Winning in the Limit: Average-Case Committee Selection with Many Candidates
Yifan Lin, Shenyu Qin, Kangning Wang, Lirong Xia
TL;DR
This work analyzes average-case committee selection under the Impartial Culture model with a very large candidate set and a fixed number of voters. It reveals sharp phase transitions for two notions of collective dominance, showing that $α$-winning sets of size $k$ exist with high probability when $α<α_{ m win}^star$ and do not when $α>α_{ m win}^star$, while $α$-dominating sets exhibit a similar threshold at $α_{ m dom}^star$. The proofs introduce a score-based IC representation suitable for large $m$, establish a key lemma ruling out universal two-voter wins, and employ duality and rounding to handle dominating sets, yielding polynomial-time constructions in positive cases. The results provide tight thresholds and improve previous worst-case bounds, bridging average-case behavior with classical Condorcet-type questions. The findings have implications for designing reliable multi-winner outcomes in elections or recommender systems when the candidate space is vast and voter counts are comparatively small.
Abstract
We study the committee selection problem in the canonical impartial culture model with a large number of voters and an even larger candidate set. Here, each voter independently reports a uniformly random preference order over the candidates. For a fixed committee size $k$, we ask when a committee can collectively beat every candidate outside the committee by a prescribed majority level $α$. We focus on two natural notions of collective dominance, $α$-winning and $α$-dominating sets, and we identify sharp threshold phenomena for both of them using probabilistic methods, duality arguments, and rounding techniques. We first consider $α$-winning sets. A set $S$ of $k$ candidates is $α$-winning if, for every outside candidate $a \notin S$, at least an $α$-fraction of voters rank some member of $S$ above $a$. We show a sharp threshold at \[ α_{\mathrm{win}}^\star = 1 - \frac{1}{k}. \] Specifically, an $α$-winning set of size $k$ exists with high probability when $α< α_{\mathrm{win}}^\star$, and is unlikely to exist when $α> α_{\mathrm{win}}^\star$. We then study the stronger notion of $α$-dominating sets. A set $S$ of $k$ candidates is $α$-dominating if, for every outside candidate $a \notin S$, there exists a single committee member $b \in S$ such that at least an $α$-fraction of voters prefer $b$ to $a$. Here we establish an analogous sharp threshold at \[ α_{\mathrm{dom}}^\star = \frac{1}{2} - \frac{1}{2k}. \] As a corollary, our analysis yields an impossibility result for $α$-dominating sets: for every $k$ and every $α> α_{\mathrm{dom}}^\star = 1 / 2 - 1 / (2k)$, there exist preference profiles that admit no $α$-dominating set of size $k$. This corollary improves the best previously known bounds for all $k \geq 2$.