  Small operators who take part in secondary wireless spectrum markets
typically have strict budget limits. In this paper, we study the bidding
problem of a budget constrained operator in repeated secondary spectrum
auctions. In existing truthful auctions, truthful bidding is the optimal
strategy of a bidder. However, budget limits impact bidding behaviors and make
bidding decisions complicated, since bidders may behave differently to avoid
running out of money. We formulate the problem as a dynamic auction game
between operators, where knowledge of other operators is limited due to the
distributed nature of wireless networks/markets. We first present a Markov
Decision Process (MDP) formulation of the problem and characterize the optimal
bidding strategy of an operator, provided that opponents' bids are i.i.d. Next,
we generalize the formulation to a Markov game that, in conjunction with
model-free reinforcement learning approaches, enables an operator to make
inferences about its opponents based on local observations. Finally, we present
a fully distributed learning-based bidding algorithm which relies only on local
information. Our numerical results show that our proposed learning-based
bidding results in a better utility than truthful bidding.
