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## Matrix Representations and Extension of the Graph Model for Conflict Resolution

##### Abstract

The graph model for conflict resolution (GMCR) provides a convenient
and effective means to model and analyze a strategic conflict.
Standard practice is to carry out a stability analysis of a graph
model, and then to follow up with a post-stability analysis, two
critical components of which are status quo analysis and coalition
analysis. In stability analysis, an equilibrium is a state that is
stable for all decision makers (DMs) under appropriate stability
definitions or solution concepts. Status quo analysis aims to
determine whether a particular equilibrium is reachable from a
status quo (or an initial state) and, if so, how to reach it. A
coalition is any subset of a set of DMs. The coalition stability
analysis within the graph model is focused on the status quo states
that are equilibria and assesses whether states that are stable from
individual viewpoints may be unstable for coalitions. Stability
analysis began within a simple preference structure which includes a
relative preference relationship and an indifference relation.
Subsequently, preference uncertainty and strength of preference were
introduced into GMCR but not formally integrated.
In this thesis, two new preference frameworks, hybrid preference and
multiple-level preference, and an integrated algebraic approach are
developed for GMCR. Hybrid preference extends existing preference
structures to combine preference uncertainty and strength of
preference into GMCR. A multiple-level preference framework expands
GMCR to handle a more general and flexible structure than any
existing system representing strength of preference. An integrated
algebraic approach reveals a link among traditional stability
analysis, status quo analysis, and coalition stability analysis by
using matrix representation of the graph model for conflict
resolution.
To integrate the three existing preference structures into a hybrid
system, a new preference framework is proposed for graph models
using a quadruple relation to express strong or mild preference of
one state or scenario over another, equal preference, and an
uncertain preference. In addition, a multiple-level preference
framework is introduced into the graph model methodology to handle
multiple-level preference information, which lies between relative
and cardinal preferences in information content. The existing
structure with strength of preference takes into account that if a
state is stable, it may be either strongly stable or weakly stable
in the context of three levels of strength. However, the three-level
structure is limited in its ability to depict the intensity of
relative preference. In this research, four basic solution concepts
consisting of Nash stability, general metarationality, symmetric
metarationality, and sequential stability, are defined at each level
of preference for the graph model with the extended multiple-level
preference. The development of the two new preference frameworks
expands the realm of applicability of the graph model and provides
new insights into strategic conflicts so that more practical and
complicated problems can be analyzed at greater depth.
Because a graph model of a conflict consists of several interrelated
graphs, it is natural to ask whether well-known results of Algebraic
Graph Theory can help analyze a graph model. Analysis of a graph
model involves searching paths in a graph but an important
restriction of a graph model is that no DM can move twice in
succession along any path. (If a DM can move consecutively, then
this DM's graph is effectively transitive. Prohibiting consecutive
moves thus allows for graph models with intransitive graphs, which
are sometimes useful in practice.) Therefore, a graph model must be
treated as an edge-weighted, colored multidigraph in which each arc
represents a legal unilateral move and distinct colors refer to
different DMs. The weight of an arc could represent some preference
attribute. Tracing the evolution of a conflict in status quo
analysis is converted to searching all colored paths from a status
quo to a particular outcome in an edge-weighted, colored
multidigraph. Generally, an adjacency matrix can determine a simple
digraph and all state-by-state paths between any two vertices.
However, if a graph model contains multiple arcs between the same
two states controlled by different DMs, the adjacency matrix would
be unable to track all aspects of conflict evolution from the status
quo. To bridge the gap, a conversion function using the matrix
representation is designed to transform the original problem of
searching edge-weighted, colored paths in a colored multidigraph to
a standard problem of finding paths in a simple digraph with no
color constraints. As well, several unexpected and useful links
among status quo analysis, stability analysis, and coalition
analysis are revealed using the conversion function.
The key input of stability analysis is the reachable list of a DM,
or a coalition, by a legal move (in one step) or by a legal sequence
of unilateral moves, from a status quo in 2-DM or $n$-DM ($n
> 2$) models. A weighted reachability matrix for a DM or a coalition along
weighted colored paths is designed to construct the reachable list
using the aforementioned conversion function. The weight of each
edge in a graph model is defined according to the preference
structure, for example, simple preference, preference with
uncertainty, or preference with strength. Furthermore, a graph model
and the four basic graph model solution concepts are formulated
explicitly using the weighted reachability matrix for the three
preference structures. The explicit matrix representation for
conflict resolution (MRCR) that facilitates stability calculations
in both 2-DM and $n$-DM ($n
> 2$) models for three existing preference structures. In addition,
the weighted reachability matrix by a coalition is used to produce
matrix representation of coalition stabilities in
multiple-decision-maker conflicts for the three preference
frameworks.
Previously, solution concepts in the graph model were traditionally
defined logically, in terms of the underlying graphs and preference
relations. When status quo analysis algorithms were developed, this
line of thinking was retained and pseudo-codes were developed
following a similar logical structure. However, as was noted in the
development of the decision support system (DSS) GMCR II, the nature
of logical representations makes coding difficult. The DSS GMCR II,
is available for basic stability analysis and status quo analysis
within simple preference, but is difficult to modify or adapt to
other preference structures. Compared with existing graphical or
logical representation, matrix representation for conflict
resolution (MRCR) is more effective and convenient for computer
implementation and for adapting to new analysis techniques.
Moreover, due to an inherent link between stability analysis and
post-stability analysis presented, the proposed algebraic approach
establishes an integrated paradigm of matrix representation for the
graph model for conflict resolution.

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##### Cite this work

Haiyan Xu
(2009).
Matrix Representations and Extension of the Graph Model for Conflict Resolution. UWSpace.
http://hdl.handle.net/10012/4539

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