We present an optimisation problem defined using a measure preserving action by a group G on a compact probability space X. We show that this problem cannot be solved in any way measurable with respect to a finitely additive measure that extends the original probability measure and such that the group G remains measure preserving. The proof involves a computer-performed analysis of a stochastic process. On the other hand, there are some non-measurable solutions that induce some paradoxical decompositions of the space X. Additionally, we show that, for some small positive epsilon, epsilon-stability also cannot be satisfied in such a measurable way. By epsilon-stability, we mean that the expected gains from local deviations considered independently cannot exceed epsilon.