In domains, such as agronomy or manufacturing, experts need to consider trade-offs when making decisions that involve several, often competing, objectives. Such analysis is complex and may be conducted over long periods of time, making it hard to revisit. In this paper, we consider the use of analytic provenance mechanisms to aid experts recall and keep track of trade-off analysis. We implemented VisProm, a web-based trade-off analysis system, that incorporates in-visualization provenance views, designed to help experts keep track of trade-offs and their objectives. We used VisProm as a technology probe to understand user needs and explore the potential role of provenance in this context. Through observation sessions with three groups of experts analyzing their own data, we make the following contributions. We first, identify eight high-level tasks that experts engaged in during trade-off analysis, such as locating and characterizing interest zones in the trade-off space, and show how these tasks can be supported by provenance visualization. Second, we refine findings from previous work on provenance purposes such as recall and reproduce, by identifying specific objects of these purposes related to trade-off analysis, such as interest zones, and exploration structure (e.g., exploration of alternatives and branches). Third, we discuss insights on how the identified provenance objects and our designs support these trade-off analysis tasks, both when revisiting past analysis and while actively exploring. And finally, we identify new opportunities for provenance-driven trade-off analysis, for example related to monitoring the coverage of the trade-off space, and tracking alternative trade-off scenarios.