  Motivated by online advertising auctions, we consider repeated Vickrey
auctions where goods of unknown value are sold sequentially and bidders only
learn (potentially noisy) information about a good's value once it is
purchased. We adopt an online learning approach with bandit feedback to model
this problem and derive bidding strategies for two models: stochastic and
adversarial. In the stochastic model, the observed values of the goods are
random variables centered around the true value of the good. In this case,
logarithmic regret is achievable when competing against well behaved
adversaries. In the adversarial model, the goods need not be identical and we
simply compare our performance against that of the best fixed bid in hindsight.
We show that sublinear regret is also achievable in this case and prove
matching minimax lower bounds. To our knowledge, this is the first complete set
of strategies for bidders participating in auctions of this type.
