Matrix Representations and Extension of the Graph Model for Conflict Resolution

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

On optimal completions of incomplete pairwise comparison matrices

An important variant of a key problem for multi-attribute decision making is considered. We study the extension of the pairwise comparison matrix to the case when only partial information is available: for some pairs no comparison is given. It is natural to define the inconsistency of a partially filled matrix as the inconsistency of its best, completely filled completion. We study here the uniqueness problem of the best completion for two weighting methods, the Eigen-vector Method and the Logarithmic Least Squares Method. In both settings we obtain the same simple graph theoretic characterization of the uniqueness. The optimal completion will be unique if and only if the graph associated with the partially defined matrix is connected. Some numerical experiences are discussed at the end of the paper

Network Analysis, Creative System Modelling and Decision Support: The NetSyMoD Approach

This paper presents the NetSyMoD approach – where NetSyMod stands for Network Analysis – Creative System Modelling – Decision Support. It represents the outcome of several years of research at FEEM in the field of natural resources management, environmental evaluation and decision-making, within the Natural Resources Management Research Programme. NetSyMoD is a flexible and comprehensive methodological framework, which uses a suite of support tools, aimed at facilitating the involvement of stakeholders or experts in decision-making processes. The main phases envisaged for the process are: (i) the identification of relevant actors, (ii) the analysis of social networks, (iii) the creative system modelling and modelling of the reality being considered (i.e. the local socio-economic and environmental system), and (iv) the analysis of alternative options available for the management of the specific case (e.g. alternative projects, plans, strategies). The strategies for participation are necessarily context-dependent, and thus not all the NetSyMod phases may be needed in every application. Furthermore, the practical solutions for their implementation may significantly differ from one case to another, depending not only on the context, but also on the available resources (human and financial). The various applications of NetSyMoD have nonetheless in common the same approach for problem analysis and communication within a group of actors, based upon the use of creative thinking techniques, the formalisation of human-environment relationships through the DPSIR framework, and the use of multi-criteria analysis through the mDSS software.Social Network, Integrated Analysis, Participatory Modelling, Decision Support

Method maximizing the spread of influence in directed signed weighted graphs

We propose a new method for maximizing the spread of influence, based on the identification of significant factors of the total energy of a control system. The model of a socio-economic system can be represented in the form of cognitive maps that are directed signed weighted graphs with cause-and-effect relationships and cycles. Identification and selection of target factors and effective control factors of a system is carried out as a solution to the optimal control problem. The influences are determined by the solution to optimization problem of maximizing the objective function, leading to matrix symmetrization. The gear-ratio symmetrization is based on computing the similarity extent of fan-beam structures of the influence spread of vertices v_i and v_j to all other vertices. This approach provides the real computational domain and correctness of solving the optimal control problem. In addition, it does not impose requirements for graphs to be ordering relationships, to have a matrix of special type or to fulfill stability conditions. In this paper, determination of new metrics of vertices, indicating and estimating the extent and the ability to effectively control, are likewise offered. Additionally, we provide experimental results over real cognitive models in support

Deriving consensus rankings via multicriteria decision making methodology

Purpose - This paper seeks to take a cautionary stance to the impact of the marketing mix on customer satisfaction, via a case study deriving consensus rankings for benchmarking on selected retail stores in Malaysia. Design/methodology/approach - The ELECTRE I model is used in deriving consensus rankings via multicriteria decision making method for benchmarking base on the marketing mix model 4P's. Descriptive analysis is used to analyze best practice among the four marketing tactics. Findings - Outranking methods in consequence constitute a strong base on which to found the entire structure of the behavioral theory of benchmarking applied to development of marketing strategy. Research limitations/implications - This study looks only at a limited part of the puzzle of how consumer satisfaction translates into behavioral outcomes. Practical implications - The study provides managers with guidance on how to generate a rough outline of potential marketing activities that can be used to take advantage of capabilities and convert weaknesses and threats. Originality/value - The paper interestingly portrays the effective usage of multicriteria decision-making and ranking method to help marketing managers predict their marketing trends

EEMCS final report for the causal modeling for air transport safety (CATS) project

This document reports on the work realized by the DIAM in relation to the completion of the CATS model as presented in Figure 1.6 and tries to explain some of the steps taken for its completion. The project spans over a period of time of three years. Intermediate reports have been presented throughout the project’s progress. These are presented in Appendix 1. In this report the continuous‐discrete distribution‐free BBNs are briefly discussed. The human reliability models developed for dealing with dependence in the model variables are described and the software application UniNet is presente