In the impartial selection problem, a subset of agents up to a fixed size $k$ among a group of $n$ is to be chosen based on votes cast by the agents themselves. A selection mechanism is impartial if no agent can influence its own chance of being selected by changing its vote. It is $\alpha$-optimal if, for every instance, the ratio between the votes received by the selected subset is at least a fraction of $\alpha$ of the votes received by the subset of size $k$ with the highest number of votes. We study deterministic impartial mechanisms in a more general setting with arbitrarily weighted votes and provide the first approximation guarantee, roughly $1/\lceil 2n/k\rceil$. When the number of agents to select is large enough compared to the total number of agents, this yields an improvement on the previously best known approximation ratio of $1/k$ for the unweighted setting. We further show that our mechanism can be adapted to the impartial assignment problem, in which multiple sets of up to $k$ agents are to be selected, with a loss in the approximation ratio of $1/2$.

翻译：在无偏选择问题中，需要根据代理人自身投出的选票从一组$n$个代理人中选出不超过固定规模$k$的子集。若选择机制中没有任何代理人能通过更改其投票影响自身被选中的概率，则该机制称为无偏的。若对于所有实例，被选子集获得的票数至少达到获得最高票数的$k$规模子集票数的$\alpha$倍，则称该机制为$\alpha$-最优的。本文研究了更具一般性的带任意权重投票场景中的确定性无偏机制，并首次给出了近似保证约$1/\lceil 2n/k\rceil$。当待选代理人数量相对于总代理人数量足够大时，该结果改进了无权重设置下先前已知最优的$1/k$近似比。我们进一步证明该机制可适用于无偏分配问题（需同时选择多个不超过$k$个代理人的子集），此时近似比损失因子为$1/2$。