In approval-based multiwinner voting, voters express approval preferences over a set of candidates, and the goal is to return a winning committee. This model captures a broad range of subset selection problems under preferences. Prior work has focused on the study of binary proportionality axioms that certify whether a given committee is proportionally representative or not. We take a more fine-grained perspective and initiate the study of explanation systems that quantify how a committee represents the electorate, i.e., how much influence each voter exerts, how this influence is allocated across selected candidates, how each candidate is backed by the voters, and why certain candidates were not chosen. Building on the notion of priceability, we propose price systems as a framework for such explanations. A price system assigns each voter an individual budget, which they can spend on selected candidates they approve, and each candidate needs to be purchased at a unit price. Since many price systems can exist for a given outcome, selecting among them requires care. We initiate an axiomatic study of price systems and propose several axioms capturing structural coherence, faithful attribution of influence, and alignment with proportionality. On the algorithmic side, we introduce a polynomial-time computable rule in which voters continuously gain and exercise influence and show that it satisfies all jointly satisfiable axioms. Experiments on synthetic and real-world instances indicate that our explanations correlate with established proportionality notions and can recover unequal influence when it is present.

翻译：在基于认可的多人胜选投票中，选民对候选人集合表达认可偏好，目标是返回一个获胜委员会。该模型涵盖了偏好条件下广泛的子集选择问题。先前的工作集中在研究能够证明给定委员会是否具有比例代表性的二元比例性公理上。我们采用更细致的视角，首次研究解释系统，该系统量化了委员会如何代表选民群体，即每位选民施加了多少影响力、这种影响力如何在当选候选人之间分配、每位候选人如何得到选民支持，以及为何某些候选人未被选中。基于价格可行性概念，我们提出价格系统作为此类解释的框架。价格系统为每位选民分配一个个人预算，选民可将预算用于其认可且当选的候选人，且每位候选人需以单位价格被购买。由于给定结果可能存在多个价格系统，选择其中之一需要谨慎。我们首次对价格系统进行公理化研究，并提出若干公理，以捕捉结构一致性、影响力忠实归属以及与比例性的对齐。在算法方面，我们引入一个可在多项式时间内计算的规则，其中选民持续获得并行使影响力，并证明该规则满足所有可共同满足的公理。在合成实例和真实实例上的实验表明，我们的解释与既定比例性概念相关，并能恢复存在的不均衡影响力。