Original Graph Model for Brinco
For example, consider the graph model for Brinco given in Figure 4.5. There are twelve states S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} and the preference relation for Brinco is:
7 8 9 11 10 12 6 1 ∼ 2 ∼ 3 ∼ 4 ∼ 5 (4.1)
1
2
4 3 5
6
7
8
10 9 11
12
Preference Ranking for Brinco is 7 ≻ 8 ≻ 9 ≻ 11 ≻ 10 ≻ 12 ≻ 6 ≻1 ∼ 2 ∼ 3 ∼ 4 ∼ 5 Preference Ranking
Figure 4.5: The Graph Model for Brinco
In Figure 4.5, all elements of the graph are represented: states by numbered nodes referring to each feasible states’, oriented arcs, which represent possible unilateral moves
for Brinco, and finally preferences, which are written at the bottom of the graph. Although the preferences are written with the graph, they seem more like an attachment to the graph rather than an original element of the graph. This leads to the proposed improvement in the graph model, which is presented in the next section.
The Preference Graph Model for Brinco
Utilizing the same illustrative example given in Section 4.4, the enhanced graph model as shown in Figure 4.6 represents the proposed developments in the graph model. Solid arcs represent preferred unilateral moves for Brinco (Mi). All solid arcs in this graph are reversible, except the moves to node 6 where a vertical line crosses the solid arc, 7→
or 9, which represent an irreversible move (Vi). The dashed arcs represent preferences but not actual moves. Thus, Brinco would prefer to move from state 9 to 8 but cannot do so unilaterally (Ui). The combined nodes of (1, 2) and (3, 4, 5) represent groups of equally preferred states, where Brinco can unilaterally move within each group. The main advantage of the enhanced graph is that preferences are embedded in the graph, helping the analyst to find states that are stable for the DM. Moreover, knowing which moves are desired but not available to the DM can be used to understand the evolution of a conflict.
4.5 Conclusions
This work provides the basis for the analysis of long-term conflicts. In long-term conflicts, there is a need to define the parameters of the original conflict in order to understand whether or not disputes over the same issue can be analyzed and attributed to the original
1,2 3,4,5
8 7
11
12 10 9
6
Figure 4.6: The Preference Graph Model for Brinco
or not. Conflict analysis requires that a conflict is studied from the beginning. Modes of the conflict provide a framework to define stages of a conflict from goal realization, to initiation and resolution. Also, an analyst should pay more attention to the macro level of a conflict for the purpose of resolution. The parameters of the conflict can define which features are unique to the attribution and which are not.
Moreover, embedding the preferences of a DM in the graph not only facilitated the understanding of unilateral improvements and desired moves, but also simplified the anal-ysis and recognition of certain individual stability conditions. However, even though the preference graph in its current form does not account for intransitive preferences, most real life conflicts have transitive preferences. The original graph model can account for intransitive preferences by providing a list of pairwise comparison of all states attached to the graph.
4.6 Chapter Summary
The fact that a conflict takes place in many rounds can obscure its nature. This research aims to answer the question of how to determine whether several rounds of a conflict happening at different points in time are connected. A conflict starts when a decision maker challenges a status quo. A decision maker can be involved in more than one conflict at a time; the features that differentiate conflicts are the objectives of the decision makers. The objectives are reflected in the decision makers’ preferences over outcomes. The ultimate goal for a decision maker in a conflict is to obtain the most preferred achievable state.
In addition, GMCR is enhanced to provide the analyst with more in-depth information about an underlying conflict. This improvement is achieved by representing decision mak-ers’ preferences within the graph, which makes it possible to infer Nash stability condition by glancing at the graph.
Chapter 5
Robustness of Equilibria in the
Graph Model for Conflict Resolution
Strategic conflicts can be formally modelled and analyzed using the Graph Model for Conflict Resolution (GMCR) (Kilgour et al.,1987;Fang et al.,1993;Hipel,2009a,b;Kilgour and Eden, 2010; Hipel et al., 2011), which utilizes various stability concepts to determine possible equilibria or resolutions to a given conflict, and thereby obtain strategic insights.
It has been observed that some real world conflicts demonstrate that it is possible for conflicts to continue evolving even after reaching an equilibrium (Matbouli et al., 2013a, 2014b). Therefore, further analysis of equilibrium robustness is desired in order to gain some insights about the sustainability of a resolution. In this research, robust equilibrium is not necessarily a binary relation; instead, robustness can be viewed as a level in which equilibria are ranked from most robust to least robust. Such a methodology, for example, will make it possible to distinguish between stable states, where, for instance, in the sense of GMCR, a specific Nash equilibrium can be more stable than another Nash equilibrium.
A new formal robustness of equilibrium analysis is introduced within GMCR, which
will provide insights for better understanding the evolution of long-term conflicts. Refined stability definitions are presented, which facilitates the evaluation of equilibrium robust-ness, thereby making conflict resolution more sustainable. In the following sections, a background on strategic long-term conflicts is summarized. Then, a formal methodology for analyzing robust stability and ranking procedures to assess equilibrium robustness are proposed. The new strategic approach is applied to a groundwater contamination dispute that took place in Elmira, Ontario, Canada. Subsequently, strategic results and insights obtained when applying robustness of equilibria analysis, are discussed. The contents of this chapter are based on the publications byMatbouli et al. (2013c, 2014a, 2015c)