Matrix representations of berge stabilities in the graph model for conflict resolution

Abstract

Conflicts occur when multiple parties interact and have different evaluations about the possible scenarios that may occur. There are conflicts not only in daily events as in parents deciding which school to choose for their kids, but also among countries deciding how to reduce global warming. The graph model for conflict resolution (GMCR) is an efficient model to represent and analyze conflicts. The GMCR models the conflict considering sequences of moves and countermoves that can be made by the decision makers (DMs) in the course of a conflict. Since DMs may present different behavior in conflicts there are several stability concepts that have been proposed in the GMCR. Matrix representations have been developed for many of these stability concepts in order to provide a more computationally tractable method to find stable states in a conflict. Berge stabilities were recently introduced into the GMCR to analyze the effects of altruistic behavior in the stability analysis of conflicts. In particular, Berge behavior occurs in disputes where DMs act benevolently anticipating a similar response from others such that, in the end, their actions are in their own self-interest. As it happens with other solution GMCR concepts, the logical definitions of Berge stabilities are hard to apply in large conflicts with a high number of states or DMs. In this work, our objective is to propose matrix representations for Berge stabilities so that it becomes computationally tractable to analyze the effect of altruistic behavior in large conflicts. To illustrate the applicability of the proposed matrix representations, we apply the method to the Elmira conflict.

Appendices

Computational codes

Matrices for the Elmira conflict

We now show the matrices that are inputs for the Berge stabilities calculations in the Elmira conflict for DM UR. First, the accessibility and strict preference matrices for DM UR are given by:

$$\begin{aligned} J_{UR} = \left[ \begin{array}{ccccccccc} 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 1\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 1\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ \end{array} \right] \;\;\hbox { and }\;\; P^+_{UR} = \left[ \begin{array}{ccccccccc} 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 0 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 1 &{}\quad 1 &{}\quad 1\\ 1 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 1\\ 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 1 &{}\quad 1 &{}\quad 1\\ 1 &{}\quad 0 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 1\\ 1 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0\\ \end{array} \right] . \end{aligned}$$

Then, the matrices that represent the legal sequence of unilateral moves and unilateral improvements for the coalition formed by the opponents of DM UR are given by:

$$\begin{aligned}{} & {} J_{N-\{UR\}} = \left[ \begin{array}{ccccccccc} 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 1 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ \end{array} \right] \;\;\hbox { and } \\{} & {} J_{N-\{UR\}}^+ = \left[ \begin{array}{ccccccccc} 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ \end{array} \right] . \end{aligned}$$

Finally, we display matrices $M_{UR}^{Berge}$, $M_{UR}^{WB}$, $M_{UR}^{SMB}$, $M_{UR}^{SMB}$, $M_{UR}^{WMB}$, $M_{UR}^{WSMB}$, whose elements of the main diagonal are equal to zero if and only if the corresponding state satisfies the appropriate Berge stability for DM UR.

$$\begin{aligned}{} & {} M_{UR}^{Berge} = \left[ \begin{array}{ccccccccc} 0 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 2 &{}\quad 1 &{}\quad 0 &{}\quad 1\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 3 &{}\quad 2 &{}\quad 0 &{}\quad 2 &{}\quad 3 &{}\quad 2 &{}\quad 1 &{}\quad 2\\ 0 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 1\\ 0 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 2 &{}\quad 1 &{}\quad 1 &{}\quad 1\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 3 &{}\quad 2 &{}\quad 0 &{}\quad 2 &{}\quad 3 &{}\quad 3 &{}\quad 1 &{}\quad 2\\ 0 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ \end{array} \right] \hbox {,}\\{} & {} M_{UR}^{WB} = \left[ \begin{array}{ccccccccc} 0 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 1\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 1\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ \end{array} \right] , \end{aligned}$$

$$\begin{aligned}{} & {} M_{UR}^{MB} = \left[ \begin{array}{ccccccccc} 0 &{}\quad 4 &{}\quad 3 &{}\quad 0 &{}\quad 2 &{}\quad 5 &{}\quad 3 &{}\quad 1 &{}\quad 3\\ 0 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 1\\ 0 &{}\quad 3 &{}\quad 2 &{}\quad 0 &{}\quad 2 &{}\quad 3 &{}\quad 2 &{}\quad 1 &{}\quad 2\\ 0 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 1\\ 0 &{}\quad 4 &{}\quad 3 &{}\quad 1 &{}\quad 3 &{}\quad 5 &{}\quad 4 &{}\quad 2 &{}\quad 3\\ 0 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 1 &{}\quad 2 &{}\quad 1 &{}\quad 1 &{}\quad 1\\ 0 &{}\quad 3 &{}\quad 2 &{}\quad 0 &{}\quad 2 &{}\quad 3 &{}\quad 3 &{}\quad 1 &{}\quad 2\\ 0 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ \end{array} \right] \hbox {,}\\{} & {} M_{UR}^{SMB} = \left[ \begin{array}{ccccccccc} 0 &{}\quad 13 &{}\quad 11 &{}\quad 0 &{}\quad 4 &{}\quad 14 &{}\quad 11 &{}\quad 2 &{}\quad 5\\ 0 &{}\quad 4 &{}\quad 4 &{}\quad 0 &{}\quad 2 &{}\quad 4 &{}\quad 4 &{}\quad 0 &{}\quad 2\\ 0 &{}\quad 6 &{}\quad 5 &{}\quad 0 &{}\quad 2 &{}\quad 6 &{}\quad 5 &{}\quad 1 &{}\quad 2\\ 0 &{}\quad 2 &{}\quad 2 &{}\quad 0 &{}\quad 1 &{}\quad 2 &{}\quad 2 &{}\quad 0 &{}\quad 1\\ 0 &{}\quad 13 &{}\quad 11 &{}\quad 1 &{}\quad 5 &{}\quad 14 &{}\quad 13 &{}\quad 3 &{}\quad 5\\ 0 &{}\quad 4 &{}\quad 4 &{}\quad 0 &{}\quad 2 &{}\quad 5 &{}\quad 4 &{}\quad 2 &{}\quad 2\\ 0 &{}\quad 6 &{}\quad 5 &{}\quad 0 &{}\quad 2 &{}\quad 6 &{}\quad 6 &{}\quad 1 &{}\quad 2\\ 0 &{}\quad 2 &{}\quad 2 &{}\quad 0 &{}\quad 1 &{}\quad 2 &{}\quad 2 &{}\quad 1 &{}\quad 1\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ \end{array} \right] , \end{aligned}$$

$$\begin{aligned}{} & {} M_{UR}^{WMB} = \left[ \begin{array}{ccccccccc} 0 &{}\quad 2 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 2 &{}\quad 1 &{}\quad 0 &{}\quad 1\\ 0 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 1\\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 1\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ \end{array} \right] \;\;\hbox { and }\\{} & {} M_{UR}^{WSMB} = \left[ \begin{array}{ccccccccc} 0 &{}\quad 5 &{}\quad 3 &{}\quad 0 &{}\quad 0 &{}\quad 5 &{}\quad 3 &{}\quad 0 &{}\quad 1\\ 0 &{}\quad 4 &{}\quad 4 &{}\quad 0 &{}\quad 2 &{}\quad 4 &{}\quad 4 &{}\quad 0 &{}\quad 2\\ 0 &{}\quad 2 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 2 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 2 &{}\quad 2 &{}\quad 0 &{}\quad 1 &{}\quad 2 &{}\quad 2 &{}\quad 0 &{}\quad 1\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ \end{array} \right] . \end{aligned}$$