Efficiency Guarantees in Auctions with Budgets

In settings where players have limited access to liquidity, represented in the form of budget constraints, efficiency maximization has proven to be a challenging goal. In particular, the social welfare cannot be approximated by a better factor than the number of players. Therefore, the literature has mainly resorted to Pareto-efficiency as a way to achieve efficiency in such settings. While successful in some important scenarios, in many settings it is known that either exactly one truthful auction that always outputs a Pareto-efficient solution, or that no truthful mechanism always outputs a Pareto-efficient outcome. Moreover, since Pareto-efficiency is a binary property (is either satisfied or not), it cannot be circumvented as usual by considering approximations. To overcome impossibilities in important setting such as multi-unit auctions with decreasing marginal values and private budgets, we propose a new notion of efficiency, which we call liquid welfare. This is the maximum amount of revenue an omniscient seller would be able to extract from a certain instance. For the aforementioned setting, we give a deterministic O(log n)-approximation for the liquid welfare in this setting. We also study the liquid welfare in the traditional setting of additive values and public budgets. We present two different auctions that achieve a 2-approximation to the new objective. Moreover, we show that no truthful algorithm can guarantee an approximation factor better than 4/3 with respect to the liquid welfare.