  Dynamic spectrum sharing is a promising technology to improve spectrum
utilization in the future wireless networks. The flexible spectrum management
provides new opportunities for licensed primary user and unlicensed secondary
users to reallocate the spectrum resource efficiently. In this paper, we
present an oligopoly pricing framework for dynamic spectrum allocation in which
the primary users sell excessive spectrum to the secondary users for monetary
return. We present two approaches, the strict constraints (type-I) and the QoS
penalty (type-II), to model the realistic situation that the primary users have
limited capacities. In the oligopoly model with strict constraints, we propose
a low-complexity searching method to obtain the Nash Equilibrium and prove its
uniqueness. When reduced to a duopoly game, we analytically show the
interesting gaps in the leader-follower pricing strategy. In the QoS penalty
based oligopoly model, a novel variable transformation method is developed to
derive the unique Nash Equilibrium. When the market information is limited, we
provide three myopically optimal algorithms "StrictBEST", "StrictBR" and
"QoSBEST" that enable price adjustment for duopoly primary users based on the
Best Response Function (BRF) and the bounded rationality (BR) principles.
Numerical results validate the effectiveness of our analysis and demonstrate
the fast convergence of "StrictBEST" as well as "QoSBEST" to the Nash
Equilibrium. For the "StrictBR" algorithm, we reveal the chaotic behaviors of
dynamic price adaptation in response to the learning rates.
