Hello! You obviously cared enough about a math fact that is scary that you clicked the title. We’re going to build up a *slight* amount of theory first, but I promise that this post requires literally no prerequisites, and hidden in this article you will find a cool picture that you can make the background of your phone.

Anyway, first I have to tell you what a “function” is in math, and I have to give you a pretty abstract definition, but things should be more clear after the few paragraphs, I promise.

Definition: A function is any “machine” that takes in certain inputs, called the domain values, and for each thing you put in, you get one (and only one!) thing back out.

A lot of people have seen functions in math that look like . For example, you can feed in the number “2” to that function, and the function returns . Surprisingly, that is the last time we will talk about numbers in a post about math.

A weirder example of a function is a vending machine. The “domain” is the code you punch in, and the “range” are the different kinds of snacks you can get out. For example, you can type “E5” into the domain and get out “Skittles” as your return value.

An even weirder example of a function, and the one we will talk about today, is a voting system:

Definition: A voting system is a function which takes in a “preference list” for every person in your “society” and returns what’s called a social preference list, that is, a ranking of all the candidates from “best for society according to this voting system” to “worst for society according to this voting system.”

Here are a bunch of examples to get your head around it! Imagine in your eighth grade class there are three people running for president, Alice, Bob, and Tom Gannon.

Example 1: In a first past the post voting system, everyone in the class ranks who they’d most want to be class president, who’d they’d want second most, and who they’d want third most to be president. The output of the first past the post voting system is “whoever gets the most number one rankings is first, and then whoever gets the second most number one rankings is second, and whoever gets the least number one rankings is third (with ties possible).

(A side note you can skip over if you want: You might think it’s weird that a voting system asks for everyone’s full list instead of just who they think is the best. But this definition of voting system is more general–you can view “just tell me who you think is best” voting systems as voting systems that take in the entire list of preferences but then only focus on what’s written in the number one spot.)

Example 2: Now imagine there is a boy in your class named Jimmy, and Jimmy’s parents made a recent large donation to your school. In a dictatorship, everyone in the class still ranks who they’d most want to be class president, who’d they’d want second most, and who they’d want third most to be president (just like all voting systems), but then the voting system just returns Jimmy’s list. So even if everyone else (rightfully) voted fed the list “Tom Gannon is first, followed by Alice, followed by Bob” if Jimmy put in the list “Alice is first, followed by Bob, followed by Tom Gannon” then the “societal preference list” that would get returned is “Alice is first, followed by Bob, followed by Tom Gannon.

Example 3. In a Tom Gannon is The Best voting system, everyone in the class still ranks who they’d most want to be class president, who’d they’d want second most, and who they’d want third most to be president (just like all voting systems), but the voting system returns the list “Tom Gannon is first, followed by Alice, followed by Bob.” This is not a commonly used term or voting system, sadly, but if everyone lobbies the government we can make this the new voting system we use!

These last two examples bring up a point that just because we have a voting system doesn’t mean it has to be a good voting system. In fact, there is a “voting system” that takes in everyone’s preferences and returns whoever got the least number of #1 votes as first, and whoever got the second least number of #1 votes as second, etc.

We are about 75% to the scary part. Keep going!

Anyway, here are two things that you would expect a “good” voting system to have. You might want more, but you can definitely expect any “good” voting system to have these two properties:

If everyone puts in the same preference list, the output list should be the same preference list. (This is called pareto.)

For example, if everyone is conspiring against me because I am not actually an 8th grader and the whole class writes the list “Alice, then Bob, then Tom Gannon” then we should expect any good voting system to return the list “Alice, then Bob, then Tom Gannon”.

We’d also want any “good” voting system to satisfy this second property, called independence of irrelevant alternatives:

2. If the same people prefer one person over another, then the output our voting system says shouldn’t depend on where other people are on the list.

This one is a bit more confusing and is best explained with an example. Let’s say Tom Gannon has convinced a few of the eighth graders that he is cooler than Bob, because Tom Gannon has run a smear campaign against Bob. (What kind of eighth grader is named Bob anyway?) Then, all other things being equal, no matter if some of the eighth graders feed in their preferences as “Tom Gannon, then Alice, then Bob” or more feed in their preference as “Tom Gannon, then Bob, then Alice,” then Tom should appear first.

Can you think of a voting system that solves both of these? (Seriously. Think about it for a second! Hint: It is one of the examples discussed above.)

It’s a dictatorship. For example, if we let our voting system be “ignore everyone else’s votes and just focus on what rich Jimmy has to say”, then if everyone returns the same list, the return list will be the same because Jimmy put that list too! Also regardless of whether Jimmy returns “Tom Gannon, then Alice, then Bob” or “Tom Gannon, then Bob, then Alice,” Tom Gannon will appear on the societal preference list before Bob.

Now people don’t tend to like dictatorships, so the question is–are there any others? You can try to find them, but here’s the scary part. In 1950, Kenneth Arrow proved that as long as you’re choosing between more than two alternatives there are no voting systems that satisfy pareto (i.e. 1) and independence of irrelevant alternatives (i.e. 2) besides dictatorships. This is called Arrow’s Impossibility Theorem.

I want to make a point very clear here. Arrow’s Impossibility Theorem does not just say “we haven’t found a voting system that works.” It says that if you have any voting system you know satisfies 1 and 2, then it has to be a dictatorship. If that doesn’t scare you, read this article again.

Now, why is this theorem true? Well, if you can ever get me alone for a few hours I can explain it to you–it’s actually not that hard to prove after you know it’s true (of course, to think of the possibility that this result might be true is incredibly hard and earned Arrow the Nobel Prize!) but I don’t have the space to explain it in this blog. You can see more about this scary theorem here, but I want to take this opportunity to grandstand a little bit.

People often come up to me and say “woah man, why would you ever care as something so abstract as the notion of a function?” But one of the many key insights that go into this sort of thing is that voting systems are functions, so everything we know about functions can be applied to understand how different voting systems can work.

Also I just learned about this theorem a few days ago, so if anyone has a remedy for the sleep I’ve lost worrying about this theorem please let me know.

(PS: If you’re thinking, “Phew! I’m glad that the elections for US presidents only has two options!” then you are definitely forgetting about third party candidates. And if you are thinking “Well there are only two candidates that really matter, you should know that our voting system inherently limits the “real” choices to two candidates or causes a candidate that less than half of the country would have preferred to become president. This is actually how Woodrow Wilson became president, and why people say “third party candidates steal votes.”)