Elections used to be so simple. As Harvard professor Pippa Norris writes, a study by the Electoral Integrity Project performed in 2012 found that experts rated American elections as the worst among all Western democracies. And regardless of which side of American politics you agree with more, if anything Americans are even less satisfied with the current voting system. If you want to read a contemporary and relatively unbiased account of our electoral system, I highly recommend James Michener's book Presidential Lottery. Even though the book was written in 1969, Michener's main arguments still resonate today.
As it turns out, there isn't a perfect solution here. No system of voting is perfect. Let's look at an example, which will hopefully make my point more clear. Let's say that the local ice cream store has agreed to provide free ice cream for the local elementary school on the last day of the year. They want to keep things simple, so they decide to have the school take a vote. The store will supply the school with whichever flavor wins the election. Let's also assume that there are 200 students in the school (it's not a big school).
Here are the results of the ice cream election:
Cookies & Cream - 50 votes
Vanilla - 45 votes
Strawberry - 40 votes
Chocolate - 65 votes
Which flavor is the most popular? Well I guess that depends on how you decide to tally the votes and determine the winner. Clearly, Chocolate has the most votes, right? If we were using plurality voting (where the most votes wins), Chocolate would be the clear winner. If the ice cream store brings Chocolate ice cream, 65 students are going to be happy, even though 135 students voted for another flavor!
Okay, what if we require the winner to have a simple majority (over 50% of all votes)? Since no one won a simple majority, we would have to hold a run-off election, perhaps with the top two vote-winners competing. That means that Chocolate will run against Cookies & Cream. Whichever flavor wins the run-off will depend on the preference order of the students who voted for Vanilla and Strawberry. So we could easily end up with a situation where Cookies & Cream wins, if more students from Vanilla and Strawberry prefer Cookies & Cream to Chocolate.
The 18th century French mathematician, the Marquis de Condorcet, proposed a different method entirely called the Condorcet method, which is a form of ranked-choice voting. Here, the winner is the flavor of ice cream that wins a majority of the votes in a head-to-head election against every other flavor. In other words, we would have to run pairwise elections between each of the four flavors (obviously this would take more time than just a simple vote, but stick with me here). Let's use a smaller subset of the group of students to look at this further.
Let's look at three students from the larger group of 200 - student A, student B, student C, and student D. In order to make things even more simple, let's drop Strawberry from consideration (Strawberry had the fewest votes overall, so I think that's fair). Here is the preference order for each student:
Student A: Chocolate > Vanilla > Cookies & Cream
Student B: Cookies & Cream > Vanilla > Chocolate
Student C: Vanilla > Chocolate > Cookies & Cream
Here are the results of our pairwise elections:
In this case, the Condorcet winner is Vanilla, because Vanilla won in two out of the three pairwise elections. Notice anything? Depending upon the method of voting we use, we have three different winners! There are several other different versions of voting systems, each with their own sets of pros and cons. However (and you will have to trust me on this), there is no system that is perfect.
The American economist Kenneth Arrow won the Nobel Prize in Economics in 1972 for his work on what is now called Arrow's Impossibility Theorem, which states that whenever there are three or more choices or options in an election, no system can be designed which satisfies the following conditions:
1. Pareto Efficiency: If every voter prefers A to B, then A must always win.
2. Transitivity: If the system prefers A to B and B to C, it must prefer A to C.
3. Independence from Irrelevant alternatives: If the system prefers A to B, they can't prefer B to A when C enters the contest.
4. Nondictatorship: The wishes of multiple voters must be taken into consideration (one person can't decide who wins).
Admittedly, I am only scratching the surface here. If you are interested in reading about voting systems from a game theoretic perspective, I would encourage you to take a look at William Poundstone's book, "Gaming the Vote: Why Elections Aren't Fair (and What We Can Do About It)".
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