Value Alignment in Participatory Budgeting

On the literal anchor requirement, this paper has no qualifying anchor. It contains Definitions 1–15, but no numbered Theorem, Lemma, Corollary, or Proposition asserting an algorithmic or complexity result. The statement in Section 6 that VAPBP is an instance of multi-objective knapsack is neither a named result nor proved as a complexity theorem; the exact algorithms are cited from elsewhere. The closest conceptual anchors are Def. 9, VAPBP, and Def. 13, the Nash product rule, both introduced here.

The strongest positive case is therefore an unanchored but faithful mirror of Defs. 9 and 13. My lead problem would be Continuous-Nash-VAPBP.

An instance contains a finite proposal set \(P\), costs \(c(p)\), budget \(b\), exclusivity and generalisation relations \(R_x,R_g\), and the government’s value data \(V,\succeq,r,\operatorname{prom}\). It also contains a finite set \(T\) of complete voter types. A type \(t\) is a ballot \(B_t\subseteq P\), or, if ballot admissibility uses additional information, the ballot together with that information. The society is a rational distribution \(\mu\in\Delta_T\), where \(\mu_t\) is the fraction of citizens of type \(t\). Thus the support density of proposal \(p\) is \(s_\mu(p)=\sum_{t\in T}\mu_t\mathbf 1[p\in B_t]\).

Define \(a(p)=\sum_{v\in V}r(v)\operatorname{prom}(p,v)\). The decision variable remains an indivisible proposal set \(S\subseteq P\), so this does not continuize the outcome space. Let \(\mathcal F\) be the sets satisfying the paper’s budget, exclusivity, and generalisation constraints. The problem is to return any \(S^\star\in\mathcal F\) maximizing \( \left(\sum_{p\in S}s_\mu(p)\right)\left(\sum_{p\in S}a(p)\right) \), with the usual nonnegative-utility normalization for the Nash interpretation.

This is a natural high-multiplicity regime for participatory budgeting: a large city has many residents but a relatively small number of recurring support profiles, produced by neighbourhood associations, unions, advocacy groups, or standard issue coalitions. The relevant comparison is \(N\gg\tau\), where \(N\) is the number of citizens and \(\tau=|T|\) is the number of distinct ballots. This is not a claim that every Barcelona or Warsaw voter has an identical ballot; it is a plausible regime for a repeated, standardized proposal catalogue. It is especially faithful to the paper because the paper already discards individual voting details and uses only the aggregate function \(vot:P\to\mathbb N\).

The bridge is exact. Given discrete type counts \(n_t\), set \(\mu_t=n_t/N\). Then \(s_\mu(p)=vot(p)/N\), and normalization multiplies the Nash objective by the same positive constant for every proposal set. Conversely, the rational continuous instance can be viewed as a high-multiplicity election after clearing denominators.

I expect Continuous-Nash-VAPBP to be Class B: hardness transfers from the proposal-selection problem rather than disappearing through population aggregation. Even with no conflict relations and one value, it contains knapsack. Take one singleton-ballot type per proposal, set \(\mu_t\) proportional to the item’s profit, and choose promotion values so that \(a(p)=s_\mu(p)\). The Nash objective then becomes the square of total selected profit subject to the budget constraint. The combinatorics live in the proposal agenda, not in the number of citizens.

This mirror covers the paper’s central VAPBP, particularly Def. 9 and its Nash selection rule in Def. 13. I would not submit additional anchors for Def. 14 or Def. 15: they are also definitions, and adding them would obscure rather than strengthen the case. Natural follow-up questions include exact Pareto-front enumeration for \((\sum s_\mu(p),\sum a(p))\), robustness of the selected budget under perturbations of \(\mu\), and the minimum mass of citizens whose ballots must change to make a target proposal set Nash-selected.

The weakest point is substantial: for the paper’s formal model, \(\mu\) may be only a reparameterization of the already-aggregated vote counts. If nearly every citizen has a distinct ballot, then \(\tau\) is not small and the high-multiplicity story loses force. So this is a credible continuous population mirror, but not a new tractability result—and, strictly speaking, not a paper-anchored computational result because the paper proves none.