Six Candidates Suffice to Win a Voter Majority

TL;DR

This work investigates Condorcet-style stability in multi-winner elections by asking whether a small committee can be guaranteed to beat any single alternative by a majority. It proves a general existence theorem: for any $k\ge2$ and $\alpha\in(0,1]$, if $\frac{\alpha}{1 - \ln \alpha} \ge \frac{2}{k+1}$, there exists an $\alpha$-undominated committee of size $k$, yielding a concrete corollary that Condorcet dimension is at most $6$. The proof combines a probabilistic method with von Neumann's minimax theorem, introducing a distribution over committees (stable lotteries), a rank-based activation function $g$, and three technical innovations to handle small $k$ and fine-grained voter accounting. The results bridge to approximate stability literature, offering improved bounds for large $k$ and providing a framework that could inform practical algorithmic approaches to selecting small, representative winning sets in elections and related decision problems. They also outline several avenues for extending the theory beyond ranked preferences and for developing constructive, scalable algorithms.

Abstract

A cornerstone of social choice theory is Condorcet's paradox which says that in an election where $n$ voters rank $m$ candidates it is possible that, no matter which candidate is declared the winner, a majority of voters would have preferred an alternative candidate. Instead, can we always choose a small committee of winning candidates that is preferred to any alternative candidate by a majority of voters? Elkind, Lang, and Saffidine raised this question and called such a committee a Condorcet winning set. They showed that winning sets of size $2$ may not exist, but sets of size logarithmic in the number of candidates always do. In this work, we show that Condorcet winning sets of size $6$ always exist, regardless of the number of candidates or the number of voters. More generally, we show that if $\fracα{1 - \ln α} \geq \frac{2}{k + 1}$, then there always exists a committee of size $k$ such that less than an $α$ fraction of the voters prefer an alternate candidate. These are the first nontrivial positive results that apply for all $k \geq 2$. Our proof uses the probabilistic method and the minimax theorem, inspired by recent work on approximately stable committee selection. We construct a distribution over committees that performs sufficiently well (when compared against any candidate on any small subset of the voters) so that this distribution must contain a committee with the desired property in its support.

Figures (5)

• Figure 1: A visualization of the ranks, in the election from \ref{['tab:cyclic']}, and $\Delta$ which chooses the committees $\{1, 4\}, \{2, 5\}, \{3, 6\}$ with probabilities 0.45, 0.35, 0.2 respectively. The ranks of candidate $1$ are shown in red lines. One way to see that at most 2 voters prefer $1$ over $\{3, 6\}$ is that the three lowest ranks for $\{3, 6\}$ have a higher total than the three highest ranks for $1$. ($1.05$ from $v_1, v_2, v_4$ vs. $0.9$ from $v_1, v_6,$ and any third.)
• Figure 2: A depiction of \ref{['eq:voter-contribution']} as a Riemann sum. In the example, we suppose that $v \in U_1, U_2$. Note that the total width of the grey boxes is precisely $w(v)$.
• Figure 3: A depiction of \ref{['eq:high-per-voter']} as a Riemann sum. In the example, $v$ has the preference $S_1 \succeq S_2 \succeq S_3$, and $v \in U_{S_2}, U_{S_3}$. Note that the total width of the grey boxes is precisely $w(v)$.
• Figure 4: The key transformation from $h$ to $\widehat{h}$ that maintains \ref{['eq:condition-h']}. After a linear transformation, the green curve becomes a shifted ReLU function.
• Figure 5: Comparison between the guarantee of \ref{['thm:stronger']}, the lower bound, and the upper bound implied by jiang2020approximately for $k\leq 100$.

Theorems & Definitions (25)

• Theorem 1
• Definition 1: Approximately stable committees jiang2020approximately
• Theorem 2
• Definition 2: $\alpha$-undominated and $\alpha$-dominated
• Theorem 3: Minimax Theorem v1928theorie
• Theorem 4
• proof : Proof of \ref{['thm:main']} assuming \ref{['thm:meat']}
• Definition 3
• Lemma 1: Candidates are low
• Lemma 2: A committee is high