New mechanism for repeated posted price auction with a strategic buyer without discounting

On ad exchange platforms the place for advertisement is sold through different kinds of auctions. However, it is not uncommon the situation where the seller repeatedly encounters only one buyer, thus the posted price auction degenerates into a monopoly-monopsony game with asymmetric information and nearly an infinite number of rounds; on each round the seller proposes the price and the buyer accepts or rejects it. I learned this problem from a discussion with members of Yandex research team and my main motivation was to find an incentive-compatible seller's strategy. In this short paper such a strategy is proposed and a corresponding distortion at the top type lower bound (Spence-Mirrlees property, actually) for the surplus of the buyer is established; this shows that the proposed strategy is the best possible. The key ingredients are the following. The main leash that the buyer has is the frequency of accepted deals. Once this frequency (as a function on the buyer's type) is fixed, the strategy randomly chooses between the {\it rewarding} price which incentivises the buyer to reveal his type (the higher the type, the more average surplus the buyer has), the {\it adaptation} price which allows the buyer to communicate that his type is higher then the current guess of the cook, and the {\it type confirmation} price which disincentivises the buyer to pretend that his type is higher than it is.Comment: completely rewritte

Optimal Pricing in Repeated Posted-Price Auctions

We study revenue optimization pricing algorithms for repeated posted-price auctions where a seller interacts with a single strategic buyer that holds a fixed private valuation. We show that, in the case when both the seller and the buyer have the same discounting in their cumulative utilities (revenue and surplus), there exist two optimal algorithms. The first one constantly offers the Myerson price, while the second pricing proposes a "big deal": pay for all goods in advance (at the first round) or get nothing. However, when there is an imbalance between the seller and the buyer in the patience to wait for utility, we find that the constant pricing, surprisingly, is no longer optimal. First, it is outperformed by the pricing algorithm "big deal", when the seller's discount rate is lower than the one of the buyer. Second, in the inverse case of a less patient buyer, we reduce the problem of finding an optimal algorithm to a multidimensional optimization problem (a multivariate analogue of the functional used to determine Myerson's price) that does not admit a closed form solution in general, but can be solved by numerical optimization techniques (e.g., gradient ones). We provide extensive analysis of numerically found optimal algorithms to demonstrate that they are non-trivial, may be non-consistent, and generate larger expected revenue than the constant pricing with the Myerson price.Comment: 29 pages, 8 figure

Reserve Pricing in Repeated Second-Price Auctions with Strategic Bidders

We study revenue optimization learning algorithms for repeated second-price auctions with reserve where a seller interacts with multiple strategic bidders each of which holds a fixed private valuation for a good and seeks to maximize his expected future cumulative discounted surplus. We propose a novel algorithm that has strategic regret upper bound of $O(\log\log T)$ for worst-case valuations. This pricing is based on our novel transformation that upgrades an algorithm designed for the setup with a single buyer to the multi-buyer case. We provide theoretical guarantees on the ability of a transformed algorithm to learn the valuation of a strategic buyer, which has uncertainty about the future due to the presence of rivals.Comment: 22 page