In random assignment, fairness is often captured by stochastic-dominance envy-freeness (SD-EF). We observe that assignments satisfying SD-EF may admit decompositions that result in each agent envying another agent with high probability. To address this, we introduce decomposition envy-freeness (Dec-EF), which is a property of a decomposition rather than of an assignment matrix. We show that an SD-EF assignment matrix always admits a Dec-EF decomposition when there are at most three agents or the agents have at most two distinct preferences.

翻译：在随机分配中，公平性通常通过随机占优无嫉妒性（SD-EF）来刻画。我们发现，满足SD-EF的分配可能允许存在分解，使得每个代理以高概率嫉妒另一个代理。为解决此问题，我们引入了分解无嫉妒性（Dec-EF），这是针对分解本身而非分配矩阵的一种性质。我们证明，当代理人数不超过三人或代理人至多拥有两种不同偏好时，SD-EF分配矩阵始终存在一个Dec-EF分解。