#### 4.1. Income Analysis Among Farmers

Analyzing the optimization schemes M1, M2, and M3, under each scheme the overall income of farmers increases, mainly because it supplements the irrigation water demand of farmers

N_{3}, thereby increasing the overall income. However, there are also losses in the optimization process of some farmers’ interests; the crop revenue before and after the optimization of irrigation quota for each farmer without any cost of water rights transaction can be obtained, as shown in

Figure 6 and

Figure 7.

It can be seen from the Figure that after using the economic irrigation quota to optimize the irrigation water consumption of farmers, the total income of farmers N_{1} and N_{3} under the optimization scheme M1 reached 134.37 thousand dollars; compared with the total income before optimization, the total income increase by 31.30 thousand dollars, the income of the farmer N_{3} increased significantly after the optimization, and the income of the farmer N_{3} was increased by 45.03 thousand dollars before the optimization, but the income of the farmer N_{1} was, however, lost; similarly, after implementing the optimization plan M2, the income of farmer N_{3} increased, but the income of farmer N_{2} was lost; when the optimization plan M3 was implemented, the overall income of the farmers was the largest, reaching 184.16 thousand dollars, and the crop incomes of N_{1}, N_{2}, and N_{3} were 59.44 thousand dollars, 44.32 thousand dollars, and 80.42 thousand dollars, respectively. At this time, the income of the farmers N_{3} also became largest, which is an increase of 46.09 thousand dollars compared to the previous one, but this would inevitably lose the income of farmers N_{1} and N_{2}.

After the implementation of the alliance decision A1~A4, the overall income of farmers also changed relatively, and there is no water rights transaction between farmers under the A1 strategy; the income of N_{3}, N_{1} and N_{2} are 34.33 thousand dollars, 68.75 thousand dollars and 47.97 thousand dollars, respectively, Under the A2 strategy, N_{3} and N_{2} conduct water rights trading, the income of N_{3} and N_{2} is 46.56 thousand dollars and 60.14 thousand dollars, respectively. Compared to previous to the water rights transaction, the income increased by 12.23 thousand dollars and 12.17 thousand dollars; under the A3 strategy, N_{3} and N_{1} conducted water rights trading, and the income of N_{3} and N_{1} was 49.98 thousand dollars and 84.40 thousand dollars, respectively; compared with before the water rights transaction, the income increased by 15.65 thousand dollars and 15.65 thousand dollars; under the A4 strategy, N_{3} conducted water rights transactions with N_{1} and N_{2}, respectively, the income of N_{3}, N_{1}, and N_{2} are 54.65 thousand dollars, 76.89 thousand dollars, and 52.63 thousand dollars, respectively; compared with before the water rights transaction, the income increased by 20.33 thousand dollars, 8.14 thousand dollars, and 4.66 thousand dollars. At this time, the income per hectare of farmers N_{1}, N_{2}, and N_{3} were 1.92, 2.39, and 1.61 thousand $/hm^{2}, respectively, which are 11.6%, 9.6%, and 59.4% higher than that before the water rights transaction. The benefits after the alliance show that when there are only two farmers who trade in water rights, the farmers who have not joined do not get the benefits. When the three farmers trade, they all get benefits as a whole. Compared with the non-alliance in different alliance decisions, the overall income increased without losing the interests of any farmer. Therefore, in the decision-making process of the alliance, by adjusting the distribution of benefits among farmers, a certain amount of compensation is given to farmers who suffer economic losses, and the corresponding compensation responsibilities are assumed for farmers whose profits increase, so as to achieve the greatest overall economic benefit.

#### 4.2. Analysis on the Price of Water Rights Transaction among Farmers

The water rights that can be traded among farmers come from water-saving projects or the amount of water saved after optimizing irrigation quotas. In this paper, through the optimization of irrigation quota and marginal benefit, the tradable water volume among different farmers is obtained. Before optimization, the initial water rights of farmers

N_{1},

N_{2} and

N_{3} are 166,700 m

^{3}, 91,700 m

^{3} and 141,700 m

^{3}, respectively, while after optimization, the water rights of farmers are 130,700 m

^{3}, 77,000 m

^{3} and 192,100 m

^{3}, respectively. When the optimized plan M1 was implemented, the farmer

N_{1} can trade 47,600 m

^{3} of water rights to the farmer

N_{3}. When the optimized plan M2 was implemented, the farmer

N_{2} could trade 36,500 m

^{3} of water rights to the farmer

N_{3}. When the optimized plan M3 was implemented, the farmers

N_{1} and

N_{2} could trade water rights to Farmer

N_{3} respectively trades water rights of 36,000 m

^{3} and 14,700 m

^{3}, as shown in

Figure 8. After the optimization of water saving, the water consumption of farmers

N_{1} and

N_{2} was reduced compared with the initial allocation of water rights, and the remaining water could be traded to other water demand farmers

N_{3}.

Driven by different optimization schemes and alliance schemes, the amount of water that can be traded between farmers was different, and the benefits under different schemes were different, resulting in different prices for water rights transactions under different schemes. The transaction price of water rights between farmers was also affected by the value of water resources, water saving costs, and market crop sales prices. This paper uses cooperative game theory to introduce Shapley value to establish the alliance relationship between farmers, and then determine the price of water rights transaction among farmers. Through model calculation, it was found that the price of water rights transaction was closely related to optimization and farmers’ income before and after the alliance. For example, when farmers N_{2} and N_{3} trade, the optimized income of N_{2} and N_{3} are 34.17 thousand dollars and 34.33 thousand dollars, respectively. The income of N_{2} and N_{3} after alliance is 60.25 thousand dollars and 46.56 thousand dollars, respectively, and the water rights transaction price is 0.737 $/m^{3}, and when farmers N_{1}, N_{2}, and N_{3} jointly establish an alliance relationship, the water rights transaction price is the lowest, and the water rights transaction price was 0.485 $/m^{3}, and the water rights transaction price was 0.565 $/m^{3}.

Based on the production function and game theory, this paper constructed a water rights transaction price model between farmers. In the model, the maximum marginal benefit of crops is considered to calculate the water rights allocated by farmers. The economic irrigation quota is used as an optimization method for the allocation of water rights between farmers. The right allocation obtained the tradable amount of water rights among farmers, and establishes decision-making schemes under different farmers’ alliances based on game theory, calculates the overall benefits of different decisions, and seeks the maximum benefits of the overall farmers after the alliance. For example, in this article, the overall income of the three farmers after the alliance is 184.16 thousand dollars, which is higher than the income of the non-alliance by 33.13 thousand dollars. At that time, the price of water rights transactions between farmers was relatively low. In this study, multiple farmers were used as the research object, and the total income of water rights distribution, water rights transactions, and different alliances between multiple farmers was analyzed, which broke through the limitations of individual farmers’ research. In the research process, because the alliance discussed under the alliance was the largest overall benefit, it couldn’t guarantee the maximum benefit of each farmer, so this scheme was suitable for the overall decision-making in the process of irrigation management. In future research, farmers’ willingness factors may also be added to the calculation process of the model, and farmers can then choose not to participate in the alliance after their interests are harmed.