Many institutions worldwide are considering how to include uncertainty about future changes in sea-levels and storm surges into their investment decisions regarding large capital infrastructures. Here we examine how to characterize deeply uncertain climate change projections to support such decisions using Robust Decision Making analysis. We address questions regarding how to confront the potential for future changes in low probability but large impact flooding events due to changes in sea-levels and storm surges. Such extreme events can affect investments in infrastructure but have proved difficult to consider in such decisions because of the deep uncertainty surrounding them. This study utilizes Robust Decision Making methods to address two questions applied to investment decisions at the Port of Los Angeles: (1) Under what future conditions would a Port of Los Angeles decision to harden its facilities against extreme flood scenarios at the next upgrade pass a cost-benefit test, and (2) Do sea-level rise projections and other information suggest such conditions are sufficiently likely to justify such an investment? We also compare and contrast the Robust Decision Making methods with a full probabilistic analysis. These two analysis frameworks result in similar investment recommendations for different idealized future sea-level projections, but provide different information to decision makers and envision different types of engagement with stakeholders. In particular, the full probabilistic analysis begins by aggregating the best scientific information into a single set of joint probability distributions, while the Robust Decision Making analysis identifies scenarios where a decision to invest in near-term response to extreme sea-level rise passes a cost-benefit test, and then assembles scientific information of differing levels of confidence to help decision makers judge whether or not these scenarios are sufficiently likely to justify making such investments. Results highlight the highly-localized and context dependent nature of applying Robust Decision Making methods to inform investment decisions.