Candidate Nomination for Condorcet-consistent Voting Rules

The strongest case is a direct high-multiplicity mirror of the paper’s Possible President problem. The population becomes continuous; the parties’ nomination decision does not. That distinction preserves the paper’s computational core.

A natural regime is a national election with many voters but relatively few preference blocs. A type is a complete ranking of all potential nominees. If two voters have the same ranking, they are indistinguishable for every operation in the paper, so they are legitimately one type. The input is therefore a finite support \(\Theta\subseteq S_C\) of rankings with rational masses \(\mu_\theta\), where \(\mu_\theta\) is the fraction of the electorate of type \(\theta\). We expect \(\tau=|\Theta|\ll N\), while \(N\) may be arbitrarily large. Parties and candidate identities remain exactly as in the paper.

For \(a,b\in C\), define the weighted pairwise support

\[ w_\mu(a,b)=\sum_{\theta:\,a\succ_\theta b}\mu_\theta. \]

A nomination is still an indivisible choice \(\eta(P)\in P\) for every party \(P\). It is not permissible for a party to nominate \(30\%\) of one candidate and \(70\%\) of another. If \(C_\eta\) is the resulting nominee set, the voters’ rankings are restricted to \(C_\eta\), exactly as in the paper.

My lead problem is \(\mathrm{Possible\ President}^{\infty}\text{-}\mathrm{Maximin}\). Its instance consists of \(C\), a party partition \(\mathcal P\), a distinguished party \(P^\star\), a finite rational distribution \(\mu\) over complete rankings of \(C\), and the usual unique-winner convention. The question is whether there is a nomination function \(\eta\) such that \(p^\star=\eta(P^\star)\) is the unique winner, where

\[ \operatorname{MM}_{\mu,\eta}(a) = \min_{b\in C_\eta\setminus\{a\}} w_\mu(a,b) \]

and

\[ \operatorname{MM}_{\mu,\eta}(p^\star) > \operatorname{MM}_{\mu,\eta}(a) \]

for every other nominee \(a\).

This is a direct mirror of the paper’s problem. Clearing denominators in \(\mu\) produces a finite election with repeated voters, and every pairwise comparison, Maximin score, tie, nomination, and unique-winner certificate is preserved exactly. Conversely, collapsing identical voters in a finite election produces the distribution. The paper’s full model uses no voter identity beyond the ranking, so no relevant information is lost.

The anchor is Theorem 4.4 (★): “There exists an algorithm that solves Possible President for the Maximin voting rule and runs in FPT time with parameter \(t\), the number of parties.” This is the authors’ own result, with the starred proof supplied in the full version [18], rather than a theorem merely cited from elsewhere.

I expect the continuous problem above to remain fixed-parameter tractable in \(t=|\mathcal P|\), with running time \(f(t)\operatorname{poly}(m,\tau,L)\), where \(L\) is the rational encoding length. The proof architecture survives almost verbatim. Every integer count \(N_E(a,b)\) is replaced by the rational support \(w_\mu(a,b)\). The guessed Maximin value can be chosen from the finitely many supports \(w_\mu(p^\star,c)\). The guessed witness party \(\delta(P)\), the deletion steps, and the labelled digraph construction are unchanged, with comparisons such as \(N_E(c,c')<s^\star\) replaced by \(w_\mu(c,c')<s^\star\). Lemma 4.5 then supplies the same polynomial-time solution of the resulting bounded-indegree Partitioned Subdigraph Isomorphism instance.

This is a genuine Class A continuation: the continuous population does not merely restate the theorem; it exposes why the algorithm works. The algorithm depends on pairwise support inequalities and a party-indexed nomination structure, not on the fact that supports happen to be integers. Further questions include whether the FPT dependence on \(t\) can be improved, whether one can obtain FPT algorithms parameterized by the number of voter types \(\tau\), and what rounding guarantee relates an arbitrary rational distribution to a finite election with a prescribed winner margin.

A second, independent mirror is \(\mathrm{Possible\ President}^{\infty}\text{-}\mathrm{Maximin}[4]\), the restriction of the same problem to at most four voter types, each of mass \(1/4\), and maximum party size \(\sigma=2\). The question and solution certificate are exactly the ones just stated; only the population domain is restricted.

The anchor is Theorem 4.3 (★): “Possible President for Maximin is NP-complete even for instances where the number of voters is a fixed constant \(n\ge4\), and the maximum party size is \(\sigma=2\).” Again, this is proved by the authors in the full version [18].

Taking \(n=4\), any instance from the theorem becomes a continuous instance with four ranking types of mass \(1/4\). The weighted supports are exactly the original pairwise counts divided by \(4\), so the Maximin ordering and unique-winner condition are unchanged. Conversely, replicating every one of the four types \(r\) times gives an election with \(4r\) voters but the same normalized distribution. Thus this is not merely a four-person model: it describes arbitrarily large electorates consisting of four repeated preference blocs.

The expected classification is NP-complete, and the hardness is Class B rather than continuum-specific. The reduction’s combinatorics live in the candidate and party nomination structure, not in the multiplicity of voters. That is valuable evidence for the mirror: continuization does not erase the paper’s hard cases indiscriminately. A natural follow-up is to determine whether Maximin becomes easier when the candidate-side structure is restricted while \(\tau\) grows, and whether there are cases whose hardness genuinely depends on having many distinct voter types.

My third mirror is \(\mathrm{Possible\ President}^{\infty}\text{-}\mathrm{Copeland}_\alpha[4]\). Fix any constant rational \(\alpha\in[0,1]\). The instance again contains candidates, parties, a distinguished party, and four equally weighted complete ranking types. For a nomination \(\eta\), define

\[ q_\alpha(a,b)= \begin{cases} 1,&w_\mu(a,b)>1/2,\\ \alpha,&w_\mu(a,b)=1/2,\\ 0,&w_\mu(a,b)<1/2, \end{cases} \]

and

\[ \operatorname{Cpl}_{\mu,\eta}^{\alpha}(a) = \sum_{b\in C_\eta\setminus\{a\}}q_\alpha(a,b). \]

The question is whether some nomination makes the candidate selected by \(P^\star\) the unique maximizer of this score.

The anchor is Theorem 3.8 (★): for every constant \(\alpha\in[0,1]\), Possible President for Copeland\({}_\alpha\) is W[1]-hard when parameterized by the number \(t\) of parties, even with only four voters. This is also an authors’ result, with the starred proof in [18].

The theorem transfers exactly: four equal masses reproduce the four-voter majority relation, including ties, and therefore reproduce every Copeland\({}_\alpha\) score. The expected continuous classification is W[1]-hard parameterized by \(t\), while remaining in XP by enumerating the at most \(\sigma^t\) nomination choices. Here too the hardness is Class B: the reduction encodes a multicolored-clique choice among candidates and parties, not a population-multiplicity phenomenon. Questions generated by this mirror include whether joint parameterization by \(t+\tau\) helps, whether hardness persists with two or three voter types, and whether arbitrary rational masses create any new continuum-specific obstruction.

The scope is deliberately limited. These mirrors cover the paper’s Possible President results for Maximin and Copeland\({}_\alpha\), especially Theorems 4.3, 4.4, and 3.8. They do not claim to continuize the cited literature, Necessary President, the Nash-equilibrium questions, or every construction in the paper.

The weakest point is that the lead result is likely a weighted extension of the authors’ algorithm rather than a radically new continuous phenomenon. A referee could reasonably say that the continuous input is an exact high-multiplicity encoding and that Theorem 4.4 should transfer after a careful proof. I regard that as a limitation of novelty, not of legitimacy. The paper’s problem is explicitly anonymous in voters, pairwise support is its operative statistic, and party nomination remains fully discrete. This is precisely the setting in which a population mirror is most faithful: it tests whether their parameterized structure is robust when a large electorate is represented by rational type masses, while also preserving the paper’s NP-hard and W[1]-hard boundaries.