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How to Make a Solution to a Territorial Dispute More Realistic: Taking into Account Uncertainty, Emotions, and Step-by-Step Approach

In many real-life situations, it is necessary to divide a disputed territory between several interested parties. The usual way to perform this division is by using Nash’s bargaining solution, i.e., by finding a partition that maximizes the product of the participants’ utilities. However, this solution is based on several idealized assumptions: that we know the exact values of all the utilities, that division is performed on a purely rational basis, with no emotions involved, and that the entire decision is made once. In practice, we only know the utilities with some uncertainty, emotions are often involved, and the solution is often step-by-step. In this paper, we show how to make a solution to a territorial dispute more realistic by taking all this into account. I. FORMULATION OF THE PROBLEM Dividing a disputed territory: a real-life problem. In many real-life situations, from conflicts between neighboring farms to conflict between states, there is a dispute over a territory, as a result of which none of the sides can use this territory efficiently. In such situations, it is desirable to come up with a mutually beneficial agreement. Current solution. The current solution is based on the work by the Nobelist J. Nash, who showed that under reasonable assumptions, the best mutually beneficial solution is the one that maximizes the product of utilities of all the sides [8], [9]. Nash’s solution is a good agreement with common sense – as formalized by fuzzy logic. The above solution – known as Nash’s bargaining solution – is in perfect agreement with informal common sense description as formalized by fuzzy logic; see, e.g., [5], [12], [15]. Indeed, we want all participants to be happy, i.e., we want the first participant to be happy and the second participant to be happy, etc. The degree of happiness of each participant can be described by his or her utility. To represent “and”, it is reasonable to use one of the most frequently used fuzzy “and”-operations – the product. Thus, the degree to which our overall objective – to make everyone happy – is satisfies can be estimated as the product of all the utilities. Out of all possible partitions, we want to select a partition for which this degree of overall happiness. This is exactly what Nash’s bargaining solution is about. Nash’s solution: from a theoretical formulation to practical recommendations. Now that we are reasonably convinced that Nash’s bargaining solution is a reasonable idea, let us recall what exactly partitions it leads to. Let ui(x) be the utility (per area) of the i-th participant at location x. Then, we should select a partition for which the