We extend the recently introduced framework of metric distortion to multiwinner voting. In this framework, n agents and m alternatives are located in an underlying metric space. The exact distances between agents and alternatives are unknown. Instead, each agent provides a ranking of the alternatives, ordered from the closest to the farthest. Typically, the goal is to select a single alternative that approximately minimizes the total distance from the agents, and the worst-case approximation ratio is termed distortion. In the case of multiwinner voting, the goal is to select a committee of k alternatives that (approximately) minimizes the total cost to all agents. We consider the scenario where the cost of an agent for a committee is her distance from the q-th closest alternative in the committee. We reveal a surprising trichotomy on the distortion of multiwinner voting rules in terms of k and q: The distortion is unbounded when $q⩽k/3$, asymptotically linear in the number of agents when $k/3<q⩽k/2$, and constant when $q>k/2$.

我们将最近引入的度量失真框架扩展到多赢者投票。在这个框架中，n 个代理和m个备选方案位于一个基础度量空间中。代理人和替代品之间的确切距离是未知的。相反，每个代理都提供备选方案的排名，从最近到最远的顺序排列。通常，目标是选择一个近似地最小化与代理的总距离的单一替代方案，最坏情况的近似比称为失真。在多赢者投票的情况下，目标是选择一个有k个备选方案的委员会，以（大约）最小化所有代理的总成本。我们考虑这样一种情况，委员会代理人的成本是她与q - 委员会中最接近的备选方案。我们在k和q方面揭示了多赢者投票规则失真的令人惊讶的三分法：失真是无限的，当$q⩽ķ/3$, 在代理数量上渐近线性，当$ķ/3<q⩽ķ/2$，并且当$q>ķ/2$.