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制定基于批准投票的参与式预算选举中比例性公理的关键前提。衡量选民满意度的两种常见但截然不同的方法分别考虑（i）获批项目的数量和（ii）获批项目的总成本。一般而言，很难确定哪种满意度度量最能反映选民的真实效用。本文研究了针对广泛类别的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，并探讨了能否实现相对于多个满意度函数均具有比例性的结果。研究表明，在考虑基于扩展正当代表权的比例性概念时，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权的概念则是可行的。针对后者，我们引入了价格可接受性的强化形式，并证明多种多项式时间可计算规则（包括等额分配法和Phragmèn顺序规则）均满足该性质。