In many causal inference problems, multiple action variables, such as factors, mediators, or network units, often share a common causal role yet lack a natural ordering. To avoid ambiguity, the scientific interpretation of a vector of estimands should remain invariant under relabeling, an implicit principle we refer to as permutation equivariance. Permutation equivariance can be understood as the property that permuting the variables permutes the estimands in a trackable manner, such that scientific meaning is preserved. We formally characterize this principle and study its combinatorial algebra. We present a class of weighted estimands that project unstructured potential outcome means into a vector of permutation equivariant and interpretable estimands capturing all orders of interaction. To guide practice, we discuss the implications and choices of weights and define residual-free estimands, whose inclusion-exclusion sums capture the maximal effect, which is useful in context such as causal mediation and network interference. We present the application of our general theory to three canonical examples and extend our results to ratio effect measures.

翻译：在许多因果推断问题中，多个行动变量（如因子、中介变量或网络单元）通常共享共同的因果角色，但缺乏自然排序。为避免歧义，估计量向量的科学解释应在重新标记下保持不变，这一隐含原则我们称之为置换等变性。置换等变性可理解为变量置换以可追踪方式置换估计量的性质，从而保持科学意义。我们正式刻画了这一原理并研究其组合代数。我们提出一类加权估计量，将非结构化潜在结果均值投影为置换等变且可解释的估计量向量，捕获所有阶交互作用。为指导实践，我们讨论权重的含义与选择，并定义无残差估计量——其容斥和捕获最大效应，这在因果中介和网络干扰等情境中具有实用价值。我们将一般理论应用于三个典型示例，并将结果推广至比率效应度量。