Rationality of reward sharing in multi-agent reinforcement learning

In multi-agent reinforcement learning systems, it is important to share a reward among all agents. We focus on theRationality Theorem of Profit Sharing5) and analyze how to share a reward among all profit sharing agents. When an agent gets adirect reward R (R>0), anindirect reward μR (μ≥0) is given to the other agents. We have derived the necessary and sufficient condition to preserve the rationality as follows;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mu < \frac{{M - 1}}{{M^W (1 - (\tfrac{1}{M})^{W_o } )(n - 1)L}}$$ \end{document} whereM andL are the maximum number of conflicting all rules and rational rules in the same sensory input,W andWo are the maximum episode length of adirect and anindirect-reward agents, andn is the number of agents. This theory is derived by avoiding the least desirable situation whose expected reward per an action is zero. Therefore, if we use this theorem, we can experience several efficient aspects of reward sharing. Through numerical examples, we confirm the effectiveness of this theorem.