Stochastic Redistribution of Indistinguishable Items in Shared Habitation: A Multi-Agent Simulation Framework

TL;DR

This work formalizes the stochastic redistribution of indistinguishable items among $N$ cohabitants using a discrete-event, multi-agent framework implemented in the SimPy environment. By distinguishing identifiable ($S_I$) and indistinguishable ($S_U$) socks and modeling per-agent state dynamics with $S_{i,I}^{own}$, $S_{i,I}^{others}$, $S_{i,U}^{own}$, and $S_{i,U}^{others}$, the authors show that cycles of mixing ($p_m$), recovery ($p_c$), and loss ($p_l$) yield emergent asymmetries and quasi-equilibria even under symmetric rules. Key findings include that the steady-state variance in holdings scales with $p_e^U/p_r^U$, and that three behavioral regimes arise depending on the relation between diffusion and restoration ($p_e^U$ vs $p_r^U$): isolation, cyclical imbalance, and diffusion. The results provide a quantitative lens on everyday social diffusion processes and highlight the role of stochasticity and limited feedback in shaping redistribution in shared systems, with implications for understanding entropy-like dynamics in decentralized environments.

Abstract

This paper presents a discrete-event stochastic model for the redistribution of indistinguishable personal items, exemplified by socks, among multiple cohabitants sharing a communal laundry system. Drawing on concepts from ecological population dynamics, diffusion processes, and stochastic exchange theory, the model captures the probabilistic mechanisms underlying item mixing, recovery, and loss. Each cohabitant is represented as an autonomous agent whose belongings interact through iterative cycles of collective washing, sorting, and partial correction. The system's evolution is characterized by random mixing events, selective recollection, and attrition over time. Implemented using the SimPy discrete-event simulation framework, the model demonstrates that even minimal exchange probabilities can generate emergent asymmetries, quasi-equilibrium distributions, and long-term disorder. The findings illustrate how stochastic processes inherent to shared domestic systems can produce persistent imbalances, offering a quantitative perspective on an everyday social phenomenon.

Figures (8)

• Figure 1: Schematic of sock exchange dynamics among $N$ cohabitants.
• Figure 2: Fluctuations of variance in the number of socks held by 2 cohabitant.
• Figure 3: Fluctuations of variance in the number of socks held by N cohabitant.
• Figure 4: Trajectories of individual sock holdings over time. Each line represents a cohabitant's sock count, illustrating the dynamic nature of the exchange process.
• Figure 5: Final distribution of sock holdings among cohabitants. Each bar represents the total number of socks held by a cohabitant at the end of the simulation.