Approximately Dominating Sets in Elections

TL;DR

This work tackles Condorcet paradoxes by proving that every election admits a small, approximately dominating committee: for any ε>0, a set of size O(1/ε^2) suffices to ensure no outside candidate beats all members by more than ε. The authors achieve this via a probabilistic construction: sample from a maximal lottery, a Nash-equilibrium-derived distribution, and show that the resulting empirical distribution has constant support that closely approximates the maximal lottery using sharp concentration tools (Massart/Dvoretzky–Kiefer–Wolfowitz bounds). The results bridge social choice and learning theory, providing a concrete method to obtain small, effective committees and revealing that certain Nash equilibria can be approximated with much smaller supports than general theory would suggest. They also connect to related work on theta-winning sets, intransitive dice, and concurrent developments offering broader insights into the structure of dominating sets in tournaments.

Abstract

Condorcet's paradox is a fundamental result in social choice theory which states that there exist elections in which, no matter which candidate wins, a majority of voters prefer a different candidate. In fact, even if we can select any $k$ winners, there still may exist another candidate that would beat each of the winners in a majority vote. That is, elections may require arbitrarily large dominating sets. We show that approximately dominating sets of constant size always exist. In particular, for every $\varepsilon > 0$, every election (irrespective of the number of voters or candidates) can select $O(\frac{1}{\varepsilon ^2})$ winners such that no other candidate beats each of the winners by a margin of more than $\varepsilon$ fraction of voters. Our proof uses a simple probabilistic construction using samples from a maximal lottery, a well-studied distribution over candidates derived from the Nash equilibrium of a two-player game. In stark contrast to general approximate equilibria, which may require support logarithmic in the number of pure strategies, we show that maximal lotteries can be approximated with constant support size. These approximate maximal lotteries may be of independent interest.

Figures (3)

• Figure 1: A visual of how the ranks change from $D_{\textnormal{ML}}$ to $\widehat{D}$ for a particular voter $v$ with ranking $a\succ b\succ c \succ d$. $D_{\textnormal{ML}}$ chooses $a, b, c, d$ with probability $0.2,0.3,0.1,0.4$ respectively, and these correspond to the widths of each segment in the top line. The yellow points represent the samples $X_1, X_2, X_3$, and the arrows indicate how the ranks of each candidate change. For example, $\mathop{\mathrm{rank}}\nolimits_v(a;D_{\textnormal{ML}}) = 0.8$ and $\mathop{\mathrm{rank}}\nolimits_v(a;\widehat{D}) = \frac{2}{3}$.
• Figure 2: A blue point at $(k^*, \alpha^*)$ implies that every election has an $\alpha^*$-dominating set of at most $k^*$ candidates (i.e, $k_\alpha \leq k^*$ for $\alpha \leq \alpha^*$). These points are derived by evaluating $\delta(k)$ directly, and the red line compares to the estimate from \ref{['lem:DKW']} that $\delta(k) \approx \sqrt{\frac{\pi}{8k}}$.
• Figure 3: A visual of the preferences for voters $v_j$ and $u_j$. Their favorite candidates are on the left, and their least favorite candidates are on the right.

Theorems & Definitions (10)

• Definition 1: Dominating sets
• Definition 2: Condorcet winning sets elkind2011choosingelkind2015condorcet
• Theorem 1
• Lemma 2
• proof : Proof of \ref{['lem:DKW']}
• proof
• Remark 1
• Remark 2
• Theorem 2
• proof