Found at: http://commons.wikimedia.org/wiki/File:Vote_with_check_for_v.svg

Does voting work? This seems like an obvious question, all you do is have a bunch of people select a candidate and then count up the ballots. But, there is actually a lot of mathematics behind figuring out the most effective method of voting so that the results accurately reflect what the population believes.

Kenneth Arrow, a prominent economist, developed a list of standards that a good voting method should have:

(1) Decisive: there should always be one winner.

(2) Pareto Principle: if all the voters vote for candidate A, then candidate A should win.

(3) Nondictatorship: no single voter should be able to decide the election

(4) Independence of Irrelevant Alternatives: If candidate A wins the election over B and C, then removing candidates B or C should not change that outcome. Candidates B and C are considered “irrelevant” because they didn’t win the election.

This list seems pretty reasonable, but once we start applying these principles to find a voting system that meets them, things get far more complicated…

**Plurality vs. Majority**

There are many different ways to determine the winner of an election. Two of them are known as plurality and majority rule. The difference between them is extremely subtle:

a) Plurality: this rule says that the winner of an election is preferred by the majority of the voters in the population. So, if there are candidates A, B, and C and candidate A gets 45% of the votes, B gets 35%, and C gets 20%, candidate A wins. This is used in the United States to determine the outcome of elections.

b) Majority: this rule says that the winner of an election is preferred *over all the other candidates* by the majority of the voters in the population. The ballot would allow the voter to rank the candidates to show what their order of preference is, instead of just voting for one candidate. The nice thing about this method is that, say you really love candidate A, but if candidate A is not going to win you would much rather candidate B win over C; this method allows you to state that in your ballot. So, if we look at candidates A, B, and C again, we get something like the following data:

45% of the voters have the following preference: A>B>C

35% of the voters think the following: B>C>A

and 20% of the voters think the following C>B>A

In this case, B is the winner of this election, because 35%+20% = 55% of the voters prefer B to A and 45%+35%=80% of the voters prefer B to C, which are all majorities. So, in this case, you look at the preferences of one candidate to another.

As you can see, it is the same election, but determining the election winner in different ways will give two different winners.

**So, how do we determine which method is better?**

How do these let us determine which rule described above (plurality or majority) is more fair? Well, Arrow’s impossibility theorem states that actually no such voting system can meet all of these principles. Here are some problems with the two rules:

a) The plurality rule tends to break rule #4 a lot; the candidates who did not win end up affecting the election a lot, as what happens in the United States elections occasionally. For example, in the Bush v. Gore election, the election in Florida was so close that had Ralph Nader not been on the ticket, Gore could have won decisively. In this case, the voting method did not meet Arrow’s principles of fair voting.

b) The majority rule falls prey to the Condorcet Paradox, which states that the preferences of a population can end up being irrational.

Say there are three voters and three candidates, our previous A, B, and C. They have the following preferences:

Voter 1: A>B>C

Voter 2: B>C>A

Voter 3: C>A>B

So now we use the majority rule by pitting each candidate against one other:

A vs. B: A is preferable to B for 2/3 of the voters, so A wins.

B vs. C: B is preferable to C for 2/3 of the voters, so B wins.

C vs. A: C is preferable to A for 2/3 of the voters, so C wins.

So, A is more preferable than B, B is more preferable than C, and C is more preferable than A, or

A>B>C>A

But wait…that doesn’t make any sense. This is an example where, even though the preferences of individual voters are very rational, when using the majority rule the preferences of the population become suddenly irrational. This is a big problem for the majority rule.

**So, does voting work?**

Each one of Kenneth Arrow’s principles seems very logical and appealing, but at least one is violated in every voting system. There are more preferable systems, but according to Arrow, there does not exist a perfect voting system that effectively takes the beliefs of all individual voters an consolidates them into one final decision. What is interesting is how the amount of parties affects these principles. What if we had only two parties? Would the voting system be fairer? What if we had no parties at all? All food for (scientific) thought…

For more information about voting, check out the previous post: https://foodforscientificthought.wordpress.com/2014/04/02/combating-data-manipulation-iv-voting/

**Works Cited**

https://www.sss.ias.edu/files/papers/econpaper93.pdf

https://www.math.hmc.edu/funfacts/ffiles/10007.8.shtml

https://www.economicsnetwork.ac.uk/iree/v4n2/stodder.htm

http://faculty.georgetown.edu/kingch/Condorcet%27sParadoxandArrow%27stheoremoverhead.pdf