We study budget aggregation, in which voters report their ideal distribution of a budget across a set of alternatives, and a mechanism aggregates these reports into an allocation. A central goal is truthfulness: voters should have no incentive to misreport their preferences. For this setting, the literature has identified a class of mechanisms called moving-phantom mechanisms, which are truthful and satisfy three basic desiderata: anonymity, neutrality, and continuity. This talk addresses three results in this setting.

(I) For two alternatives, moving-phantom mechanisms are precisely the mechanisms satisfying all four properties mentioned above. Whether additional mechanisms exist for more alternatives has been a recurring open question. We answer this question affirmatively by presenting a class of truthful, anonymous, neutral, and continuous mechanisms that are not moving-phantom mechanisms.

(II) The breadth of the moving-phantom class makes choosing among its members nontrivial. Surprisingly, we show that most moving-phantom mechanisms can nevertheless be linearly ordered by their social welfare guarantees.

(III) Moving-phantom mechanisms have been shown to offer limited fairness guarantees. We show that these limitations extend to truthful mechanisms more generally and present a non-truthful mechanism satisfying strong proportionality properties.

This is joint work with Mark de Berg, Javier Cembrano, Rupert Freeman, and Markus Utke.