Truthful aggregation of budget proposals with proportionality guarantees

We study a participatory budgeting problem, where a set of strategic agents wish to split a divisible budget among different projects, by aggregating their proposals on a single division. Unfortunately, the straightforward rule that divides the budget proportionally is susceptible to manipulation. Recently, a class of truthful mechanisms has been proposed, namely the moving phantom mechanisms. One such mechanism satisfies the proportionality property, in the sense that in the extreme case where all agents prefer a single project to receive the whole amount, the budget is assigned proportionally.

While proportionality is a naturally desired property, it is defined over a limited type of preference profiles. To address this, we expand the notion of proportionality, by proposing a quantitative framework that evaluates a budget aggregation mechanism according to its worst-case distance from the proportional allocation. Crucially, this is defined for every preference profile. We study this measure on the class of moving phantom mechanisms, and we provide approximation guarantees. For two projects, we show that the Uniform Phantom mechanism is optimal among all truthful mechanisms. For three projects, we propose a new, proportional mechanism that is virtually optimal among all moving phantom mechanisms. Finally, we provide impossibility results regarding the approximability of moving phantom mechanisms.