Service Preservation from Matching Non-Matching Socks Under Stochastic Loss

Socks are produced and replaced at a massive scale, yet their paired use makes them unusually vulnerable to service loss, as the disappearance of a single sock can leave usable wear-capacity stranded and create sockless days even when functional socks remain available. In this study, we examine whether pairing non-matching \say{orphan} socks can preserve daily sock service under stochastic loss, and how this benefit trades off against perceived social discomfort. We formalize sock ownership as a sequential decision problem under uncertainty in which socks wear out and disappear stochastically during laundering, while public exposure induces a person-specific mismatch penalty. We conducted an in-person study to estimate mismatch sensitivity and diversity preference, linking behavioural heterogeneity to interpretable mixing policies. Using these results, an exact benchmark on small tractable instances, and a simulation-based evaluation of pairing policies, we show that strict matching can appear resource-frugal largely because it generates many sockless days. In contrast, controlled tolerance for mismatch sustains service and reduces stranded wear-capacity across loss regimes. The ecological-cost term is treated as a proportional embodied-cost proxy rather than an independent life-cycle assessment measure, so the environmental interpretation is suggestive and mechanism-based rather than a direct estimate of environmental savings. This study establishes the feasibility and limitations of matching non-matching socks as a simple strategy for preserving service from already-owned garments.