Social Optimum, Eigen-Equilibrium and Symmetric Externalities

This work presents a theory for games with symmetric externali-ties, where the externalities by any one player imposed on another is same as the vice versa. The theory connects a social optimum outcome with its eigen-equilibrium, a Nash equilibrium of a game that is attained by transforming a social optimum. An implication of the theory is that social optimum could be implemented in a way with complete ignorance of individual players' utility functions. Our result has relevance to the recent trend of designing distributed controls and algorithms for resource sharing with in mind of selfish users, as to provide a basis for achieving social optimum at selfish users' equilibrium.