This paper, founding on the discussions by S. F. Nadel, intends to explicate the role concept using digraphs. Nadel captured the role, focusing on its internal structure, as the hierarchy including 3 grades (that is, pivotal, relevant, and peripheral). We can formulate this internal structure of role by simple digraphs, if we translate Nadel's “attributes” as “nodes” and “implicitness” as “arcs”, furthermore if we permit the procedure of condensation by strong components. Acquired these role digraphs, innumerable non-role digraphs can be classified into three types, and finally three patterns of structural change of roles will be taken out.

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     This paper tries to compose a model of an imaginary society which have a certain rules of resource acquisition. In the first, we describe the structure of this society and in the second, we consider the stability of the resource distribution pattern which is formed at this society.
     Quantity of resources that an actor acquisite in this imaginary society is prescribed by the order of resource acquisition. But, it is supposed that the resource distribution can be changed locally by a lower rank actor's ‹fight› to an higher rank actor. And also, it is supposed that such a local resource redistribution may spread to the whole society.
     We derivate the following results from this model. The condition of a local resource redistribution changes regularly according to the distance from actors who change their resource distribution first. The influence of a resource distribution change is limited to a comparatively small range. The way of the spread to a lower rank actor is different from the way of the spread to a higher rank actor and it's pattern changes according to the value of parameters in the model.

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     In this paper, we investigate and formalise self-placement and distribution of self-placements in a class system, and develop mathematical models according to Fararo's axioms. The following points are our findings.
     (1) In image of stratifications, an actor perceives his subjective class as a minority.(2) As dimensions and ranks of stratification structure increase, his subjective class is perceived as a minority.(3) The distribution curve of self-placements is symmetrical and opens downwards.(4) As dimensions increase, the distribution curve concentrates toward center.(5) Extending Fararo's axioms, we generalize scanning processes.(6) We develop a mathematical model of the reversal of comparative positions, and find paradoxical relationship between objective class and subjective class identification.

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     Some relationship among the individual choice competence, the number of individuals in the group, its decision rule and the collective choice competence was initially investigated in Condorcet's Jury Theorem. Although several researchers has developed the theorem in different directions, they in common assume implicitly one collective choice area and introduce only the individual choice competence for the area. In reality, however, there may exist multiple collective choice areas such as ‘management strategy and staff distribution’ and the individual choice competencies may be different for the corresponding area such that ‘he/ she is good at management strategy but not staff distribution’. Under these existence and difference, how does the group assign what members in it to what collective choice area? This “group decomposition problem” can not be treated within the collective choice models under Jury Theorem and its developed results because one collective choice area and one individual choice competence for it are assumed. This paper extends the basic model under Jury Theorem to the model of two collective choice are as and formulates in general the group two-decomposition problem. It also gives a solution to this problem under the complete homogeneity assumption that all of the members in the group have the same competence of individual choice for all of the collective choice areas. Finally, some unsolved problems are showed.

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     The purpose of this paper is to formalize Merton's famous idea of the self-fulfilling prophecy. First, we will check the insufficient points in his discussion, and clarify the vicious circles which are characteristic of the self-fulfilling prophecies. Then we will construct a mathematical model to specify the mechanism of the self-fulfilling prophecy. From this formal analysis we can suggest five different situations under which a prophecy leads to its own fulfillment in the case of racial conflict. In particular, we can point to the possibility that even if “the feelings of whites toward Negroes are favorable” and “Negroes respect the traditions of trade unionism,” the prophecy may successfully lead to its own fulfillment. We cannot understand clearly the mechanism of the self-fulfilling prophecy, until we undertake such a formalization thoroughly and completely and surpass the Mertonian “theory of middle range.”

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     We classify the past theories about “authority” into “basis outside society” theory, “basis inside society” theory and “self-sustain” theory. But these three theories have each fault. We try to investigate “authority” by referring to Frame Problem. Firstly we survey “Frame-Problem” and define “Generalized Frame-Problem” according to Matsubara. Secondly we confirm that “Generalized Frame-Problem” cannot be soluted by man or artificial intelligence. Therefore, we should investigate “false solution” of “Generalized Frame-Problem”.
     Finaly we present our model. We assume a hypothesis of “norm-anxiety”. If we suppose this hypothesis, probability that “authority-bubble” (area of reliability of an authority) comes about increases. We assert that this “authority-bubble” model can be a solution of a problem about authority and that this model can be an answer to the question how people false-solute Generalized Frame-Problem.

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