People often face choices that have several uncertain parts. This paper introduces a new way to describe how a decision maker evaluates such “multidimensional” risky options. The authors call the main idea the structured multidimensional expected utility (SMEU) representation. It uses a simple tree diagram to say which parts are judged together, which are judged separately, and which are judged only after some other parts are observed.

To build that diagram, the authors use what they call a rooted clustered tree (RCT). An RCT groups the different dimensions of a choice into clusters (nodes of the tree). Dimensions in the same cluster are evaluated jointly. A cluster can be evaluated conditional on the realized outcomes of clusters above it in the tree. If two dimensions sit on different branches, they are treated separately. The tree therefore captures three common evaluation styles: first-aggregation-then-expectation (FATE), first-expectation-then-aggregation (FETA), and recursive evaluation that conditions one part on the realized value of another.

At a high level the SMEU works by a step-by-step, recursive calculation. For each cluster in the tree, and for each possible outcome in that cluster, the decision maker combines that local outcome with the evaluations coming from its child clusters. That combined value is then averaged (taken in expectation) conditional on the values of ancestor clusters. The process repeats until the whole lottery—meaning the full distribution of outcomes across all dimensions—is assigned a single utility value. The authors show this framework extends earlier models used for risky choices over time and nests the FATE and FETA approaches as extreme cases.

The paper also shows why this matters for concrete problems. One application is inequality: when a policymaker evaluates uncertain incomes for many people or generations, the tree can represent ex post concern (outcomes judged jointly), ex ante concern (expectations judged first), or intergenerational mobility preferences (recursive conditioning). Another application is multisource income, where income sources are grouped into “brackets.” The model lets the authors characterize how such bracketing affects whether one risky income distribution dominates another (stochastic dominance) and when a decision maker prefers income coming from a single source versus spread across sources.