Elsevier

Applied Mathematics and Computation

Volume 274, 1 February 2016, Pages 372-382
Applied Mathematics and Computation

State transition time analysis in the Graph Model for Conflict Resolution

https://doi.org/10.1016/j.amc.2015.11.020Get rights and content

Highlights

  • New conflict analysis methodologies are developed.

  • The methodologies allow to deal with state transition time in a conflict.

  • New stability concepts are proposed and their interrelationships are examined.

  • The new methodologies are applied to a modified prisoners’ dilemma situation.

  • The application demonstrates how the newly proposed methodologies work.

Abstract

New conflict analysis methodologies which allow the analysts to deal with state transition time in a conflict are developed within the framework of the Graph Model for Conflict Resolution. The methodologies include new variety of stability concepts, whose interrelationships with the existent standard stability concepts are verified in the propositions in this paper. A modified version of the Prisoners’ Dilemma situation is analyzed as a simple example of conflicts in order to demonstrate how the newly proposed methodologies work.

Introduction

In the framework of the Graph Model for Conflict Resolution (GMCR) [1], [2], [13], a conflict is described as a state transition model, where the possible states of a conflict are provided as a set of vertices of a graph and the possible state transitions in the conflict are expressed by a set of arcs of the graph. Moreover, in order to describe a conflict, the framework of the GMCR requires the information of the set of decision makers (DMs) who are involved in the conflict and that of the DMs’ preferences on the possible states of the conflict.

The framework of the GMCR has been extended in many ways. Among them, the extensions with respect to the DMs’ preference information have been concentrated by many authors. Strength of DMs’ preferences [5], preference uncertainty [15], and preference change [12], [19] are essential examples of the extensions of the framework of the GMCR in terms of the DMs’ preference information. Moreover, attitude-based analysis [8] in the framework of the GMCR can be included in this way of extensions. Another way of extensions of the GMCR framework is coalition analysis [9], [10], [14]. This way of extensions can be regarded as the ones with respect to the information of DMs, because a coalition is considered as a DM in the coalition analysis methodologies. Moreover, the preference uncertainty extension and the coalition analysis methodologies are integrated recently [16].

Despite of the many extensions with respect to the DMs’ preference information and the information of DMs, the framework of the GMCR has not been extended with respect to the state transition. In a conflict in the real world, however, the length of the time which a DM needs to complete a state transition in the conflict strongly affects the evolution and the result of the conflict. For example, a prisoner needs to communicate with the sheriff in order to make a state transition in the Prisoners’ Dilemma situation, and the length of the time which the prisoners take to communicate with the sheriff may be different because one prisoner may speak faster than the other. Then, a state transition which takes longer time may be blocked by another state transition which requires shorter time, and thus, such a slow state transition should not be considered as a feasible one in the situation. The framework of the GMCR should be extended so that the analysts can take into account such state transition time in the stability analysis of a conflict in it. In fact, in Section 3 of [6], which is a review paper on crimes and conflicts from the view point of complexity science and information systems, recent availability of large datasets on both crime and conflict is discussed, and it is pointed out that such data has allowed us to analyze statistically location and timing of crimes and conflicts. Such data will also contributes to provide realistic and precise information of timing of conflicts to the methodologies of state transition time analysis, which are newly developed in this paper.

In this paper, the author proposes new stability concepts, which allow the analysts to take into account state transition time in the stability analysis of a conflict in the framework of the GMCR. The author also verifies propositions which show interrelationships among the newly proposed and the existent standard stability concepts in the framework of the GMCR. The author, moreover, shows the results of the analysis of a modified version of the Prisoners’ Dilemma situation, by which the readers can understand how the newly proposed stability concepts work in the stability analysis of a conflict.

The standard framework of the GMCR is introduced in the next section, Section 2, and then, in Section 3, the framework for dealing with the state transition time is newly proposed extending the standard GMCR framework. Also, new stability concepts taking the state transition time into account are defined. In Section 4, some propositions on the interrelationships among the newly proposed and the existent standard stability concepts are verified. The last section, Section 5, is devoted for concluding remarks.

Section snippets

The Graph Model for Conflict Resolution

In this section, the standard framework of the GMCR are presented.

State transition time

In this section, the framework for dealing with the state transition time is newly proposed extending the standard framework of the GMCR.

Propositions

In this section, propositions which show interrelationships among the newly proposed stability concepts and standard stability concepts are verified.

We can easily verify the following interrelationships among Nash, tNash, and tNash-c by using Proposition 1, in particular (ii), that is, the inclusion relationships that tRi,T+(s)tcRi,T+(s)Ri+(s) for iN, sS, and TN\{i}.

Proposition 3 Nash, tNash, and tNash-c

For iN, SiNashSitNashcSitNash.

Proof

Because tcRi,T+(s)Ri+(s) from Proposition 1 (ii), if Ri+(s)= ∅ then tcRi,T

Conclusions

In this paper, a framework for dealing with the state transition time was newly proposed expanding the standard GMCR framework, and new stability concepts with the consideration on state transition time were defined. Moreover, some propositions on the interrelationships between the newly proposed stability concepts and the standard ones are verified. Combining the framework proposed in this paper and those for coalition analysis [9], [10], [14], for attitude analysis [8], [11], and for

Acknowledgment

A part of this work is supported by KAKENHI (26282081).

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