  We study a budget-aggregation setting in which a number of voters report
their ideal distribution of a budget over a set of alternatives, and a
mechanism aggregates these reports into an allocation. Ideally, such mechanisms
are truthful, i.e., voters should not be incentivized to misreport their
preferences. For the case of two alternatives, the set of mechanisms that are
truthful and additionally meet a range of basic desiderata (anonymity,
neutrality, and continuity) exactly coincides with the so-called moving-phantom
mechanisms, but whether this space is richer for more alternatives was
repeatedly stated as an open question. We answer this question in the
affirmative by presenting a class of truthful mechanisms that are not
moving-phantoms but satisfy the three properties. Since moving-phantom
mechanisms can only provide limited fairness guarantees (measured as the
worst-case distance to a fair share solution), one motivation for broadening
the class of truthful mechanisms is the hope for improved fairness guarantees.
We dispel this hope by showing that lower bounds holding for the class of
moving-phantom mechanisms extend to all truthful, anonymous, neutral, and
continuous mechanisms.
