In pursuit of participatory budgeting (PB) outcomes with broader fairness guarantees, we initiate the study of lotteries over discrete PB outcomes. As the projects have heterogeneous costs, the amount spent may not be equal ex ante and ex post. To address this, we develop a technique to bound the amount by which the ex-post spend differs from the ex-ante spend -- the property is termed budget balanced up to one project (BB1). With respect to fairness, we take a best-of-both-worlds perspective, seeking outcomes that are both ex-ante and ex-post fair. Towards this goal, we initiate a study of ex-ante fairness properties in PB, including Individual Fair Share (IFS), Unanimous Fair Share (UFS) and their stronger variants, as well as Group Fair Share (GFS). We show several incompatibility results between these ex-ante fairness notions and existing ex-post concepts based on justified representation. One of our main contributions is a randomized algorithm which simultaneously satisfies ex-ante Strong UFS, ex-post full justified representation (FJR) and ex-post BB1 for PB with binary utilities.

翻译：为寻求具有更广泛公平保障的参与式预算结果，我们率先研究离散型参与式预算结果的抽签机制。由于项目成本存在异质性，事前与事后支出金额可能不相等。针对这一问题，我们开发了一种技术来约束事后支出偏离事前支出的程度——该性质称为"预算平衡至多一个项目"。在公平性方面，我们采用"两全其美"视角，寻求同时满足事前与事后公平性的结果。为此，我们首次研究了参与式预算中的事前公平性质，包括个人公平份额、一致公平份额及其更强变体，以及群体公平份额。我们证明了这些事前公平概念与基于合理代表性的事后公平概念之间存在若干不相容性结果。本文主要贡献之一是提出一种随机算法，该算法在二元效用参与式预算中同时满足事前强一致公平份额、事后完全合理代表性及事后预算平衡至多一个项目。