Experiments in economics generally elicit a choice function, whereas most of the theoretical literature use a choice correspondence as a primitive. I build a methodology for eliciting choice correspondences in an incentive-compatible manner, pay-for-certainty. I identify the conditions on the preferences and the behavior of the decision maker to get partial and full identification of the choice correspondence. I illustrate pay for certainty with a laboratory experiment. Subjects chose between different paid tasks. With choice correspondences, 40% of observed choices are singletons, and 3% of subjects always chose singletons. I partially identify the choice correspondence of 72% of subjects and fully identify it for 26% of the subjects. 46% of choice correspondences and 57% of choice functions verify the Weak Axiom of Revealed Preferences. One significant difference between choice correspondences and choice functions is the ability to elicit indifference. It matters in practice, choice correspondences which verify WARP have on average 50% of the comparisons between alternatives which are indifference. I use this new data to study two ‘model-free’ approaches from behavioral welfare economics and show that the transitive core performs well. I also explore relaxations of WARP which require choice correspondences. Intransitive indifference and incomplete preferences rationalize 49% of choice correspondences. Menu dependent choices rationalize up to 93% of observed choice correspondences. These results show how fruitful the methodology developed can be and pave the way for exploring various behavioral models with a unified methodology.