Efficiency of what, though?
Arrow's Impossibility Theorem states that there is no way of aggregating a given set of preference vectors that is complete, transitive, independent of irrelevant alternatives and a non-dictatorship (no uniquely decisive individual); see:
http://www.math.cornell.edu/twiki/pub/SMI/SocialChoice/JohnGArrow.pdf. Given exactly the same sets of preferences, the method you use to aggregate them will return a different outcome. If the sole point of democracy is to aggregate preferences, then you're stuck because aggregation of preferences is ultimately dependent on which aggregation mechanism you use and you have no way of determining which one that would be. As an example, given the exact same set of preferences, AV and FPTP return different selections. Which one do we choose and why? What principles do we use to determine what our democracy looks like?
Not to mention you're already talking in philosophical outcomes when you're talking about outcomes that are "at least as fair", whatever that means; and you're already talking in philosophical terms when you're saying "morally better" - is fairness necessarily morally better? Might there not be a situation which is less fair but more legitimate? How do we compare the two? The philosophy precedes the maths.