On consistency of optimal pricing algorithms in repeated posted-price auctions with strategic buyer

We study revenue optimization learning algorithms for repeated posted-price auctions where a seller interacts with a single strategic buyer that holds a fixed private valuation for a good and seeks to maximize his cumulative discounted surplus. For this setting, first, we propose a novel algorithm that never decreases offered prices and has a tight strategic regret bound in Θ(log log T ) under some mild assumptions on the buyer surplus discounting. This result closes the open research question on the existence of a no-regret horizon-independent weakly consistent pricing. The proposed algorithm is inspired by our observation that a double decrease of offered prices in a weakly consistent algorithm is enough to cause a linear regret. This motivates us to construct a novel transformation that maps a right-consistent algorithm to a weakly consistent one that never decreases offered prices. Second, we outperform the previously known strategic regret upper bound of the algorithm PRRFES, where the improvement is achieved by means of a finer constant factor C of the principal term C log log T in this upper bound. Finally, we generalize results on strategic regret previously known for geometric discounting of the buyer’s surplus to discounting of other types, namely: the optimality of the pricing PRRFES to the case of geometrically concave decreasing discounting; and linear lower bound on the strategic regret of a wide range of horizonindependent weakly consistent algorithms to the case of arbitrary discounts. ∗16, Leo Tolstoy St., Moscow, Russia, 119021 (www.yandex.com) †Lomonosov Moscow State University, Faculty of Mechanics and Mathematics; GSP-1, 1 Leninskiye Gory, Main Building, Moscow, Russia, 119991 1 ar X iv :1 70 7. 05 10 1v 1 [ cs .G T ] 1 7 Ju l 2 01 7