Subtree perfectness, backward induction, and normal-extensive form equivalence for single agent sequential decision making under arbitrary choice functions

TL;DR

This paper explores the concept of subtree perfectness in single agent sequential decision making, extending existing theories to arbitrary choice functions without relying on probabilities or utilities, and establishes its relation to backward induction.

Contribution

It extends Hammond's characterization to the normal form and arbitrary choice functions, and demonstrates the equivalence between subtree perfectness and normal-extensive form equivalence.

Findings

Subtree perfectness is equivalent to normal-extensive form equivalence.

Subtree perfectness is sufficient but not necessary for backward induction.

The framework does not require probabilities or utilities.

Abstract

We revisit and reinterpret Selten's concept of subgame perfectness in the context of single agent normal form sequential decision making, which leads us to the concept of subtree perfectness. Thereby, we extend Hammond's characterization of extensive form consequentialist consistent behaviour norms to the normal form and to arbitrary choice functions under very few assumptions. In particular, we do not need to assume probabilities on any event or utilities on any reward. We show that subtree perfectness is equivalent to normal-extensive form equivalence, and is sufficient, but, perhaps surprisingly, not necessary, for backward induction to work.