d4-conf-aaai-2026-2003

I’d make a focused case around continuous-population online Nash-welfare selection, anchored by Theorem 6. Theorem 6, proved in this paper, says that Online Nash Rule returns a feasible committee whose expected Nash Welfare is at least \((1-1/e)/7\) of the offline maximum. The paper’s offline NP-hardness statement for Nash Welfare is attributed to Ramezani and Endriss (2009), rather than proved in a numbered result here, so I would not use it as an anchor or claim this mirror dissolves a hardness result.

The scenario is a large membership organization selecting a fixed-size slate from candidates shown in a randomized order. Members have cardinal evaluations, but many share the same complete evaluation profile. For example, a platform might have millions of members and a few dozen recurring preference types. This is a regime claim about that organization, not about hiring committees in general; the paper’s two-person committee example is too small to motivate a population limit.

Here is the mirror I would hand to a prover. There are \(m\) candidates, a committee limit \(k\), and \(\tau\) voter types. Type \(t\) has rational mass \(\mu_t\), with \(\sum_t\mu_t=1\), and a complete cardinal ballot \(u_t:C\to\mathbb{Q}_{\ge 0}\). Candidates arrive in a uniformly random order. When candidate \(c\) arrives, the policy sees \(u_t(c)\) for each type and must irrevocably accept or reject \(c\), selecting at most \(k\) candidates. For a committee \(W\), define continuous Nash Welfare by \(\mathrm{NW}_\mu(W)=\sum_t\mu_t\log(1+\sum_{c\in W}u_t(c))\), and let \(\mathrm{OPT}_\mu=\max_{|W|\le k}\mathrm{NW}_\mu(W)\). The problem, Continuous-Population Online Nash-Welfare Selection, is to find a bit-polynomial online policy with \(\mathbb{E}[\mathrm{NW}_\mu(W)]\ge (1-1/e)\mathrm{OPT}_\mu/7\), where the expectation is over arrival order and any policy randomness. The policy’s solution is an online accept/reject rule; the committee remains integral.

This is a direct high-multiplicity version of the paper’s problem. If a finite electorate has \(n_t\) members of each type and \(\mu_t=n_t/n\), then \(\mathrm{NW}_\mu(W)\) is exactly the paper’s Nash Welfare divided by \(n\). Thus the approximation ratio is unchanged, and grouping repeated ballots preserves the objective exactly. The population is what becomes continuous; candidates, arrival decisions, and committee slots do not. I’d expect the tractable side of the trichotomy: Online Nash Rule’s welfare calculations can be written as type-weighted sums, so the theorem’s argument appears to carry over. The natural follow-up questions are whether the guarantee holds for arbitrary rational masses in the standard bit model, what the best achievable ratio is as a function of \(\tau\), and how much the result depends on random arrival.

The authors should recognize this as their problem: it keeps cardinal ballots, irrevocable online selection, committee size, and the Nash objective intact. The paper also presents Online Nash Rule as a polynomial-time algorithm, so this is a meaningful algorithmic mirror rather than a fractional relaxation of the committee problem. My weakest point is that the paper’s result already gives an approximation for finite electorates. The continuous formulation may mainly provide a concise type-weighted representation and a clearer high-multiplicity regime, rather than expose a new complexity boundary. The required scenario also assumes a small set of complete, shared evaluation profiles and randomized candidate order, both of which need defending in an application. This case covers Theorem 6 only.