Water scarcity, population growth and lack of proper resource allocation mechanisms tend to cause regional instability [
1]. A typical example concerns the northern African countries within Nile basin located in the most arid region of the world, where an unfair distribution of water resources has been present for a long time. Introducing a fair mechanism for water allocation can help the region’s economy and political stability. So far, the centralised system by a central planner (CP) has been a standard water management approach, by which the whole water basin is modelled as a centralised system and then water is distributed for maximising the total benefit of users. Centralised system techniques assume that all agents will allocate the water among each other such that their aggregate welfare is maximised [
2,
3,
4]. In this mechanism, the water is allocated to achieve the equal marginal return for all the users. This leads to an ambiguous interpretation of the aggregated problem. In [
5], it is argued that the aggregated problem (
i.e., CP) carries strong institutional assumptions, presupposing either central planning or perfectly functioning of the water market. In fact, the aggregated formulation (a) does not recognise the asymmetric accessibility of the water to users (e.g., from upstream to downstream); (b) ignores the selfishness of competing water users; and (c) assumes the best solution to the system would be accepted completely by all the participants. Therefore, the standard aggregated approach (
i.e., CP) is not practical when it is used to deal with sharing the water resource. To overcome the above issues, decentralised planning (DC) is introduced. In [
6] a priority based sequential algorithm for upstream-downstream water reallocation is implemented. Once the upstream user maximises its benefit, its decision (solution) is imposed to its immediate downstream user as predefined status; this continues until all the individual problems are solved in sequence. The applicability of multi-agent systems have also been investigated in the field of environmental and natural resource management as reported in [
7,
8]. In this type of approach, each user is autonomous by itself and exchange information with its neighbour users within a system. An example of using a multi-agent system is developed in [
9], and is further extended in allocation of water in the Yellow river basin [
10] and is used to compare administrative and market based water allocation [
11]. This approach considers all users as individual agents making decisions by interacting with each other and a coordinator who resolves the users’ conflict in later stages. The method implements the modified penalty-based nonlinear programme with a two-step problem. The first step finds a solution to agents individually with a possibility of constraint infeasibility and the second step is an optimisation model which reduces the constraint violation at the system level. In application, constraint infeasibility is explained as either the deficit or as an agent behavioural adjustment indicator for reducing the constraint violation [
9]. From a game theoretical perspective, non-cooperative approaches have been examined in the systems in which users involve in a game to increase their pay-off, knowing that their decisions affect those of the other users. The approach provides insights for understanding water conflicts and is often implemented for the games with qualitative information about the users’ payoffs [
12]. Another approach to the above problems is developed in [
5]. They use the multiple complementarity problems to express spatial externalities resulting from asymmetric access to water use for water right pricing. The individual optimisation problem is formulated for each user with the inflow quantity given as exogenous value to each problem as opposed to being a decision variable in the centralised formulation,
i.e., aggregated welfare maximisation. The price of the demanded water is used to clear the output market and the uniform wage rate is used to clear the labour market formulated as complementary constraints to the problem. To this framework, introducing extra coupling constraints changes the formulation to a more general problem framework namely, quasi variational inequality problem (
i.e., a complementarity problem with shared constraints amongst the users [
13]). The convergence of the algorithm is guaranteed upon the convexity assumption and continuously differentiable functions with diagonally dominant Jacobians [
14].
Fair Resource Allocation
Although the above decentralised tools and techniques satisfy the selfishness of each agent in maximising its utility function to achieve higher revenue, they lead to an inefficient solution from CP perspective. That is, it is possible within a water basin that the most inefficient agents located upstream use water up to their operating capacity, and leave very limited units of water to the most efficient firms located downstream; a situation explained in [
5,
15]. Therefore, it is desirable to allocate the water based on the efficient CP solution, but to re-distribute the achieved revenue to the agents in a
fair way—considering, of course, that the revenue is transferable between agents. Different allocation approaches in the literature considers different ways to address the fairness [
16]. The distance based methods, namely, least square solution, maximin (minimax) and compromise programming are some mathematical methods which generally evaluate the performance of solutions based on their distance from ideal solution. These are some reliable indicators to be used to quantify the dissatisfaction level of a user within a shared system. To account for fairness, in this paper, a notion of fairness based on each agent’s contribution on achieving the CP solution is defined. A unique solution is calculated with some favourable properties which guarantees the cooperation maintenance. To find the agent’s impact on CP solution, as will be discussed in the next sections, the best response of each agent on the action of the other group of agents and
vice versa should be known, simultaneously. Therefore, as a major contribution of this paper, an evolutionary algorithm is developed solving interrelated optimisation problems in parallel guiding the search towards a feasible solution in a distributed manner so that the impact of agents on CP solution is realised. This will guarantee that the contribution of each agent is properly captured for fair revenue distribution considering the conflicts within such a shared resource system.