Strength of preference in graph models for multiple-decision-maker conflicts

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Abstract

Models of strength of preference are incorporated into the graph model for conflict resolution to study realistically multi-objective decision making situations in disputes with any finite number of participants. A decision maker’s preference is expressed using a triplet of binary relations that allows preference of one state over another to be strong or weak, and also allows indifference. Four basic stability definitions used in the graph model to represent human behavior in conflicts are extended to the context of the new triplet preference structure, producing strong and weak stabilities. Theorems governing the relationships among standard, strong, and weak stabilities are presented. Finally, application of the new preference representation to an international water resource management conflict confirms the practical utility of the triplet structure and the significance of the strategic insights it can provide.

Section snippets

Introduction and motivation

A conflict is a clash of interests, values, or actions. Conflicts arise naturally in human relationships at any level [1]. Strategic conflicts involve choices of action by several participants. The ubiquity of strategic conflicts in engineering management, business, and elsewhere in real life has motivated the development of formal conflict resolution methodologies to support the decision making process [2], [3]. In turn, computerized Decision Support Systems (DSSs) have been designed to enable

Expanded preference structure

Traditionally, a pair of binary relations {≻, ∼} on S expresses each DM’s relative preferences over states in the graph model, where s1  s2 indicates that s1  S is strictly preferred to s2  S and s1  s2 means that s1  S is indifferent to s2  S. The binary relation called “greatly preferred to”, ≫, indicates a DM’s strong preference for one state over another in such a way that the paired preference structure {≻, ∼} usually used in the graph model is extended to a triplet preference structure {≫, >, ∼},

General stability definitions

Fang et al. [14] define the solution concepts Nash, GMR, SMR, and SEQ for 2-DM and n-DM (n > 2) conflict models. The definitions presented next are for n-DM conflicts, and are built on the new notations introduced in Section 2.

Stability in an n-DM model requires that the possible responses by other DMs to a move by DM i  N be examined. We begin by introducing the notion of a sanction sequence, which is relevant to the stability definitions GMR, SMR, and SEQ.

Suppose that H  N, H  ∅, is any non-empty

Strong and weak solution concepts for n-decision-maker models

With strength of preference taken into account, stability can be defined as strong or weak according to the level of sanctioning. A unilateral improvement (UI) of a focal DM i is sanctioned strongly if it could result in a greatly less preferred state relative to i’s initial state (GMR and SEQ), and this sanction cannot be avoided by a counterresponse (SMR). Sanctions that produce mildly less preferred scenarios (GMR and SEQ), or that can be escaped from to mildly less preferred states (SMR),

The interrelationships of stability definitions

In a real-world conflict, knowledge of the theoretical links among different solution concepts can be very enlightening. For example, it has been proven that a state s satisfies sSiNash, also satisfies sSiGMR, sSiSMR and sSiSEQ [14]. Relationships among the Nash, GMR, SMR and SEQ solution concepts and their strong and weak extensions in the graph model framework are developed in this section.

Decision makers, options, and state transitions

The environmental dispute over the Garrison Diversion Unit (GDU) project took place between Canada and the United States in the 1970s. Fang et al. [14] carried out a strategic study of this dispute using the graph model, but did not take strength of preference into account.

The Garrison Diversion Unit (GDU) was a proposed irrigation project in the American state of North Dakota that involved the transfer of an immense amount of water from the Missouri River Basin to the Hudson Bay Basin, for use

Conclusions

The Graph Model for Conflict Resolution and its associated decision support system GMCR II constitute a flexible set of decision tools for addressing strategic issues arising in any field where humans make interactive decisions such as in engineering management, business, and policy analysis. In this paper, the classical binary preference structure is extended to a triplet of relations that accounts for strength of preference between states in an n-DM conflict situation which can be

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