Prompt mechanisms for online auctions

We study the following online problem: at each time unit, one of m identical items is offered for sale. Bidders arrive and depart dynamically, and each bidder is interested in winning one item between his arrival and departure. Our goal is to design truthful mechanisms that maximize the welfare, the sum of the utilities of winning bidders. We first consider this problem under the assumption that the private information for each bidder is his value for getting an item. In this model constant-competitive mechanisms are known, but we observe that these mechanisms suffer from the following disadvantage: a bidder might learn his payment only when he departs. We argue that these mechanism are essentially unusable, because they impose several seemingly undesirable requirements on any implementation of the mechanisms. To crystalize these issues, we define the notions of prompt and tardy mechanisms. We present two prompt mechanisms, one deterministic and the other randomized, that guarantee a constant competitive ratio. We show that our deterministic mechanism is optimal for this setting. We then study a model in which both the value and the departure time are private information. While in the deterministic setting only a trivial competitive ratio can be guaranteed, we use randomization to obtain a prompt truthful Theta(1/log m)-competitive mechanism. We then show that no truthful randomized mechanism can achieve a ratio better than 1/2 in this model.