In the last entry on voting paradoxes, I mentioned that things are different if preferences are single peaked. Let’s look at another scenario.
voter X: A B C
voter Y : C A B
voter Z : C B A
In this case, no matter what order is undertaken, C always ends up being the law that is enacted. These preferences are single peaked. ( if you graph them, you will find that for each individual they will have a top choice ( a peak point), and as you move further away from that choice ( in the A-B-C spectrum) they prefer the other choices less and less. In the previous example, voter Y did not have single peaked preferences and that is what caused the cycling or order dependent outcomes.
With single peaked preferences there is a new problem. With single peaked preferences, the median point of the preference distribution will elicit the most votes. Only those laws or candidates with a centrist twist will get the majority of the votes. Only those voters with centrist views will be happy, and it makes it very difficult for candidates to be elected if they want to bring about major reforms. This phenomenon is referred to as the ‘median voter theorem.’