It is well-known that with three agents and more, Bayesian monotonicity and the selfselection condition are both necessary and sufficient for Bayesian implementation in a general environment. Matsushima (1993) identified a condition, Condition 1, which is equivalent to Bayesian monotonicity under the strict self-selection (SSS) condition in an environment with side payments. This paper identifies a condition that is equivalent to Bayesian monotonicity under the self-selection condition in an environment with side payments when the prior probability is independently distributed. This condition identified here is weak in that its combination with the self-selection condition is weaker than the SSS condition.