Mathematics is not known to be everybody's favorite subject. For myself, it really took me some time and a couple of unintentional moments to convince me that mathematics is actually a subject I love. Some of you know that I am a mathematician by trade, and for those of you who didn't, well, you do now! And even though I am no longer a researcher, I can't help but look at the world through the lens of this deep and dense language I have spent so much of my life becoming fluent in.
You see, mathematics is a language for understanding the world around us. Like all other languages, it is not perfect. It does not provide the truth as black and white, as many have grown up to believe. Really, math gives us a way to describe, discuss, and evaluate subjects in life where our intuition might not be enough to really understand what is happening. That all being said, of course mathematics is not a language that provides us with value statements. Rather, it is a language that provides us with logical frameworks needed to hold complex arguments and thoughts so that we can then, if we wish, place value.
With that, there is a result in mathematics that I think about very often in my daily life, especially when scrolling through social media: Arrow's Theorem.
A theorem in mathematics is a statement that is proven to be true... or in other words, there is a very solid argument based on the foundations of mathematical logic for why the statement is true. Arrow's theorem says the following:
Take any society that makes decisions based on a voting system in which you rank your preferences from most favored to least, with ties allowed. So say you can choose between candidates A, B, or C and you submit your vote by ranking them from best to worse, with ties allowed. So then a vote could be A>B=C (A is preferred first and B and C are tied for second), or B>C>A (B is preferred first, C second, and A third), for instance.
This voting system respects "transitivity" (if A>B and B>C, then we must have A>C) - this is like saying that there is a clear order in the ranking. This voting system also respects "independence of irrelevant alternatives" (Say A>B. Then it does not matter how A or B related to C.... it does not change the fact that A>B no matter where C is placed in the ranking). And this voting system respects "unanimity" (if every person in the society votes A>B then the society as a whole votes that A>B - it's a unanimous decision).
A society with such a voting system is a dictatorship.
That is Arrow's Theorem.
Let that sink in.
And how is this bold statement a true one? Well, I will do my best to make the argument for you in a way that is accessible. That being said, mathematics is the densest language I know of... and it takes time to digest each sentence. You might want to take a long time with each sentence even, to think about it, draw pictures, make up examples, until you feel that you understand it and agree. And if you don't agree, then talk to somebody about it, even try to prove it wrong! But don't read the next sentence until you are satisfied. It is work, but it is worth it.
Before I go into the proof, or argument on how you can always get a dictatorship with such a voting system, I want to tell you an important observation I have gathered from understanding this result: in reading the proof of Arrow's theorem, you see that the way you get a dictatorship is by polarizing views of the population. And that even more directly, all you need to do it is to polarize the population on one issue in order to form a dictator who can control the outcome of the final vote.
For me, this is the key take-away point of Arrow's Theorem: the more polarized we are in our views in a society the more control we lose as a society and the more vulnerable we are to the will of a dictator. We can see this pattern in our history and we can see it happening now. The more we seek to fight against each other rather than understand each other, the more we lose our freedom to truly have any say in how our society operates. The more freedom we actually give to the real tyrants lurking in the background, making the decisions while we fight with our neighbors. And this is why the value statement I conclude from my understanding of Arrow's theorem is that polarization of views leads to the destruction of democratic society and if we wish to work within a democratic framework then we must all seek love and understanding, rather than hate and defensiveness.
And now I will present an argument for why Arrow's theorem is true to the best of my ability in order to make it understandable for all. And if you do not find something I say understandable, please feel free to google "Arrow's theorem" and you will find a lot of different proofs and explanations. Mathematics is about understanding, and there are many ways to understand - do the work to find a way that works for you.
We will make the argument for why Arrow's theorem is true in four steps:
First, let's show that if every voter puts alternative B either first or last in their ranking then society must also. To show that this is true, we will assume that the the opposite is true and show that we come to a contradiction, meaning that the opposite cannot be the case and so our original statement must be true. In mathematics, this is what we call a "proof by contradiction". So, lets assume that every voter puts B either first or last in their ranking, but society does not. Let's say the society vote ends up being A>B and B>C. By "independence of irrelevant alternatives", this should hold even if every individual moved C above A, since B occupies an extreme position in everybody's votes. In other words, everybody has a vote of either A>C>B, C>A>B, B>A>C, B>C>A, A=C>B, A>C=B, B=A>C, B>A=C, or B=A=C; so, even if somebody were to switch their preference of A and C, it would not change B's position as either first or last. So let's say that everybody changes their vote to have C>A, meaning that by "unanimity" that society must have C<A. We can make this assumption because it is a case that must be considered as a possibility that can hold true. But by "transitivity", we have that since society concludes that A>B and B>C, then A>C. A contradiction! So we must have that if every voter puts B as either first or last then B must end up either first or last in society's conclusive vote.
Next, let's show that there exists a voter n who is extremely pivotal in the sense that by changing their vote at some profile they can move B from the bottom of the social ranking to the top. Whoa. That is quite powerful, let's discuss how this can happen. Let's say that every voter puts B at the bottom of their ranking. By "unanimity" we must have that society puts B at the bottom of the ranking. Let N be the total number of voters, and let's say we order the voters 1 through N and we let each voter one-by-one move B from the bottom of their rankings to the top of their rankings. Let n* be the first voter whose change causes the ranking of society to change (we know by "unanimity" that this must happen at the latest when n*=N, so it will eventually happen). Let profile I be the list of voters who have not yet moved B to the top of their rankings and let profile II be the list of voters who have moved B from the bottom of their ranking to the top. Now we have that every voter either has B at the top or at the bottom of their ranking, and that B is no longer at the bottom of the social ranking... well then we know from 1. that B must then have shifted to the top of society's vote! So, if society is completely polarized on issue B, then there exists an individual who can change the outcome of society's vote by simply changing their ranking of B from one extreme position to the other. Again, whoa.
Next, let's show that n* is a dictator over any pair AC that does not involve issue B. OK, let's choose one issue, say A, from the pair AC. Let's construct a profile III from profile II by letting n* move A above B, so that n*'s ranking is A>B>C, and by letting all other voters that are not n* in profile II arbitrarily rearrange their relative rankings of A and C while leaving B in its extreme position. We do this to show that it is irrelevant how anybody else but n* ranks A and C. This would put A>B as society's preference since profile I and n* together have A>B by "independence of irrelevant alternatives". We would also have B>C since profile II and n* together has B at the top. If society concludes that A>B and B>C, then by "transitivity" we would have A>C. And so we would have that the social ranking over AC must agree with n*'s ranking by "independence of irrelevant alternatives".
Finally, let's show that n* is also a dictator over every pair AB. This is the final generalization which should that we have our dictator. Full stop. Let's take any two issues A and B. Then we know from 2. that we can find a third alternative C to put at the bottom of the ranking. We then know from 3. that we can find a voter, call them M, who is a dictator for any pair that does not involve C. We also know from 2. that we can construct an n* that can affect society's AB ranking through the construction we used to prove 2. and so we must then have that M=n* - the dictator is the n* that we find in 2. by ordering the voters one by one and who is the first one to cause society's vote to flip. And there you have it, our little dictator.
Now, now of course we need to recognize that there takes some setting up of this perfect storm in order to get a dictator. And that in this proof it is about forming a dictator for each specific issue at hand, and so on. So like everything in our world, even in mathematics, Arrow's theorem is not an absolute truth. But it does point out an incredibly important flaw to be recognized. And this flaw is for all rank voting systems that fill a couple common-sense principles that such voting systems usually follow. And the proof shows us that it can be exploited in two ways, really: the first way is to take an issue society is completely polarized on and there is a way to leverage the voters to change the favor of society... and second, if you have a particular ranking you would like to win, like A>B, then all you need to do is pick a sure-fire issue, say C, that will completely polarize the population and then there is a way you can leverage the voters so that A>B. It's like presenting the population with such an extreme choice that suddenly the original issues at stake are no longer as dramatically polarizing and can be swayed in the dictator's favor. At this point, I hope everybody's eyes are wide open and stomachs are a little bit uncomfortable. Because this kind of game has been going on for years, generations. Why is it that every time in history when a population's views become highly polarized that dictatorships arise? I am not saying that Arrow's theorem is the absolute proof, but I think it lays the foundation of an interesting discussion, no?
And, well, what can we do? I don't know really. But I certainly would say that falling way to polarizing views on any "side" is probably not very productive. Those of you who know me know that I personally am a fairly radical person. But I do not want to speak in terms of "us" vs. "them". I do not think that setting myself opposed to an opposite is productive. Polarization will only lead to further oppression. And it's really a role of the die as to which side of the coin will be the oppressor... and if history has told me anything, it won't be those I personally identify with who will come out on top. And so, selfishly even, I do my best to seek understanding rather than opposition. I know it is easier said than done, trust me I know, but I do think that the best we can all do is to listen and understand each other and the needs of all life on this planet we live on. And to not be swayed in our views, but to stick by them, respectfully, and with grace, while at the same time being open to other perspectives and the possibility of compromise and change. As I said, easier said than done, but certainly better to strive for than reliving the destructive spiral of polarization over and over again.
