Sustaining Cooperation with Multiple Relations: a Theory of Multiplexity

Abstract : People are embedded in multiple social relations. These relationships are not isolated from each other. This paper provides a framework to analyze the multiplex of networks. We present a model in which each pair of agents may form more than one relationship. Each relationship is captured by an infinitely repeated prisoner’s dilemma with variable stakes of cooperation. We show that multiplexity, i.e. having more than one relationship on a link, boosters incentives as different relationships serve as social collateral for each other. We then endogenize the network formation and ask: when an agent has a new link to add, will she multiplex with a current neighbor or link with a stranger? We find the following: (1) There is a strong tendency to multiplex, and the “multiplexity trap” can occur. That is, agents may keep adding relationships with the current neighbor(s), even if it is more compatible to cooperate with a stranger. (2) Individuals tend to multiplex when the current network (a) has a low degree dispersion (i.e., all individuals have similar numbers of friends), or (b) is positively assortative. We also find that when relationships differ in their importance, agents tend to multiplex when the new relationship is less important and link with a stranger when it’s more important. Lastly, we find empirical evidence that supports our theoretical findings.

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Rulebooks in Relational Contracts

Abstract : Firms can accrue large benefits by fostering worker initiative, but standardized work rules are still widely used. We present a model of relational incentives where the use of rules fluctuates as the firm faces shocks to its credibility. Worker initiative in adapting to local information can ensure production efficiency but requires strong incentives. As shocks weaken relational incentives, the firm may adopt rules that yield satisfactory (though suboptimal) performance. Rules help the relationship survive the shocks, but the relationship becomes less efficient in the future. While the relationship may recover, its ability to weather future shocks deteriorates, and, over time, it becomes more reliant on rules.

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Modular Organization

Abstract : The defining products of our times are made up of modules, groups of components that are highly connected to each other but only loosely connected to those in other groups. We explore the implications of modular production for the emergence of modular organizations. To this end, we develop a team-theoretic model with costly communication. Production is described by a production network where each node represents a decision and a state, and the weight of the link between two nodes captures the need to coordinate the two decisions. Each node has an agent who observes the state, communicates with other agents in a designed communication network, and makes the decision. The optimal communication network trades off the cost of an additional link with the gain from better coordination. To explore modular production, we assume production has a community structure where each decision belongs to one community. The need for coordination is higher for decisions in the same community. We show the optimal organization will only be modular if the properties of the production function are moderate: there are few modules, none is large, and the need for coordination within modules is neither too high nor too low. We extend the model to allow for an overlapping community structure in which each decision belongs to two communities, and we generate testable predictions for the emergence of commonly observed organizational structures, such as M-form, U-form, and matrix organizations.

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