  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.
