Mathematical models of stable jurisdiction partitions: A survey of results and new directions

The human society consists of many large and small groups. Why do these groups emerge? How does their structure form? What does their dynamics look like? Is there always a stable configuration that does not change anymore? In the models described in this survey it is assumed that agents live in some geographical or virtual space and groups are formed to produce some club good located in the same space and available only inside a group. There are two opposite forces that influence the size of the groups. On the one hand, in larger groups the agents enjoy economy of scale, so the good costs less. On the other hand, in smaller and homogeneous groups the good may be located close to all members. Do these forces always balance each other in some stable configuration and what does this configuration look like? The answer depends on the model details and on the notion of equilibrium. In this survey we present different frameworks and main known results about (non-)existence of equilibria and their properties, as well as some new directions of research.