Payoff Polarization and Garbage Concentration in a Stochastic Garbage Disposal Game
Abstract
We study a stochastic garbage disposal game on a finite graph in which agents transfer garbage through pairwise interactions governed by payoff-dependent Fermi probabilities. We investigate the long-term behavior of both payoffs and garbage distributions. We prove that payoff dispersion grows at least linearly in time almost surely, leading to payoff polarization, with the maximum and minimum payoffs diverging to positive and negative infinity, respectively. Moreover, the payoff range grows at least on the order of the square root of time. For garbage dynamics, we establish almost-sure concentration on a single agent in the two-agent case for any positive disposal efficiency. When disposal is complete and the underlying graph is connected, we further prove that all garbage becomes concentrated on a single agent after a finite random time, although the agent holding the garbage may change over time. These results characterize the emergence of payoff inequality and garbage concentration in stochastic competitive interactions on networks.
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BibTeX
@misc{li2026payoff,
title = {{Payoff Polarization and Garbage Concentration in a Stochastic Garbage Disposal Game}},
author = {Hsin-Lun Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27842},
eprint = {2610.27842},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27842},
}
RIS
TY - UNPB
TI - Payoff Polarization and Garbage Concentration in a Stochastic Garbage Disposal Game
AU - Hsin-Lun Li
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27842
UR - https://arxiv.org/abs/2610.27842
AB - We study a stochastic garbage disposal game on a finite graph in which agents transfer garbage through pairwise interactions governed by payoff-dependent Fermi probabilities. We investigate the long-term behavior of both payoffs and garbage distributions. We prove that payoff dispersion grows at least linearly in time almost surely, leading to payoff polarization, with the maximum and minimum payoffs diverging to positive and negative infinity, respectively. Moreover, the payoff range grows at least on the order of the square root of time. For garbage dynamics, we establish almost-sure concentration on a single agent in the two-agent case for any positive disposal efficiency. When disposal is complete and the underlying graph is connected, we further prove that all garbage becomes concentrated on a single agent after a finite random time, although the agent holding the garbage may change over time. These results characterize the emergence of payoff inequality and garbage concentration in stochastic competitive interactions on networks.
ER -