Monotonicity of one-dimensional PMDP values

Prove that the limit values of the one-dimensional Bernoulli PMDPs with action sets \(\{1,m\}\) are nondecreasing in \(m\), namely, establish \(v_\infty^{1,m}\leq v_\infty^{1,m+1}\) for every \(m\geq1\).

Background

The dimensional lifting theorem proves that the values of the one-dimensional PMDPs with action sets {1,m}\{1,m\} converge as m→∞m\to\infty to the value of a two-dimensional PMDP. The paper compares the sequence numerically and observes behavior consistent with monotonicity.

A general proof of monotonicity is not provided; the authors explicitly conjecture the inequality and note that proving it appears difficult.

References

We conjecture that $v_{\infty}{1, m} \leq v_{\infty}{1, m+1}$, although proving this inequality appears to be surprisingly difficult.

— Percolation Markov Decision Processes  (2609.19905 - García et al., 17 Sep 2026) in Section 7.4, discussion surrounding Figure 2