We give new bounds for the single-nomination model of impartial selection, a problem proposed by Holzman and Moulin (Econometrica, 2013). A selection mechanism, which may be randomized, selects one individual from a group of $n$ based on nominations among members of the group; a mechanism is impartial if the selection of an individual is independent of nominations cast by that individual, and $\alpha$-optimal if under any circumstance the expected number of nominations received by the selected individual is at least $\alpha$ times that received by any individual. In a many-nominations model, where individuals may cast an arbitrary number of nominations, the so-called permutation mechanism is $1/2$-optimal, and this is best possible. In the single-nomination model, where each individual casts exactly one nomination, the permutation mechanism does better and prior to this work was known to be $67/108$-optimal but no better than $2/3$-optimal. We show that it is in fact $2/3$-optimal for all $n$. This result is obtained via tight bounds on the performance of the mechanism for graphs with maximum degree $\Delta$, for any $\Delta$, which we prove using an adversarial argument. We then show that the permutation mechanism is not best possible; indeed, by combining the permutation mechanism, another mechanism called plurality with runner-up, and some new ideas, $2105/3147$-optimality can be achieved for all $n$. We finally give new upper bounds on $\alpha$ for any $\alpha$-optimal impartial mechanism. They improve on the existing upper bounds for all $n\geq 7$ and imply that no impartial mechanism can be better than $76/105$-optimal for all $n$; they do not preclude the existence of a $(3/4-\varepsilon)$-optimal impartial mechanism for arbitrary $\varepsilon>0$ if $n$ is large.

翻译：我们给出了单提名公正选择模型的新界，该问题由Holzman和Moulin（Econometrica，2013）提出。选择机制（可能带有随机性）根据群体内成员之间的提名从$n$个个体中选出一人；若机制对个体的选择独立于该个体所投出的提名，则称其为公正的；若在任何情况下，被选个体获得的期望提名数至少是任意个体所得提名数的$\alpha$倍，则称其为$\alpha$-最优的。在允许多提名的模型中（个体可投任意数量的提名），所谓的置换机制是$1/2$-最优的，且此界是最优的。在单提名模型中（每个个体恰好投出一票），置换机制表现更好，且在此工作之前已知其是$67/108$-最优的，但不会优于$2/3$-最优。我们证明该机制对所有$n$实际上都是$2/3$-最优的。这一结果是通过对任意最大度$\Delta$的图（我们使用对抗性论证证明）给出该机制的紧界而获得的。然后我们证明置换机制并非最优；事实上，通过结合置换机制、另一种称为“得票次高者”的机制以及一些新思路，对所有$n$可实现$2105/3147$-最优性。最后，我们给出了任意$\alpha$-最优公正机制的新的上界。这些上界对所有$n\geq 7$改进了现有上界，并意味着没有任何公正机制能对所有$n$优于$76/105$-最优；它们不排除存在对任意$\varepsilon>0$的$(3/4-\varepsilon)$-最优公正机制（当$n$足够大时）的可能性。