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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 optimal among all moving phantom mechanisms. Finally, we provide impossibility results regarding the approximability of moving phantom mechanisms.

We study the problem of impartial selection, a topic that lies at the intersection of computational social choice and mechanism design. The goal is to select the most popular individual among a set of community members. The input can be modeled as a directed graph, where each node represents an individual, and a directed edge indicates nomination or approval of a community member to another. An impartial mechanism is robust to potential selfish behavior of the individuals and provides appropriate incentives to voters to report their true preferences by ensuring that the chance of a node to become a winner does not depend on its outgoing edges. The goal is to design impartial mechanisms that select a node with an in-degree that is as close as possible to the highest in-degree. We measure the efficiency of such a mechanism by the difference of these in-degrees, known as its additive approximation.Following the success in the design of auction and posted pricing mechanisms with good approximation guarantees for welfare and profit maximization, we study the extent to which prior information on voters' preferences could be useful in the design of efficient deterministic impartial selection mechanisms with good additive approximation guarantees. We consider three models of prior information, which we call the opinion poll, the a priori popularity, and the uniform model. We analyze the performance of a natural selection mechanism that we call approval voting with default (AVD) and show that it achieves a O ( √ ln ) additive guarantee for opinion poll and a O (ln 2 ) for a priori popularity inputs, where is the number of individuals. We consider this polylogarithmic bound as our main technical contribution. We complement this last result by showing that our analysis is close to tight, showing an Ω(ln ) lower bound. This holds in the uniform model, which is the simplest among the three models.

Impartial selection has recently received much attention within the multi-agent systems community. The task is, given a directed graph representing nominations to the members of a community by other members, to select the member with the highest number of nominations. This seemingly trivial goal becomes challenging when there is an additional impartiality constraint, requiring that no single member can influence her chance of being selected. Recent progress has identified impartial selection rules with optimal approximation ratios. Moreover, it was noted that worstcase instances are graphs with few vertices. Motivated by this fact, we propose the study of additive approximation, the difference between the highest number of nominations and the number of nominations of the selected member, as an alternative measure of the quality of impartial selection.Our positive results include two randomized impartial selection mechanisms which have additive approximation guarantees of Θ( √ n) and Θ(n 2/3 ln 1/3 n) for the two most studied models in the literature, where n denotes the community size. We complement our positive results by providing negative results for various cases. First, we provide a characterization for the interesting class of strong sample mechanisms, which allows us to obtain lower bounds of n − 2, and of Ω( √ n) for their deterministic and randomized variants respectively. Finally, we present a general lower bound of 2 for all deterministic impartial mechanisms.

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