Truthful Learning Mechanisms for Multi-Slot Sponsored Search Auctions   with Externalities
  Sponsored search auctions constitute one of the most successful applications of microeconomic mechanisms. In mechanism design, auctions are usually designed to incentivize advertisers to bid their truthful valuations and to assure both the advertisers and the auctioneer a non-negative utility. Nonetheless, in sponsored search auctions, the click-through-rates (CTRs) of the advertisers are often unknown to the auctioneer and thus standard truthful mechanisms cannot be directly applied and must be paired with an effective learning algorithm for the estimation of the CTRs. This introduces the critical problem of designing a learning mechanism able to estimate the CTRs at the same time as implementing a truthful mechanism with a revenue loss as small as possible compared to an optimal mechanism designed with the true CTRs. Previous work showed that, when dominant-strategy truthfulness is adopted, in single-slot auctions the problem can be solved using suitable exploration-exploitation mechanisms able to achieve a per-step regret (over the auctioneer's revenue) of order $O(T^{-1/3})$ (where T is the number of times the auction is repeated). It is also known that, when truthfulness in expectation is adopted, a per-step regret (over the social welfare) of order $O(T^{-1/2})$ can be obtained. In this paper we extend the results known in the literature to the case of multi-slot auctions. In this case, a model of the user is needed to characterize how the advertisers' valuations change over the slots. We adopt the cascade model that is the most famous model in the literature for sponsored search auctions. We prove a number of novel upper bounds and lower bounds both on the auctioneer's revenue loss and social welfare w.r.t. to the VCG auction and we report numerical simulations investigating the accuracy of the bounds in predicting the dependency of the regret on the auction parameters. 
Nicola Gatti, Alessandro Lazaric, Marco Rocco, Francesco Trov\`o
cs.GT