  We consider the problem of computing Nash Equilibria of action-graph games
(AGGs). AGGs, introduced by Bhat and Leyton-Brown, is a succinct representation
of games that encapsulates both "local" dependencies as in graphical games, and
partial indifference to other agents' identities as in anonymous games, which
occur in many natural settings. This is achieved by specifying a graph on the
set of actions, so that the payoff of an agent for selecting a strategy depends
only on the number of agents playing each of the neighboring strategies in the
action graph. We present a Polynomial Time Approximation Scheme for computing
mixed Nash equilibria of AGGs with constant treewidth and a constant number of
agent types (and an arbitrary number of strategies), together with hardness
results for the cases when either the treewidth or the number of agent types is
unconstrained. In particular, we show that even if the action graph is a tree,
but the number of agent-types is unconstrained, it is NP-complete to decide the
existence of a pure-strategy Nash equilibrium and PPAD-complete to compute a
mixed Nash equilibrium (even an approximate one); similarly for symmetric AGGs
(all agents belong to a single type), if we allow arbitrary treewidth. These
hardness results suggest that, in some sense, our PTAS is as strong of a
positive result as one can expect.
