Before revision:  It is satisfied, for instance, when the choice set variation is the result of either a search process or unobserved feasibility. Using variation of the uncovered choice sets we show how to do ordinal (nonparametric) welfare comparisons. When one is willing to make additional assumptions about preferences , we show how to nonparametrically identify the ranking over average utilities in the standard multinomial choice setting .
After revision:  It is satisfied, for instance, when the choice sets are nested, or when they form a partition. Our estimation procedure is computationally fast and uses mixed-integer optimization to recover the sparse support of choice sets. Analyzing the ready-to-eat cereal industry using a household scanner dataset, we find that ignoring the unobservability of choice sets can lead to biased estimates of preferences due to significant latent heterogeneity in choice sets .