  We consider an incomplete information network game in which agents'
information is restricted only to the identity of their immediate neighbors.
Agents form beliefs about the adjacency pattern of others and play a
linear-quadratic effort game to maximize interim payoffs. We establish the
existence and uniqueness of Bayesian-Nash equilibria in pure strategies. In
this equilibrium agents use local information, i.e., knowledge of their direct
connections to make inferences about the complementarity strength of their
actions with those of other agents which is given by their updated beliefs
regarding the number of walks they have in the network. Our model clearly
demonstrates how asymmetric information based on network position and the
identity of agents affect strategic behavior in such network games. We also
characterize agent behavior in equilibria under different forms of ex-ante
prior beliefs such as uniform priors over the set of all networks, Erdos-Renyi
network generation, and homophilic linkage.
