When Bad Things Happen to Good Choices Part 1

At the present moment, I'm taking a break from my social choice theory homework to bring you a blog post.  I have justified this discontinuity in my studies by constructing the following quasi-ramble on the subject of social choice theory.  As such, I have helpfully filed this post under "Not Game Theory" for those who might abjure such digressions.

I am not, by nature, particularly socially inclined.  Some of this, however, is not my fault!  Deciding what a group ought to do, as it turns out, is very difficult.  If you are not yet convinced, allow me to present the following seemingly benign situation.

You and two of your friends wish to eat lunch together at a restaurant.  While there are many restaurants available, you have narrowed your field down to three, which happen to be named a, b, and c.  (Incidentally, you and your friends are named 1, 2, and 3.  These are fine names with long and glorious histories.)  Unfortunately, you are not members of the hive mind, so each of you has a different opinion of which restaurant to patronize.  Here are your preferences (higher up is better).

1 2 3 a b c b c a c a b

Now, there are quite a few ways that the three of you could potentially decide.  You, 1, would probably like it if you could have your way.  If you were a dictator, that is exactly how it would be.  When dictatorship is the method of social choice, someone is the dictator and determines the social preferences by their social preferences.  This is no time for delusions of grandeur however, as your friends are willing to resist your tyranny by any means necessary and you cannot overpower them both.  

Far more reasonable, or at least less likely to result in a violent uprising, is majority rule.  This should be quite familiar.  Something is majority preferred to something else if just over half of the people in a group prefer it to that other thing.  Here, a is majority preferred to b.  An alternative is optimal in majority rule if nothing else is majority preferred to it.  This is pretty simple.  So simple in fact that you and your friends decide to adopt majority rule to determine where you go for lunch.  Unfortunately, you are now paralyzed by indecision.  Oh no!

Your indecision happens to be the product of a phenomenon called Condorcet's Paradox.  Notice that a is majority preferred to b (I'll write this statement in the form aP*b from now on), bP*c, and cP*a.  Nothing is optimal, and your social preference forms a cycle.  This is not a good thing.  

However, it is not yet time to resort to dictatorship.  Other options remain.  

Another common method is plurality rule.  Plurality rule counts the number of times an alternative appears at the top of individuals' lists, and ranks alternatives by their resulting score.  All three restaurants appear at the top once.  You are now hungrier, but no closer to a solution.

Unfortunately for you, your particular profile of preferences makes an effective social choice quite difficult.  Given enough time, it would be possible to formulate an effective strategy, but lunch hour is only so long.  Under the circumstances, perhaps it would be better to solicit somebody else's advice (I am always more than willing to offer an opinion).  

We'll look at some less indecisive situations in Part 2.  For anybody who still remains unconvinced that group decisions are difficult, I might suggest that you have thus far been quite lucky, or are a dictator.  Either way, you read the rest of my post.  Mission accomplished.