Order-Sensitive Sequential Interventions on Ideal Lattices

TL;DR

This paper develops a theoretical framework for understanding order-sensitive sequential interventions within ideal lattices, providing insights into path equivalences, potential functions, and planning algorithms under prerequisite constraints.

Contribution

It introduces a local-to-global theory of order sensitivity, characterizes path-independence via diamond curvature, and establishes conditions for path models and planning on ideal lattices.

Findings

Any two admissible paths with the same endpoints differ by elementary diamond swaps.

Path-independence is equivalent to vanishing diamond curvature for edge-additive valuations.

Exact planning bounds and dynamic programming methods are derived for ideal lattice interventions.

Abstract

We study sequential interventions under prerequisite constraints. In this setting, admissible intervention sequences are paths in the ideal lattice of a finite prerequisite poset rather than unconstrained action strings. We give an exact local-to-global theory of order sensitivity on this state space. First, we prove that any two admissible paths with the same endpoints differ by a finite sequence of elementary diamond swaps. Second, for edge-additive path valuations, we show that path-independence is equivalent to vanishing diamond curvature, yielding an endpoint potential with a canonical M\"obius parameterization on the ideal lattice. Third, we prove that a local diamond field is induced by an edge-based path model if and only if it satisfies cube consistency, with uniqueness after fixing a reference-tree gauge. Under reduced-state longitudinal assumptions, supported reference paths identify reference-path scores, whereas local order effects require two-sided support of both orders on each diamond. These results yield exact planning consequences, including an order-insensitivity bound and dynamic programming on the truncated ideal lattice.