Improved Efficiency Guarantees in Auctions with Budgets

TL;DR

This paper introduces new auction mechanisms that improve efficiency guarantees for distributing divisible goods among budget-constrained agents, achieving better approximation ratios for liquid welfare in both public and private budget scenarios.

Contribution

It presents a novel truthful auction with a 1.618 approximation ratio for public budgets and the first constant ratio of 34 for private budgets, applicable to any valuation function.

Findings

Achieved a 1.618 approximation ratio for liquid welfare with public budgets.

Designed the first constant approximation auction (ratio 34) for private budgets.

Extended applicability to any valuation function, surpassing previous bounds.

Abstract

We study the efficiency guarantees in the simple auction environment where the auctioneer has one unit of divisible good to be distributed among a number of budget constrained agents. With budget constraints, the social welfare cannot be approximated by a better factor than the number of agents by any truthful mechanism. Thus, we follow a recent work by Dobzinski and Leme (ICALP 2014) to approximate the liquid welfare, which is the welfare of the agents each capped by her/his own budget. We design a new truthful auction with an approximation ratio of $\frac{\sqrt{5}+1}{2} \approx 1.618$, improving the best previous ratio of $2$ when the budgets for agents are public knowledge and their valuation is linear (additive). In private budget setting, we propose the first constant approximation auction with approximation ratio of $34$. Moreover, this auction works for any valuation function. Previously, only $O(\log n)$ approximation was known for linear and decreasing marginal (concave) valuations, and $O(\log^2 n)$ approximation was known for sub-additive valuations.