  In revenue maximization of selling a digital product in a social network, the
utility of an agent is often considered to have two parts: a private valuation,
and linearly additive influences from other agents. We study the incomplete
information case where agents know a common distribution about others' private
valuations, and make decisions simultaneously. The "rational behavior" of
agents in this case is captured by the well-known Bayesian Nash equilibrium.
  Two challenging questions arise: how to compute an equilibrium and how to
optimize a pricing strategy accordingly to maximize the revenue assuming agents
follow the equilibrium? In this paper, we mainly focus on the natural model
where the private valuation of each agent is sampled from a uniform
distribution, which turns out to be already challenging.
  Our main result is a polynomial-time algorithm that can exactly compute the
equilibrium and the optimal price, when pairwise influences are non-negative.
If negative influences are allowed, computing any equilibrium even
approximately is PPAD-hard. Our algorithm can also be used to design an FPTAS
for optimizing discriminative price profile.
