Improving Randomized Metric Distortion to 2.3282

In metric social choice, voters rank candidates by distance in an unknown metric space, while a randomized voting rule must use only these rankings. The paper introduces random-size stable lotteries and proves that, by mixing a suitably chosen random-size stable lottery with Integrated Veto, one obtains a randomized voting rule with metric distortion at most $11641/5000=2.3282$. This improves the previous best upper bound of $2.5$. The proof combines infinite-dimensional conic linear-programming duality, heuristic nonlinear optimization, and exact rational verification using polynomial nonnegativity in the Bernstein basis.

The paper proves that there exists a randomized voting rule using only ordinal rankings with metric distortion at most $11641/5000=2.3282$. This improves the previous best upper bound of $2.5$ and closes about $44\%$ of the gap to the known asymptotic lower bound of approximately $2.1126$. It does not determine the optimal randomized metric distortion: for $m\ge4$, the exact optimum and its asymptotic limit remain open.