We consider the problem of a single seller repeatedly selling a single item
to a single buyer (specifically, the buyer has a value drawn fresh from known
distribution D in every round). Prior work assumes that the buyer is fully
rational and will perfectly reason about how their bids today affect the
seller's decisions tomorrow. In this work we initiate a different direction:
the buyer simply runs a no-regret learning algorithm over possible bids. We
provide a fairly complete characterization of optimal auctions for the seller
in this domain. Specifically:
- If the buyer bids according to EXP3 (or any "mean-based" learning
algorithm), then the seller can extract expected revenue arbitrarily close to
the expected welfare. This auction is independent of the buyer's valuation D,
but somewhat unnatural as it is sometimes in the buyer's interest to overbid. -
There exists a learning algorithm A such that if the buyer bids
according to A then the optimal strategy for the seller is simply
to post the Myerson reserve for D every round. - If the buyer bids according
to EXP3 (or any "mean-based" learning algorithm), but the seller is restricted
to "natural" auction formats where overbidding is dominated (e.g. Generalized
First-Price or Generalized Second-Price), then the optimal strategy for the
seller is a pay-your-bid format with decreasing reserves over time. Moreover,
the seller's optimal achievable revenue is characterized by a linear program,
and can be unboundedly better than the best truthful auction yet simultaneously
unboundedly worse than the expected welfare