Water scarcity is becoming more prominent, with the intensification of global climate change and the acceleration of industrialization and urbanization (Salman 2007). In the past decades, human activities have caused the globally available water resources to decrease at a rate of about 100 billion \(m^3/year\) (Mueller Schmied et al. 2014; Wang et al. 2018). At the same time, global water consumption has increased six times in the past 100 years and is still growing steadily at a rate of approximately 1.0% per year (WWAP 2020). Uncontrolled water withdrawal and increased demand for fresh water are main causes of water shortages. Therefore, putting forward an equitable, reasonable, and sustainable water resource allocation manner is an effective way to solve the shortage of water resources.
Water resources management in a basin has changed from a single-goal problem into more complex multi-criteria decision making (MCDM) problems, which involve multiple features, multiple aspects, and multiple stakeholders (Huang et al. 2011; Gebre et al. 2021). Water is a fundamental resource for economic development, social welfare, and environmental protection. The allocation of water resources is a complex process that needs to meet the basic needs of agriculture, industry, and domestic use, as well as to maintain the balance of the ecosystem. At the same time, different water users have different preferences and characteristics, coordinating the conflicts of interests and demands among watershed stakeholders is a challenge for decision-makers.
Researchers try to solve the MCDM problems of water resources by using various methods, but defects are accompanied. The classical tools transform the MCDM problem into a single objective function and solve it through optimization algorithms (Harou et al. 2009). Although those methods can obtain optimal results theoretically, they still face low implement ability in practice because of their complex algorithms and abundant assumptions. Therefore, Game Theory (GT) has been introduced to describe the relationship between the individual and/or group rationality and to analyze the global equilibrium (Kaveh 2009; Thomson 2003). Even GT can better reflect the reality and provide foreseeable consequences, the reliable scenario requires accurate and large data, and proper determination of utility functions, which are often difficult to quantify (Kaveh 2009; Yu et al. 2019; Li et al. 2019; Lee 2012). Therefore, to maximize the comprehensive benefits of water resources allocation under MCDM, the following questions need to be answered: How to raise reasonable and realistic alternatives and how to choose them fairly and effectively, when data is scarce or utility functions are difficult to obtain?
The Bankruptcy problems, coming from enterprise bankruptcy scenario, mainly study on how to distribute the remaining assets E, which is less than the claims C, among shareholders and creditors (O’Neill 1982). Distribution rules under Bankruptcy theory can offer equitable and reasonable solutions under limited resources, which has been widely used in many areas (Brink et al. 2013; Gimenez-Gomez and Penis 2014; Dietzenbacher et al. 2021). In water resource management, when the available water cannot meet the demands from basin users, how to efficiently and reasonably allocate water has a similar scenario with bankruptcy problems.
Several classic Bankruptcy rules have been proposed, under various interpretations of equity, which includes: Proportional rule (PRO), Constrained equal awards (CEA), Constrained equal losses (CEL), Piniles rule (Pin), the Talmud rule (TAL), Constrained egalitarian (CE), Adjust proportional (AP), Random arrival (RA) rule, and so on (Curiel et al. 1987; Mianabadi et al. 2014; Madani et al. 2014b; Thomson 2003). In addition to classic rules, two branches of Bankruptcy problems can be roughly extended: 1) weighted-based; 2) sequential-based. Considering the contribution and corresponding claims, scholars re-determine the weight of each stakeholder by introducing coefficients or vectors from different standards (Mianabadi et al. 2015), like marginal contribution to the coalition (Degefu et al. 2016), willingness to pay criterion (Sechi and Zucca 2015), multiple hydrological constraints (Yong et al. 2017), to adjust equitable consequences. Meanwhile, other scholars have considered spatial variability of river basins users, Ansink and Weikard (2012) transfer a basin-based bankruptcy problem to a linear order two-agent sharing problem, and Goetz et al. (2008) define two different sequential allocation rules that respect asymmetry. In recent years, many studies integrate Bankruptcy theory with other game-based theories to explore new allocation methods: Degefu et al. (2016) systematically combine Bankruptcy framework with the Bargaining theory, Yuan et al. (2017) construct a cooperation bankruptcy game model, and Yazdian et al. (2021) develop a non-cooperative optimal management scenario under bankruptcy conditions. In practice, we found that current Bankruptcy rules mostly sets water allocation weights when facing economic factors, while insufficiently considering the details and differences of participants that are reflected by the factors. Failure to consider the characteristics and constraints of the sectors (agriculture, industry, domestic, etc.) in each region may lead to a gap between theory and reality.
To solve this problem, we propose a novel distribution rule, the Adjusted minimal overlap rule (AMO), based on the Bankruptcy theory, which takes into account the different characteristics of participants while ensuring fairness. Then, we propose a new restriction, the Security Restriction, which considers the influence of different economic factors to determine whether the alternatives are feasible.
Applying Bankruptcy rules to water resources allocation has the following advantages: 1) Bankruptcy rules provide fair and reasonable allocations to the riparian stakeholders. 2) They are game-theoretic-based methods, which can reflect the individual preference and group rationality of stakeholders. 3) They are well understood, easily implemented, which is more valuable for solving actual water conflict.
Social Choice Theory (SCT) studies the relationship between individual preferences and group choices, which can be considered as a voting technique that belongs to MCDM (Madani et al. 2014b). Due to few requirements and a concise voting process, SCT has been widely accepted in scenarios with incomplete information or unknown utility functions. By designing a voting process, individual preferences are aggregated into a collective decision, and the “win” alternative is selected (Feldman and Serrano 2006).
Water resources are managed by different stakeholders who have different characteristics and interests. Considering the heterogeneity of stakeholders, centralized optimization models cannot well reflect the individual preferences, and game-based models insufficient consideration of the group decision-making process, which reduces the motivation of agents to participate and leads to deviations. SCT can evaluate and rank different water resources allocation alternatives based on the preferences of stakeholders. Although the result may not be Pareto optimal, SCT can aggregate consensus among stakeholders and reach an acceptable and implementable solution (Read et al. 2014).
The advantages of SCT in water resources management can be summarized as follows: 1) Relatively simple and clear rules, which suitable for MCDM problems. 2) Concise voting process does not rely on detailed data and utility functions, which is particularly attractive when information is uncertain. 3) Well participation of stakeholders provides better acceptability and stability, which is especially valuable for resolving conflicts under scarcity scenarios.
Innovation and Structure
In response to the questions raised previously, this research aims to make water resource allocation decisions in an equitable, reasonable, and sustainable manner, in the case of insufficient data or the utility function is unavailable. The highlights of this paper can be summarized as follows:
Propose a model that can solve the above problems, which mainly includes three steps: raising, filtering, and choosing alternatives (see Fig. 1 for details).
Based on the Bankruptcy theory, we propose a novel distribution rule (the AMO rule) that takes into account the different characteristics of participants while ensuring fairness.
Propose a new constrain measure, the Security Restriction, to find the feasible solutions, together with the “Core” Solution in the Cooperative Game Theory (CGT).
Five voting methods, base on SCT, are launched to aggregate preferences and to obtain a “win” alternative in different situations
Apply this model to water resource planning problems of Ezhou City, Hubei Province, China as a case study. This study provides a concise and efficient decision-making solution for the multi-agent decentralized MCMD problem under the condition of insufficient information.
This paper organizes in the following structure: Sect. 2 mainly defines and describes the model, which consists of three parts. The first part describes the basic rules of Bankruptcy and proposes the new rules, the second part takes the economic factors as constraints to ensure feasibility, the last part introduces the aggregating process under SCT. A case study application of the three parts is described in Sect. 3. The results and discussion of the model application will be presented in Sect. 4. The last Sect. 5 presents a summary of the study.