An Introductory Note

The solution I put forth in the article below is a thought experiment. It’s also intended to be partly satirical. But whether you agree with the conclusion or not, I should note that both the data and the science I explore are real.

Here we go:

Cookies and Democracy

Imagine a third grade class in which each week ends with a cookie celebration. The students vote on which variety of cookies to eat; majority wins. Of the 20 students in the class, 11 favor chocolate chip. And so it doesn’t matter what the other nine want, because, predictably, chocolate chip wins, with slightly more than half the popular vote, every single time.

Despite being democratic, this cookie election process sure seems to ignore the demands of anyone who doesn’t fall into the majority. For one or two weeks, that might be fine. Yet over a long enough period of time, something seems off when 9 out of 20 students never get to eat their cookie of choice.

If this sounds familiar (and you live in the United States or another federal republic), it might be because representative democracies tend to work the same way.

The problem at play is neither a liberal or a conservative one, but a root problem that affects all of us. To show that, I’ll offer two somewhat arbitrarily chosen states that fall on either side of the political divide in America. Let’s dig into some data about recent congressional elections.

On the one hand, we have New York, which has only had Democratic Senators for over two and a half decades. On the other hand, we have Tennessee, which has only had Republican Senators for around three decades. For an entire generation, the legislators elected to fight for the population of these states only represented the interests of the left and the right respectively.

In those past few Senate elections, the results have gotten as close as a 54–46 split. This means that a third to nearly half of every eligible voter in these states has had zero representation of their political ideology in the United States Senate for over two decades. That is staggering.

Even if you’re a Democrat living in New York, or a Republican living in Tennessee, you should be alarmed by the fact that your fellow Americans in any red or blue state might feel completely unrepresented by their leaders and have no recourse to fix that, simply because of where they live. This is a system that structurally favors a certain group of people because of their geography, or more explicitly, because of the fact that their neighbors happen to share their political ideology.

If there existed a state that had 90+% membership to a single party, one could accept having the Senators from that state be from that party all the time. But, as the party split moves closer to the halfway mark, it becomes harder to stomach. Keep in mind that there are two Senators per state. Doesn’t it seem like, in a region with a 54–46 distribution of political views, there should be one Senator from each party most of the time? That seems conceptually correct, but virtually never happens in practice.

This analysis focuses on the Senate, because it’s the easiest way to see this problem manifest itself with real world results. In reality, this same logic can be linked to many parts of the American system: at the federal, state, and local levels; in both houses of congress, the presidency, and even the courts; with the electoral college, gerrymandering, and generally a two party system.

Gradients

Let’s set aside cookies and politics for a moment and consider a seemingly different topic (that is actually the same thing in disguise). Let’s draw a gradient transitioning from black to white that looks like this:

And now imagine that we had a retro printer that could only print things in pure black or pure white. No gray, no subtlety. This printer (made in the USA!) can’t make sense of those pixels in the middle of the gradient, so it rounds them up or down (to black or white, depending on which represents the “majority” of the pixel’s color). This should sound familiar; each pixel is changing according to the results of a winner-take-all election.

The printer prints this out:

Clearly, something has been lost. We took what was complex and nuanced, and we forced it to be something simple and artificial.

This retro printer has a limitation called quantization, a term that comes from the field of digital signal processing. You’ve probably heard the word quantum before, meaning the smallest discrete amount that something can be. Quantization occurs when a certain amount of information is crammed into fewer options than can actually represent it. In the case of the printer, quantization occurs when every shade between black and white is pigeonholed into one of those two choices.

We’ve also seen how quantization occurs when voting on cookies in a classroom. Most often, quantization happens when you attempt to take data which is analog (like a smooth gradient of grays, or a diverse set of cookie preferences, or political opinions) and reduce it to something digital, binary, multiple choice.

Ever since I learned about quantization, I see it everywhere. Its impact on us is ubiquitous, not only at the level of printed gradients and Cookie Fridays. It even rises all the way up to the level of American Democracy. And frankly, I don’t understand how more of us aren’t talking about this problem.

Make Some Noise

If there are analogies between the problems, perhaps there might be analogies between the solutions? Or phrased differently, if scientists have figured out how to solve quantization to serve their needs, might we apply their logic to solve it for our country’s needs?

We can. And it’s a wild idea.

Let’s travel back in time to World War II, when Allied military engineers developed mechanical computers to help planes compute bomb trajectories.

The engineers were bewildered to discover that the calculations were more accurate in practice on the planes themselves than they were in the labs. How odd, given that perfect lab conditions should make conditions ideal for testing and development. Wouldn’t you expect the opposite?

The problem, as it turns out, was quantization, arising from the machines’ components sticking together in the labs. But up in the air, there was a consistent rattle of parts due to the vibration of the plane during flight. The parts would move around and stick less, thereby making calculations more accurate.

This accidental discovery led to one of the most beautiful and counterintuitive breakthroughs in engineering history. It’s how mankind learned that if you introduce a base level of random noise to a bunch of data, it can cause the aggregation of that data to be more accurate, not less.

This concept is called dithering, the process of adding noise to reduce the errors introduced by quantization. To see how it works, let’s revisit the gradient we looked at earlier:

Let’s introduce a whole bunch of random noise. That means that for every pixel, we’re going to randomly make them darker or lighter. And the result looks like this:

Not as nice as our original smooth gradient! But watch what happens when we tell the retro printer (the one that must round every pixel to black or white) to print this new version. In other words, what gets outputted when we quantize the original gradient with added noise?

Isn’t that amazing? The outputted photo is more accurately a reflection of the original. And this was accomplished by first making it less accurate (by adding noise). Zoomed out, your eye sees something akin to the gradient pattern we are after, all because individual pixels were allowed to randomly deviate from their neighbors.

The principle of dithering states that introducing random errors to an input prior to quantization can actually result in an output that is more representative of the original input.

The applications of dithering are typically seen in digital signal processing. But once you realize the similarities between the quantization problems of gradients and those of our elections, you can naturally see the similarities between their solutions.

A Modest Proposal

Let’s explore one such possible solution. Imagine every election had a two step process. First, as now, the votes get tallied. But then, the results actually get modified randomly, with a systematic methodology that is equally applies across all states, via an algorithm agreed upon in advance. In the short term, every election will be less accurate. But in the long term, the aggregate of all elections will be closer to the true distribution of political views within the constituency.

Take New York as an example. If in each of the past two decades of elections to choose Senators, there was a random adjustment to the vote such that the outcome had a slight chance of changing each and every time, then on a long enough timeline, the distribution of political party representation would more closely represent the ratio of all votes cast for either party across all of the elections.

For example, to keep it simple, suppose that each of the past 100 elections in State X had a 51–49 Republican-Democrat breakdown. The Republican would win every time in the real world. But in this dithered world, suppose we introduced a randomly selected adjustment between -2% and +2% to the vote after each Election Day. Statistically, that would mean that roughly 25% of the elections would have an adjustment that would bring the Republican vote down far enough below the Democratic vote. Or put differently, that would mean that 25 of the 100 elections would be won by a Democrat. That seems to be a bit closer to the “will of the people”, does it not?

What if there was even more noise? In a -4% to +4% adjustment, 37 of the 100 elections would favor the Democrat. We’re getting closer to 51–49.

Of course, if the gap between the two candidates were much wider, it would be significantly harder for a noise adjustment to bridge the gap. So the likelihood of a flip to the results of the election would go down. So the Republican would continue to win most of the time, which makes sense.

The suggestion is that we will be more accurately represented by our leaders over a long-term timeline if we are willing to accept that each individual election might seem completely unfair, because the real winner lost and the loser won.

It’s important to clarify the parallel to dithering in the digital signal processing world. The election is not the whole gradient. The election is one single dot in the gradient. All of the elections over many years are the gradient. The goal is to get that overall picture to be more accurate, and we do so by introducing random noise at each pixel, each election.

The above example assumes that we exist in a pure two party system. One of the real benefits of an election process like this, however, is that it gives a real incentive for third party candidates to enter the race. It raises their likelihood of winning the adjusted vote.

I intentionally am not being prescriptive about how much noise should be applied to each election to keep this fair and real. The more noise we add, the more we run the risk of disappointing single elections. And at a certain point, the scales tip so much such that the overall picture itself is distorted in unintended ways. I don’t know where that fine line is that renders the dithering helpful without being harmful. But it seems clear that, whatever the methodology is, it has to remain consistent over many elections.

This idea requires one fundamental rule to work, one that humans are notoriously bad at adhering to. It requires follow-through. It requires that the disappointing outcome of elections here and there not cause us to deviate from this long term strategy, because this long term strategy does theoretically work. Yet it only works if it’s truly long term. It’s the Law of Big Numbers; it takes large quantities of data to result in a truly accurate result.

So…

I imagine one common opposition to this idea would sound something like this: “But I have one vote, and my vote should count for something. You’re suggesting making my vote worth less.” But I would argue the opposite: If you only have one vote, why have it be drowned out by the existing process? Might this proposal actually make your vote worth more, if you’re willing to think about it on a long enough timescale?

This new process may in fact encourage more people to vote, even to vote for third parties, because it would mean that there is a multiplier effect, however small, that might tip the outcome in their favor. For the party in charge, there is a chance, however small, of losing its position of authority. So voters that identify with the majority are more incentivized to vote to preserve that majority’s control. Most importantly, all elected officials are more incentivized to avoid extremist politics, to grab a larger share of the electorate, and not rely on winning by a slim majority.

It’s the same reason why people like to gamble or play the lottery. Hey, you never know.