  We develop a game-theoretic framework for the study of competition between
firms who have budgets to "seed" the initial adoption of their products by
consumers located in a social network. The payoffs to the firms are the
eventual number of adoptions of their product through a competitive stochastic
diffusion process in the network. This framework yields a rich class of
competitive strategies, which depend in subtle ways on the stochastic dynamics
of adoption, the relative budgets of the players, and the underlying structure
of the social network.
  We identify a general property of the adoption dynamics --- namely,
decreasing returns to local adoption --- for which the inefficiency of resource
use at equilibrium (the Price of Anarchy) is uniformly bounded above, across
all networks. We also show that if this property is violated the Price of
Anarchy can be unbounded, thus yielding sharp threshold behavior for a broad
class of dynamics.
  We also introduce a new notion, the Budget Multiplier, that measures the
extent that imbalances in player budgets can be amplified at equilibrium. We
again identify a general property of the adoption dynamics --- namely,
proportional local adoption between competitors --- for which the (pure
strategy) Budget Multiplier is uniformly bounded above, across all networks. We
show that a violation of this property can lead to unbounded Budget Multiplier,
again yielding sharp threshold behavior for a broad class of dynamics.
