Decisions over Sequences

TL;DR

This paper develops a formal framework for decision rules over infinite sequences, establishing their equivalence, computability, and behavioral properties, with implications for modeling sequential decision-making processes.

Contribution

It introduces decision rules and their subclasses, proves their equivalence, links computability to finite automata, and explores behavioral heuristics in sequential decision contexts.

Findings

Decisions over sequences can be modeled with decision rules and their subclasses.

Computable decision rules are implementable with finite automata.

Continuity of rules implies their computability.

Abstract

This paper introduces a class of objects called decision rules that map infinite sequences of alternatives to a decision space. These objects can be used to model situations where a decision maker encounters alternatives in a sequence such as receiving recommendations. Within the class of decision rules, we study natural subclasses: stopping and uniform stopping rules. Our main result establishes the equivalence of these two subclasses of decision rules. Next, we introduce the notion of computability of decision rules using Turing machines and show that computable rules can be implemented using a simpler computational device: a finite automaton. We further show that computability of choice rules -- an important subclass of decision rules -- is implied by their continuity with respect to a natural topology. Finally, we introduce some natural heuristics in this framework and provide their behavioral characterization.