Inequality, cooperation, collective action, and delayed marital unions: papers from the Sixth Joint Japan–US Conference on Mathematical Sociology and Rational Choice

The papers in this special issue were presented at the Sixth Joint Japan–US Conference on Mathematical Sociology and Rational Choice held on August 2016 in Seattle, WA. The conference was cosponsored by two sections of the American Sociological Association (Mathematical Sociology and Rationality and Society) and by the Japanese Association for Mathematical Sociology. Professors Jun Kobayashi (Seikei University), Masayuki Kanai (Senshu University), Kikuko Nagayoshi (Tohoku University), John Skvoretz (University of South Florida), and Douglas Heckathorn (Cornell University) were the conference organizers. The four papers in this special issue are a subset of the papers that were invited for the issue, and these in turn were a subset of all the papers presented. Invited papers were selected by the organizers with an eye toward the generality of problem and depth of mathematical content. All submitted papers were peer reviewed as per the standard review procedures of the journal.

Three of the papers are very much in the tradition of rational choice: “Late Marriage and Transition from Arranged Marriages to Love Matches” (Kezuka), “The Survival of Inefficient and Efficient Norms” (Kira), and “Self-organizing Collective Action” (Obayashi). Actors are assumed to gain utility from their actions but how much depends on the actions of others and on parametric factors of theoretical interest. Actors are presumed to maximize utility. Their behavioral strategies include first-order actions (such as cooperate or not) and possible higher-order actions that are sanctioning reactions to lower-order behaviors by others. The aim of analysis, generally, is to derive equilibrium conditions. The problems addressed in the Kira and Obayashi papers are quite broad, namely, collective action and the survival of norms of cooperation. The Kezuka paper is motivated by an empirical puzzle in developed countries with the specific example of Japan, in which there is increasing delay in first marriage. The fourth paper, “What Can You and I Do to Reduce Income Inequality?” (Jasso), differs from the others in several ways. There is no formal model of actors nor functional specification of the factors on which their utility depends. The point of paper is not to assume some action set available to actors and look for equilibrium conditions expressed as stable probabilities over strategies, but rather to develop an understanding of what actions are available to actors if they were to seek to reduce income inequality.

In “Late Marriage and Transition from Arranged Marriages to Love Matches,” Kezuka links the delay to a change in the basis of marriage from arranged matches to love matches, a change that is driven in turn by a change in value system from traditional to individualistic. An important background element is the division of the actor population into different social classes because arranged matches can only occur between actors of the same class. Love matches can occur even if the classes of the actors differ. The analysis has two steps: a decision-making step which occurs within periods as single individuals consummate matches (or not) seeking to make an arranged marriage or a love marriage depending on whether they have traditional or individualistic preferences. Between periods, replicator dynamics are applied to the mix of traditional versus individualistic preference holders in the population. The proportions are changed in response to the expected satisfaction achieved by an actor. It is assumed that staying single brings less satisfaction than being in an arranged match, and the latter less than being in a love match. Expected satisfaction depends on the likely success of a search for a mate, which in turn depends on the type of match sought as determined by the searcher’s value system and the associated costs of search but also the current mix of traditionalists and individualists. The model is a complicated one because it has the two levels of within-period play and between-period evolution of preference mix. Keszuka’s analysis is necessarily done through numerical simulation of the implied processes over 3,000 to 5,000 periods.

Each of the next two papers presents motivating examples to illustrate their general problem and focus their analysis on the general issues the examples illustrate.

In “The Survival of Inefficient and Efficient Norms,” Kira explores cooperative equilibria in social dilemma games under three strategies that respond to a first-order deviation from cooperation. The first two strategies, the tit-for-tat (TFT) strategy and the meta-norm (MN) strategy, are familiar from previous literature. The third strategy, proposed in the article, is the punishment-or-opportunism (PO) strategy. As is well known, the TFT strategy punishes first-order deviation by withholding the cooperation on the next round (a first-order deviation to punish the offender). The other two strategies make use of costly punishments that are inflicted on first-order deviations. The difference between the MN strategy and the PO strategy lies in the reaction to a second-order deviation, that is, a failure to comply with the norm to punish first-order deviations. The MN strategy relies on a second-order norm to punish those who do not punish first-order offenders (hence its name as the MN strategy). The PO strategy, instead, calls for ceasing first-order cooperation as a reaction to a failure to punish a first-order offender. Thus, the penalty for failing to punish first-order deviation is not a cost incurred because others punish you for failing to punish the first-order offender but the loss of first-order benefits from their cooperation. This difference is the key insight of the article. The PO strategy is effective only so long as the benefits from first-order cooperation exceed the individual costs of that action, that is, has a positive externality for all actors. If instead first-order cooperation has a negative externality, withholding the “cooperative” action is actually a benefit since received costs are reduced. Thus, it is impossible for the PO strategy to sustain an inefficient norm, one that calls for a first-order action that adds cost to all actors. The article shows that this is not true for the MN strategy under certain conditions. The use of that strategy can sustain first-order “cooperation” even under negative externality and thus sustain an “inefficient” first-order norm.

In “Self-organizing Collective Action,” Obayashi also examines cooperation but in the context of the collective action problem. The actions available to individuals in a group are simply to cooperate or “do nothing.” There is a constant cost to cooperation no matter how many group members cooperate and cooperation is a social dilemma with the cost to an actor if he or she is the only cooperator exceeding the benefit of the action. The analytical focus is on the size of the group for which cooperation returns benefits to its members. Outsiders to the group receive no benefits. Time enters into the analysis as the size of the group changes overtime in response to group members leaving (at a constant rate) and new members joining at a rate that reflects the success of the group’s collective action (based on the number of cooperators at the previous time point). The article demonstrates that the strategy to always cooperate when one is a member of the group (ALLC) can be an equilibrium strategy (without assuming first-order deviations are punished) if group size is sensitive to collective reputation. Collective reputation refers to the perception of outsiders as to the group’s effectiveness. The group’s effectiveness, in turn, is driven by the prevalence of cooperation in the group, that is, how effective collection action is at returning benefits to group members. Two types of collective reputation effects are investigated. In the first case, the number of new members increases in proportion to the rate of cooperators in the group, that is, the proportion of group members that cooperate. In the second case, the number of new members increases in proportion to the raw number of cooperators in the group. In the first case, the rate of joining (the parameter calibrating the idea of reputational impact) and the rate at which old members exit jointly determine an equilibrium group size at which the ALLC strategy can be sustained. In the second case, the dynamics are more complex, but there are clear regions of the parameter space in which the ALLC strategy can be sustained without assuming first-order deviation is punished. The implications of this analysis are worth careful attention to capture the intuition behind the formal proofs. The importance of exit from the group is highlighted as being, in effect, an alternative mechanism to punishment, keeping group members in line. Equally important is the assumption that rewards from cooperation increase with the number of cooperators since then larger groups can create a payoff difference that sustains cooperation.

Finally, in “What Can You and I Do to Reduce Income Inequality?,” Jasso uses basic mathematics to understand the levers individuals can use to reduce income inequality. The first insight is that common measures of inequality (such as the Gini coefficient) implicitly identify the kinds of mechanisms that will increase or decrease income inequality—for instance, it is easy to prove that adding a constant amount to all shares (concretely, a constant bonus to all employees) reduces inequality as measured by the common coefficients. The second insight is that simple behavioral patterns that influence inequality can be altered (although perhaps not easily) to produce less inequality. For instance, positive assortative mating by income level (rich marrying rich and poor marrying poor) is a common societal pattern that provably exacerbates inequality among households (and for many baseline distributions of income at the individual level). From these two insights flow five different levers by which individuals could reduce inequality: (1) make a rank-preserving transfer of resources to a poorer unit (person or household); (2) distribute windfalls or surpluses in equal absolute shares; (3) marry a person whose relative rank in the income distribution is either one above or one below one’s own rank; (4) use multiple worker characteristics to set wages, ideally, characteristics that are not correlated with each other; and (5) in other circumstances, use input from multiple raters to set wages and obtain this input through voting procedures that (a) seek to increase the number of voters, their diversity of perspectives, and (b) are conducted by secret ballot to avoid correlated judgments through social influence.