3.1. Level of Decision-Making for Basin Authority
Total economic benefits of the society: As an upper-level decision maker, the basin authority represents the public and considers the public interest that benefits the region and society as a whole (Lu et al., 2009 [
23]; Tu et al., 2015 [
24]). In the bi-level optimization problem, the basin authority takes action first, considering ecological sustainability, formulating a fair and effective distribution plan, and assigning initial water rights to each sub-region to achieve the goal of the greatest overall social benefit
(10
10 Yuan).
The objective function consists of three parts: the first part is the ecological benefit
, where
(10
7 m
3)is the public ecological water right to meet the ecological needs of the basin,
(Yuan/m
3) is the ecological benefit parameter of the basin unite ecological water use; the second part is the payment of the sub-areas’ water costs, where
(
) is the index of sub-area in the basin,
(
) is the index of water user,
(10
7 m
3) is the decision variable for the sub-area,
(Yuan/m
3) is the unit water cost for user
(
Appendix A). For municipal, agricultural and industrial water users, they need to pay water fees to the basin authority for their utilization of water resources, which is the cost of water for the sub-region managers, while for the basin authority, it is the income from their distribution of water resources for the use of water users. The third part is the total economic benefit of the sub-region,
indicates the total economic benefit of the region without cooperation, where
(10
10 Yuan) is the objective function of sub-region
. In the case of cooperation, the regional total economic benefits include two parts, the benefits of the alliance (i.e.,
) and the rest of the non-cooperative areas’ total benefit (i.e.,
).
Water supply constraints: The water supply of the river basin is limited, and the total available water is
(10
7 m
3/annual). At the same time, the basin authority should consider the principle of efficiency when allocating water and make full use of available water. Therefore, the total available water is allocated to sub-areas and ecological use as much as possible, where
(10
7 m
3) is the initial water rights of the sub-area
and is the decision-making variable of the superior.
Demand constraints: The basin authority must consider environmental sustainability when developing a water allocation plan. Therefore, public ecological water rights should not be lower than the minimum water demand of the river basin’s ecological environment (i.e.,
). The amount of water allocated to each sub-area, excluding the loss of water delivery, and other sources of water in the sub-area, should meet the minimum requirements of that sub-area (i.e.,
). Where
represents the water-transfer loss rate:
Storage capacity constraints: Each sub-area has a reservoir for storing water. The sum of the amount of water allocated by the basin authority to each sub-area (excluding water loss) and other water sources cannot exceed the maximum storage capacity of the area (i.e.,
).
Water allocation constraints: The upper decision variables have a non-negative limit because the initial water rights assigned to each region cannot be negative.
3.3. Water Allocation Model under Cooperation and Non-Cooperation
Based on relevant information of sub-areas and available water of the basin, the basin authority determines the initial allocation of water rights to maximize economic benefits to society under the constraints of the existing water supply, demand, storage capacity, and non-negative constraints. The user’s optimal water withdrawal decision is then made in each sub-area to maximize their own economic benefits (non-cooperation) based on the initial water rights of the upper level and other water sources. In the case of cooperation between sub-regions, the sub-regional managers within the alliance, under constraints (10), (12), and (13), jointly make decisions to maximize the overall economic benefits of the alliance (9). Members outside the alliance decide on the optimal water withdrawal right under constraints (10), (11), and (13) to maximize their own economic benefits.
There is an interactive relationship between upper and lower level decision makers in the regional water resource allocation system, which is represented by the Stackelberg game. In order to show the relationship between decision makers, the corresponding objective function and constraints were integrated to obtain the bi-level regional water resource allocation model under non-cooperation (i.e., (14)), cooperative information asymmetry (i.e., (15), and cooperative information symmetry (i.e., (18)):
Solving this system, the optimal decision and maximum benefit value of basin authority and sub-regions can be obtained as .
Case 2: Cooperation
Cooperative information asymmetry:
In this case, the basin authority does not know the formation of alliances between sub-regions, and still develops a water allocation plan according to the model without cooperation described above (i.e., (14)). The optimal solutions of the basin authority and sub-areas outside the coalition remain unchanged, and the optimal initial water rights remain
for all sub-areas. Suppose sub-regions 1 and 2 now form an alliance. After the basin authority makes the optimal decision, the two sub-regions 1 and 2 of the alliance pursue the maximum overall economic benefits of the alliance, withdrawal of users for the allocated water rights of the alliance is
. Sub-regions 1 and 2 obtain the maximum economic benefits of the alliance according to the following model:
According to the maximum economic benefit value of sub-areas 1 and 2 obtained by model (14) (i.e.,
) and the maximum economic benefit value of the alliance obtained by model (15) (i.e.,
), the incremental benefit under asymmetry of information sharing can be calculated:
If the incremental benefit is greater than zero, sub-regions 1 and 2 will have a motivation for cooperation. Taking into account the principle of fairness, sub-regions 1 and 2 are equally distributed with incremental benefits, and the maximum economic benefit value under the asymmetry of cooperative information is obtained:
Cooperative information symmetry:
In this case, the basin authority is aware of the formation of alliances between certain sub-regions and knows other information, including water demand, and own-sourced water volume. When deciding on the initial allocation of water rights, the basin authority managers, based on the amount of water available for distribution, make decisions to maximize the overall benefits to society, while meeting ecological water requirements, minimum requirements for each sub-region, and maximum water storage constraints. For the sub-areas that form the alliance, the alliance is directly assigned a total amount of water, and the sub-areas within the alliance are no longer allocated separately. Suppose sub-regions 1 and 2 form alliance C, and there is a public reservoir for storing water resources allocated by the basin authority (i.e., ). The maximum water storage capacity of the public reservoir is , the rate of water loss is , and the minimum demand for alliance C is (), is the other water source that alliance C can allocate to the water users, then the total amount of water available for the alliance C is . Sub-areas 1 and 2 are rational subjects, and under the condition of meeting the minimum demand of water users, jointly determine the water withdrawal of the water users to achieve the maximum economic benefit to the alliance. The remaining sub-regions still make decisions based on maximizing their respective economic benefits.
Therefore, the bi-level water resource allocation model under cooperative information symmetry is as follows:
Solving this system, the optimal decision and maximum benefit value of regional authorities and sub-regions under cooperative information symmetry can be obtained: .
According to the maximum economic benefit value of sub-areas within the alliance under non-cooperation obtained by system (14) (i.e.,
) and the maximum economic benefit value of the alliance obtained by model (18) (i.e.,
), the incremental benefit under the symmetry of cooperative information can be calculated:
If the incremental benefit is greater than zero, sub-regions within the alliance will be motivated to cooperate. Taking into account the principle of fairness, they are equally distributed with incremental benefits, and the maximum economic benefit value under the symmetry of cooperative information is therefore: