Math Is Democracy III: A Short Rant about Voting Theory

I have been full tilt exploring the relationship between mathematics and democracy for the last at least year and a half. E.g., check out my TED talk.

And I’ve been thinking about this relationship, in some form, for the last 8 or 9 years, ever since Jay Gillen said to me, “Everything you love about math is what free people love about democracy.” (See this previous post.)

Given this, it may be surprising that until fairly recently, the well-established mathematical subfield that explicitly addresses democracy, voting theory (aka social choice theory), never grabbed my interest. FWIW, this has changed, but I think my journey around it is indicative of something worth keeping in mind for mathematical people who want to use math to think about how society works. (E.g., me, and hopefully you.)

Voting theory asks: given a large number of people forming a collective (e.g., a nation or state!), and a choice the collective is faced with (e.g., electing a political leader!), what are the possible ways one could aggregate individual preferences into a collective decision? (E.g., everybody vote for one candidate, and the candidate with the most votes is elected, aka plurality vote, aka how most political elections work in practice; but other possibilities too, like instant runoff, the Borda count, approval voting, and score voting.) And what are their properties? (E.g., in plurality voting, there is a potential spoiler effect when similar candidates split the vote of the majority, which is mitigated in approval and score voting since they do not force voters to choose only one candidate to support.)

It’s sort of a puzzle: given how much I’ve always cared about math, and how much I’ve always cared about democracy, why wouldn’t I want to understand this developed and beautiful theory exploring this most fundamental democratic operation — how does a collective make a decision?

Well, I’ll tell you.

The first and loudest thing I heard about voting theory, beyond just what it’s about, was Arrow’s impossibility theorem. This is a “foundational” result in the theory. It is often glossed as asserting that “there is no perfect voting system.” (Do not take this interpretation at face value. More below.)

From this, I got the impression that the point of voting theory was for us to brain out about the possibilities, and then conclude (with mathematical certainty!) that “we the people” can’t win no matter what. I’m supposed to be excited about this?

Was the whole thing just an intellectual exercise? Many years ago, I listened to a mathematician excitedly describe the Borda count to some graduate students. I remember thinking, with some confusion, “Where is the excitement coming from?” It seemed to me that the mathematician was excited only intellectually, about the mathematical properties of this system, and was totally disengaged from the question of whether anybody had used it or would use it or should use it to make any collective decision.

In this context, voting theory seemed almost grotesque to me. I love math for math’s sake — my PhD work is in pure algebra. But taking the urgent moral and political problem of building a system of government that adequately reflects the will of the people, and using it as loose inspiration for some math for math’s sake — ick.

Now, I take full responsibility for this impression. Many practitioners of voting theory believe the system can be radically improved, and are much more engaged with the realities of elections than I was giving them credit for. Steven Brams, now a mentor of mine, was involved in the adoption of approval voting for leadership elections in several professional societies. (Steve’s attempts to push something similar with political elections have mostly been frustrated so far, although advocacy by the Center for Election Science, an organization Steve has advised, played a role in the recent adoption of approval voting by Fargo, ND.) And Warren Smith has been developing a website to spread information about score voting since 2005. Just as two examples.

Now that I understand that voting theory is meaningfully engaged with actual elections, I’ve done a 180, and I’ve been reading up on both the foundations and recent research, and am looking for ways to contribute myself.

That said, I retain the feeling (with hopefully due humility that I could be wrong again) that the field would benefit from greater accountability to the problems it’s trying to speak to. The imperatives of mathematical exploration are different than the imperative of improving the mechanisms by which the will of the people expresses itself, and these differing imperatives can be in tension.

I’m reading one of Donald Saari’s books on voting theory. Saari has developed a beautiful, geometric way of looking at election systems that offers great clarity in making sense of some of the counterintuitive things that can happen. This is beautiful and important math. He also tends to portray the Borda count as kind of the best voting system, and I cannot shake the feeling that this is more because of the mathematical beauty of its properties, when viewed through a particular mathematical lens, than a sober assessment of its value to society.

So. I want to see and be part of a voting theory that is responsive to the problem it involves itself in. We can talk about voting systems using the tools of academic discourse, but we must remember the stakes. This is not an academic exercise.


I promised above a little explanation of my comment that Arrow’s theorem should not be taken as stating that “there is no perfect voting system”. Lemme get into that for a sec. Arrow’s theorem says a voting system of a very specific type cannot obey a certain very specific list of prima facie desirable properties all at once. My basic objection is that the theorem is too specific for this broad an interpretation. My more fine-grained caveat is that there is nothing God-given about the particular set of desirable features Arrow chose to analyze, so deviation from them is just one definition of “imperfection”, and there are plenty of others.

In either case, I don’t blame Arrow for this, but I definitely thought I was being told that making the system better isn’t on the table. My real motivation here is to disabuse you of that misimpression if you had it too.


7 thoughts on “Math Is Democracy III: A Short Rant about Voting Theory

  1. This reminds me of a tweet I saw a while back. Someone was visiting Arlington Cemetery and was amazed by the huge number of gravestones, all ordered according to some principle. And this person couldn’t help it…their mind went to the question: how many?

    And isn’t there something quite wrong with that? There’s a children’s book by the guy who wrote Stinky Cheese Man, called The Math Curse. I think there’s something like that going on here.

    What is it about mathematical culture that enables mathematicians do take something like voting theory — it’s about VOTING for god’s sake — and lose the reality of voting in the process? Is that part of the lineage of the 20th century consensus that math is about abstract structure rather than anything in the natural or human world?

    1. This question has a personal/psychological dimension and a cultural/historical dimension. I think the answer has both these parts, and additionally has a structural/institutional dimension. I have the feeling as I contemplate entering the field of voting theory that there will be, for me personally, a tension I’ll have to navigate between (a) working on questions that allow me to tap my particular expertise and to prove theorems that could be published in journals, and (b) working on questions directly motivated by the desire to make democracy work better. It already seems useful to me to acknowledge that these two things are not the same.

      On the personal/psychological front, I feel like one dimension of it is not just about getting distracted from the human-values context of the inquiry by the math itself, but also of mistaking values and aesthetics coming from the math for values coming from the context. Or maybe, treating mathematical modeling decisions as neutral and ignoring their values implications, and thus concluding that the output of particular models is more definitive than it is. For example, what I was saying about Donald Saari and the Borda count, which is probably not fair to Donald Saari, who knows 1000x more about voting theory than I do, but I just cannot shake the feeling I described above. Actually, I could say something similar about Warren Smith, and again, I say this with hopefully due humility because he is a serious effing expert and I do not want to step to him, and also as you’ll see if you spend any time on his website, he is very directly engaged with what really happens in real elections. But still, when I read his website, he communicates profound conviction that score voting is like the absolute best, and I have the impression that this conviction is grounded primarily in studies based on a particular measure of effectiveness called Bayesian regret. Now I think this type of evidence is pretty interesting and compelling, but it’s just one particular mathematical construct, among many others. (Actually, if my understanding of Bayesian regret is correct, then it is a construct that is particularly adapted to pick up on the strengths of score voting in relation to other systems, and therefore it is no surprise to me that it picks out score voting as the best system. In a very similar way, it seems to me that Saari’s geometric point of view is particularly adapted to draw out the positive features of the Borda count.) I want to trust that voting theory as a whole stays loyal to actual humans’ sense of being well-represented by these choices, and stays curious and open about how to measure this, rather than being loyal to closed views of what’s best, based on particular intellectual approaches.

      The tie between the structural/institutional angle in the first par. and this personal/psychological angle just described, is that, okay again I’m talking out of my *ss here, but it seems to me that the institutional frame supports and rewards the thing I was just saying I don’t like. I.e. if you’re Donald Saari, you come up with this particular very illuminating geometric point of view, and you are celebrated and rewarded by developing it further and further over the course of decades. Many people take your ideas and develop them further, attributing the general program they’re participating in to you. Meanwhile, this mathematical framework points you toward the Borda count, so you favor that. The structural/institutional incentives are lining up behind loyalty to your intellectual paradigm over loyalty to whether people will actually feel effectively represented by implementation of the ideas.

      p.s. I didn’t know Stinky Cheese Man but I found the first few pages of The Math Curse on Amazon and was delighted. And yes, this resonates.

      1. If I understand you correctly, you’re saying:

        1. There is a danger of getting lost in your research perspective. Especially easy because you can formalize your values via various measures. These measures allow you to close the circle — the value-laden measure tells you that your value-laden voting system is superior to all others.

        2. But isn’t that just good research practice? You’re saying that in the Darwinian academic world, what survives is that which opens up problems for generations of researchers. That doesn’t happen unless you work on your perspective for decades, essentially making it a paradigm overflowing with puzzles for others to solve. You have to get lost in your own perspective to survive, encouraging you to get lost in your (single) research perspective.

        The second of these definitely isn’t unique to mathematics, and probably the first issue isn’t unique either. So maybe this is just an issue with academia, not unique to math?

  2. @Michael — I think there’s something general here, but also something somewhat math-specific, or maybe applied-math-specific, or maybe more generally, specific to fields involving abstract modeling. (So, perhaps structuralist anthropology, but not post-structuralist anthropology.) It’s the thing I said above about mistaking values coming from the math for values coming from the context. All applied math involves making modeling decisions. My broad and probably totally unjustified generalization is that mathematical modelers as a whole are underdeveloped in thinking about the values implications of their modeling choices. This is something Cathy O’Neil has been beating a drum about for some time. We make the model and then we treat the model as reality; our choices in putting it together are sort of invisible to us.

    This makes me think of a criticism Pierre Bourdieu leveled at linguistics (which, ok, not exactly applied math, but I trust you’ll see the connection), in one of the most memorable sentences I read as an undergraduate. “In fact, as long as they are unaware of the limits that constitute their science, linguists have no choice but to search desperately in language for something that is actually inscribed in the social relations within which it functions, or to engage in a sociology without knowing it, that is, with the risk of discovering, in grammar itself, something that their spontaneous sociology has unwittingly imported into it.” – Pierre Bourdieu, Language and Symbolic Power, p. 38. (The course was The Politics of Language and Identity, and the professor was Dr. Kira Hall. I remember struggling mightily to understand this sentence, but once I understood it, it stuck with me forever.)

    1. Yeah, this is interesting and I need to think more about it. Two examples of mathematical modeling gone wrong come to mind, and neither fits in neatly with the “modeling ignores the values” story.

      1. Long-Term Capital Management, which I blogged a bit about. What I took out of that story was that the model was applied too broadly. The thought was the model would apply in contexts beyond what they had checked out carefully. Plus they just got greedy and dysfunctional. So the sin wasn’t ignorance but arrogance, greed, lack of caution. But unless “we can wait out any short-term loss and our long-term strategy is sound” is taken to be a value, this is some kind of other destructive behavior.

      2. VAM – Value-added modeling is super controversial and has all sorts of limitations, and yet it’s a totally valuable research tool. I don’t know whether individual economists were invisible of its values and limitations — probably many were, as the idea that everything that’s good for a kid is measurable on tests is precisely the kind of invisible value that I think you’re describing. And yet that’s not just an issue with the modelers — that’s something that is pervasive in the way we all talk about education. (Even progressives have a hard time admitting that something might be bad for measurable learning but still worth doing.) And the issues really only came up when this research tool was weaponized against teachers in service of a political agenda. So the story of VAM isn’t exactly a story of invisible values on the part of abstract modelers. It’s more a story of values invisible more broadly, and then a research tool brought in as a political weapon.

      There are a lot of examples in O’Neill’s book but none of them is coming to mind as an issue of value-invisibility. For example, AI being trained on data sets that reflect current reality (e.g. white people have lower crime rates) and therefore predicting the data it was trained on (e.g. tada! white people tend to do less crime). That doesn’t seem like a story about invisible values to me.

      1. I just clicked through re: Long-Term Capital Management. I totally buy the greed/arrogance interpretation and the connection to thinking you’re smarter than everybody. Actually, to me that is not really in any tension with what I’m trying to articulate here, so maybe I’m doing a bad job articulating it. Thinking we’re smarter than everybody and mistaking our models for reality are very interconnected it seems to me. (Either way you view the sin, humility is the warranted remedy.)

        And I also totally buy the way you’re talking about VAM, and it does make me want to tell a richer story. As you say, the phenomenon there of mistaking the model for reality is happening more broadly and sociologically-complexly than just “inside the minds of the developers of the model”.

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