Determining Winners in Elections with Absent Votes

The strongest honest positive case is a narrow but very clean one: the paper’s Theorem 7 has a direct continuous-population mirror, and that mirror is tractable for exactly the structural reason the theorem is tractable.

My lead anchor is Theorem 7, proved in this paper: “For any \(\ell\ge2\) and \(\vec s_\ell\) such that \(a_2=\cdots=a_\ell\), WAV-\(\vec s_\ell\) can be determined in polynomial time.”

The natural regime is a large ranked-choice election with a fixed number \(m\) of candidates and fixed ballot length \(\ell\), but many voters. A type is a top-\(\ell\) ranking \(R\): an ordered list of \(\ell\) candidates, with all others tied below them. There are \(\tau=\frac{m!}{(m-\ell)!}\) possible types. In a national or statewide election, it is entirely plausible that \(n\gg\tau\): millions of voters may fall into a few hundred or few thousand repeated top-\(\ell\) ballot classes. Delayed mail ballots, polling uncertainty, or batches whose contents are not yet available provide a credible source of absent mass. The absence status is observation data; the type itself remains the complete preference description used by the voting rule.

I would call the continuous problem Continuous Top-\(\ell\) Absent-Mass Winner, or \(\mathrm{C\text{-}WAV}_{\vec s_\ell}\). Its instance consists of a candidate set \(M\), a fixed \(\ell\ge2\), a scoring vector \(\vec s_\ell=(a_1,a_2,\ldots,a_\ell)\) with \(a_2=\cdots=a_\ell=A\), a target candidate \(c\), a rational absent mass \(\alpha\in[0,1]\), and a rational known-mass distribution \(\mu\) over \(T_\ell(M)\) satisfying \(\sum_R\mu_R=1-\alpha\). The question is whether there exists a distribution \(\nu\) over \(T_\ell(M)\), with \(\sum_R\nu_R=\alpha\), such that \(c\) is the winner of the society \(\mu+\nu\), using the paper’s tie-breaking convention. A solution is the mass assignment \(\nu\); in the decision version, the objective is feasibility.

This is not a softened manipulation problem. It is exactly the paper’s problem after replacing integer profile counts by proportions: \(P\) becomes \(\mu\), \(t\) becomes \(\alpha\), and the unknown profile \(P'\) becomes \(\nu\). The voting rule, top-truncation convention, target candidate, and winner condition are unchanged. Only the anonymous electorate has been put into its high-multiplicity form.

The continuous version should be in Class A. Let \(S_\mu(a)\) be candidate \(a\)’s score from the known mass. For this scoring vector, any successful completion may assume that \(c\) is ranked first throughout the absent mass: moving \(c\) upward increases \(c\)’s score and does not increase anyone else’s score. The remaining \((\ell-1)\alpha\) mass consists of lower-ranked slots. Introduce variables \(z_{d,a}\), for \(d=2,\ldots,\ell\) and \(a\ne c\), where \(z_{d,a}\) is the mass placing \(a\) in position \(d\). They satisfy

\[ \sum_{a\ne c}z_{d,a}=\alpha \]

for each lower position \(d\), and

\[ \sum_{d=2}^{\ell}z_{d,a}\le\alpha \]

for every candidate \(a\), since a candidate can appear at most once in one absent ballot. The winner constraints are

\[ S_\mu(a)+A\sum_{d=2}^{\ell}z_{d,a} \le S_\mu(c)+\alpha a_1 \]

for every \(a\ne c\), with the appropriate strict or weak version if the tie-breaking order matters.

These are precisely fractional matching or flow constraints. A feasible flow decomposes into a distribution over valid top-\(\ell\) ballots, so it gives an actual \(\nu\), not merely inconsistent marginal statistics. Thus the continuous problem has a polynomial-size flow or linear-programming formulation. Its rational solutions also preserve the high-multiplicity bridge: multiplying by a common denominator produces a finite election with integer ballot counts.

The expected result is therefore a polynomial-time algorithm for \(\mathrm{C\text{-}WAV}_{\vec s_\ell}\), extending Theorem 7 from integer absent votes to rational absent mass. It also yields natural optimisation variants: the minimum absent mass that can make \(c\) win, the maximum unknown mass compatible with \(c\)’s victory, and a margin version requiring \(c\) to beat every rival by at least \(\delta\). The unresolved general-scoring case becomes a meaningful boundary question: when \(a_2,\ldots,a_\ell\) are not equal, does the continuous completion problem still admit a compact flow or configuration-LP formulation?

The paper’s authors should recognise this as their problem. Their voting rule is anonymous and unweighted; no individual identity, ballot-specific price, or voter-specific budget appears. The only information used about the electorate is how much mass has each ballot type. The continuous formulation therefore removes an irrelevant granularity while preserving the substantive question: can the missing portion of the electorate be completed so that the target wins?

The weakness is that this mirror covers only the paper’s special positional-scoring case. It does not yet cover the NP-completeness results for STV, Maximin, or Copeland. I would not claim hardness transfer for those theorems without further work: their RXC3 reductions encode discrete set selection, and fractional absent mass might satisfy the corresponding coverage constraints through uniform fractional mixtures even when no exact cover exists. Moreover, with fixed \(\ell\), the number of top-\(\ell\) types is already polynomial in \(m\), so the continuous formulation does not expose the exponential pricing phenomenon central to the broader ChoCo programme.

That limitation is real, but it does not undermine the mirror itself. Theorem 7 supplies a precise, plausible, genuinely population-continuous problem whose tractability follows from a structural insight plus continuous optimisation. For this paper, that is the strongest defensible positive case.