  In economic theory, an agent chooses from available alternatives -- modeled
as a set. In decisions in the field or in the lab, however, agents do not have
access to the set of alternatives at once. Instead, alternatives are
represented by the outside world in a structured way. Online search results are
lists of items, wine menus are often lists of lists (grouped by type or
country), and online shopping often involves filtering items which can be
viewed as navigating a tree. Representations constrain how an agent can choose.
At the same time, an agent can also leverage representations when choosing,
simplifying his/her choice process. For instance, in the case of a list he or
she can use the order in which alternatives are represented to make his/her
choice.
  In this paper, we model representations and decision procedures operating on
them. We show that choice procedures are related to classical choice functions
by a canonical mapping. Using this mapping, we can ask whether properties of
choice functions can be lifted onto the choice procedures which induce them. We
focus on the obvious benchmark: rational choice. We fully characterize choice
procedures which can be rationalized by a strict preference relation for
general representations including lists, list of lists, trees and others. Our
framework can thereby be used as the basis for new tests of rational behavior.
  Classical choice theory operates on very limited information, typically
budgets or menus and final choices. This is in stark contrast to the vast
amount of data that specifically web companies collect about their users'
choice process. Our framework offers a way to integrate such data into economic
choice models.
