Performance Bounds for Nash Equilibria in Submodular Utility Systems with User Groups

We consider two types of grouping among users in utility systems. The first type of grouping is from Chen et al. (2014), where each user belongs to a group of users having social ties with it. For this type of utility system, each user’s strategy maximizes its social group utility function, giving rise to the notion of social-aware Nash equilibrium. We prove that, for a valid utility system, if the social utility function is submodular, then any social-aware Nash equilibrium achieves at least 1/2 of the optimal social utility, subject to a functiondependent additive term. We further prove that if the social utility function is nondecreasing and submodular, then any social-aware Nash equilibrium achieves at least 1/(1 + c) of the optimal social utility, where c is the curvature of the social utility function. For the second type of grouping we consider, the set of users is partitioned into l disjoint groups, where the users within a group cooperate to maximize their group utility function, giving rise to the notion of group Nash equilibrium. We prove that, for a valid utility system, if the social utility function is submodular, then any group Nash equilibrium achieves at least 1/2 of the optimal social utility, subject to a function-dependent additive term. By defining the curvature cki associated with group i with ki users, we show that if the social utility function is nondecreasing and submodular, then any group Nash equilibrium achieves at least 1/(1+max1≤i≤l cki) of the optimal social utility. Finally, we present an example of a utility system for database assisted spectrum access to illustrate our results.