Decisions over Sequences: Computability and Choice

We develop a framework to study situations where decision makers face alternatives sequentially. Within this framework, we focus on endogenous stopping behavior using two broad classes of decision rules: stopping rules and bounded stopping rules. We establish the equivalence of these two classes and examine two of its implications. First, focusing on the procedural aspects of decision making, we define computable rules using the model of a Turing machine. Our equivalence result enables us to show that computable rules are implementable by finite automata. Second, we extend the setup of abstract choice theory beyond choice from sets and finite lists, to that from infinite sequences of alternatives. The equivalence result allows us to derive testable implications of choice behavior. We develop a revealed-preference ``toolkit'' and use it to characterize a threshold-based and a satisficing choice procedure.