On Infinite Versions of the Prisoner Problem

We investigate infinite analogues of the well-known 100 prisoner problem, replacing the finite setting with a countably infinite number of prisoners and boxes. In this setting, randomness is no longer central—deterministic constraints, cost structures, and combinatorial strategies take precedence. We analyze several natural infinite variants, focusing on conditions under which infinitely many prisoners (or all but finitely many) can succeed in finding their labels using pointer-following algorithms. We provide necessary and sufficient conditions in terms of weighted box-access costs and explore how rearrangement inequalities govern the success of allocation strategies. Our results highlight the delicate interplay between infinite series and deterministic constraints in this infinite combinatorial framework.