Population Heterogeneity and Between-Group Substitutability and Complementarity of Social Actions

Abstract

This article extends a foundation for sociological theories of the micro-macro linkage in collective action. Key elements of the foundation are population heterogeneity and a model of interdependence of actions. By “population heterogeneity” I mean differences among people in the payoffs of their alternative choices. This article introduces the concepts of interpersonal substitutability and complementarity of actions between groups for analyzing their collective outcomes by extending similar concepts and methods previously introduced in the literature for interdependence of actions within a group. The article applies these theories and methods (1) to evaluate Tsebelis's theory of the relationship between crime and punishment and present distinct conclusions on the effectiveness of alternative crime control methods depending on the conditions of population heterogeneity, and (2) to clarify the role of population heterogeneity as a modifier for the effects of an increase in the gender equality of job opportunities on men's and women's gender role attitudes about the division of household labor.

Keywords:

• collective dynamics
• rational choice
• social influence

Acknowledgments

I am grateful to special-issue editors and anonymous referees for their helpful comments. A voluntary assistance of Kohki Yamaguchi in depicting graphs is also appreciated.

Notes

¹One reviewer argues that if we incorporate a learning model of “fictitious play” (Fudenberg and Levine, 1998), which was applied to this game by Rauhut (2009), we need not assume knowledge of individual payoffs for the criminal to learn about the law-enforcement rate or for the police officer to learn about the crime rate. The point, however, is that it is rational for each criminal to employ a pure strategy, rather than a mixed strategy, for a given law-enforcement rate. It is also rational for each police officer to employ a pure strategy for a given crime rate. While the law-enforcement rate and the crime rate are not fixed and change as a result of the choices of each group of individuals, the rational adjustment of actors' pure strategies yields the equilibrium of the two rates described in this article, rather than Tsebelis's (1990) solution.

²This implies that a realized state does not jump around to approach an equilibrium state but moves smoothly to reach it.

³A reviewer pointed out that if we assume a homogeneous population, there is also a mixed-strategy equilibrium, as in the discoordination game. However, in the coordination game, this is an unstable equilibrium, a small deviation from which in the mixing probability leads to one of the two stable pure-strategy Nash equilibria.

⁴This is shown graphically by alternating the collective choice between one group and the other group, by an iterative approach toward a convergence, starting from any point in Figure 5. The logic is also applicable below to explanations of which equilibrium will be attained.

⁵This should yield at least one stable equilibrium. Since Φ(Q) runs across the horizontal Q axis from a value greater than 0 at Q = 0, (P, Q) = (a, 0), a > 0, to a value smaller than 1 at Q = 1, (P, Q) = (b, 1), 1 > b > a, and Γ(P) runs across the vertical P axis from a value greater than 0 at P = 0, (P, Q) = (0, c), c > 0 to a value smaller than 1 at P = 1, (P, Q) = (1, d), 1 > d > c, they must intersect at least once as is shown in various cases depicted in Figure 6, and since the average dP/dQ is greater for Γ(P) than for Φ(Q), at least one intersection must satisfy the condition of a stable equilibrium for which dP/dQ is greater for Γ(P*) than for Φ(Q*).