  We study competition between wireless devices with incomplete information
about their opponents. We model such interactions as Bayesian interference
games. Each wireless device selects a power profile over the entire available
bandwidth to maximize its data rate. Such competitive models represent
situations in which several wireless devices share spectrum without any central
authority or coordinated protocol.
  In contrast to games where devices have complete information about their
opponents, we consider scenarios where the devices are unaware of the
interference they cause to other devices. Such games, which are modeled as
Bayesian games, can exhibit significantly different equilibria. We first
consider a simple scenario of simultaneous move games, where we show that the
unique Bayes-Nash equilibrium is where both devices spread their power equally
across the entire bandwidth. We then extend this model to a two-tiered spectrum
sharing case where users act sequentially. Here one of the devices, called the
primary user, is the owner of the spectrum and it selects its power profile
first. The second device (called the secondary user) then responds by choosing
a power profile to maximize its Shannon capacity. In such sequential move
games, we show that there exist equilibria in which the primary user obtains a
higher data rate by using only a part of the bandwidth.
  In a repeated Bayesian interference game, we show the existence of reputation
effects: an informed primary user can bluff to prevent spectrum usage by a
secondary user who suffers from lack of information about the channel gains.
The resulting equilibrium can be highly inefficient, suggesting that
competitive spectrum sharing is highly suboptimal.
