Interrelationships Among Attitude-Based and Conventional Stability Concepts Within the Graph Model for Conﬂict Resolution

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 _ _ vertices are one at most. For s, t ∈ S, (s, t) ∈ Ai means that DM i can shift the state of the conflict from state s to state t. 1 2 i ∈ N s ∈ S i s _ _ For and , DM ’s reachable list from state 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, ). ⎨ DPij if eij =+ i Accordingly, the set of all unilateral improvements of DM RP(e)ij = APij if eij = − s {x ∈ Ri(s) | x i s}⊆Ri(s) ⎩ from state is the set , called 1 if eij =0, DM i’s unilateral improvement list from state s, and is denoted by R+(s). For coalition H ⊆ N and s ∈ S,theimprovement where 1 denotes the ordering on S which is defined as for all i list of coalition H from state s is defined inductively, under s, s ∈ S, s1s . the restriction in which a DM can only move once at a time, The definition of RP reflects the assumptions on the + as the set RH (s) that satisfies the next two conditions: (i) if relationships between attitudes and DMs’ behavior, which are + + i ∈ H and s ∈ Ri (s), then s ∈ RH (s), and (ii) if i ∈ H shown in Table I. + + + S and s ∈ RH (s) and s ∈ Ri (s ), then s ∈ RH (s). φi (s) Definition 9 (Totally Relational Preference (TRP) on ): denotes the set of all states that are equally or less preferred The totally relational preference of DM i ∈ N at e, denoted TRP(e) s, s ∈ S sTRP(e) s to state s by DM i, that is, {x ∈ S | s i x}. by i, is defined as for , i if and Definition 1 (Nash Stability): For i ∈ N, state s ∈ S is only if sRP(e)ij s for all j ∈ N. Nash i e Nash stable for DM i, denoted by s ∈ Si , if and only if The totally relational preference of DM at is a relation on + S (RP(e) ) Ri (s)=∅. which is consistent with the list ij j∈N of relational Definition 2 (General Metarationality): For i ∈ N, state preferences RP(e)ij of DM i to DM j at e for all j ∈ N. s ∈ S is generally metarational for DM i, denoted by 2) Attitude-Based Replies and Stability Concepts: GMR + s ∈ Si , if and only if for all s ∈ Ri (s), RN\{i}(s ) ∩ Definition 10 (Totally Relational Reply (TRR) List): The i ∈ N e s ∈ S φi (s) = ∅. totally relational reply list of DM at from is Definition 3 (Symmetric Metarationality): For i ∈ N, state defined as the set {x ∈ Ri(s) ∪{s}|xTRP(e)is}, denoted s ∈ S is symmetrically metarational for DM i, denoted by by TRR(e)i(s). SMR + TRR(e) (s) i s ∈ Si , if and only if for all s ∈ Ri (s), there exists s The totally relational reply list i of DM at e s ∈ RN\{i}(s ) ∩ φi (s) such that s ∈ φi (s) for all s ∈ from in the attitude analysis within GMCR serves as the Ri(s ). irreflexive reachable list Ri(s) of the DM i from s in the Definition 4 (Sequential Stability): For i ∈ N, state s is standard analysis within GMCR. SEQ sequentially stable for DM i, denoted by s ∈ Si , if and Definition 11 (Totally Relational Reply List of Coalition): + + H ⊆ N e only if for all s ∈ Ri (s), RN\{i}(s ) ∩ φi (s) = ∅. The totally relational reply list of coalition at from s ∈ S is defined inductively, under the restriction C. Attitude-Based Stability Concepts in which a DM can only move once at a time, as the set In this section, a framework incorperating the notion of TRR(e)H (s) that satisfies the next two conditions: (i) if attitudes into the GMCR is presented based on [19], [22]. i ∈ H and s ∈ TRR(e)i(s), then s ∈ TRR(e)H (s), and 1) Attitude-Based Preferences: (ii) if i ∈ H and s ∈ TRR(e)H (s) and s ∈ TRR(e)i(s ), Definition 5 (Attitudes): For i ∈ N, the attitude of DM i is then s ∈ TRR(e)H (s). ei =(eij )j∈N , where eij ∈{+, 0, −} for j ∈ N. eij is called Definition 12 (Rφ (e)i(s)): For i ∈ N and s ∈ S, the attitude of DM i to DM j. Rφ (e)i(s) is defined {x ∈ S | x = s or ¬(xTRP(e)is)}, Note that given a list e =(ei)i∈N of attitudes ei of DM i where ¬ denotes “not.” for each i ∈ N, one can specify such a valued, directed, and Rφ (e)i(s) is the set of all states which are not preferred to complete graph with loops as in Figures 1, 2, 3, and 4. s by DM i at e with respect to the totally relational preference A list e =(ei)i∈N of attitudes ei of DM i for each i ∈ N TRP(e)i of DM i ∈ N at e. This serves in the attitude is said to be totally positive, if and only if eij =+for all analysis within GMCR as φi (s) in the standard analysis within i, j ∈ N (see Figure 1). Similarly, e =(ei)i∈N is said to be GMCR. totally negative, if and only if eij = − for all i, j ∈ N (see Employing the foregoing definitions, attitude-based stability Figure 2). e =(ei)i∈N is said to be discrete, if and only if concepts can be defined as an extension of standard stability eii =+for all i ∈ N and eij =0for all i, j ∈ N such that concepts. i = j (see Figure 3), and is said to be totally neutral, if and Definition 13 (Relational Nash Stability): For i ∈ N, state only if eij =0for all i, j ∈ N (see Figure 4). s ∈ S is relational Nash stable at e for DM i, denoted by S RNash(e) Definition 6 (Devoting Preference (DP) on ): The devot- s ∈ Si , if and only if TRR(e)i(s)={s}. ing preference of DM i ∈ N to DM j ∈ N, denoted by DPij , Definition 14 (Relational General Metarationality): For is defined as for s, s ∈ S, sDPij s if and only if s j s . i ∈ N, state s ∈ S is relational general metarational at e S RGMR(e) Definition 7 (Aggressive Preference (AP) on ): The ag- for DM i, denoted by s ∈ Si , if and only if for all i ∈ N j ∈ N gressive preference of DM to DM , denoted s ∈ TRR(e)i(s)\{s}, RN\{i}(s ) ∩ Rφ (e)i(s) = ∅. AP s, s ∈ S sAP s by ij , is defined as for , ij if and only if Definition 15 (Relational Symmetric Metarationality): For s s j . i ∈ N, state s ∈ S is relational symmetric metarational at

1158 RSMR(e) Corollary 5 (Corollary of Propositions 1 and 2): Assume e for DM i, denoted by s ∈ Si , if and only if for s ∈ TRR(e) (s)\{s} s ∈ R (s) ∩ that DMs’ attitudes e =(ei)i∈N are discrete and DM i’s all i , there exists N\{i} RNash(e) S = SNash Rφ (e)i(s) such that s ∈ Rφ (e)i(s) for all s ∈ Ri(s ). preference i is anti-symmetric. Then, i i . Definition 16 (Relational Sequential Stability): For i ∈ N, 2) RGMR and GMR: state s ∈ S is relational sequential stable at e for DM i, denoted Lemma 2: Assume that DMs’ attitudes e =(ei)i∈N are RSEQ(e) i i ∈ N by s ∈ Si , if and only if for all s ∈ TRR(e)i(s)\{s}, discrete and DM ’s preference i is complete for . Rφ(e) (s) ⊆ φ(s) s ∈ S (TRR(e)N\{i}(s ) \{s }) ∩ Rφ (e)i(s) = ∅. Then, i i for all . Proof: s ∈ Rφ (e)i(s) implies that s = s or NTERRELATIONSHIPS AMONG TABILITY ONCEPTS III. I S C ¬(s TRP(e)is). By Lemma 1, this is equivalent to s = s Consider a graph model (N,S,(Ai)i∈N , (i)i∈N ) of a or ¬(s i s). By the completeness of i, this implies that conflict and a list e =(ei)i∈N of attitudes ei of DM i for s = s or s i s , which is followed by s i s . Thus, s ∈ i ∈ N. {x ∈ S | s i x} = φi (s). Lemma 3: Assume that DMs’ attitudes e =(ei)i∈N are A. Discrete Cases discrete and DM i’s preference i is anti-symmetric for i ∈ N. 1) RNash and Nash: Then, φi (s) ⊆ Rφ (e)i(s) for all s ∈ S. Lemma 1: Assume that DMs’ attitudes e =(ei)i∈N are Proof: φi (s) is defined as the set {x ∈ S | s i x} = discrete. Then, for all i ∈ N, TRP(e)i = i, that is, for all {x ∈ S | s ∼ x}∪{x ∈ S | s x} i i , and anti-symmetry s, s ∈ S, sTRP(e)is if and only if s i s . {x ∈ S | s ∼ x} = {s} s of i implies that i . Therefore, Proof: By Definition 9, we have that sTRP(e)is if and ∈ φ(s) s = s s s i implies that or i , which means that only if sRP(e)ij s for all j ∈ N. That is, sRP(e)iis and s = s ¬(s s) s = s or i . This implies, by Lemma 1, that sRP(e)ij s for all j ∈ N such that j = i. sRP(e)iis is ¬(sTRP(e) s) s ∈ Rφ(e) (s) or i , which is equivalent to i . equivalent to sDPiis by Definition 8 and the assumption that Corollary 6 (Corollary of Lemmas 2 and 3): Assume that eii =+, and this is equivalent to s i s by Definition 6. e =(e ) i DMs’ attitudes i i∈N are discrete and DM ’s pref- We have, moreover, that sRP(e)ij s for all j ∈ N such that i ∈ N erence i is complete and anti-symmetric for . Then, j = i if and only if s1s , which means that sRP(e)ij s for Rφ (e)i(s)=φi (s) for all s ∈ S. all j ∈ N such that j = i is always true by Definition 8 and Proposition 3: Assume that DMs’ attitudes e =(ei)i∈N the assumption that eij =0if j = i. Thus, we have that for are discrete and DM i’s preference i is complete for i ∈ N. all i ∈ N, sTRP(e)is if and only if s i s . RGMR(e) GMR Then, Si ⊆ Si . Corollary 1 (Corollary of Lemma 1): Assume that DMs’ RGMR(e) s ∈ S e =(e ) TRR(e) (s)= Proof: Let i . Then, by Definition 14, we attitudes i i∈N are discrete. Then, i + have that for all s ∈ TRR(e)i(s)\{s}, RN\{i}(s ) ∩ R (s) ∪{x ∈ Ri(s) | x ∼i s}∪{s}. i Rφ(e) (s) = ∅ R+(s) Corollary 2 (Corollary of Lemma 1): Assume that DMs’ i . By Corollary 3, we have that i e =(e ) i ⊆ TRR(e)i(s)\{s}. Moreover, by Lemma 2, we see that attitudes i i∈N are discrete and DM ’s preference i + Rφ (e)i(s) ⊆ φ (s), which implies that RN\{i}(s ) ∩ is anti-symmetric for i ∈ N. Then, TRR(e)i(s)=R (s) ∪ i i Rφ(e) (s) ⊆ R (s) ∩ φ(s) {s} i N\{i} i . Thus, we have that for . + all s ∈ Ri (s), RN\{i}(s ) ∩ φi (s) = ∅, which means that Corollary 3 (Corollary of Corollary 1): Assume that GMR + s ∈ Si . DMs’ attitudes e =(ei)i∈N are discrete. Then, Ri (s) ⊆ Proposition 4: Assume that DMs’ attitudes e =(ei)i∈N TRR(e)i(s)\{s}. i Corollary 4 (Corollary of Corollary 2): Assume that are discrete and DM ’s preference i is anti-symmetric for i ∈ N SGMR ⊆ SRGMR(e) DMs’ attitudes e =(ei)i∈N are discrete and DM i’s preference . Then, i i . s ∈ SGMR s ∈ TRR(e) (s)\{s} i is anti-symmetric for i ∈ N. Then, TRR(e)i(s)\{s} = Proof: Let i .If i , then + s ∈ S+(s) R (s) ∩ R (s). i by Corollary 4. Then, we have that N\{i} i φ(s) = ∅ s ∈ SGMR φ(s) ⊆ Rφ(e) (s) Proposition 1: e =(ei)i∈N i by i , and that i i by Assume that DMs’ attitudes RNash(e) Nash Lemma 3. Therefore, RN\{i}(s ) ∩ Rφ (e)i(s) = ∅, which are discrete. Then, for i ∈ N, Si ⊆ Si . RNash(e) s ∈ SRGMR(e) Proof: Let s ∈ Si . Then, by Definition 13, we implies that i . have that TRR(e)i(s)={s}. We also have, by Corollary 1, Corollary 7 (Corollary of Propositions 3 and 4): Assume + e =(ei)i∈N that TRR(e)i(s)=Ri (s) ∪{x ∈ Ri(s) | x ∼ s}∪{s}. that DMs’ attitudes are discrete and DM + + i i Since s/∈ Ri (s), these imply that Ri (s)=∅, which implies ’s preference is complete and anti-symmetric. Then, Nash SRGMR(e) = SGMR that s ∈ Si by Definition 1. i i . Proposition 2: Assume that DMs’ attitudes e =(ei)i∈N 3) RSMR and SMR: are discrete and DM i’s preference i is anti-symmetric for Proposition 5: Assume that DMs’ attitudes e =(ei)i∈N Nash RNash(e) i i ∈ N i ∈ N. Then, for i ∈ N, Si ⊆ Si . are discrete and DM ’s preference i is complete for . Nash + RSMR(e) SMR Proof: If s ∈ Si , then we have that Ri (s)=∅. Then, Ri ⊆ Ri . TRR(e) (s)=R+(s) ∪ RSMR(e) By Corollary 2, moreover, that i i Proof: Let s ∈ Si . Then, by Definition 15, we {s}. These imply that TRR(e)i(s)={s}, which means that have that for all s ∈ TRR(e)i(s)\{s}, there exists s RNash(e) s ∈ Si by Definition 13. ∈ RN\{i}(s ) ∩ Rφ (e)i(s) such that s ∈ Rφ (e)i(s)

1159 for all s ∈ Ri(s ). This is equivalent to for all s ∈ that for all s ∈ TRR(e)i(s)\{s}, (TRR(e)N\{i}(s ) \{s }) TRR(e) (s)\{s} s ∈ R (s) ∩ Rφ(e) (s) SEQ i , there exists N\{i} i ∩ Rφ (e)i(s) = ∅, which means that s ∈ Si . such that Ri(s ) ⊆ Rφ (e)i(s). Corollary 9 (Corollary of Propositions 7 and 8): Assume + + By Corollary 1 and s/∈ Ri (s), we have that Ri (s) ⊆ that DMs’ attitudes e =(ei)i∈N are discrete and DM i’s + TRR(e)i(s)\{s}. Thus, we have that for all s ∈ Ri (s), there preference i is complete and DMs’ preferences (i)i∈N are RSEQ(e) SEQ exists s ∈ RN\{i}(s ) ∩ Rφ (e)i(s) such that Ri(s ) ⊆ anti-symmetric. Then, S = S . i i Rφ (e)i(s). By Lemma 2, we have Rφ (e)i(s) ⊆ φi (s). So, + we have that for all s ∈ Ri (s), there exists s ∈ RN\{i}(s ) B. Totally Neutral Cases ∩ φ(s) R (s) ⊆ Rφ(e) (s) i such that i i . Again, by Lemma 2, Lemma 4: Assume that DMs’ attitudes e =(ei)i∈N are Rφ(e) (s) ⊆ φ(s) we have that i i . This implies that for all totally neutral. Then, TRR(e)i(s)=Ri(s)∪{s} for all i ∈ N s ∈ R+(s) s ∈ R (s) ∩ φ(s) i , there exists N\{i} i such that and s ∈ S. R (s) ⊆ φ(s) s ∈ SSMR i i , which means i . Proof: By Definition 10, TRR(e)i(s)={x ∈ Ri(s) ∪ e =(e ) Proposition 6: Assume that DMs’ attitudes i i∈N {s}|xTRP(e)is}. Since e is totally neutral, then we have, i are discrete and DM ’s preference i is anti-symmetric for by Definition 8 and 9, that for all s, s ∈ S, s TRP(e)is,this i ∈ N SSMR ⊆ SRSMR(e) . Then, i i . implies TRR(e)i(s)=Ri(s) ∪{s}. s ∈ SSMR s ∈ R+(s) Proof: Let i . Then, for all i , there Proposition 9: Assume that DMs’ attitudes e =(ei)i∈N s ∈ R (s) ∩ φ(s) R (s) ⊆ φ(s) exists N\{i} i such that i i . are totally neutral. Then, for all i ∈ N and all s ∈ S,wehave RNash(e) Since we have φi (s) ⊆ Rφ (e)i(s) by Lemma 3, we have s ∈ S R (s)=∅ + that i if and only if i . s ∈ R (s) s ∈ R (s) RNash(e) that for all i , there exists N\{i} s ∈ S Proof: By Definition 13, we have that i if ∩ Rφ (e)i(s) such that Ri(s ) ⊆ Rφ (e)i(s). Because and only if TRR(e)i(s)={s}, which is equivalent to Ri(s)∪ TRR(e) (s)\{s}⊆R+(s) i i by Corollary 4, we see that for {s} = {s} s/∈ R (s) by Lemma 4. Since i , this is equivalent all s ∈ TRR(e)i(s)\{s}, there exists s ∈ RN\{i}(s ) ∩ R (s)=∅ to i . Rφ (e)i(s) such that Ri(s ) ⊆ Rφ (e)i(s), which implies RSMR(e) Lemma 5: Assume that DMs’ attitudes e =(ei)i∈N are that s ∈ S . i totally neutral. Then, Rφ (e)i(s)={s} for all i ∈ N and Corollary 8 (Corollary of Propositions 5 and 6): Assume s ∈ S. that DMs’ attitudes e =(ei)i∈N are discrete and DM Proof: Let s ∈ Rφ (e)i(s). Then, by Definition 12, we i’s preference i is complete and anti-symmetric. Then, have that s ∈{x ∈ S | x = s or ¬(xTRP(e)is)}. Since e SRSMR(e) = SSMR i i . is totally neutral, then we have, by Definition 8 and 9, that 4) RSEQ and SEQ: for all s, s ∈ S, s TRP(e)is, this implies that s ∈{x ∈ S | e =(e ) Proposition 7: Assume that DMs’ attitudes i i∈N x = s}, which means s = s. i are discrete. Also, assume that DM ’s preference i is Proposition 10: Assume that DMs’ attitudes e =(ei)i∈N complete for i ∈ N and DMs’ preferences (i)i∈N are anti- i ∈ N s ∈ S RSEQ(e) SEQ are totally neutral. Then, for all and all ,we symmetric. Then, S ⊆ S . RGMR(e) i i have that s ∈ Si if and only if for all s ∈ Ri(s), s ∈ SRSEQ(e) Proof: Let i . Then, by Definition 16, we have s ∈ RN\{i}(s ). that for all s ∈ TRR(e)i(s)\{s}, (TRR(e)N\{i}(s ) \{s }) s ∈ SRGMR(e) Proof: By Definition 14, we have that i ∩ Rφ (e)i(s) = ∅. s ∈ TRR(e) (s)\{s} R (s) + + if and only if for all i , N\{i} By Corollary 1 and s/∈ Ri (s), we have that Ri (s) ⊆ ∩ Rφ(e) (s) = ∅ + i . By Lemmas 4 and 5, we have that TRR(e)i(s)\{s}. Thus, we have that for all s ∈ Ri (s), TRR(e) (s)=R (s) ∪{s} Rφ(e) (s)={s} i i and i , respec- (TRR(e)N\{i}(s ) \{s }) ∩ Rφ (e)i(s) = ∅. By Lemma 2, s ∈ R (s) R (s) tively. Therefore, we have that for all i , N\{i} we have Rφ (e)i(s) ⊆ φi (s). So, we have that for all ∩{s} = ∅ s ∈ R (s) + , which is equivalent to for all i , s ∈ R (s) (TRR(e)N\{i}(s ) \{s }) ∩ φ (s) = ∅ i , i . More- s ∈ RN\{i}(s ). over, since we have, by Corollary 4, that TRR(e)j (s)\{s}⊆ Proposition 11: Assume that DMs’ attitudes e =(ei)i∈N R+(s) j ∈ N\{i} s ∈ R+(s) j for all , we have that for all i , are totally neutral. Then, for all i ∈ N and all s ∈ R,wehave R+ (s) ∩ φ(s) = ∅ s ∈ SSEQ RSMR(e) N\{i} i , which means that i . that s ∈ Si if and only if Ri(s)=∅. Proposition 8: Assume that DMs’ attitudes e =(ei)i∈N RSMR(e) Proof: By Definition 15, we have that s ∈ Si if are discrete. Also, assume that DMs’ preferences (i)i∈N are s ∈ TRR(e) (s)\{s} s ∈ SEQ(e) RSEQ and only if for all i , there exists S ⊆ S anti-symmetric. Then, i i . RN\{i}(s ) ∩ Rφ (e)i(s) such that s ∈ Rφ (e)i(s) for all s ∈ SSEQ Proof: Let i . Then, by Definition 4, we have s ∈ Ri(s ). Since we have that TRR(e)i(s)=Ri(s) ∪ s ∈ R+(s) R+ (s) ∩ φ(s) = ∅ that for all i , N\{i} i . Since {s} and Rφ (e)i(s)={s} by Lemmas 4 and 5, respectively, + we have, by Corollary 4, that TRR(e)i(s)\{s} = Ri (s), we equivalently have for all s ∈ Ri(s), there exists s ∈ + we have that for all s ∈ TRR(e)i(s)\{s}, RN\{i}(s ) ∩ RN\{i}(s ) ∩{s} such that Ri(s ) ⊆{s}. φ(s) = ∅ φ(s) ⊆ Rφ(e) (s) RSMR(e) i . We have, moreover, that i i by If Ri(s)=∅, then s ∈ Si is logically true. Consider s ∈ TRR(e) (s)\{s} Lemma 3, so it is satisfied that for all i , the case in which Ri(s) = ∅ and assume that s ∈ Ri(s).In R+ (s) ∩ Rφ(e) (s) = ∅ N\{i} i . Additionally, because we have, this situation, we have that there exists s ∈ RN\{i}(s ) ∩ + by Corollary 4, that TRR(e)i(s)\{s} = Ri (s), it is implied {s} such that Ri(s ) ⊆{s}. s ∈ RN\{i}(s ) ∩{s} implies

1160 that s = s, and thus, we have that Ri(s) ⊆{s}, but since [12] N. Howard, “Drama theory and its relation to game theory. Part 2: formal model of the resolution process,” Group Decision and Negotiation, vol. s/∈ Ri(s), this means Ri(s)=∅, which is a contradiction. RSMR(e) 3, pp. 207-235, 1994. Thus, we have that s ∈ Si if and only if Ri(s)=∅. [13] N. Howard, “n-person ‘soft’ games,” Journal of Operational Research Proposition 12: Assume that DMs’ attitudes e =(ei)i∈N Society, vol. 49, pp. 144-150, 1998. i ∈ N s ∈ R [14] N. Howard, P. Bennett, J. Bryant, and M. Bradley, “Manifesto for a are totally neutral. Then, for all and all ,we theory of drama and irrational choice,” Journal of Operational Research RSEQ(e) have that s ∈ Si if and only if for all s ∈ Ri(s), s ∈ Society, vol. 44, no. 1, pp. 99-103, 1993. [15] T. Inohara, “Relational dominant strategy equilibrium as a generalization RN\{i}(s ). RSEQ(e) of dominant strategy equilibrium in terms of a social psychological Proof: By Definition 16, we have that s ∈ Si if aspect of decision making,” European Journal of Operational Research, and only if for all s ∈ TRR(e)i(s)\{s}, (TRR(e)N\{i}(s ) vol. 182, pp. 856-866, 2007. [16] T. Inohara, “Relational Nash equilibrium and interrelationships among \{s }) ∩ Rφ (e)i(s) = ∅. By Lemmas 4 and 5, we relational and rational equilibrium concepts,” Applied Mathematics and have that TRR(e)i(s)=Ri(s) ∪{s} and Rφ (e)i(s)= Computation, vol. 199, no. 2, pp. 704-715, 2008. {s}, respectively. Thus, we equivalently have that for all [17] T. Inohara and K. W. Hipel, “Coalition analysis in the graph model for conflict resolution,” Systems Engineering, vol. 11, no.4, pp. 343-359, s ∈ Ri(s), (TRR(e)N\{i}(s ) \{s }) ∩{s} = ∅. Moreover, 2008. since we have, by Lemma 4, that TRR(e)N\{i}(s ) \{s }) [18] T. Inohara and K. W. Hipel, “Interrelationships among noncooperative = RN\{i}(s ), this is equivalent to for all s ∈ Ri(s), and coalition stability concepts,” Journal of Systems Science and Systems Engineering, vol. 17, no. 1, pp. 1-29, 2008. RN\{i}(s ) ∩{s} = ∅, which means that for all s ∈ Ri(s), [19] T. Inohara, K. W. Hipel, and S. Walker, “Conflict analysis approaches for s ∈ RN\{i}(s ), because s/∈ RN\{i}(s ). investigating attitudes and misperceptions in the War of 1812,” Journal of Corollary 10 (Corollary of Propositions 9 and 11): Systems Science and Systems Engineering, vol. 16, no. 2, pp. 181-201, 2007. Assume that DMs’ attitudes e =(ei)i∈N are totally neutral. RNash RSMR [20] T. Inohara and B. Nakano, “Properties of ‘soft’ games with mutual Then, Si = Si for all i ∈ N. exchange of inducement tactics,” Information and Systems Engineering, Corollary 11 (Corollary of Propositions 10 and 12): vol. 1, no. 2, pp. 131-148, 1995. [21] T. Inohara, S. Takahashi, and B. Nakano, “Credibility of information Assume that DMs’ attitudes e =(ei)i∈N are totally neutral. RGMR RSEQ in ‘soft’ games with interperception of emotions,” Applied Mathematics Then, Si = Si for all i ∈ N. and Computation, vol. 115, no. 2&3, pp. 23-41, 2000. [22] T. Inohara, S. Yousefi and K. W. Hipel, “Propositions on Interrelation- IV. CONCLUSIONS ships among Attitude-Based Stability Concepts,” 2008 IEEE Interna- tional Conference on Systems, Man and Cybernetics, Singapore, October This paper verified some inclusion relationships among 12-15, 2008, pp. 2502-2507. attitude-based and conventional stabilty concepts. One of the [23] D. M. Kilgour, K. W. Hipel, X. Peng, and L. Fang, “Coalition analysis topics to be considered in future research on attitude-based sta- in group decision support,” Group Decision and Negotiation, vol. 10, pp. 159-175, 2001. bility conceps is their interrelationships with coalition stability [24] D. Krech, R. S. Crutchfield, and E. L. Ballachey, Individual in Society: concepts [17], [18]. A Textbook of Social Psychology, New York, McGraw-Hill, 1962. [25] K. W. Li, K. W. Hipel, D. M. Kilgour, and L. Fang, “Preference uncer- ACKNOWLEDGMENT tainty in the graph model for conflict resolution,” IEEE Transactions on Systems, Man, and Cybernetics, Part A, vol. 34, pp. 507-520, 2004. A part of this work is supported by KAKENHI (19310097). [26] J. F. Nash, “Equilibrium points in n-person games,” Proceedings of National Academy of Sciences of the USA, vol. 36, pp. 48-49, 1950. REFERENCES [27] J. F. Nash, “Non-cooperative games,” Annals of Mathematics, vol. 54, pp. 286-295, 1951. [1] P. Bennett, “Confrontation analysis as a diagnostic tool,” European [28] M. J. Osborne and A. Rubinstein, A Course in Game Theory, The MIT Journal of Operational Research, vol. 109, pp. 465-482, 1998. Press, Cambridge, Massachusetts, 1994. [2] P. Bennett and N. Howard, ”Rationality, emotion and preference change [29] H. Sakakibara, N. Okada, and D. Nakase, “The application of robustness - drama theoretic model of choice,” European Journal of Operational analysis to the conflict with incomplete information,” IEEE Transactions Research, vol. 92, pp. 603-614, 1996. on Systems, Man, and Cybernetics, Part C, vol. 32, pp. 14-23, 2002. [3] J. W. Bryant, “Confrontations in health service management: Insights [30] H. F. Taylor, “Balance and change in the two-person group,” Sociometry, from drama theory,” European Journal of Operational Research, vol. 142, vol. 30, no. 3, pp. 262-279, 1967. no. 3, pp. 610-624, 2002. [31] H. F. Taylor, Balance in Small Groups, New York, Litton Educational [4] L. Fang, K. W. Hipel, and D. M. Kilgour, “Conflict models in graph Publishing Inc., 1970. form: Solution concepts and their interrelationships,” European Journal [32] S. Walker, K. W. Hipel, and T. Inohara, “Strategic Analysis of the Kyoto of Operational Research, vol. 41, pp. 86-100, 1989. Protocol,” 2007 IEEE International Conference on Systems, Man and [5] L. Fang, K. W. Hipel, and D. M. Kilgour, Interactive Decision Making: Cybernetics, Montreal, Canada, October 7-10, 2007, pp.1806-1811. The Graph Model for Conflict Resolution. Wiley, New York, 1993. [33] S. Walker, K. W. Hipel and T. Inohara, “Attitudes and Coalitions [6] N. M. Fraser and K. W. Hipel, “Solving complex conflicts,” IEEE within Brownfield Redevelopment Projects,” 2008 IEEE International Transactions on Systems, Man, and Cybernetics, vol. 9, pp. 805-816, Conference on Systems, Man and Cybernetics, Singapore, October 12- 1979. 15, 2008, pp. 2901-2906. [7] N. M. Fraser and K. W. Hipel, Conflict Analysis: Models and Resolu- [34] S. Yasui and T. Inohara, “A graph model of unanimous decision systems,” tions. North-Holland, New York, 1984. 2007 IEEE International Conference on Systems, Man and Cybernetics, [8] L. Hamouda, D. M. Kilgour, and K. W. Hipel, “Strength of preference Montreal, Canada, October 7-10, 2007, pp.1752-1757. in the Graph Model for Conflict Resolution,” Group Decision and [35] F. C. Zagare, “Limited-move equilibria in 2 × 2 games,” Theory and Negotiation, vol. 13, pp. 449-462, 2004. Decision, vol. 16, pp. 1-19, 1984. [9] N. Howard, Paradoxes of Rationality: Theory of Metagames and Political [36] D. Z. Zeng, L. Fang, K. W. Hipel, and D.M. Kilgour, “Policy stable Behaviour. MIT Press, Cambridge, Massachusetts, 1971. states in the graph model for conflict resolution,” Theory and Decision, [10] N. Howard, “ ‘Soft’ game theory,” Information and Decision Technolo- vol. 57, pp. 345-365, 2004. gies, vol. 16, no. 3, pp. 215-227, 1990. [37] D. Z. Zeng, L. Fang, K. W. Hipel, and D.M. Kilgour, “Generalized [11] N. Howard, “Drama theory and its relation to game theory. Part 1: metarationalities in the graph model for conflict resolution,” Discrete dramatic resolution vs. rational choice,” Group Decision and Negotiation, Applied Mathematics, vol. 154, pp. 2430-2443, 2006. vol. 3, pp. 187-206, 1994.

1161