Interrelationships Among Attitude-Based and Conventional Stability Concepts Within the Graph Model for Conflict Resolution
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Proceedings of the 2009 IEEE International Conference on Systems, Man, and Cybernetics San Antonio, TX, USA - October 2009 Interrelationships among Attitude-Based and Conventional Stability Concepts within the Graph Model for Conflict Resolution Takehiro Inohara Keith W. Hipel Department of Value and Decision Science Department of Systems Design Engineering Tokyo Institute of Technology, Tokyo, Japan University of Waterloo, Waterloo, Ontario, Canada [email protected] [email protected] Abstract—Propositions are presented on interrelationships in [18]. The concept of policies, which is similar to that of among attitude-based and conventional stability concepts within strategies in game theory [28], is introduced to the graph the paradigm of Graph Model for Conflict Resolution. In fact, model framework in [36], [37]. Research on the topic of the authors verify the following properties: if decision makers’ attitudes are discrete and decision makers’ preferences are incomplete preference information is also carried out [25], complete and anti-symmetric, then i) relational Nash stability is [29]. Cooperative decision situations are also analyzed within equivalent to Nash stability, ii) relational general metarationality GMCR [34]. is equivalent to general metarationality, iii) relational symmetric In a conflict, the attitudes of the various DMs can signifi- metarationality is equivalent to symmetric metarationality, and iv) relational sequential stability is equivalent to sequential cantly influence the outcome of the conflict. Thus, analyzing stability; under totally neutral attitudes of decision makers, i) the conflict as well as the accompanying attitudes is useful for relational Nash stability is equivalent to relational symmetric better understanding a given conflict. Attitude-based analysis metarationality, and ii) relational general metarationality is allows decision analysts to examine a lot of potential conflicts equivalent to relational sequential stability. without needing to reevaluate DMs’ preference. Index Terms—The Graph Model for Conflict Resolution; Attitudes; Stability Concepts Attitude-based stability concepts are furnished and applied to the analysis of the War of 1812 in [19]. Such attitude-based I. INTRODUCTION stability concepts in [19] as relational Nash stability, relational The purpose of this research is to provide propositions on general metarationality, relational symmetric metarationality, interrelationships among attitude-based [19], [22], [33] and and relational sequential stability are natural generalizations conventional stability concepts within the Graph Model for of the relational equilibrium concepts proposed in [15], [16] Conflict Resolution (GMCR) [4], [5]. GMCR is a flexible within the game theoretic paradigm and the standard stability framework for describing and analyzing a conflict. In fact, concepts within the GMCR paradigm. Some interrelation- GMCR allows one to describe and analyze a conflict with ships among attitude-based stability concepts are investigated consideration of infeasibility of outcomes, irreversibility of in [22]. choices of actions, countermoves against a Decision Maker Attitude is defined by Krech et al. [24] as “an enduring (DM)’s unilateral moves, and so on [4], [5]. Consequently, this system of positive or negative evaluations, emotional feelings framework has been utilized to model and analyze realworld and pro and con action tendencies, with respect to a social conflicts such as Cuban Missile Crisis, Garrison Diversion object.” Following the framework in [22], the authors treat Unit conflict, Normandy Invasion, Fall of France, Zimbabwe three types of attitudes, that is, positive, negative, and neutral conflict, Watergate Tapes controversy, Poplar River conflict, attitudes, as in [30], [31] in which a DM’s attitudes toward and Holston River negotiations [7]. Additionally, coalition her/himself as well as toward others are considered. More- formation [23] and strength of DMs’ preferences [8] have been over, it is assumed in this paper that positive, negative, and incorporated into the framework of GMCR. The coalition anal- neutral attitudes of a DM toward others derive “altruistic,” ysis framework in [23] is expanded in [17], [18] and applied “sadistic,” and “apathetic” behaviors, respectively, and those to the analysis of environmental issues regarding the Kyoto toward her/himself derive “selfish,” “masochistic,” and “self- Protocol [32]. Definitions of such coalition stability concepts less” behaviors, respectively. Table I shows these assumptions as coalition Nash stability, coalition general metarationality, on the relationships between attitudes and DMs’ behavior. coalition symmetric metarationality, and coalition sequential These assumptions imply that a DM modeled in this paper stability are provided in [17], and their interrelationships to is not “rational” in the classical game-theoretic sense, but are such standard stability concepts as Nash stability [26], [27], consistent with those of DMs’ emotions in the ‘soft’ game general metarationality [9], symmetric metarationality [9], and theory [10], [13], [20], [21], drama theory [2], [3], [11], [12], sequential stability [6], [7] within GMCR are investigated [14], and confrontation analysis [1]. 1156 978-1-4244-2794-9/09/$25.00 ©2009 IEEE TABLE I 0 ASSUMPTIONS ON RELATIONSHIPS BETWEEN ATTITUDES AND 0 BEHAVIORS [22] 1 2 0 Attitudes 0 types toward others toward her/himself Fig. 4. Totally neutral attitudes [22] for DM 1 and DM 2 positive altruistic selfish negative sadistic masochistic neutral apathetic selfless is equivalent to relational symmetric metarationality, and ii) relational general metarationality is equivalent to relational sequential stability. In [22], two specific types of attitudes of DMs, that is, The definitions of the concepts within the GMCR frame- the kinds in which all of the DMs’ attitudes are positive (see work employed in this paper are presented in the next section. Figure 1) and negative (see Figure 2), respectively, are dealt Then, in Section III, the propositions on interrelationships with, and propositions on the equivalence between relational among attitude-based and conventional stability concepts are Nash stability and relational sequential stability under those provided. The concluding remarks are furnished in the last types of DMs’ attitudes are verified. section. In this paper, two other kinds of attitudes of DMs are treated: the type in which the attitudes from one DM to II. ATTITUDE ANALYSIS WITHIN GMCR: DEFINITIONS her/himself are positive and those from one DM to another This section gives the definitions of the concepts within the are neutral, and the type in which all of the DMs’ attitudes GMCR framework employed in this paper. are neutral. In this paper, these attitudes are called discrete and totally neutral, respectively. These types of attitudes are A. The Graph Models for Conflict Resolution — A General depicted in Figures 3 and 4, respectively, and their precise Definition definitions are provided in Section II-C. A graph model of a conflict is a 4-tuple (N,S,(Ai)i∈N , (i ) ) N + i∈N . is the set of all decision makers (DMs), where + |N|≥2. S is the set of all states of the focal decision making 1 2 situation, where |S|≥2.Fori ∈ N, (S, Ai) constitutes DM + + i’s graph,forwhichS is the set of all vertices and Ai ⊆ S ×S (S, A ) Fig. 1. Totally positive attitudes [22] for DM 1 and DM 2 is the set of all arcs. It is assumed that i is a directed graph with no loop or multiple arcs, that is, (s, s) ∈/ Ai for each s ∈ S, and the number of arcs between two distinct s, t ∈ S (s, t) ∈ A _ _ vertices are one at most. For , i means that DM i can shift the state of the conflict from state s to state t. 2 1 For i ∈ N and s ∈ S, DM i’s reachable list from state s is _ _ defined as the set {t ∈ S | (s, t) ∈ Ai}, denoted by Ri(s). i Fig. 2. Totally negative attitudes [22] for DM 1 and DM 2 is the preference of DM i ∈ N on S. For i ∈ N and s, s ∈ S, s i s means that s is more or equally preferred to s by DM i. s ∼i s means that s i s and s i s, that is, s is equally preferred to s by DM i. 0 + s i s means that s i s and “not (s i s), ” that is, s is 1 2 strictly more preferred to s by DM i. + 0 DM i’s preferences i is said to be anti-symmetric, if and only if for all s, s ∈ S,ifs i s and s i s then s = s . i Fig. 3. Discrete attitudes [22] for DM 1 and DM 2 is said to be complete, if and only if for all s, s ∈ S, s i s or s i s. The objective of this paper is to verify some inclusion relationships among attitude-based and conventional stabilty B. Stability Concepts within GMCR concepts. More specifically, the authors verify that if decision Given a graph model (N,S,(Ai)i∈N , (i)i∈N ) of a con- makers’ attitudes are discrete and decision makers’ preferences flict, one can define Nash stability [26], [27], general meta- are complete and anti-symmetric, then i) relational Nash stabil- rationality [9], symmetric metarationality [9], and sequential ity is equivalent to Nash stability, ii) relational general metara- stability [6], [7] as follows: tionality is equivalent to general metarationality, iii) relational For coalition H ⊆ N and s ∈ S,thereachable list of symmetric metarationality is equivalent to symmetric metara- coalition H from state s is defined inductively, under the tionality, and iv) relational sequential stability is equivalent to restriction in which a DM can only move once at a time, as the sequential stability. Also, the authors show that under totally set RH (s) that satisfies the next two conditions: (i) if i ∈ H neutral attitudes of decision makers, i) relational Nash stability and s ∈ Ri(s), then s ∈ RH (s), and (ii) if i ∈ H and 1157 s ∈ RH (s) and s ∈ Ri(s ), then s ∈ RH (s).Aunilateral Definition 8 (Relational Preference (RP) on S): The rela- improvement s of DM i from state s is defined as a state tional preference RP(e)ij of DM i ∈ N to DM j ∈ N at that is in DM i’s reachable list from s (that is, s ∈ Ri(s)), e is defined as follows: ⎧ i s s s i s and DM strictly prefers state to state (that is, ).