  We consider a participatory budgeting problem in which each voter submits a
proposal for how to divide a single divisible resource (such as money or time)
among several possible alternatives (such as public projects or activities) and
these proposals must be aggregated into a single aggregate division. Under
$\ell_1$ preferences -- for which a voter's disutility is given by the $\ell_1$
distance between the aggregate division and the division he or she most prefers
-- the social welfare-maximizing mechanism, which minimizes the average
$\ell_1$ distance between the outcome and each voter's proposal, is incentive
compatible (Goel et al. 2016). However, it fails to satisfy the natural
fairness notion of proportionality, placing too much weight on majority
preferences. Leveraging a connection between market prices and the generalized
median rules of Moulin (1980), we introduce the independent markets mechanism,
which is both incentive compatible and proportional. We unify the social
welfare-maximizing mechanism and the independent markets mechanism by defining
a broad class of moving phantom mechanisms that includes both. We show that
every moving phantom mechanism is incentive compatible. Finally, we
characterize the social welfare-maximizing mechanism as the unique
Pareto-optimal mechanism in this class, suggesting an inherent tradeoff between
Pareto optimality and proportionality.
