Collective Preferences and Voting

I am loving my econ class. I just wanted to state that again.

A fun question brought up in class: "Can a voting mechanism always find a socially-optimal allocation?"

Let's begin with the following voter preference scenario...



A brief (and probably obvious) explanation. Person 1 prefers "policy option" A to policy option B. And person 1 prefers policy option B to policy option C. And so on for persons 2 and 3.

Basic logic dictates that if someone prefers A to B and B to C, then through the transitive property that person should prefer A to C. But look at how the above preferences when processed through a voting mechanism yield an illogical outcome.

If policy options A and B were on a ballot A would win because persons 1 and 3 would vote for it while only person 2 would vote for B. Likewise, if options B and C were on the ballot, B would win because persons 1 and 2 would vote for it while only person 3 would vote for C. So clearly we have established democratically that collectively the group prefers A to B and B to C.

But look what happens if you put options A and C on the ballot. C wins because persons 2 and 3 would vote for it while only person 1 would vote for A. This now has given us the following illogical result.

Collectively, the group prefers A to B and B to C. But it prefers C to A.

As it turns out, the only way for a voting mechanism to yield a socially-optimal (and logical) result is if the individuals have an identical preference ordering.