A matrix approach to status quo analysis in the graph model for conflict resolution

An algebraic method is developed to carry out status quo analysis within the framework of the graph model for conflict resolution. As a form of post-stability analysis, status quo analysis aims at confirming that possible equilibria, or states stable for all decision-makers, are in fact reachable from the status quo or any other initial state. Although pseudo-codes for status quo analysis have been developed, they have never been implemented within a practical decision support system. The novel matrix approach to status quo analysis designed here is convenient for computer implementation and easy to employ, as is illustrated by an application to a real-world conflict case. Moveover, the proposed explicit matrix approach reveals an inherent link between status quo analysis and the traditional stability analysis and, hence, provides the possibility of establishing an integrated paradigm for stability and status quo analyses

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

A matrix-based approach to searching colored paths in a weighted colored multidigraph

An algebraic approach to finding all edge-weighted-colored paths within a weighted colored multidigraph is developed. Generally, the adjacency matrix represents a simple digraph and determines all paths between any two vertices, and is not readily extendable to colored multidigraphs. To bridge the gap, a conversion function is proposed to transform the original problem of searching edge-colored paths in a colored multidigraph to a standard problem of finding paths in a simple digraph. Moreover, edge weights can be used to represent some preference attribute. Its potentially wide realm of applicability is illustrated by a case study: status quo analysis in the graph model for conflict resolution. The explicit matrix function is more convenient than other graphical representations for computer implementation and for adapting to other applications. Additionally, the algebraic approach reveals the relationship between a colored multidigraph and a simple digraph, thereby providing new insights into algebraic graph theory

An integrated multiple criteria preference ranking approach to the Canadian west coast port congestion conflict

An integrative conflict analysis approach, incorporating an Analytic Hierarchy Process (AHP) based preference ranking method into the Graph Model for Conflict Resolution (GMCR), is employed to investigate the Canadian west coast port congestion dispute. The Canadian west coast has historically been an important gateway connecting North America to Asia thanks to its specific geographical and strategic location. Despite successful operations and maintenance of the port facilities to handle international trade during the past few decades, the west coast is now facing increasing congestion problems, resulting in significant delays in transporting goods from the west coast to other parts of Canada and the USA. The strategic analyses carried out in this research suggest potential resolutions in which Canada would expand port facilities at various locations, encouraging traders to continue choosing the Canadian west coast as one of their trade gateways to North America

The Process of Conflict Resolution: A Case Study of Ichinose Disaster Management Conflict, Tottori Prefecture, Japan

ゲーム理論に基づいたコンフリクト解決のグラフモデルを、鳥取県智頭町市瀬集落において進行中のコンフリクト分析に適用した。本コンフリクトは、県が災害緩和のための行動をとるよう命じたのに対して、地域企業がこれを拒絶したときに生じた。対立が生じた原因と、議論の進展の過程を理解するためにこのコンフリクトを2段階に分けて分析した。得られた均衡解は、本ゲームにおいて可能な解決シナリオを明示するのに有用である。効果的なコミュニケーションとプレイヤーの能力が、状況を改善するために不可欠の要因であることが示された。Graph model for conflict resolution, a game theoretic based approach is used to analyze the on going conflict in Ichinose community, Chizu Town, Tottori Prefecture, Japan. The conflict arose when the local company refused to take a possible action for disaster mitigation as ordered by the prefecture government. Two different phases of this conflict are analyzed in order to understand why the confrontation occurred and how the disputes evolved over time. The equilibria help to manifest the possible solution scenario in this game. It is recommended that the effective communication, player’s capability are essential factors which can help to move the situation in a better way.ゲーム理論に基づいたコンフリクト解決のグラフモデルを、鳥取県智頭町市瀬集落において進行中のコンフリクト分析に適用した。本コンフリクトは、県が災害緩和のための行動をとるよう命じたのに対して、地域企業がこれを拒絶したときに生じた。対立が生じた原因と、議論の進展の過程を理解するためにこのコンフリクトを2段階に分けて分析した。得られた均衡解は、本ゲームにおいて可能な解決シナリオを明示するのに有用である。効果的なコミュニケーションとプレイヤーの能力が、状況を改善するために不可欠の要因であることが示された。Graph model for conflict resolution, a game theoretic based approach is used to analyze the on going conflict in Ichinose community, Chizu Town, Tottori Prefecture, Japan. The conflict arose when the local company refused to take a possible action for disaster mitigation as ordered by the prefecture government. Two different phases of this conflict are analyzed in order to understand why the confrontation occurred and how the disputes evolved over time. The equilibria help to manifest the possible solution scenario in this game. It is recommended that the effective communication, player’s capability are essential factors which can help to move the situation in a better way

Conflict Resolution. Risk-As-Feelings Hypothesis

Mathematical model of a goal-oriented thinking with feedback is described. Basic notions: decision graph, feed-back hierarchy and self-duality are introduced and explained. A source of the conflict in our approach is the ignorance about the importance of decision maker's goals. In contrast to Shar, Simonson & Tversky [4] and Deutsch [2] conflict resolution is modeled as a problem of finding a fixed point of a self-assessment operator, i.e. without adding or removing any decision element from decision hierarchy.multi-criteria decision making, duality, self-ranking, preference graph, potential method, risk-as-feeling model.

Statistical mechanics of voting

Decision procedures aggregating the preferences of multiple agents can produce cycles and hence outcomes which have been described heuristically as `chaotic'. We make this description precise by constructing an explicit dynamical system from the agents' preferences and a voting rule. The dynamics form a one dimensional statistical mechanics model; this suggests the use of the topological entropy to quantify the complexity of the system. We formulate natural political/social questions about the expected complexity of a voting rule and degree of cohesion/diversity among agents in terms of random matrix models---ensembles of statistical mechanics models---and compute quantitative answers in some representative cases.Comment: 9 pages, plain TeX, 2 PostScript figures included with epsf.tex (ignore the under/overfull \vbox error messages