^{LS}< l

^{NA}< l

^{FSCC}.

_{1}

^{LCNA}< m

_{1}

^{NA}< m

_{1}

^{LS}. For the e-business enterprise, it is in the order m

_{2}

^{LS}< m

_{2}

^{LCNA}< m

_{2}

^{NA}; m

_{2}

^{LS}< m

_{2}

^{ES}.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Table 1.**Operational performance for the 3PL-e-business enterprise service system in five different game scenarios.

Variable | Scenario1 | Scenario2 | Scenario3 | Scenario4 | Scenario5 |
---|---|---|---|---|---|

l | $\frac{a(D-d{c}_{2}-d{c}_{1})}{8\tau d-{a}^{2}}$ | $\frac{a(D-d{c}_{2}-d{c}_{1})}{8\tau d-2{a}^{2}}$ | $\frac{a(D-d{c}_{2}-d{c}_{1})}{6\tau d-{a}^{2}}$ | $\frac{a(D-d{c}_{2}-d{c}_{1})}{9\tau d-{a}^{2}}$ | $\frac{a(D-d{c}_{2}-d{c}_{1})}{4\tau d-{a}^{2}}$ |

m_{1} | $\frac{\tau (D-d{c}_{2}-d{c}_{1})}{2\tau d-\frac{{a}^{2}}{4}}$ | $\frac{\tau (D-d{c}_{2}-d{c}_{1})}{4\tau d-{a}^{2}}$ | $\frac{\tau (D-d{c}_{2}-d{c}_{1})}{3\tau d-\frac{{a}^{2}}{3}}$ | $\frac{\tau (D-d{c}_{2}-d{c}_{1})}{3\tau d-\frac{{a}^{2}}{3}}$ | $\frac{\tau (D-d{c}_{2}-d{c}_{1})}{3\tau d-\frac{{a}^{2}}{2}}$ |

m_{2} | $\frac{\tau (D-{c}_{2}d-{c}_{1}d)}{4\tau d-\frac{{a}^{2}}{2}}$ | $\frac{D-d{c}_{1}-d{c}_{2}}{2d}$ | $\frac{\tau (D-d{c}_{1}-d{c}_{2})}{3\tau d-\frac{{a}^{2}}{2}}$ | $\frac{\tau (D-d{c}_{1}-d{c}_{2})}{3\tau d-\frac{{a}^{2}}{3}}$ | |

Π_{l} | $\frac{(8d{\tau}^{2}-\tau {a}^{2})\text{\hspace{0.05em}}{(D-d{c}_{1}-d{c}_{2})}^{2}}{{(8\tau d-{a}^{2})}^{2}}$ | $\frac{(4d{\tau}^{2}-\tau {a}^{2})\text{\hspace{0.05em}}{(D-d{c}_{1}-d{c}_{2})}^{2}}{{(8\tau d-2{a}^{2})}^{2}}$ | $\frac{(4d{\tau}^{2}-\tau {a}^{2})\text{\hspace{0.05em}}{(D-d{c}_{1}-d{c}_{2})}^{2}}{{(6\tau d-{a}^{2})}^{2}}$ | $\frac{(9d{\tau}^{2}-\tau {a}^{2})\text{\hspace{0.05em}}{(D-d{c}_{1}-d{c}_{2})}^{2}}{{(9\tau d-{a}^{2})}^{2}}$ | -- |

Π_{e} | $\frac{4d{\tau}^{2}\text{\hspace{0.05em}}{(D-d{c}_{1}-d{c}_{2})}^{2}}{{(8\tau d-{a}^{2})}^{2}}$ | $\frac{\tau \text{\hspace{0.05em}}{(D-d{c}_{1}-d{c}_{2})}^{2}}{8\tau d-2{a}^{2}}$ | $\frac{4d{\tau}^{2}\text{\hspace{0.05em}}{(D-d{c}_{1}-d{c}_{2})}^{2}}{{(6\tau d-{a}^{2})}^{2}}$ | $\frac{9d{\tau}^{2}\text{\hspace{0.05em}}{(D-d{c}_{1}-d{c}_{2})}^{2}}{{(9\tau d-{a}^{2})}^{2}}$ | -- |

Π_{l+e} | $\frac{(12d{\tau}^{2}-\tau {a}^{2})\text{\hspace{0.05em}}{(D-d{c}_{1}-d{c}_{2})}^{2}}{{(8\tau d-{a}^{2})}^{2}}$ | $\frac{(12d{\tau}^{2}-3\tau {a}^{2})\text{\hspace{0.05em}}{(D-d{c}_{1}-d{c}_{2})}^{2}}{{(8\tau d-2{a}^{2})}^{2}}$ | $\frac{(8d{\tau}^{2}-\tau {a}^{2})\text{\hspace{0.05em}}{(D-d{c}_{1}-d{c}_{2})}^{2}}{{(6\tau d-{a}^{2})}^{2}}$ | $\frac{(18d{\tau}^{2}-\tau {a}^{2})\text{\hspace{0.05em}}{(D-d{c}_{1}-d{c}_{2})}^{2}}{{(9\tau d-{a}^{2})}^{2}}$ | $\frac{(4d{\tau}^{2}-\tau {a}^{2})\text{\hspace{0.05em}}{(D-d{c}_{1}-d{c}_{2})}^{2}}{{(4\tau d-{a}^{2})}^{2}}$ |

q | $\frac{({a}^{2}-6d\tau )\text{\hspace{0.05em}}(D-d{c}_{1}-d{c}_{2})}{8\tau d-{a}^{2}}+D-d{c}_{1}-d{c}_{2}$ | $\frac{({a}^{2}-2d\tau )\text{\hspace{0.05em}}(D-d{c}_{1}-d{c}_{2})}{8\tau d-2{a}^{2}}+\frac{D-d{c}_{1}-d{c}_{2}}{2}$ | $\frac{({a}^{2}-4d\tau )\text{\hspace{0.05em}}(D-d{c}_{1}-d{c}_{2})}{6\tau d-{a}^{2}}+D-d{c}_{1}-d{c}_{2}$ | $\frac{({a}^{2}-6d\tau )\text{\hspace{0.05em}}(D-d{c}_{1}-d{c}_{2})}{9\tau d-{a}^{2}}+D-d{c}_{1}-d{c}_{2}$ | $\frac{({a}^{2}-2d\tau )\text{\hspace{0.05em}}(D-d{c}_{1}-d{c}_{2})}{4\tau d-{a}^{2}}+D-d{c}_{1}-d{c}_{2}$ |

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**MDPI and ACS Style**

Guo, J.; Ma, S.
Game Analysis of Low Carbonization for Urban Logistics Service Systems. *Math. Comput. Appl.* **2017**, *22*, 12.
https://doi.org/10.3390/mca22010012

**AMA Style**

Guo J, Ma S.
Game Analysis of Low Carbonization for Urban Logistics Service Systems. *Mathematical and Computational Applications*. 2017; 22(1):12.
https://doi.org/10.3390/mca22010012

**Chicago/Turabian Style**

Guo, Jidong, and Shugang Ma.
2017. "Game Analysis of Low Carbonization for Urban Logistics Service Systems" *Mathematical and Computational Applications* 22, no. 1: 12.
https://doi.org/10.3390/mca22010012