# Comparison of Transboundary Water Resources Allocation Models Based on Game Theory and Multi-Objective Optimization

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

^{2}. The Huaihe River basin contains Henan province, Anhui province, and Jiangsu province (three provinces). The problem of water resources shortage is prominent, and the inter-provincial water resources conflict has become a serious problem. Though the study area is not international, an inter-provincial basin is effectively equivalent to an international basin as long as its boundaries do not match political boundaries. The stakeholders in the process of the Huaihe River basin water resources allocation are the watershed management agency, Henan province, Anhui province, and Jiangsu province. The spatial distribution and location of the associated province of Huaihe River basinis described in Figure 1.

#### 2.2. Transboundary Water Resources Allocation Based on Asymmetric Nash–Harsanyi Leader–Follower Game Model and Its Solution

- Let ${x}_{0}$ be the initial feasible solution of the optimization problem, and the corresponding objective function value is $f({x}_{0})$;
- The nonlinear objective function is linearized by using the first order descriptive form point $({x}_{0},f({x}_{0}))$ of Taylor series, and the optimal solution $({x}_{1}^{\prime},f({x}_{1}^{\prime}))$ of linearized objective function in the neighborhood $({x}_{0}-\delta ,{x}_{0}-\delta )$ is obtained by using the linear programming method;
- Taking the optimal solution $({x}_{1}^{\prime},f({x}_{1}^{\prime}))$ as a new feasible solution, the nonlinear objective function is linearized again by using the first order description point $({x}_{1}^{\prime},f({x}_{1}^{\prime}))$ of Taylor series, and the optimal solution $({x}_{2}^{\prime},f({x}_{2}^{\prime}))$ of linearized objective function in the neighborhood $({x}_{1}-\delta ,{x}_{1}-\delta )$ is obtained by using the linear programming method;
- Repeat the above step 3 for the optimal solution and continue to iterate until the obtained optimal solution and the optimal objective function meet the condition of iteration termination.

#### 2.3. Transboundary Water Resources Allocation Based on Multi-Objective Optimization Model and Its Solution

#### 2.3.1. Transboundary Water Resources Allocation Based on Multi-Objective Optimization Model

#### 2.3.2. Multi-Objective Optimization Method

- An initial population ${P}_{l}$ of size $M$ is randomly generated, save the population ${P}_{l}$ to an external document $E$, and multi-objective optimization method based on objective decomposition is used to generated a set of reference points $R$ with a scale of ${M}_{R}$;
- A mating pool selection strategy based on the enhanced inverted generational distance with noncontributing solution detection is used to select the initial population ${P}_{l}$, mating pool ${p}_{l}^{\prime}$ is selected and the offspring population ${P}_{{}_{l}}^{\prime}$ is generated;
- The archive $E$ and the adapted reference point set ${R}^{\prime}$ are updated through the offspring population ${P}_{{}_{l}}^{\prime}$, then the next generation population ${P}_{l+1}$ is selected by the enhanced inverted generational distance with a noncontributing solution detection selection strategy;
- The above steps 1, 2, and 3 are repeated for the new initial population ${P}_{l+1}$ until a termination criterion is reached.

- An initial population ${p}_{l}$ of size $M$ is randomly generated, and the initial population is stratified by non-inferiority, and the genetic operator (selection operator, crossover operator, mutation operator) is used to obtain an offspring population ${q}_{l}$. Then, the initial population ${p}_{l}$ and the offspring population ${q}_{l}$ are mixed together to form a new population ${N}_{l}$ of size $2M$;
- The new population ${N}_{l}$ of size $2M$ is rapidly sorted to form a non-dominated set ${E}_{l}$, then the crowding degree of the individuals in the non-dominated set ${E}_{l}$ is calculated. According to the non-dominant relationship and the crowding degree of individuals, a new initial population ${p}_{l+1}$ with the appropriate individual composition size of $M$ is selected;
- The above steps 1 and 2 are repeated for the new initial population ${p}_{l+1}$ until a termination criterion is reached.

#### 2.3.3. Performance Metrics for Multi-Objective Optimization Evolutionary Algorithms

#### 2.4. Data of Nash–Harsanyi Leader–Follower Game Model and Multi-Objective Optimization Model

## 3. Results

#### 3.1. Water Resources Allocation Based on Asymmetric Nash–Harsanyi Leader–Follower Game Model and Its Solution in Huaihe River Basin

^{3}.

^{3}, the model of the three provinces is as follows:

^{3}, 97.5 hundred million m

^{3}, and 107.4 hundred million m

^{3}, respectively, and the satisfaction of rates in the corresponding provinces were 68.6%, 61.6%, and 65.6%, respectively. In addition, the benefit of the three provinces were 734.6 hundred billion yuan, 600.8 hundred million yuan, and 683.3 hundred million yuan; the total benefit was 2018.7 hundred million yuan.

#### 3.2. Water Resources Allocation Based on Multi-Objective Optimization Model and Its Solution in Huaihe River Basin

^{3}. The objective function of the watershed management agency in transboundary water resources allocation can be converted into constraint conditions, then four objective functions converted into three objective functions. Water resources allocation based on the multi-objective optimization model is established as follows:

## 4. Discussion

#### 4.1. Comparison of Water Resource Allocation Results Based on NSGA-II Method and AR-MOEA Method

- The convergence metric (GD) and composite metrics (IGD, HV) pass a 5% significance level; this means that there is no significant difference in convergence metric (GD) and composite metrics (IGD, HV). Therefore, we think that the two methods have almost the same effect on convergence metric (GD) and composite metrics (IGD, HV);
- The diversity metric ($\Delta $) cannot pass a 5% significance level; this means that there is a significant difference in the diversity metric ($\Delta $). As for the diversity metric ($\Delta $), when $\Delta $ tends to zero, it indicates that the non-dominated solution set calculated by the multi-objective optimization algorithm has good diversity. The average value of the diversity metric ($\Delta $) in the AR-MOEA method is lower than that in the NSGA-II method. Therefore, the AR-MOEA method is better than the NSGA-II method in terms of the diversity metric ($\Delta $).

#### 4.2. Comparison of Water Resource Allocation Results Based on Multi-Objective Optimization Model and Game Model

- The water demand satisfaction rate of the three provinces of the Huaihe River basin is not balanced in the multi-objective optimization solution under the principle of efficiency, so the fairness of water resources allocation result in the multi-objective optimization solution under the principle of efficiency is ignored. The water demand satisfaction rate of Henan province is higher than that of Anhui province and Jiangsu province, which makes Anhui province and Jiangsu province unable to accept the water resources allocation scheme;
- The multi-objective optimization solution under the principle of fairness satisfies the fairness of water resources allocation result in the three provinces, but the total benefit of the multi-objective optimization solution under the principle of fairness is lower than the water resources allocation result solved by the asymmetric Nash–Harsanyi Leader–Follower game model. The above analysis makes the three provinces unable to accept the water resources allocation scheme;
- The water resources allocation result of the asymmetric Nash–Harsanyi Leader–Follower game model is a compromise solution between two kinds of multi-objective optimization solutions under different preference information (focusing on overall comprehensive benefit or overall fairness). The efficiency and fairness are considered comprehensively in the game model, which makes it easier for the three provinces to accept the water resources allocation scheme.

## 5. Conclusions

- The AR-MOEA method and the NSGA-II method were applied to solve the multi-objective optimization model of the Huaihe River basin water resources allocation. The results show that the AR-MOEA method is better than the NSGA-II method in terms of the diversity metric ($\Delta $);
- The solution of the asymmetric Nash–Harsanyi Leader–Follower game model is a non-dominated solution, and the asymmetric Nash–Harsanyi Leader–Follower game model can obtain the same water resources allocation scheme of the multi-objective optimal allocation model under a specific preference structure;
- After the multi-objective optimization model obtains the Pareto front, it still needs to construct the preference information of the Pareto front for the second time to make the optimal solution of multi-objective decision, while the game model can directly obtain the water allocation scheme in one time by participating in the negotiation.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**Pareto front of multi-objective optimization based on the NSGA-II method in the Huaihe River basin.

**Figure 5.**Pareto front of multi-objective optimization based on the AR-MOEA method in the Huaihe River basin.

Parameter | Henan Province | Anhui Province | Jiangsu Province |
---|---|---|---|

$Q$ | 400.5 (Hundred million m^{3}) | ||

${s}^{\ast}$ | 100.1 (Hundred million m^{3}) | ||

$\eta $ | 0.667 | ||

$(1-\eta )$ | 0.333 | ||

${r}_{i}$ | 126.4 (Hundred million m^{3}) | 135.2 (Hundred million m^{3}) | 137.3 (Hundred million m^{3}) |

${\lambda}_{i}$ | 27.7 (Hundred million m^{3}) | 37.0 (Hundred million m^{3}) | 50.4 (Hundred million m^{3}) |

${I}_{i}$ | 27.9 (Hundred million m^{3}) | 37.0 (Hundred million m^{3}) | 50.4 (Hundred million m^{3}) |

${d}_{i}$ | 273.443 (Hundred million yuan) | 289.476 (Hundred million yuan) | 389.663 (Hundred million yuan) |

${D}_{i}$ | 666.8 (m^{3}/Hundred million yuan) | 1035.2 (m^{3}/Hundred million yuan) | 1034.7 (m^{3}/Hundred million yuan) |

${\alpha}_{i}$ | 0.423 | 0.288 | 0.289 |

${u}_{i}({w}_{i})$ | $\begin{array}{l}{u}_{1}({w}_{1})=-0.0302{\tilde{w}}_{1}^{2}+10.5478{\tilde{w}}_{1}\\ +2.6678\end{array}$ | $\begin{array}{l}{u}_{2}({w}_{2})=-0.0254{\tilde{w}}_{2}^{2}+8.6364{\tilde{w}}_{2}\\ +0.1733\end{array}$ | $\begin{array}{l}{u}_{3}({w}_{3})=-0.024{\tilde{w}}_{3}^{2}+8.9395{\tilde{w}}_{3}\\ +0.0763\end{array}$ |

**Table 2.**The average value and significance test of algorithm performance metrics of the two methods running for 30 times.

Metrics | Average Value | Significance Test | |
---|---|---|---|

NSGA-II Method | AR-MOEA Method | ||

Convergence metric GD | 17.18 | 16.99 | h = 0 ^{1} |

Diversity metric $\Delta $ | 2.74 | 1.97 | h = 1 ^{2} |

Composite metric IGD | 40.94 | 40.93 | h = 0 |

Composite metric HV | 3.22 | 3.30 | h = 0 |

^{1}indicates that significance test does not reject the null hypothesis at the 5% significance level, this means that there is no significant difference in the values of algorithm performance metrics of two methods;

^{2}indicates that significance test rejects the null hypothesis at the 5% significance level, this means that there is significant difference in the values of algorithm performance metrics of two methods.

Province | Parameters | Multi-Objective Optimization Solution | Asymmetric Nash–Harsanyi Leader–Follower Game Model | |
---|---|---|---|---|

Efficiency | Fairness | |||

Henan | w_{1} (Hundred million m^{3}) | 107.8 | 92.2 | 95.5 |

F_{1} (Hundred million yuan) | 788.8 | 718.4 | 734.6 | |

water demand satisfaction rate (%) | 81.1 | 65.3 | 68.6 | |

Anhui | w_{2} (Hundred million m^{3}) | 90.5 | 101.1 | 97.5 |

F_{2} (Hundred million yuan) | 573.7 | 613.7 | 600.8 | |

water demand satisfaction rate (%) | 54.5 | 65.3 | 61.6 | |

Jiangsu | w_{3} (Hundred million m^{3}) | 102.1 | 107.1 | 107.4 |

F_{3} (Hundred million yuan) | 662.6 | 682.2 | 683.3 | |

water demand satisfaction rate (%) | 59.5 | 65.3 | 65.6 | |

Total benefit (Hundred million yuan) | 2025.1 | 2014.3 | 2018.7 |

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**MDPI and ACS Style**

Fu, J.; Zhong, P.-A.; Xu, B.; Zhu, F.; Chen, J.; Li, J. Comparison of Transboundary Water Resources Allocation Models Based on Game Theory and Multi-Objective Optimization. *Water* **2021**, *13*, 1421.
https://doi.org/10.3390/w13101421

**AMA Style**

Fu J, Zhong P-A, Xu B, Zhu F, Chen J, Li J. Comparison of Transboundary Water Resources Allocation Models Based on Game Theory and Multi-Objective Optimization. *Water*. 2021; 13(10):1421.
https://doi.org/10.3390/w13101421

**Chicago/Turabian Style**

Fu, Jisi, Ping-An Zhong, Bin Xu, Feilin Zhu, Juan Chen, and Jieyu Li. 2021. "Comparison of Transboundary Water Resources Allocation Models Based on Game Theory and Multi-Objective Optimization" *Water* 13, no. 10: 1421.
https://doi.org/10.3390/w13101421