Can Selfish Groups be Self-Enforcing?

Abstract

Algorithmic graph theory has thoroughly analyzed how, given a network describing constraints between various nodes, groups can be formed among these so that the resulting configuration optimizes a \emph{global} metric. In contrast, for various social and economic networks, groups are formed \emph{de facto} by the choices of selfish players. A fundamental problem in this setting is the existence and convergence to a \emph{self-enforcing} configuration: assignment of players into groups such that no player has an incentive to move into another group than hers. Motivated by information sharing on social networks -- and the difficult tradeoff between its benefits and the associated privacy risk -- we study the possible emergence of such stable configurations in a general selfish group formation game. Our paper considers this general game for the first time, and it completes its analysis. We show that convergence critically depends on the level of \emph{collusions} among the players -- which allow multiple players to move simultaneously as long as \emph{all of them} benefit. Solving a previously open problem we exactly show when, depending on collusions, convergence occurs within polynomial time, non-polynomial time, and when it never occurs. We also prove that previously known bounds on convergence time are all loose: by a novel combinatorial analysis of the evolution of this game we are able to provide the first \emph{asymptotically exact} formula on its convergence. Moreover, we extend these results by providing a complete analysis when groups may \emph{overlap}, and for general utility functions representing \emph{multi-modal} interactions. Finally, we prove that collusions have a significant and \emph{positive} effect on the \emph{efficiency} of the equilibrium that is attained.