McKelvey’s Chaos Theorem

I would like to explain what McKelvey’s Chaos theorem is. Yesterday I gave the information about the implication of McKelvey’s Chaos theorem that when there are three or more voters and at least two policy dimensions in a majority voting system, any policy outcome is theoretically possible; today I want to provide a simple explanation about how McKelvey has come to this theorem. Because of the majority voting system, to design a policy that wins the election, politicians do not have to make everyone to be happy, they only need to make the majority like their ideas; therefore, they can put different weights on different groups according to their sizes, they can ignore the interests of the minority groups and focus on the interests of the majority groups and win the election. Under such ideology, the politicians can always make their policies a bit more extreme and a bit more discriminated, through a series of such policy making, policies could be very extreme.

Let’s use some mathematical way to further explain the theorem. Firstly, we can make the question a bit simple that we only have three voters and two dimensions. Then we model the system by a triangle, each of the angles represents a different voter, the distance represents their preferences, the shorter the distance is, more favour the voter has. The ideal policy point is in the middle of the triangle; however, we are able to find another point that is preferred by two of the voters over the middle policy and the point could be outside the triangle. In addition, if we want to get a specific policy on the policy plane, we can choose a different point from the policy point at that moment that is preferred by the majority for each period and eventually we are able to deviate the current policy to the policy we want.

Therefore, McKelvey suggests any policy is possible as long as we have more than two voters and more than one policy dimension.