3 problems

Let $\rho\in(0,1)$ and $n\geq k!+k-1$. Consider partitions $\Omega_1,\ldots,\Omega_k$ of $\mathbb{R}^n$ with Gaussian measure $\gamma_n(\Omega_i)=1/k$ that maximize the ranked-choi…

Let $S_k$ be the set of rankings of $k$ candidates, and let $f\colon S_k^n\to\Delta_k$ be a ranked-choice voting method, where $\Delta_k$ is the simplex of randomized outcomes on t…

A ranked vote is a ranking of $k$ candidates, and a voting method assigns a winning candidate to each profile of voters' rankings. A method is fair when candidates are treated symm…