Two Observations on Metric Distortion and Condorcet Winning Sets

In this research note we briefly connect two well-studied topics in computational social choice: metric distortion and the selection of undominated committees. In particular, we show that undominated committees are (in some sense) both necessary and sufficient to achieve bi-criteria metric distortion below $3$ (Banishashem et al., 2026). First, we show that any $\alpha$-undominated committee with $\alpha \le 0.5-\mathrm{\Omega }(1)$ has a bi-criteria metric distortion strictly of $3-\mathrm{\Omega }(1)$. In particular, this implies that a committee of size $5$ with distortion at most $2.7384$ exists. Secondly, we show that if a committee has a bi-criteria metric distortion strictly of $3-\mathrm{\Omega }(1)$, then it must also be $1-\mathrm{\Omega }(1)$-undominated.