Efficiency Guarantees in Auctions with Budgets

In settings where players have a 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 then 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 incentive-compatible auction that always outputs a Pareto-efficient solution, or that no truthful mechanism can always guarantee a Pareto-efficient outcome. Traditionally, impossibility results can be avoided by considering approximations. However, Pareto-efficiency is a binary property (is either satisfied or not), which does not allow for approximations. In this paper we propose a new notion of efficiency, called \emph{liquid welfare}. This is the maximum amount of revenue an omniscient seller would be able to extract from a certain instance. We explain the intuition behind this objective function and show that it can be 2-approximated by two different auctions. Moreover, we show that no truthful algorithm can guarantee an approximation factor better than 4/3 with respect to the liquid welfare, and provide a truthful auction that attains this bound in a special case. Importantly, the liquid welfare benchmark also overcomes impossibilities for some settings. While it is impossible to design Pareto-efficient auctions for multi-unit auctions where players have decreasing marginal values, we give a deterministic $O(\log n)$-approximation for the liquid welfare in this setting.