Abstract
A water distribution problem in the Mexican Valley is modeled first as a three-person noncooperative game. Each player has a five-dimensional strategy vector, the strategy sets are defined by 15 linear constraints, and the three payoff functions are also linear. A nonlinear optimization problem is first formulated to obtain the Nash equilibrium based on the Kuhn-Tucker conditions, and then, duality theorem is used to develop a computational procedure. The problem can also be considered as a conflict between the three players. The non-symmetric Nash bargaining solution is suggested to find the solution. Multiobjective programming is an alternative solution concept, when the water supply of the three players are the objectives, and the water authority is considered to be the decision maker. The optimal water distribution strategies are determined by using these solution concepts and methods.
Highlights
Game theory is the most commonly applied methodology in decision making problems, when the decision makers have conflicting interests
Mexico City with its 19 million inhabitants is considered the most populated city in the world. It is located in the Mexican valley where the very limited water resources are distributed between three users: agriculture, industry and domestic users
Since the three applied solution concepts are based on different types of instituting water distribution, our results show the consequences of different water distribution mechanisms on the optimal solutions
Summary
Game theory is the most commonly applied methodology in decision making problems, when the decision makers have conflicting interests. A comprehensive summary is given for example, in [1], where a combination of the Kuhn-Tucker conditions and nonlinear optimization is discussed in detail We will apply this method in our case study, when an alternative algorithm is developed based on duality theory. Nash equilibrium assumes that each decision maker wants to maximize its benefit without any consideration to the others This problem can be considered as a conflict between the users, so conflict resolution methodology is a reasonable alternative approach, in which the decision makers select a Pareto optimal solution that satisfies certain fairness conditions. In our case study the non-cooperative Nash equilibrium of the three player game of the water users in the Mexican Valley will be first determined.
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