Entropy production rate as a criterion for inconsistency in decision theory

TL;DR

This paper introduces a novel criterion based on entropy production rate to assess inconsistency in pairwise comparison matrices used in decision-making, linking them to Markov processes and providing a new consistency measure.

Contribution

It establishes a connection between PCMs and Markov processes, identifying entropy production as a new metric for inconsistency and proposing methods for incomplete matrices.

Findings

Entropy production rate quantifies PCM inconsistency.

Consistent PCMs correspond to detailed balanced Markov processes.

The approach handles incomplete pairwise comparison matrices.

Abstract

Individual and group decisions are complex, often involving choosing an apt alternative from a multitude of options. Evaluating pairwise comparisons breaks down such complex decision problems into tractable ones. Pairwise comparison matrices (PCMs) are regularly used to solve multiple-criteria decision-making (MCDM) problems, for example, using Saaty's analytic hierarchy process (AHP) framework. However, there are two significant drawbacks of using PCMs. First, humans evaluate PCMs in an inconsistent manner. Second, not all entries of a large PCM can be reliably filled by human decision makers. We address these two issues by first establishing a novel connection between PCMs and time-irreversible Markov processes. Specifically, we show that every PCM induces a family of dissipative maximum path entropy random walks (MERW) over the set of alternatives. We show that only `consistent' PCMs correspond to detailed balanced MERWs. We identify the non-equilibrium entropy production in the induced MERWs as a metric of inconsistency of the underlying PCMs. Notably, the entropy production satisfies all of the recently laid out criteria for reasonable consistency indices. We also propose an approach to use incompletely filled PCMs in AHP. Potential future avenues are discussed as well. keywords: analytic hierarchy process, markov chains, maximum entropy