A feasible social state is irreducible if and only if, for any non-trivial partition of individuals with two groups, there exists another feasible social state at which every individual in the ﬁrst group is equally well-oﬀ and someone strictly better-oﬀ. Competitive equilibria decentralize irreducible Pareto optimal social states
Ghosal, S. and Polemarchakis, Heracles M., "Exchange and Optimality" (1996). Cowles Foundation Discussion Papers. 1381.