  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.
