# Can one prove Pareto efficiency in an n-party system by showing all Pareto improvements between any two given parties are made?

I've made a proof of Pareto efficiency of a funding system that I've developed. There are effectively four types of actors. I've shown all Pareto improvements are made between any two given parties are made (including between two actors within the same party). I believe that this proof by cases is valid, but I'm not 100% certain. My one concern is that maybe there is some tricky three-way or four-way trade that is somehow not captured in my proof. Thoughts?

• Yes, you do have to consider those 3-way and 4-way cases in general. – Herr K. Jun 28 '17 at 17:07

@HerrK. got it right in his comment (he should have deleted the somewhat confusing "yes" from the beginning and then posted it as an answer) It is possible that no pairwise improvements are possible but general Pareto-improvements are still possible. A simple counterexample for three actors and three goods is as follows. Let the utility functions be the same for all actors, that is $$U_A(x,y,z) = U_B(x,y,z) = U_C(x,y,z) = \min(x,y,z).$$ Let actor $A$ have 3 units of good $x$, and 0 units of the other goods, whereas actor $B$ has 3 units of good $y$, and 0 units of the other goods, and actor $C$ has 3 units of good $z$, and 0 units of the other goods. No pairwise exchange can increase utilities above zero. But the goods could be distributed so that everyone has a unit of each. This would increase all utilities to one, making the original distribution not Pareto-efficient.