A Hotelling-Downs Game for Strategic Candidacy with Binary Issues

There is a credible continuous mirror, and the paper’s strongest one is unusually direct: its own Section 3.2 already represents voters by a distribution over hypercube positions. The mirror replaces integer voter counts by rational population masses while preserving the candidates, their strategy sets, Hamming distance, plurality rule, tie-breaking, and local deviations exactly.

The regime is an election with many voters answering the same fixed battery of \(K\) binary issues. A voter type is a complete issue vector \(p\in H:=\{0,1\}^{K}\). Voters with the same \(p\) are indistinguishable for everything this game uses, so a society is a rational distribution \(\mu=(\mu_p)_{p\in H}\), where \(\mu_p\) is the fraction of voters of type \(p\). A natural instance representation is a sparse list of the nonzero masses. This fits, for example, a national election with millions of voters but a much smaller number of repeated issue profiles, or organized partisan and demographic blocs answering the same questionnaire. The candidates remain a small set of individually strategic agents; only the voter population is continuized.

For a candidate-position profile \(s=(s_1,\ldots,s_m)\), let \(M_s(p)\) be the candidates minimizing Hamming distance from \(p\). Candidate \(c_i\)'s score is \( \operatorname{sc}^{\mu}_s(c_i)=\sum_{p\in H}\mu_p\,\mathbf{1}[i\in M_s(p)]/|M_s(p)| \), and \(F_\mu(s)\) is the highest-scoring candidate under the paper’s fixed tie-breaking order. A \(t\)-local deviation is exactly a change from \(s_i\) to some \(s_i'\in H_i\) with \(\operatorname{dist}(s_i,s_i')\le t\). Thus the continuous model changes population counts into mass, not the strategic-candidacy question.

My lead anchor is Theorem 4.4, proved in this paper. It states that deciding whether a \(t\)-local equilibrium exists is NP-hard for \(t\in\{2,\ldots,K\}\), even with narcissistic preferences. The corresponding continuous problem is:

Continuous BSC \(t\)-Local-Equilibrium Existence. Given \(K\), a sparse rational distribution \(\mu\) over \(\{0,1\}^{K}\), candidates \(C\), explicitly represented strategy sets \(H_i\subseteq\{0,1\}^{K}\), truthful positions \(p_{c_i}\in H_i\), narcissistic candidate preferences, a tie-breaking order, and \(t\), decide whether there exists \(s\in\prod_iH_i\) such that no candidate has an improving deviation \(s_i'\in H_i\) with Hamming distance at most \(t\). A YES solution includes such a state \(s\).

I expect this problem to be Class B: hardness transfers. The X3C reduction in Theorem 4.4 already uses repeated voter positions with weighted multiplicities. Replace every listed voter count by a proportional rational mass and normalize the total mass to \(1\). The reduction has only \(3q+4\) voter types: the \(3q\) unit vectors and four special positions \(p_1,p_2,p_3,p_4\). All scores and score comparisons are merely scaled by the same normalization factor, so winners, improving deviations, and equilibrium existence are unchanged. The combinatorics live in the issue coordinates and the subset-candidates’ strategy sets, not in named voter identities. Consequently, the continuous problem remains NP-hard even for a sparse high-multiplicity society.

This is a particularly good mirror because the authors themselves say that the voters may equivalently be represented by a distribution \(f_N:H\to\mathbb N\). Normalizing that distribution is not a change of political meaning. A finite discrete electorate gives \(\mu_p=f_N(p)/n\), while a rational \(\mu\) can be expanded into a finite clone electorate after clearing denominators. The continuous problem is therefore the high-multiplicity version of exactly their game.

A second, independent anchor is Theorem 4.7, proved here. It states that when \(m=2\) and \(H_2\subseteq H_1\), a \(1\)-local equilibrium always exists and can be found in polynomial time. Its continuous counterpart is:

Continuous Nested-Strategy \(1\)-Local Equilibrium. Given a rational voter distribution \(\mu\) over \(\{0,1\}^{K}\), two candidates with explicitly represented strategy sets \(H_2\subseteq H_1\), truthful positions contained in those sets, the paper’s two-candidate preferences and tie-breaking rule, output a state \(s\in H_1\times H_2\) with no profitable unilateral Hamming-distance-one deviation.

I expect this to be Class A. The paper’s sequence of improving deviations uses only the hypercube geometry, the inclusion \(H_2\subseteq H_1\), and comparisons of candidate scores. Replacing integer counts by real masses preserves every such comparison. Each score is computed by summing over the support of \(\mu\), so the constructive proof gives an exact polynomial-time algorithm in the sparse-distribution input size and the strategy-set representation.

The natural questions are whether the guarantee extends to \(2\)-local equilibrium under the same inclusion condition—the paper explicitly leaves this open—and whether one can optimize among the guaranteed equilibria, for example by minimizing total candidate movement or preserving the truthful winner.

A third anchor is Theorem 4.8, also proved here. It states that when \(m=2\) and both candidates’ strategy sets are Hamming balls of radius one, a \(1\)-local equilibrium always exists and can be found in polynomial time. The continuous problem is:

Continuous Radius-One BSC Equilibrium. Given \(K\), a rational distribution \(\mu\) over \(\{0,1\}^{K}\), two truthful candidate positions \(p_{c_1},p_{c_2}\), with \(H_i=\{p:\operatorname{dist}(p,p_{c_i})\le1\}\), candidate preferences, and tie-breaking, output a \(1\)-local equilibrium in \(H_1\times H_2\).

This is again Class A. The proof’s matching and counting arguments remain valid for masses: statements such as “more than half of the voters lie on one side” become the corresponding inequalities between masses. There is also a direct polynomial algorithm: each ball has \(K+1\) positions, so there are only \(O(K^2)\) states, each with \(O(K)\) possible local deviations, and every score can be evaluated by summing over the support of \(\mu\).

This anchor raises a useful boundary question rather than merely reproducing a positive theorem. Proposition 4.9 shows that \(2\)-local equilibrium may fail even for radius-one balls. Its normalized rational-mass version therefore gives a sharp continuous comparison between \(1\)-local tractability and \(2\)-local nonexistence. One can ask whether larger-radius \(1\)-local games remain tractable, whether the answer changes for more than two candidates, and whether other voting rules preserve the boundary.

The mirror covers only these BSC equilibrium results. It does not claim to continuize the paper’s experiments, mean-field dynamics, judgment-aggregation extensions, or models with individually idiosyncratic voter information. It also does not claim that the positive theorems become novel merely because counts are written as masses. The value is that the same strategic-candidacy questions are now posed for a high-multiplicity society, with exact rational shares and a clean discrete-to-continuous dictionary.

The weakest point is that this may look almost too easy: the paper already has a histogram representation of voters, and all three proofs lift by replacing sums of counts with weighted sums. A referee could reasonably say that the continuous mirror exposes no new geometry and that the positive results are inherited rather than generated by continuous optimization. I would concede that. The case for the mirror does not rest on claiming a deep new Class A technique here. It rests on fidelity: the voter distribution is already the paper’s natural object, the high-multiplicity regime is realistic, Theorem 4.4’s hardness survives with only a few repeated types, and the positive and negative equilibrium boundaries can be studied exactly in the population-continuous formulation.