Performance bounds for Nash equilibria in submodular utility systems with user groups

In this paper, we consider variations of the utility system considered by Vetta, in which users are grouped together. Our aim is to establish how grouping and cooperation among users affect performance bounds. We consider two types of grouping. The first type is from \cite{Zhang2014}, 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 \emph{social-aware Nash equilibrium}. We prove that this social utility system yields to the bounding results of Vetta for non-cooperative system, thus establishing provable performance guarantees for the social-aware Nash equilibrium. For the second type of grouping, 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 \emph{group Nash equilibrium}. In this case, each group can be viewed as a new user with vector-valued actions, and a 1/2 bound for the performance of group Nash equilibrium follows from the result of Vetta. But as we show tighter bounds involving curvature can be established. By defining the group curvature $c_{k_i}$ associated with group $i$ with $k_i$ users, we show that if the social utility function is nondecreasing and submodular, then any group Nash equilibrium achieves at least $1/(1+\max_{1\leq i\leq l}c_{k_i})$ of the optimal social utility, which is tighter than that for the case without grouping. As a special case, if each user has the same action space, then we have that any group Nash equilibrium achieves at least $1/(1+c_{k^*})$ of the optimal social utility, where $k^*$ is the least number of users among the $l$ groups. Finally, we present an example of a utility system for database assisted spectrum access to illustrate our results.