Winning in the Limit: Average-Case Committee Selection with Many Candidates

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 $\alpha$. We focus on two natural notions of collective dominance, $\alpha$-winning and $\alpha$-dominating sets, and we identify sharp threshold phenomena for both of them using probabilistic methods, duality arguments, and rounding techniques. We first consider $\alpha$-winning sets. A set $S$ of $k$ candidates is $\alpha$-winning if, for every outside candidate $a \notin S$, at least an $\alpha$-fraction of voters rank some member of $S$ above $a$. We show a sharp threshold at \[ \alpha_{\mathrm{win}}^\star = 1 - \frac{1}{k}. \] Specifically, an $\alpha$-winning set of size $k$ exists with high probability when $\alpha \alpha_{\mathrm{win}}^\star$. We then study the stronger notion of $\alpha$-dominating sets. A set $S$ of $k$ candidates is $\alpha$-dominating if, for every outside candidate $a \notin S$, there exists a single committee member $b \in S$ such that at least an $\alpha$-fraction of voters prefer $b$ to $a$. Here we establish an analogous sharp threshold at \[ \alpha_{\mathrm{dom}}^\star = \frac{1}{2} - \frac{1}{2k}. \] As a corollary, our analysis yields an impossibility result for $\alpha$-dominating sets: for every $k$ and every $\alpha > \alpha_{\mathrm{dom}}^\star = 1 / 2 - 1 / (2k)$, there exist preference profiles that admit no $\alpha$-dominating set of size $k$. This corollary improves the best previously known bounds for all $k \geq 2$.