The ability to measure the satisfaction of (groups of) voters is a crucial prerequisite for formulating proportionality axioms in approval-based participatory budgeting elections. Two common - but very different - ways to measure the satisfaction of a voter consider (i) the number of approved projects and (ii) the total cost of approved projects, respectively. In general, it is difficult to decide which measure of satisfaction best reflects the voters' true utilities. In this paper, we study proportionality axioms with respect to large classes of approval-based satisfaction functions. We establish logical implications among our axioms and related notions from the literature, and we ask whether outcomes can be achieved that are proportional with respect to more than one satisfaction function. We show that this is impossible for the two commonly used satisfaction functions when considering proportionality notions based on extended justified representation, but achievable for a notion based on proportional justified representation. For the latter result, we introduce a strengthening of priceability and show that it is satisfied by several polynomial-time computable rules, including the Method of Equal Shares and Phragm\`en's sequential rule.

翻译：衡量（群体）选民满意度的能力，是基于批准的参与式预算选举中制定比例性公理的关键前提。两种常见但截然不同的衡量选民满意度的方法，分别考虑（i）批准项目的数量和（ii）批准项目的总成本。通常，很难判断哪种满意度衡量最能反映选民的真正效用。本文研究了针对大类基于批准的满意度函数的比例性公理。我们建立了所提出的公理与文献中相关概念之间的逻辑蕴含关系，并探讨能否实现同时针对多个满意度函数具有比例性的结果。我们证明，在考虑基于扩展正当代表的比例性概念时，对于两种常用的满意度函数而言，这是不可能的；但对于基于比例正当代表的概念，则是可实现的。针对后一种结果，我们引入了可价格性的一种强化形式，并证明多种多项式时间可计算的规则（包括等额分配法和Phragmèn序贯规则）满足该性质。