≉㞟ㄽᩥ Does Mathematical Contribute to the Progress of Sociology?

Yoshimichi SATO Graduate School of Arts and Letters, Tohoku University Sendai, Miyagi 980-8576, JAPAN

Abstract I propose that mathematical sociology should seek to formalize and explain the differentiation of roles in order for it to prevail in sociology. Mathematical sociology has established a niche within its parent discipline by demonstrating its explanatory power in such areas as the study of cooperation, , and social networks. However, it has not seriously examined the emergence, maintenance, and collapse of social order, a core sociological concept. Because social order is an abstract concept, I focus on in this paper, role structure in particular, and propose an agent-based modeling framework to explain the differentiation of roles. If mathematical sociologists successfully develop models along these lines, they will succeed in explaining social order, and, as a result, mathematical sociology will proliferate in the .

Key words and phrases: social order, role, agent-based model

1. Proliferation of Mathematical Sociology in Sociology?

Mathematical sociology seems to be flourishing in sociology. Conventional and evolutionary game theory have been applied to the study of the emergence of cooperation in a Prisoner’s Dilemma game (e.g., Axelrod 1984). analysis has been widely used in such fields as (e.g., Buskens and van de Rijt 2008). Agent-based modeling is also prevalent in mathematical sociology and is frequently used in the study of emergent social properties such as trust (e.g., Macy and Skvoretz 1998). However, it is another question as to whether mathematical sociology prevails in sociology by contributing to the progress of the discipline. My judgment of the current status of mathematical sociology is rather pessimistic. Mathematical sociology has established a niche in sociology, but has not prevailed in the field at large. Concretely speaking, the membership of the Japanese Association for Mathematical Sociology is around 300, while the Japan Sociological has more than 3600 members. Most of the articles published in Japanese Sociological Review, the official journal of the Japan Sociological Society, are not related to the mathematical branch of the field.

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Saburo Yasuda, a leading mathematical sociologist in Japan in the 1970s, expected that mathematical sociology would disappear as a discrete field once sociology made further advances to incorporate its key premises (Yasuda 1973).1) He argued, by way of example that “” does not exist in the field of physics, as it is the norm to use in physics. Likewise, it was supposed that mathematical sociology would be subsumed as mathematics became a common tool in sociology. More than three decades have passed since Yasuda’s prediction, but the term “mathematical sociology” still exists, though it is not prevalent in sociology. Why is this so? A possible answer would be that studying mathematical sociology is costly. Mathematical sociologists need to master mathematical skills in addition to acquiring sociological knowledge. If the benefits of attaining these skills outweighed the costs, the number of mathematical sociologists would increase. However, we have not observed an escalation in this regard. It is probable that most sociologists estimate that the costs exceed the benefits. The outlay is difficult to reduce, but we could increase the gains if we succeed in demonstrating that mathematical sociology significantly contributes to the progress of sociology itself. I will explore this possibility in this paper and propose a measure to spread mathematical sociology throughout the discipline.

2. Diversity in Sociology

Diversity in sociology could provide a key to answering the above-mentioned question. Currently, the field of sociology exhibits a remarkable diversity of theories, methodologies, and themes. A critical division in sociological theory is that between micro and macro approaches. While interpretive sociology is influential in the study of social interactions and processes between actors, neo-institutionalism points out the importance of social as an explanatory variable. With regard to methodologies, we witness a divide between quantitative and qualitative approaches, although the importance of multiple methods has recently been emphasized. Themes covered in the name of sociology show a wide variety. Some sociologists study sports, others social movements, still others culture, and so on. Reflecting these aspects of diversity in sociology, the number of sections of the American Sociological Association is 49, ranging from aging to family to theory.2) Further, the International Sociological Association houses 55 research committees.3) This great diversity in sociology, I would argue, prevents any particular theory or methodology from prevailing in the discipline, and mathematical sociology is no exception. Then, is the diversity in sociology solely responsible for the minor status of mathematical sociology? My answer to this question is in the negative: Mathematical sociology is also responsible for its relative insignificance. I will clarify this point in the next section.

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3. Strengths and Weaknesses of Mathematical Sociology

Mathematical sociology has contributed to explaining the emergence of behavioral patterns. Axelrod’s seminal study on the emergence of mutual cooperation is a classic example highlighting the strengths of mathematical sociology (Axelrod 1984). He demonstrated the strength of a Tit-for-Tat strategy in two computer-simulated tournaments using repeated Prisoner’s Dilemma games. Then he proved the strength of the strategy with the help of evolutionary game theory. We could examine many mathematical models that explain the emergence of behavioral patterns. Macy and Skvoretz (1998) and Macy and Sato (2002) built agent-based models to account for the emergence of trust between strangers. Schelling (1971) proposed a simple model to explain residential segregation, and Bruch and Mare (2006) furthered this approach. Watts and Strogatz (1998) demonstrated how a “small world” emerges using a model in which actors change their ties to other actors. I could list many more mathematical models that have contributed to the advances of sociological knowledge, but that would avert attention from my main point. My central argument is that mathematical sociology is proliferating in particular subfields of sociology. However, at the same time this exposes the weakness of the sub-discipline. That is, mathematical sociology has not surmounted the boundaries of its niche. Particularly, it has not succeeded in explaining the emergence of social order. As pointed out above, the current diversity in sociology is astonishing, and social order has been a central concept throughout. Thus, if it could succeed in explaining the emergence of social order, mathematical sociology would make a substantive contribution to most areas of the discipline. However, social order is more than stable behavioral patterns. Rather, it consists of two elements, stable behavioral patterns and actors’ expectations of them (Parsons 1951), that work to reinforce each other. Actors who observe stable behavioral patterns strengthen their expectations of them. They then fortify their tendency to follow said patterns. Thus, focusing only on stable behavioral patterns would overlook the other facet of social order. In other words, there is a conceptual discrepancy between conventional theories of social order and mathematical sociology that tends to study stable behavioral patterns paying little attention to actors’ expectations. Resolving this discrepancy would lead to the proliferation of mathematical sociology in sociology. In the following sections I will explore this possibility.4)

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4. Focusing on Social Structure as a Bridgehead

Social order is such an abstract, ambiguous concept that we need a more concrete example to study and formalize with the help of mathematical sociology. I would argue that social structure is a good candidate for this purpose as it has been relatively clearly defined among concepts embodying social order. Roughly speaking, sociologists have studied two types of social structure. The first type is that of social networks, and mathematical sociology has made significant contributions to the study of the emergence, maintenance, and collapse of social structures of this type. However, its contribution to the study of the second type of social structure is not particularly substantive. The second type of social structure is that of roles and the allocation of actors and resources to them, à la Parsons (1951). Mathematical sociology finds it difficult to formalize social structure of this type. This is because a role is more complex than an actor (or an agent in agent-based modeling) and his/her actions. A role is a bundle of expectations held by incumbents of other roles (Parsons 1951). Take the role of professor, for example. Students expect a professor to teach courses; colleagues expect him/her to conduct research; and the university expects he/she will perform administrative duties. The first task of mathematical sociology in the study of the second type of social structure is to explain the differentiation of roles. This is because the differentiation of roles is the core mechanism in the emergence of the second type of social structure. Allocation of actors and resources to roles becomes possible only after roles are differentiated and socially defined. For example, nobody enters roles of professor and student, and no resources such as money, authority, and power are allocated to the roles before the role of professor and that of student are differentiated. How, then, can we formalize the differentiation of roles? Let us explore this problem with the help of agent-based modeling. Conventionally, an agent chooses a strategy in an agent-based model. For example, an agent chooses cooperation or defection in a Prisoner’s Dilemma game. In more complex models, an agent chooses more than one strategy. In the model proposed by Macy and Sato (2002), for example, an agent chooses staying in his/her neighborhood or entering the market,5) cooperating with his/her partner or not, and reading telltale signs exhibited by his/her partner or not. However, these strategies do not constitute a role, because they are assigned to an agent and not to a role. A combination of entering the market, cooperating, and reading telltale signs does not make sense as a role. Conversely, a combination of teaching, conducting research, and performing administrative duties constitutes the role of professor.

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5. A Possible Solution

As noted above, an agent chooses a strategy in conventional agent-based models (Figure 1). However, a role consists of more than one strategy. The role of professor, for example, consists of three strategies at least, that is teaching, conducting research, and performing administrative duties.

Agent 1 Agent 2 Agent 3

Strategy 1 Strategy 2 Strategy 3

Figure 1 Agents and Strategies in Conventional Agent-Based

Furthermore, an agent (or an actor) usually performs more than one role. A person could be a professor, a father, and a community leader. Considering the relationship between strategies and roles and that between roles and agents, we should modify Figure 1 by adding roles. Figure 2, the modified figure, represents the basic structure of an agent-based model that captures the core of the second type of social structure. Two mechanisms must be identified for the structure to be set in motion. The first is the way in which multiple strategies are assigned to a role, and the second is the way an agent takes a set of roles. I focus on the first mechanism in this paper as the second mechanism follows the same logic described below. Thus I assume below that an agent takes only one role.

Agent 1 Agent 2 Agent 3

Role 1 Role 2 Role 3 Role 4

Strategy 1 Strategy 2 Strategy 3 Strategy 4 Strategy 5

Figure 2 Connection between Agents, Roles, and Strategies Source: Sato (2003: Figure 3) 247 ⌮ㄽ࡜᪉ἲ

Structural-functionalists would argue that strategies are assigned to a role to enable it to fulfill a functional performance. However, agent-based modelers would not make this assumption, because their modeling is based on evolutionary or learning processes that do not necessarily lead to optimal functional performance. Rather, they focus on the interaction of agents and evolution or learning as the result of the interaction. Thus a possible mechanism that assigns strategies to a role consists of three stages (Figure 3). In the first stage, the initialization period, a universal set of strategies is defined. This set contains all the actions (behaviors) available to agents in society. The set is then randomly divided into sub sets, which are defined as roles.6) This is the beginning of society without institutions. Randomly created roles will change through a learning process in the third stage.

1. Initialization 1-1. A universal set of strategies is defined. 1-2. The universal set is divided into sub sets.

2. Interaction between agents

3. Update of expectations 4. Mutation (Optional)

Figure 3 Stages of the Agent-Based Model

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The division is not necessarily mutually exclusive. Teaching, for example, is a strategy chosen not only by a professor but also by a high-school teacher. As mentioned above, an agent is assumed to take only one role, a subset of the universal set of strategies. The above-mentioned assumptions of the agent-based model are formalized as follows.  ! Suppose that U is a universal set that contains all strategies. That is, 21 sssU l },,,{ , where si

(i=1,2,…,l) is a strategy. Then a subset Ri (i=1,2,…,m) of U is defined as a role. This means that there are m roles in society. As the union of the subsets contains all strategies,  '"'' 21 RRRU m . Subsets are not necessarily mutually exclusive, as mentioned in the    above paragraph. Thus the model allows cases where ,, RRji ji . The assignment of agents to roles is a mapping from the set of agents N  ! n},,2,1{ to the set of roles (or the  ! family of the subsets of U) 21 RRRR m},,,{ . In the second stage, agents choose strategies in their roles and interact with other agents based on the strategies they chose. A professor chooses teaching and a student chooses attending his/her class, for example. The choice of strategies is based on the agent’s intentions and expectations of his/her role performance held by other agents. Agent i’s expectation of a role held by agent j is formalized as follows. Suppose that agent j  " has a role Rj that consists of k strategies, that is, j ,21 sssR k },,{ . Then agent i’s expectation  " of the role is a subjective over Rj, that is, pij 21 ppp k ),,,( , where

k  " ppp  1,,,0 and B p  1. It is also assumed that agent i has expectations of all 21 k h1 h roles in society. If he/she completely meets the expectations, the agent is homo sociologicus. If he/she completely ignores the expectations, the agent is not socialized at all.7) In real situations, the agent chooses strategies to optimize his/her utility under the expectations.8) A professor, for example, may reduce his/her teaching load in order to increase the amount of time for his/her research. However, he/she cannot do this infinitely. His/her decision is constrained by expectations held by incumbents of other roles, such as students and the dean. In the third stage agents update their expectations to roles. If an agent witnessed another agent meeting his/her expectations, he/she would not change them. However, if the target agent does not meet his/her expectations, he/she has two alternatives: Enforcing the performance with sanctions and changing his/her expectations. Structural-functionalists would emphasize the first alternative, while interpretive sociologists might focus on the second. Agents could be assumed to compare the cost of the sanctions with that of changing their expectations. If the former exceeds the latter, they change their expectations resulting in a change in role structure. Then the process returns to the second stage.

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We could add a phase for “mutation” to the third stage. In this stage some roles are adopted and sets of strategies attached to a role become disbanded. Then disbanded sets are newly banded to create new roles. Thus, completely new roles are created in this stage. In conventional agent-based modeling, this type of “mutation” or “perturbation” is often included. However, it is difficult for me to figure out empirical examples of role “mutation”. Thus I propose this stage as an option. The system may reach an equilibrium after iterations where role structure does not change or makes such a circle as role structure A b role structure B b role structure C b role structure A. In this equilibrium role expectations and role performances reinforce each other, which is the realization of the differentiation of roles or the emergence of social structure.

6. Conclusion

I have explored the possibility of formalizing the emergence of the second type of social structure, focusing on roles, with the help of agent-based modeling. If mathematical sociologists succeed in developing models along this line, the subfield will proliferate beyond its current niche in sociology. This is because demonstrating mathematical sociology’s success in explaining role structure would increase the benefits of using it in sociology and, therefore, enable it to prevail in sociology. However, the agent-based modeling process mentioned in the previous section provides but a rough sketch, and far more work is needed to elaborate it. First, we need to expand the concept of role. Role in sociology is difficult to formalize because many complicated factors are attached. However, a few mathematical formalizations of the concept of role have recently been proposed. Montgomery (2005) formalizes the concept using fuzzy . Misumi (2007) applies Boolean to formalize roles. Gleave et al (2009) attempt to integrate structural and interpretive aspects of roles. Elaborating and formalizing the concept of role based on these works would enrich the framework behind the agent-based model proposed in this paper. Second, the learning process that occurs in the third stage must be particularized. This process is more complex than the learning process involved in choices of strategies in conventional agent-based modeling. The latter process usually updates only choices of strategies, while the former process updates both choices of strategies and expectations. A possible process that captures this “dual” renewal is fictitious play. In fictitious play an agent updates his/her expectations based on his/her observation of the role performance of other agents. Then he/she chooses a strategy that he/she thinks optimizes his/her utility, based on the updated expectations. Thus fictitious play properly captures the dynamics of the relationship between role expectation and performance.

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To conclude, I propose that mathematical sociologists change their direction. To my knowledge, they have tended to formalize social phenomena that they find it easy to formalize. However, they should instead explore the core phenomenon of social order, the differentiation of roles, with their powerful mathematical tools and frameworks.

Acknowledgement Previous versions of the paper were presented at the Invited Workshop on Social Science and Social Computing: Steps to Integration organized by Sun-Ki Chai, Section on Rationality and Society Invited Session: New Approaches to the Micro-Macro Link at the 105th Annual Meeting of the American Sociological Association organized by Pamela E. Emanuelson, and the Special Symposium on Mathematical Sociology in Sociology: Verification of Its Validity at the 50th Semiannual Meeting of the Japanese Association for Mathematical Sociology organized by Hiroshi Hamada. I thank the organizers for giving me opportunities to elaborate ideas for the paper as well as the audiences for their thought-provoking comments. Constructive comments from two anonymous reviewers are also gratefully appreciated. They substantively improved the quality of this paper.

Notes 1) See also Naoi (1991) for Yasuda’s argument on the relationship between mathematical sociology and sociology. 2) http://www.asanet.org/sections/list.cfm, retrieved on March 5, 2011. A section is a group of people who share the same research agenda. 3) http://www.isa-sociology.org/rc.htm, retrieved on March 5, 2011. A research committee of the International Sociological Association is functionally equivalent to a section of the American Sociological Association. 4) The following sections are based on Sato (2003). 5) The difference between a neighborhood and the market in their model relates to the possibility of being cheated. When an agent makes a transaction with a neighbor, the neighbor does not cheat him/her. In contrast, an agent meets a stranger in the market. The stranger proposes goods that are more attractive than goods available in his/her neighborhood. The stranger, however, may be untrustworthy and cheat him/her. 6) The model should make clear how newly created roles are labeled. For example, it should assume a mechanism on how a set of strategies is labeled as a professor. The study of this mechanism is beyond the scope of this paper, so the model simply assumes that labels are randomly assigned to roles. 7) The model should also make a clear assumption on how an agent is socialized or how he/she is responsive to role expectations held by other agents. A possible process is that agents learn to respect role expectations based on their interactions with other agents. This process can be included in the role-emerging process described in Figure 3. 8) It is an open question how an agent chooses strategies under the expectations. A possible mechanism is as follows. An agent sums up expectations held by other agents with some weights. For example, if

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the agent thinks that he/she has to pay more attention to agent j’s expectation than that of agent k, he/she adds a larger weight to the former. Then the agent chooses a set of strategies that he/she thinks maximizes his/her expected utility with the summed expectations.

References Axelrod, Robert. 1984. The Evolution of Cooperation. New York: Basic Books. Bruch, Elizabeth E. and Robert D. Mare. 2006. “Neighborhood Choice and Neighborhood Change.” American Journal of Sociology 112: 667-709. Buskens, V. and A. van de Rijt. 2008. “Dynamics of Networks: If Everyone Strives for Structural Holes.” American Journal of Sociology 114: 371-407. Gleave, Eric, Howard T. Welser, Thomas M. Lento, and Marc A. Smith. 2009. “A Conceptual and Operational Definition of ‘Social Role’ in Online Community.” Proceedings of the 42nd Hawaii International Conference on System Sciences-2009:1-11. Macy, Michael W. and John Skvoretz. 1998. “The Evolution of Trust and Cooperation Between Strangers: A Computational Model.” American Sociological Review 63: 638-660. Macy, Michael W. and Yoshimichi Sato. 2002. “Trust, Cooperation, and Market Formation in the U.S. and Japan.” Proceedings of the National Academy of Sciences 99(Suppl. 3): 7214-7220. Misumi, Kazuto. 2007. A Formal Theory of Roles. Fukuoka: Hana-Syoin. Montgomery, James D. 2005. “The Logic of Role Theory: Role Conflict and Stability of the Self-Concept.” Journal of Mathematical Sociology 29: 33-71. Naoi, Atsushi. 1991. “Between Mathematical Sociology and Fundamental Sociology.” Sociological Theory and Methods 6(1): 114-120. (In Japanese) Parsons, Talcott. 1951. The Social System. New York: Free Press. Sato, Yoshimichi. 2003. “Can Evolutionary Game Theory Evolve in Sociology? Beyond Solving the Prisoner’s Dilemma.” Sociological Theory and Methods 18(2): 185-196. Schelling, Thomas C. 1971. “Dynamic Models of Segregation.” Journal of Mathematical Sociology 1: 143-186. Watts, Duncan J. and Steven H. Strogatz. 1998. “Collective Dynamics of ‘Small-world’ Networks.” Nature 393: 440-442. Yasuda, Saburo. 1973. “Introduction” in Mathematical Sociology, edited by Saburo Yasuda. Tokyo: University of Tokyo Press: 1-14. (In Japanese)

(Received: April 10, 2011 /Accepted: July 15, 2011)

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