Local Aggregative Games
Aggregative games provide a rich abstraction to model strategic multi-agent interactions. We introduce local aggregative games, where the payoff of each player is a function of its own action and the aggregate behavior of its neighbors in a connected digraph. We show the existence of a pure strategy -Nash equilibrium in such games when the payoff functions are convex or sub-modular. We prove an information theoretic lower bound, in a value oracle model, on approximating the structure of the digraph with non-negative monotone sub-modular cost functions on the edge set cardinality. We also define a new notion of structural stability, and introduce γ-aggregative games that generalize local aggregative games and admit -Nash equilibrium that is stable with respect to small changes in some specified graph property. Moreover, we provide algorithms for our models that can meaningfully estimate the game structure and the parameters of the aggregator function from real voting data.
Summary: 
The authors consider games among n players over a digraph G=(V,E), where each player corresponds to a node in V, the payoff function of each player depends on the choices of its own and those of neighbors. A further assumption is that the payoff function is Lipschitz and convex (or submodular). For games with sufficiently smooth payoff functions, the authors show that an approximate PSNE exists. Also, the authors prove that there exists a digraph for which any randomized algorithms approximating PSNE needs exponentially many value-queries for a submodular payoff function. 
Then the authors consider a variant of the game called gamma-local aggregative game, for which the payoff function depends on not only neighbors but all other players under the condition that other players choices affect the payoff in the magnitude of gamma^(distance from the player). For such games, the authors show a sufficient condition that an approximate PSNE exists. They further consider a learning problem for the games where, given approximate PSNEs as examples, the learner’s goal is to predict the underlying digraph. An optimization problem for the learning problem is proposed. Finally, the authors show experimental results over real data sets.
Comments: 
The proposed game is a generalization of the aggregative game to a graph setting. The technical contribution of the paper is to derive sufficient/necessary conditions for the existence of approximate PSNEs for the games. On the other hand, the theoretical results for learning graphs from approximate PSNEs is yet to be investigated in the sense that no sample complexity is shown. However, the proposed formulation provides interesting results over real data sets. 
I read the authors' rebuttal comments and I would appreciate if the authors reflect your comments to me in the final version.