Structural Stability of a Family of Spatial Group Formation Games

We introduce and study a group formation game in which individuals form groups so as to achieve high collective strength. This strength could be group identity, reputation, or protection, and is equally shared by all group members. The group's strength is derived from its access to resources possessed by its members, and is traded off against the geographic dispersion of the group; spread-out groups are costlier to maintain. We seek to understand the properties of stable groupings in such a setting. We define several types of equilibria, where a member wishing to join a new group requires the acceptance of that group, and may further require permission to leave its current group. We show that under natural assumptions on the group utility functions, some of these equilibria always exist, and that any sequence of improving deviations by agents (or subsets of agents in the same group) converges to an equilibrium. In characterizing the properties of these equilibria, We show that an “encroachment” relationship — which groups have members in the territory of other groups — always gives rise to a directed acyclic graph (DAG). We relate our model to observations of well-established groups in a real-world dataset.