Multi-Armed Bandit Mechanisms for Multi-Slot Sponsored Search Auctions

In pay-per click sponsored search auctions which are currently extensively used by search engines, the auction for a keyword involves a certain number of advertisers (say k) competing for available slots (say m) to display their ads. This auction is typically conducted for a number of rounds (say T ). There are click probabilities μij associated with each agent-slot pairs. The goal of the search engine is to maximize social welfare of the advertisers, that is, the sum of values of the advertisers. The search engine does not know the true values advertisers have for a click to their respective ads and also does not know the click probabilities μijs. A key problem for the search engine therefore is to learn these click probabilities during the T rounds of the auction and also to ensure that the auction mechanism is truthful. Mechanisms for addressing such learning and incentives issues have recently been introduced and are aptly referred to as multi-armed-bandit (MAB) mechanisms. When m = 1, characterizations for truthful MAB mechanisms are available in the literature and it has been shown that the regret for such mechanisms will be O(T 2 3 ). In this paper, we seek to derive a characterization in the realistic but non-trivial general case when m > 1 and obtain several interesting results. Our contributions include: (1) When μijs are unconstrained , we prove that any truthful mechanism must satisfy strong pointwise monotonicity and show that the regret will be O(T ) for such mechanisms. (2) When the clicks on the ads follow a certain click precedence property , we show that weak pointwise monotonicity is necessary for MAB mechanisms to be truthful. (3) If the search engine has a certain coarse pre-estimate of μij values and wishes to update them during the course of the T rounds, we show that weak pointwise monotonicity and weakly separatedness are necessary and sufficient conditions for the MAB mechanisms to be truthful. (4) If the click probabilities are separable into agent specific and slot specific terms, we provide a characterization of MAB mechanisms that are truthful in expectation.