  We consider the problem of allocating $m$ indivisible items to a set of $n$
heterogeneous agents, aiming at computing a proportional allocation by
introducing subsidy (money). It has been shown by Wu et al. (WINE 2023) that
when agents are unweighted a total subsidy of $n/4$ suffices (assuming that
each item has value/cost at most $1$ to every agent) to ensure proportionality.
When agents have general weights, they proposed an algorithm that guarantees a
weighted proportional allocation requiring a total subsidy of $(n-1)/2$, by
rounding the fractional bid-and-take algorithm. In this work, we revisit the
problem and the fractional bid-and-take algorithm. We show that by formulating
the fractional allocation returned by the algorithm as a directed tree
connecting the agents and splitting the tree into canonical components, there
is a rounding scheme that requires a total subsidy of at most $n/3 - 1/6$.
