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On the Complementarity of Expectations: Coupling Parsons with Balance Theory Kazuto Misumi Graduate School of Social and Cultural Studies, Kyushu University 4-2-1 Ropponmatsu, Chuo-ku, Fukuoka 810-8560 Japan [email protected] Abstract When we are to introduce the interpretative dynamic role process against 'homo sociologicus,' we are confronted with the problem how to capture the various reality systematically. By coupling the 'complementarity of expectations' with balance theory, we formally propose a cognitive framework to analyze the double contingency role relationship. Through conceptual discussions, we introduce two basic signed graphs: the unit graph T that presents the in-personal role regulation and the double contingency graph G in which two unit graphs are connected by the two inter-unit lines. While the signs of inter-unit lines, that inter-personally connect role- expectations, determine the complementarity in G, the signs of G's cycles determine the balance of G. With regard to G's balance, two theorems are derived. The first states that T's imbalance is the sufficient condition of G's imbalance, and the second states that one positive cycle in G larger than 3-cycle is the sufficient condition of the balance of G that is composed of balanced T. Utilizing these theorems, we systematically classify the role relationship represented by G, and discuss about the substantial meaning that some paradoxical cases imply. Key words and phrases: Complementarity of expectations, Balance theory, Role 1. Introduction Parsons and Shils (1951:15) proposed the concept of complementarity of expectations "not in the sense that the expectations of the two actors with regard to each other's action are identical, but in the sense that the action of each is oriented to the expectations of the other." In addition, "in an integrated system, this orientation to the expectations of the other is reciprocal or complementary" (Parsons and Shils 1951:105). In the framework of social system, a key concept that mediates between individual and society is role- expectation. The logic is fundamentally parallel. Following Parsons and Shils (1951:19- 20) again, "once an organized system of interaction between ego and alter becomes stabilized, they build up reciprocal expectations of each other's action and attitudes which are the nucleus of what may be called role-expectations. .... The pattern of expectations of many alters .... constitutes in a social system the institutionalized definition of ego's roles in specified interactive situations." These descriptions evoke a paradox discussed by Dahrendorf (1968:25): "At this point where individual and society intersect stands homo sociologicus, man as the bearer of socially predetermined roles. To a sociologist the individual is his social roles, but these -1- roles, for their part, are the vexatious fact of society." In other words, by constructing 'homo sociologicus,' sociology might develop a rational understanding of society; however, it has excluded as a result individuality and freedom of the human being. Against homo sociologicus, some researchers see the dialectical possibility of liberation from the viewpoint of real actions (Tenburuck 1961, Berger and Pullberg 1965, Yamaguchi 1975, Kurioka 1980, Tanaka 1984, Nomura 1987). strong assumptions of homo sociologicus. Others point out unrealistic and too For example, Habermas (1973; also see Nomura 1982: 240-245) asserts that homo sociologicus shall satisfy following three theorems: 1) Integration theorem (high degree of the complementarity of expectations) 2) Identity theorem (coincidence between role definition and role interpretation) 3) Conformity theorem (realization of normative contents into actions) Analogously, Levinson (1959) and Morris (1971) assert that homo sociologicus requires unrealistic high level coincidence among three aspects; role-expectations, role-conception, and role-behavior. These assertions imply that the complementarity never be rigid and stable in the real world; rather, it shall be inherently flexible and unstable, being founded on interpretative processes. Although temporary stable interaction may appear through ritual actions and role-playing (Goffman 1959,1967), the complementarity is no longer a sufficient (even a necessary) condition for the stability. Our standpoint is between the extremes. As Parsonians assert, the institutionalization shall be a basic general process and, resulting in the complementarity to some extent, it conditions the stability of interaction. However, in the real world, not only the process, but also the other interpretative processes condition the stability as anti-Parsonians assert. Among those that are mostly irregular, we focus on a relatively general process, the cognitive balance. As we will see later, the balance theoretical perspective has primitively presented in role theory, but the tradition has stopped before formalization.1) The purpose of this paper is to develop a graphical model that takes in the viewpoint of complementarity and balance at the same time, and through it, to examine conditions for the stability of role relationship from multiple angles. 2. Conceptual Framework Before going forward formal analysis, we introduce our conceptual framework of role. As already cited, some role theorists developed three concepts (Levinson 1959, Morris 1971, Funatsu 1976:187-188, Watanabe 1981): ・Role-expectations .... Normative expectations from alter with regard to ego's action. (Its referent point is functional requirements of a social system.) ・Role-conception .... Ego's normative expectation with regard to action of him/herself. (Its referent point is functional requirements of a personality system.) ・Role-behavior .... Ego's action actually taken in interaction. Following Watanabe (1981: 114), we can consider the relationship among the three. ・Role-consensus (⇔role-disensus) .... Compatibility between role-expectations and role- -2- conception in regard to normative contents. ・Role-adaptation (⇔role-deviation) .... Compatibility between normative contents of roleexpectations and role-behavior. ・Role-identification (⇔role-distance) .... Compatibility between normative contents of role-conception and role-behavior. Morris (1971) represented the relationship as a triple graph, and Watanabe (1981)discussed the substantial meaning of each compatibility pattern. compatible and " － " be incompatible. adjustment process. Let "+" be Suppose that the sign indicates the result of As three graphs that have only one negative line carry logical problems, the following five are valid (Figure 1). Figure 1. Triple Graphs in Morris=Watanabe Framework This graphic presentation is of great help to classify the results of actor's internal role regulation, although some questions arise. Suppose that an actor whose state is a) and another whose state is ß) encounter. Then, certain inter-personal adjustment should occur between them; however, in this framework, we can neither describe the process, nor judge whether the complementarity is satisfied or not between them. Moreover, the state of a) could represent 'complementarity,' but only when the same role-expectations are shared by alter. In order to take these interactive viewpoints into the framework, two triple graphs shall be linked. Kobayashi (1983), after defining 'role-knowledge' as the typification knowledge based on social expectations with regard to actions, insists that it exists not only as 'stock,' but also as 'flow.' The knowledge as flow means messages of expectations that are exchanged between actors through role-behavior. In the sense, role-behavior is an output from ego's role-regulation and, at the same time, an input to alter ego's role-regulation. As role- interaction concerns plural roles that make a pair, the input is interpreted by alter as a message with regard to the role on the alter ego's side. When ego decides role-behavior to be taken, he/she anticipates what expectation it should reciprocally indicate toward the role of alter. In other words, ego anticipates how his/her role-behavior will be interpreted and, as a result, what reaction will be taken by alter. This is the point when Parsons says, "the action of each is oriented to the expectations of the other." 2) Focusing on this aspect of role-behavior, we refine the triple graphs as in Figure 2. We replace 'role-behavior' with 'role-expectation toward the other,' and designate actor i's roleexpectation toward the other actor j as Eij. We also replace original 'role-expectation' with 'role-expectation toward self,' and designate actor i's role-expectation toward self as Eii. -3- 'Role-conception' is the same; however, as it is strictly 'role-conception of own role,' we designate actor i's role-conception as Cii. This refinement makes the graph to describe consistently the cognitive relationships. More importantly, it clarifies that the role regulation is not only on one role, but also on (at least) two roles that make a pair. While the point of the former is the compatibility between Eii and Cii in the sense of coincidence of normative contents, that of the latter is the anticipated reciprocity between Eii and Eij, as well as between Cii and Eij. Figure 2. Triple Graphs in Our Framework With regard to a'), 'conformity' is better than 'complementarity' because it describes only in-personal harmonious state. as Figure 1. The meaning of the other types can be interpreted as same Suppose that there are two normative contents, 'working' and 'keeping house,' that are not compatible with each other, but are reciprocal. For a husband (actor 1), if both E11 and C11 are 'working' in regard to the role of 'husband,' the line E11-C11 is positive (compatible). Moreover, if his E12 toward his wife (actor 2) is 'keeping house' in regard to the role of 'wife,' both E11-E12 and C11-E12 are positive (reciprocal). We can judge that this cognitive state is a') in Figure 2. However, if his C11 is 'keeping house,' not only the compatibility between E11 and C11, but also the reciprocity between C11 and E12, is broken. Then, both E11-C11 and C11-E12 turn negative, so that we can judge the state as d'), and so on. Notice that our purpose at present is not to describe and predict what role-behavior will be actually taken by ego and whether or not alter ego will react against it reciprocally. Rather, given a series of concrete interaction, we are interested in finding a cognitive state that could give guidance to it, and therefore explains the observation consistently. Suppose that we observed a husband to work till late and his wife to welcome him preparing dinner everyday, moreover, we could know they had felt no difficulty in the situation. the relationship among Eii, Cii, and Eij shall be, that is our question. Then, how More generally, it is our purpose to examine under what cognitive states, interaction shall be complementary and stable, whatever the actual behavioral contents. Figure 2 will answer it, but only partially. In order to get a complete answer, as we already suggested, two triple graphs shall be linked together. It is plausible to assume that they are linked by two lines between Eii and Eji (i,j=1,2 and i ≠ j), and that the 'complementarity of expectations' directly depends on the compatibility between the expectations, that is, whether both Eii-Eji lines are positive or not. 3) -4- According to the extended framework, a wife (actor 2) is assumed to receive E12 reciprocally through her husband's (actor 1's) role-behavior and to check it with E22. for her husband (actor 1). The reverse process is assumed Notice again that, at present, the complementarity is judged from a third party's viewpoint. In the previous example, it is neither the wife nor her husband, but a third party (a researcher), to judge the sign of E12-E22, even though the former themselves might be in fact able to do it. 4) Figure 3 summarizes the conceptual framework that we have finally proposed, and Figure 4 is its simplified graphical representation. This framework represents the general cognitive relationship between two actors, but only with regard to a focused aspect of a role-relation. Generally, not only that plural role-relations (e.g. husband-wife and doctor- nurse) sometimes overlap, but also that each relation contains plural pairs of normative contents that are essentially incompatible with each other, but reciprocal (e.g. 'working''keeping house,' 'working'-'bringing up children,' and 'representing family'-'acting behind,' in husband-wife relation). In a real process of role-interaction, such relations and contents might change one after another, so that a graph in Figure 4 shall be connected to another in succession, making an infinite chain graph as a whole. However, if we focus only on one pair of contents in one role-relation, the process circles on a graph in Figure 4, as far as each actor never change his/her role-expectations and role-conception. The framework, as well as the graph, only captures a limited unchangeable aspect as such. Figure 3. Conceptual Framework of the Cognitive Role Relationship Figure 4. Graph of the Cognitive Role Relationship -5- We might have to draw one more line between E11 and E22 to check the reciprocity between them (i.e. institutionalization). However, logically thinking, we can judge that the institutionalization is successful only if all lines are positive in either of Eii-Eij-Ejj (i,j=1,2 and i≠j). 3. Formalization by Balance Theory Figure 2 and Figure 4 call up the discussions of cognitive organization by Heider (1946) and its extended formalization by Cartwright and Harary (1956), that is, balance theory. By introducing the concept of balance, we can have another analytical viewpoint that is different from the complementarity itself in order to examine the stability (or instability) of role relationship. The following assumptions and definitions are required for graphical formalization of our framework developed in the preceding section. Assumption 1. The in-personal role regulation of actor i is represented by the triple relationship among three aspects: role-expectation toward self (Eii), role-conception (Cii), and role-expectation toward the other actor j (Eij). (i,j=1,2 and i≠j.) Each relation between them is either positive or negative. Assumption 2. Actor i who received Eji from the other checks it with his/her own Eii. The relation between them is either positive or negative. Definition 1. 'Unit graph,' T, is an undirected signed triple graph that connects three nodes: Eii, Cii, and Eij. (See Figure 2). Actor i's unit graph is designated by Ti. Definition 2. 'Double contingency graph,' G, is an undirected signed graph in which two unit graphs are connected by two lines between Eij and Ejj. (See Figure 4). We call the lines 'inter-unit lines.' Definition 3. If both signs of the inter-unit lines are positive, the complementarity of expectations is satisfied. Additionally, we confirm some graphical terminologies and an assumption with regard to the stability/instability of G. Definition 4. A cycle is a path (sequence of lines) that returns to its node of origin without passing the same node or line twice. The sign of a cycle equals the product of the signs of the lines it contains. Definition 5. G (or its sub-graph, T) is balanced if all cycles in G (or T) are positive, and is imbalanced, otherwise. Assumption 3. When G is imbalanced, there occurs some pressure toward a balanced state. In that sense, imbalanced G is unstable, although balanced G is stable. By coupling logically the complementarity with balance, we get a typification of the role relationship. Our next task is to examine balance of G in each type. (Ⅰ) Perfect complementarity: The signs of all lines in G are positive. -6- It is always stable. (Ⅱ) Temporary complementarity: Both of the inter-unit lines are positive, but Ti contains at least one negative line. (Ⅱ-1) Stable －: G is balanced. (Ⅱ-2) Unstable －: G is imbalanced. (Ⅲ) Half broken complementarity: Only either of the inter-unit lines is negative. (Ⅲ-1) Stable －: G is balanced. (Ⅲ-2) Unstable －: G is imbalanced. (Ⅳ) Broken complementarity: Both of the inter-unit lines are negative. (Ⅳ-1) Stable －: G is balanced. (Ⅳ-2) Unstable －: G is imbalanced. 4. Theorems for the Balance of G At first, it is noticed that imbalance of G is determined by imbalance of T, regardless of the role relationship type. This is almost self-evident through Definition 5; however, we confirm it as a theorem. Theorem 1. In G, if both or at least either of Ti is imbalanced, G is imbalanced. ◇Proof： A unit graph T has only a 3-cycle (call it c), and as T is a sub-graph of G, c is also a cycle of G. If one of Ti is imbalanced, then c must be negative through Definition 5; so that, G necessarily has one negative 3-cycle and therefore is imbalanced. ◇ In Figure 2, imbalanced T is only e), a dissolved case where all three lines are negative, although Theorem 1 holds even for cases in which only one line is negative. Anyway, if either of the in-personal state of actors (represented by Ti) is dissolved, the dyadic role relationship (represented by G) is imbalanced and unstable. This is true, not only for types (Ⅲ-2) and (Ⅳ-2) where the complementarity is broken, but also for type (Ⅱ-2)where it is satisfied temporarily. When both of Ti are balanced, balance judgment of G seems not so simple. practically examine G that are possible in each role relationship type. Let us It is apparent that G is balanced if both actors have a), 'conformity' (i.e. for type [Ⅰ]), because all lines (therefore all cycles) in G are positive. Graphically, this is a special case of the 'positive symmetry' where both inter-unit lines are positive. In Figure 2, balanced T is exhaustively covered by the first four, a) through d). As both inter-unit lines are fixed as positive under the positive symmetry condition, possible G is given by 16 combinations between Ti as in Table 1. Here, a positive line is indicated by a solid line, and a negative one by a broken line. Also, balanced G is marked by ○, and imbalanced G by ×. balanced. Table 1 shows that 8 out of 16 are These are type (Ⅱ-1), and its special case for type (Ⅰ) (upper-left corner of Table 1). Table 2 is for type (Ⅳ), broken complementarity. Type (Ⅳ) is understood as the opposite of type (Ⅰ) and (Ⅱ); however, graphically, it is also symmetric in a sense that -7- Table 1. Balance of G Composed of Balanced Ti : Positive Symmetry Condition *) Figures are case number. ○ is balanced, and × is imbalanced. A solid line is positive, and a broken line is negative. (And so on for the following tables.) 'Positive symmetry' means that both inter-unit lines are positive. Table 2. Balance of G Composed of Balanced Ti : Negative Symmetry Condition *) 'Negative symmetry' means that both inter-unit lines are negative. Table 3. Balance of G Composed of Balanced Ti : Asymmetry Condition *) 'Asymmetry' means that the signs of inter-unit lines are opposite to each other. The cases in which positive inter-unit line is lower are omitted, as the balance judgment is the same. -8- both inter-unit lines are negative. In fact, in Table 2, balanced cases (type [Ⅳ-1]) show the same combination pattern as Table 1 even under this 'negative symmetry' condition. (Needless to say, this does not mean that the graphs are identical in the corresponding combination.) From the viewpoint of balance, the opposite of type (Ⅰ) and (Ⅱ), as well as of (Ⅳ), is type (Ⅲ), half broken complementarity. Type (Ⅲ) is a unique case of 'asymmetry' where the signs of inter-unit lines are different from each other. It is easy to confirm that, in Table 3, balanced combinations (type [Ⅲ-1]) reveals the reversed pattern. Thus, the 'symmetry' seems critical for balance of G that is composed of balanced unit graphs. Balance of G depends on the signs of its cycles, and the signs depend on the number of negative lines. Here, as the number is fixed as even in T, G' s balance depends on the signs of larger cycles, which depend on the number of negative inter-unit lines. Of course, the symmetry means that the number of negative inter-unit lines is 0 or 2 (even), and the asymmetry means it is 1 (odd). The idea of symmetry/asymmetry is of help to regulate the relationship between the types. Especially, it is suggestive that the types which are substantially located oppositely have the same pattern of balanced combinations, formally. However, as balance of G is not determined only by the signs of inter-unit lines, this idea is not enough to discriminate balanced cases in each of the combination tables. More careful examination of the tables leads next theorem that makes balance judgment very easy. Theorem 2. In G where both of Ti are balanced, all cycles other than T's 3-cycle have the same sign. Which means that the existence of only one positive cycle larger than or equal to 4-cycle is the necessary and sufficient condition for the balance of G. Proof：Generally, in G, there are following four cycles other than T. Let us examine these graphs under the symmetry condition where the signs of two inter-unit lines, E12-E22 and E21-E11, are identical. Suppose that cycle a) is positive, being based on the identical sign between C11-E11 and C11-E12. This implies as a rule that E21-E22 as well as E12-E11 must be positive, and that E21-C22 and C22-E22 must have the identical sign. Thus, the sign of each line in G is automatically almost determined as graph e) below. In this graph, apparently, cycles b)～d) are all positive, that is, they have the same sign as cycle a). Next, suppose that cycle a) is positive, but either of C11-E11 or C11-E12 is negative. This implies that E21-E22 as well as E12-E11 must be negative, and that either E21-C22 or C22-E22 must be negative, which results in graph f). Again, cycles b)～d) have the same positive sign as cycle a). -9- Such logic comes from a fact that one of the three lines in each T cannot take the sign freely because T is fixed as balanced, here. Therefore, when cycle a) is negative, through the same logic, all the four cycles must have the same negative sign, too. Moreover, even when we start from another cycle other than a), and even under the asymmetry condition, the same logic is applicable as long as T is fixed as balanced.◇ It is apparent that, through Definition 5, finding only one negative cycle is enough for the judgment of G's imbalance. Similarly, Theorem 2 guarantees that finding only one positive cycle (larger than 3-cycle) is enough for the judgment of G's balance. These together result that the sign of only one cycle (larger than 3-cycle) is critical for the judgment of balance/imbalance of G. 5. Implications The substantial implication of Theorem 1 is relatively clear. As is already mentioned, we can say that if either of the actors' in-personal states is dissolved, the dyadic role relationship is imbalanced and unstable regardless of the complementarity. Reversely speaking, it is not possible that dyadic role relationship is stable although either or both of the actors are unstable in-personally, even if the relationship is complementary. On the other hand, Theorem 2 covers the cases where both actors are in-personally stable. Some of them are paradoxical in the sense that G is complementary but unstable, or inversely, G is not complementary but stable. Theorem 2 implies that all the paradoxical cases (as well as all the other cases) can be consistently explained based on just one cycle in G. As we already saw in the proof of Theorem 2, there are possibly four such cycles. If we pull out one out of them through certain reasonable assumption, we could consistently and generally explain the double contingency situations based on just two cycles in G, that is, T and the extracted one. We have noticed that our framework stands on a viewpoint of the third party. It is apparent that, for a researcher who observes role-interaction and plans to analyze it through our framework, the minimum cycle is the best as far as it guarantees the same balance judgment as larger ones. Needless to say, a cycle that satisfies the condition is c)in the proof of Theorem 2. We call this cycle the 'minimum contingency cycle' (MCC) and we confirm its validity as the balance standard in the following corollary. Corollary of Theorem 2. In G, the minimum contingency cycle (MCC) is a cycle, by the sign of which balance/imbalance of G (composed of balanced Ti) is determined. - 10 - The sign of MCC depends on two components: the signs of inter-unit lines, Eij-Ejj, and the signs of Eii-Eij (for i,j=1,2 and i≠j). Coupling the two components with each other, as in Table 4, we can compactly regulate all the combinations in Table 1 through Table 3, keeping correspondence to the typification of role relationship. Now, we go back to the paradoxical cases previously mentioned. where G is complementary but unstable. Table 4. The first is type (Ⅱ-2) In this type, the signs of Eii-E ij must be Complementarity and Stability of G Composed of Balanced Ti different from each other, therefore the MCC takes a form: This MCC presents that, while the complementarity is kept, role-expectations toward self and toward the other are not reciprocal on either side. It is also noticed that E11 and E22 are reciprocal because all lines are positive in either E11-E12-E22 or E22-E21-E11 (This is an institutionalized situation, in that sense.) Suppose again two normative contents, 'working' and 'keeping house,' that are not compatible with each other, but reciprocal. It is impossible in the MCC above to arrange them with no logical contradiction. For example, under the condition of negative E11-E12 and positive E22-E21, if E11 is 'working,' E12 and then E22 shall be 'working,' however, as E21 shall be also 'working,' E22 and E21 never be reciprocal (E22-E21 never be positive). It is probable that this case is actually observed with no contradiction, only if it is possible to devise a kind of 'quasi-category' that satisfies the compatibility and reciprocity at the same time, between either of original categories. For example, 'part-time working' and '(full- time) working' can be understood as compatible with each other and reciprocal, at the same time. Similarly 'keeping house' could be replaced by 'part-time housework.' Another point of this case is that role-expectation toward the other never be realized either in half or at all, even if contradictions could be avoided by utilizing a quasi-category. Suppose that actor 1 (husband) is 'rebellion' (see Figure 2), therefore E11-E12 is negative. - 11 - He rejects '(full-time) working' (E11) and rebelliously expects his wife to perform it (E12). Actor 2 (wife) interprets the expectation as 'part-time working' in a compatible range of 'working.' Here, E12 will be realized, but only in half, because the wife will not actually engage in full-time work. On the contrary, the wife reciprocally expects her husband to perform 'full-time working' (E21), but her expectation will not be realized at all because her husband rejects it. Thus, instability of this case implies mutual dissatisfaction. We can understand this paradoxical case as the unsatisfied complementarity kept by quasi-categories. The next paradoxical case is type (Ⅳ-1) where G is not complementary but stable. In this type, the signs of Eii-Eij must be identical, and the MCC takes a form: When both Eii-Eij are positive, this MCC presents that role-expectation toward self and toward the other are reciprocal in each in-personal state, even though the complementarity is broken. This reciprocity may come from 'conformity,' or may be just role-playing founded on 'suppression.' Anyway, as E11 and E22 are not reciprocal (a negative line is included both in E11-E12-E22 and E22-E21-E11), the actors are conformable, but only in-personally. A probable situation is that both actor 1 (husband) and actor 2 (wife) have 'working' for Eii, and, obeying it, expect to perform 'keeping house' each other. exactly founded on the separated relationship. The stability of this case is Their expectations completely keep missing each other; however, they keep their social identities, respectively. Thus, we can call this case, which is probable in multi-cultural situations, the stable separation founded on social identities. When both Eii-Eij are negative, all relations in the MCC are broken, and E11 and E22 are neither reciprocal (one and more negative lines are included in both E11-E12-E22 and E22-E21-E11). If both actors are 'rebellion,' a probable situation is that actor 1 (husband), rejecting 'working' (E11), rebelliously expects his wife to perform it; on the contrary, actor 2 (wife), rejecting 'keeping house' (E22), rebelliously expects her husband to perform it. (Though E11 and E22 are apparently reciprocal, it is not a result of institutionalization, but an accidental result through the role regulation.) As same as the previous case, the separated relationship brings about the stability; however, in this more disordered case, the separation is founded on personal identity rather than social identity. In fact, each actor keeps his/her isolated but unshakable internal world irrespective of expectations from the society and his/her partner. In the sense, this case is the stable separation founded on personal identities. On the other hand, if both actors are 'impracticability,' the situation implies the existence of external obstacles to role performance. For example, both E11 and C11 are 'working' for actor 1 (husband), but some obstacle (e.g. unemployment) not only prevents - 12 - him from performing it, but also makes his E12 'working' unwillingly. Similarly his wife neither be able to perform 'keeping house' following E22 and C22, and she inevitably shows 'keeping house' for E21. The stability of this case clings to the internal consistency between Eii and Cii in each actor. We can call the case the stable but inevitable separation caused by external obstacles. Likewise, the other situations are also able to be explained thoroughly based on T and the MCC. Having this compact analytical framework, we could systematically find not only the graphical regularity among various G, but also the comprehensive meaning underlying identical cases from the viewpoint of complementarity and balance. 6. Concluding Remarks The discussions of social system and homo sociologicus make sense to understand the mechanism of stabilization of a society. On the other hand, the criticism against them also makes sense to understand the dynamic reality which is going on even under the stability. An important theoretical problem is that, as soon as we introduce the latter standpoint and say that the complementarity of expectations shall be seen as inherently flexible and unstable, we are confronted with the infinitely various world. We believe that the previous analysis could suggest a possibility to grasp such world systematically and to link the two theoretical streams that have confronted each other. We have considered the role relationship as a cognitive framework in which actual role interaction is explained from a viewpoint of the third party. In order that we stand on a viewpoint of the actor and follow actual interaction under the framework, some specific problems must be resolved (see also note 4]). The cognitive instability of G due to its imbalance probably reduces an actor's utility, therefore influences interaction. On the other hand, irrational interaction will not be interrupted, but will be continued through the revision of role relationship toward balance. It is open to further discussion whether we see the aspect in a unified utility formation process or in a dual framework of action. Additionally, we will be confronted with the problem how to compare objective payoff with mental comfort and distress. With regard to the cognitive revision, the dynamic process of G's change is also our further subject. From a formal point of view, we will be able to utilize the 'line index' of balance to examine which G is easier to be transformed into which. Perhaps, it is also required to investigate the relationship between our model and 'expectation states theory' (Berger et al. 1974,1977; Fararo and Skvoretz 1986; also see note 1]). 5) Our graphical formulation can easily be extended to triad and, possibly, to a general interaction system that is composed of plural actors. Based on the extended G, as exemplified in Figure 5, we will be able to re-examine Simmel's discussion, the concept of role-set, and empirical findings in sociology of family as well as in ethnomethodology and symbolic interactionism. - 13 - Figure 5. Examples of Extended Triad Graph There have been so many graphical approaches to role theory (especially in social network researches); however, most of the discussions have been concentrated on the 'structural equivalence' and on objective role-relations. We hope that this paper stimulates another graphical approach that incorporates the process of in-personal role regulation into the inter-personal double contingency relations. Acknowledgements An earlier version of this paper was presented at European Japanese Conference on Rational Choice and Formalization (Leipzig, Oct. 2001) and 102nd Conference of Japan Sociological Association for Social Analysis (Tokuyama, Dec. 2001). I appreciate all the productive comments. Notes 1) 'Expectation states theory' developed by Berger et al. (1974,1977) should be seen as a formal extension of this tradition. It is directly concerned with organization of status characteristics through task performance and evaluation in a task-oriented group; however, 'task' may be analogously replaced by 'role.' The theory also includes balance theoretical formulation. It is expected that our formulation, that focuses on dyad and actors' internal states as well, can be linked together. 2) According to Kobayashi, the process consists of two aspects: 1) ego refers to role knowledge as an interpretative code in expecting alter ego's reaction, and 2) ego refers to it as a code switch between social role-expectations and role-conception of him/herself. (Also see the discussion on 'relevance' by Schutz [1970], and the distinction between 'logic-in-use' and 'reconstructed logic' by Fukazawa [1990, 1994]). In the conceptual framework of Watanabe (1981:111), on which we developed our framework in Figure 2, the latter aspect was incorporated as 'role negotiation process'; however, the former was neglected. Our refinement is capturing the aspect. 3) In this paper, the term 'complementarity' is limited to this inter-personal aspect, and is distinguished from the 'anticipated reciprocity' in the individual recognition previously mentioned. be synonymous with the anticipated 'complementarity.' The latter should However, through this conceptual distinction, we can clearly describe those unsuccessfully institutionalized cases as the broken complementarity, in which Eii-Eij is positive, but Eij-Ejj is negative. 4) If we stand on the actors' viewpoint, we have to consider even judgment discrepancies between them. Such more complex situations where directed graphs are differently defined for the same role relationship is within our scope of analysis, but in the future. 5) Also see Misumi (1991) for a graphical analysis of in-personal change of the role knowledge. - 14 - The process of cognitive revision should be consistently related to the mechanism that determines compatibility (or reciprocity) between the nodes in G. A Boolean role model developed by Misumi (2001, 2002) is suggestive on that point. References Berger, Peter and Stanley. Pullberg 1965. "Reification and the sociological critique of consciousness." History and Theory 9(2): 196-211. Berger, Joseph, M. Hamit Fisek, Robert Z. Norman, and Morris Zelditch,Jr. 1977. Status Characteristics and Social Interaction: An Expectation-States Approach. New York: Elsevier. Berger, Joseph, Thomas L. Conner, and M. Hamit Fisek (eds.) 1974. Expectation States Theory: A Theoretical Research Program. Washinton,D.C.: Winthrop Publishers. 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