There are two types of mathematical model in sociological study i.e. “Phenomenon Explanation Model” (= PEM) and “Social Architecture Understanding Model” (= SAUM). PEM aims to explain data of specific social phenomenon by describing the phenomenon as a mechanism. SAUM aims to understand part of social architecture as a mechanism, where the architecture is more abstract, extensive and lasting social phenomenon. Although the relation of PEM and SAUM is mutually complemented, some researchers avoid SAUM because they think it to be not grounded. But first, SAUM makes us understand society more deeply by discovering part of unobserved social architecture. Second, because of the understanding, SAUM contributes to our society through reconstructing it toward better directions. Therefore SAUM has the value of discovering and the value of ethic. In addition empirical rightness of the understanding is supported by the fact that “normative propositions” SAUM produced give us some kind of obligations. Normative propositions avoid SAUM being armchair theory. SAUMs are open to extensive interests and various uses because SAUMs clear mechanisms of social architecture that are more universal than mechanisms of specific phenomena. Then SAUMs have common language property. And Mathematics is probably a unique common language bridging various academic disciplines. Therefore the two common language properties make SAUMs worth exploring.

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This paper examines complementarity between "open relation" and "closed relation" in a repeated prisoners' dilemma game in which payoffs are stochastically determined. In the ``open relation'', each player is randomly matched with another one, and there is an opportunity for them to receive higher payoffs than in the ``closed relation''. However, because their relation is one-shot, their rational behaviors prevent them from the realization of cooperation. In the ``closed relation'', in contrast, we assume that paired players' relation is fixed and that both of them are committed to cooperative behaviors. But, their payoffs are fixed by definition so that there is no opportunity to get higher payoff staying in the relation. Hence, players in the ``closed relation'' incur higher opportunity cost. If only one of the two relations exists in society, the social efficient outcome is not attained. However, if both of them coexist in society and there is mobility between them, then the social outcome improves. This is because the "open relation" has a role as decreasing opportunity costs, the "closed relation" has a role as keeping cooperation, and the dynamics between them functions as endogenous pressures to improve payoffs. But in order to keep the endogenous pressures, it needs that the rate of random breaking of ``closed relation'' is sufficiently low.

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Diekmann and Preisendörfer (1998) showed that there exist substantial inconsistencies between individuals' environmental attitudes and their behaviors. They also identified three cognitive strategies that help individuals to harmonize and reconcile these seemingly incongruent behaviors and attitudes. However, they did not specify sufficiently the generative mechanism or process that leads individuals to these cognitive strategies, and their empirical analyses of social survey data are inadequate for testing their theoretical arguments. In this paper, first, we develop Diekmann and Preisendörfer's (1998) idea further and construct a two-stage model of decision-making and justification in a potential “social dilemma” situation, focusing on the problem of recycling. The model that we propose here is a coupling of the idea of rational choice and that of cognitive dissonance reduction. We deduce several propositions from our model and translate some of them into falsifiable or empirically testable predictions, which include those on the “false consensus effect,” the association between perceived “efficacy” and environmental behaviors, and the association between normative consciousness and the behaviors. The next step for us is to analyze social survey data in order to examine whether these predictions are supported or not.

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In Kimura and Shinoki (2007), after criticizing Diekmann and Preisendörfer's (1998; 2003) argument on the cognitive strategies to harmonize the discrepancy between attitudes and behaviors, we constructed a two-stage model of decision-making (rational choice) and justification (cognitive dissonance reduction) in a potential “social dilemma” situation, focusing on the problem of recycling. We deduced several propositions from our model and translated some of them into falsifiable predictions. In this paper, we analyze data from a social survey conducted in Sendai, Japan, in order to test these predictions: (1) On average, contributors' estimate of others' contribution is greater than non-contributors' estimate of others' contribution (the “false consensus effect”); (2) The proportion of the contributors who think that their marginal contribution plays a significant role in the well-functioning of the recycling system is greater than that of the non-contributors who think that their marginal contribution plays a significant role in the well-functioning of the recycling system; (3) On average, the normative consciousness concerning environmental problems is stronger among contributors than among non-contributors. We find that our predictions are almost supported. We also conduct a “critical experiment” in the natural settings on the mechanisms for the false consensus effect. The result reveals that our model is better than the selective exposure hypothesis in explaining it. We examine some possible objections to our model in order to consider what kind of research we need and to clarify the policy implication of our model.

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The purpose of this paper is to clarify social effects of mutual caring, by using game theory. We treat only 2x2 symmetric games that are twelve types in all. We define “way of caring” by “altruistic” and “egalitarian” parameters. We pick up thirteen ways of caring, for example altruism, competition, egalitarianism etc. Caring about the other transforms objective payoff structure into subjective payoff structure. As a result of analyzing this transformation we obtain following findings. (1) Almost all the payoff-structures can be transformed into all the payoff-structures in 2x2 symmetric games by changing the values of “altruistic” and “egalitarian” parameters. (2) It depends on game type whether a way of caring causes Pareto-inefficiency or not. For example, “poor loser” in other words the anti-altruistic and egalitarian causes Pareto-inefficiency in prisoners' dilemmas and chicken games. (3) The most frequent about causing Pareto-inefficiency are “competition” and “sacrifice” (7 games in 12 games). The second most frequent is “anti-egalitarianism” (6 games). On the other hand “egalitarianism” is less frequent (2 games), so that “egalitarianism” is good not only in equality but also in Pareto-efficiency. In fact both “egoism” and “altruism” are least frequent (1 game) except “maxmin”, “maxmax”, and “sum-ranking” (0 games). We clarify the ranking of “ways of caring” about causing Pareto-inefficiency in 2x2 symmetric games.

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There are a lot of literatures about Prisoner's dilemma game, because the game is a simple model to describe social dilemma situation. As many literatures conclude, if there is long-term relationship among players, then they can cooperate in the Prisoner's dilemma game. However, the result depends on the assumption that they are forced to keep relationship with their opponents. In the real world, they can change partners for games and this fact maybe harmful for cooperation in the game. Therefore, we study the game in which they can change partners. We got the result that cooperation can emerge from the initial setting that everybody defects, if the distribution of the available partners changes.

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