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：为了在参与式预算（PB）中获得更广泛的公平性保障，我们首次研究了离散PB结果上的抽签机制。由于项目成本异质，事前与事后的支出金额可能不相等。为此，我们开发了一种技术来约束事后支出与事前支出之间的差异——该属性被称为"预算平衡至多一个项目（BB1）"。在公平性方面，我们采用"两全其美"的视角，寻求同时满足事前公平与事后公平的结果。围绕这一目标，我们首次研究了PB中的事前公平属性，包括个体公平份额（IFS）、一致公平份额（UFS）及其更强变体，以及群体公平份额（GFS）。我们证明了这些事前公平概念与基于合理表征的现存事后概念之间存在若干不相容性。主要贡献之一是一种随机算法，该算法在二元效用PB中同时满足事前强UFS、事后完全合理表征（FJR）与事后BB1。