Demonic variance and a non-determinism score for Markov decision processes

TL;DR

This paper introduces the concept of demonic variance to quantify non-determinism in Markov decision processes, providing a new score and analyzing its properties and computational aspects.

Contribution

It proposes the novel notion of demonic variance and a non-determinism score for MDPs, along with properties, bounds, and algorithms for computing these measures.

Findings

Demonic variance is between 1 and 2 times the maximal variance in an MDP.

The non-determinism score measures how much non-determinism influences the variability of outcomes.

Algorithms for computing maximal and demonic variance are developed for specific random variables.

Abstract

This paper studies the influence of probabilism and non-determinism on some quantitative aspect X of the execution of a system modeled as a Markov decision process (MDP). To this end, the novel notion of demonic variance is introduced: For a random variable X in an MDP M, it is defined as 1/2 times the maximal expected squared distance of the values of X in two independent execution of M in which also the non-deterministic choices are resolved independently by two distinct schedulers. It is shown that the demonic variance is between 1 and 2 times as large as the maximal variance of X in M that can be achieved by a single scheduler. This allows defining a non-determinism score for M and X measuring how strongly the difference of X in two executions of M can be influenced by the non-deterministic choices. Properties of MDPs M with extremal values of the non-determinism score are established. Further, the algorithmic problems of computing the maximal variance and the demonic variance are investigated for two random variables, namely weighted reachability and accumulated rewards. In the process, also the structure of schedulers maximizing the variance and of scheduler pairs realizing the demonic variance is analyzed.