Single Item Auctions with Discrete Action Spaces

An implicit assumption in truthful mechanism design is that revelation of one’s true type is always feasible. Indeed, this is not a problem in standard mechanism design setups, where it is up to the designer to determine the action spaces. However, this assumption fails to hold in many practical scenarios, where there are natural, exogenous constraints on the set of possible actions. For example, in combinatorial auctions [5] where bidders have combinatorial preferences on bundles of items, truthful revelation of such a preference requires a bidder to communicate a value on each subset of items, results in an exponential blow up in communication complexity. A practical combinatorial auction often imposes constraints on the number of package bids a bidder can place. The above observations motivate an active line of research that concerns the expressiveness of mechanisms [1, 2, 3, 4]. We consider a practical single-item auction design setting with restricted expressiveness. In particular, the action space of all bidders is restricted to a set of discrete bid levels, while the values are in continuous spaces. With this interface, truth revelation is not feasible and the revelation principle fails to hold (We will address this point later). Tailored for this setting, we put forward an auction, coined the extended second price auction (ESP). Our auction resembles the second price auction when there are multiple winners tied at the highest bid; however, when there is a unique winner, our auction charges a bit more than the second price auction. We show that, our auction satisfies the following desirable properties: