Choice by lexicographic semiorders

In Tversky's (1969) model of a lexicographic semiorder, a preference is generated via the sequential application of numerical criteria by declaring an alternative x better than an alternative y if the first criterion that distinguishes between x and y ranks x higher than y by an amount exceeding a fixed threshold. We generalize this idea to a fully fledged model of boundedly rational choice. We explore the connection with sequential rationalizability of choice (Apesteguia and Ballester 2010, Manzini and Mariotti 2007) and we provide axiomatic characterizations of both models in terms of observable choice data.