Abstract
We put forward a formal model of participatory budgeting where projects can incur costs with respect to several different resources, such as money, energy, or emission allowances. We generalise several well-known mechanisms from the usual single-resource setting to this multi-resource setting and analyse their algorithmic efficiency, the extent to which they are immune to strategic manipulation, and the degree of proportional representation they can guarantee. We also prove a general impossibility theorem establishing the incompatibility of proportionality and strategyproofness for this model.
Keywords
- Computational social choice
- Participatory budgeting
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Notes
- 1.
Note that negative costs can be appropriate as well (e.g., planting trees has “negative environmental cost”). We shall occasionally comment on the effects of doing so.
- 2.
Observe that for condition (i) it is important to count the number of voters who vote for S exactly rather than those who vote for a (not necessarily proper) superset of S. Indeed, weakening the conditions for the applicability of the axiom in this sense would immediately render it impossible to satisfy in general. To see this, consider a single-resource scenario in which we need to divide a budget of \(b=2\) amongst three projects of cost 1, and in which there are two voters, with approval ballots \(A_1 = \{p_1,p_2\}\) and \(A_2 = \{p_3\}\). Then each project forms a singleton set S for which \(n \cdot \frac{c(S)}{b} = 1\), while \(|\{i \in N : A_i \supseteq S\}| = 1\). But we cannot select all three projects.
- 3.
Note that dropping condition (ii) would render this axiom unsatisfiable in general, since sets satisfying the first condition can exceed the budget for some \(k\not \in R\).
- 4.
We are able to circumvent the need for this additional efficiency requirement because we do not impose exhaustiveness (which in multiwinner voting is an implicit part of the basic model). This gives us more freedom for the inductive lemmas we need to prove. At the same time, our result is weaker than that of Peters in other respects: his proportionality axiom is subtly weaker (as it needs to be imposed only for so-called party-list profiles) and his result applies even under subset preferences.
- 5.
We found these 14 profiles and the derivation of Table 1 by first encoding the requirements of F as a set of clauses in propositional logic, and then applying a SAT-solver to that set to compute a minimally unsatisfiable set exhibiting the impossibility of finding a mechanism of the required kind. For an introduction to this approach, the reader may wish to consult the expository article of Geist and Peters [8].
- 6.
Observe that \(F'\) might not be exhaustive, with the implications discussed above.
- 7.
The question of whether these constraints can be relaxed is of some technical interest, but arguably less relevant to practice. Indeed, we would want our mechanism to work for arbitrary numbers of voters (including those that are multiples of a budget limit).
- 8.
When all resources are relevant (in the single-resource case for instance ), there is a trivial mechanism of this kind: simply return the union of all singletons satisfying condition (i) in the definition of proportionality. To see this, recall that condition (ii) is vacuous if there are only relevant resources.
- 9.
Recall that the approval score of a set S for a given profile \(\boldsymbol{A}\) is defined as \(s_{\boldsymbol{A}}(S) = \sum _{i\in N} |S\cap A_i|\), and that \(F_{\textsc {m}}\) seeks to maximise that score.
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Acknowledgements
This paper grew out of a student project that started in the context of a course on computational social choice at the University of Amsterdam in 2020. We acknowledge the contribution of Gerson Foks during that initial stage. We also would like to thank several anonymous reviewers for their feedback.
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Motamed, N., Soeteman, A., Rey, S., Endriss, U. (2022). Participatory Budgeting with Multiple Resources. In: Baumeister, D., Rothe, J. (eds) Multi-Agent Systems. EUMAS 2022. Lecture Notes in Computer Science(), vol 13442. Springer, Cham. https://doi.org/10.1007/978-3-031-20614-6_19
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