Evaluating changes to social programs requires predicting the effects on program take-up. However, potential applicants often have incomplete information on their eligibility for or amount of program benefits and researchers may not know their information set, generating bias in estimates from standard discrete choice models. We develop a moment inequality framework to: (1) estimate preferences, allowing potential applicants to have heterogeneous willingness to pay for the program benefits, and (2) simulate policy counterfactuals and evaluate intervention targeting, without fully specifying individuals’ information. We apply this model to the Supplemental Nutrition Assistance Program (SNAP), a program with complex eligibility and benefit rules and concerns of limited take-up.