Person-specific biomedical data are now widely collected, but its sharing raises privacy concerns, specifically about the re-identification of seemingly anonymous records. Formal re-identification risk assessment frameworks can inform decisions about whether and how to share data; current techniques, however, focus on scenarios where the data recipients use only one resource for re-identification purposes. This is a concern because recent attacks show that adversaries can access multiple resources, combining them in a stage-wise manner, to enhance the chance of an attack’s success. In this work, we represent a re-identification game using a two-player Stackelberg game of perfect information, which can be applied to assess risk, and suggest an optimal data sharing strategy based on a privacy-utility tradeoff. We report on experiments with large-scale genomic datasets to show that, using game theoretic models accounting for adversarial capabilities to launch multistage attacks, most data can be effectively shared with low re-identification risk.