In multi-agent reinforcement learning systems, it is important to share a r
eward among all agents. We focus on the Rationality Theorem of Profit Shari
ng(5)) and analyze how to share a reward among all profit sharing agents. W
hen an agent gets a direct reward R (R > 0), an indirect reward muR (mu gre
ater than or equal to 0) is given to the other agents. We have derived the
necessary and sufficient condition to preserve the rationality as follows;
mu < M-1/M-W(1 - (1/M)(W)(0))(n - 1)L'
where M and L are the maximum number of conflicting all rules and rational
rules in the same sensory input, W and W-0 are the maximum episode length o
f a direct and an indirect-reward agents, and n is the number of agents. Th
is theory is derived by avoiding the least desirable situation whose expect
ed reward per an action is zero. Therefore, if we use this theorem, we can
experience several efficient aspects of reward sharing. Through numerical e
xamples, we confirm the effectiveness of this theorem.