In The Red Queen and the Purple President, I pondered about a weird phenomenon. In most animal species (and many non-animals) there are two sexes. In American politics, and similar winner-take-all electoral systems, there are two parties. These dichotomies are extremely stable over time. This observation may seem trivial, since it’s so entrenched in our everyday lives. However, it’s actually difficult to explain. Why aren’t intermediate forms more successful? Where are all the hermaphrodites? And centrists? And centrist hermaphrodites? I mused that there are no good simple mathematical models to explain this paradox.

Then the 2016 election happened. American politics seemed less like a curious abstraction and more like getting hit by a train. I never followed up on my post. But now that we’ve had a chance to at least catch our breath, let’s take up my challenge. Let’s create a model. You have to think about a little math first, but then we’ll get to some sweet animations.

Imagine a set of organisms. These could be critters competing for mates, or politicians competing for votes. Either way, each organism has a “fitness” that determines how successful it is. Fitness, in turn, is determined by two opposing forces.

The first force is a stabilizing one. Assume that there is a single optimal phenotype. Biologists often depict this as a “peak” in a “fitness landscape”. The closer an organism is to the peak, the higher their fitness. To be clear, this isn’t a real physical mountain. This is “morphospace,” an imaginary way of illustrating variation. An organism’s position in morphospace has nothing to do with actual geography. Instead, it depends on characteristics of the organism’s body, like weight or hairiness. Based on ecological factors, and/or voter preferences, there is an ideal amount of heft and hair for maximum success. You can also picture it like the bullseye on a target. The closer you are to the center, the higher your score. Let’s call this score “compliance.” All else being equal, organisms will tend to congregate around the bullseye to maximize compliance. Compliance can be represented by a bell-shaped curve, the so-called standard normal distribution. As you move farther away from the central optimum, compliance decreases down toward zero. Compliance is thus a “pulling” force that brings you back to the midpoint. The compliance distribution is a fixed feature of the environment. It doesn’t depend on where anybody else happens to be.

The second force is the destabilizing one. Assume there is an advantage to being different from others. Being different gets you attention and makes you sexier. There is pressure to move away from everyone else, even if they’re all perched on the coveted bullseye. Now, assume further that attractiveness increases exponentially the farther away you are from someone else. In other words, being super close to someone in morphospace doesn’t affect you too much one way or another. But being far away really ramps up your appeal. Let’s say the farther away you are from the crowd is called “defiance.” Defiance is a “pushing” force whose effect increases with the square of the distance between any given pair of organisms. Squaring the distance means that defiance has more impact the farther away you get. This is the opposite of, say, gravity, a “pulling” force whose effect decreases with the square of the distance. An individual’s defiance is the average of these squared distances to everyone else. Thus, if everyone occupies the exact same spot in morphospace, they all share a defiance of zero. If a single radical moves far away from the herd, their defiance goes way up. Simultaneously, everyone else’s defiance increases by a much smaller amount. And so on. Defiance doesn’t just depend on where you are in morphospace. It depends on where everyone else is, too.

Fitness, then, is simply the product of compliance and defiance. That’s it. The fate of the population is guided by these two forces. The gif below shows how the position of individual A affects fitness (for other individuals) across morphospace:

Potential mates/voters are not modeled directly. You have to assume they are out there somewhere, rewarding whoever achieves the highest fitness. Presumably the best strategy is to be a little bit compliant, but not too much, and a little bit defiant, but not too much. That’s what everyone wants. But how will it play out?

We can simulate this setup over multiple generations. An organism’s fitness directly determines the proportion of offspring it will leave in the next generation. The population size is kept constant, so what matters is relative fitness compared to everyone else. The offspring appear near the parent’s location, but randomly scattered slightly to simulate chance effects. Again, these could be literal offspring or new politicians copying a winning platform. The math is the same either way.

A typical outcome is shown in the gif below. Starting from an initial clump, the dots wriggle free of each other and eventually form two distinct clusters. The two clusters serve as each other’s foils. Although compliance would be highest in the middle, they find optimal fitness farther out where they can defy each other.

Here the organisms are initially more spread out, but condense into the familiar clumps.

And here they start off to the side. Slowly, they return.

Why does this matter? It matters because the starting point for explaining a complex system should be the simplest model that matches the data. This is the principle of parsimony, also known as Occam’s Razor. It’s easy to make up convoluted tales about why the world is the way it is. But if there is a more straightforward explanation, then the burden of proof is on you to justify the extra details. Maybe it really is more complex, but you can’t conclude so without evidence.

We often hear just-so stories about male and female biology. Men are larger in order to hunt mammoths. Women have more fat around their waists to trick potential husbands into thinking they have a larger birth canal. And so on. But according to the simulations, the two clusters have no innate properties at all. They exist only to be different from each other. The particular ways in which they differ are completely arbitrary. If the graph axes represent weight and body hair, then females could just as easily end up the heavier and hairier sex. Or vice versa. Sexes are not clear-cut categories, but shifting amorphous blobs. Now, I’m not saying every sexually dimorphic trait is so arbitrary in reality. But that’s the default null hypothesis. To say otherwise requires data.

Politics may be similar. How is it that the same party that freed the slaves is now shunned by most African Americans for its racist policies? It makes sense if you think of political parties as just drifting around each other randomly in morphospace. Who cares what your platform is, so long as you are different from the other guys? I don’t mean to sound cynical. Quite the contrary. I think understanding the dynamics of political systems is an important step for positive change. Only then can we take charge of them, and ensure they reflect the will of the people. I’m not saying this model provides comprehensive insight into democracy. But it’s a good place to start.

If you want to run your own simulations, the code is available here. There’s a lot of room to explore these ideas further. And regardless, I hope you’ll remember these clouds of dots, billowing around the bullseye, when you contemplate diversity in all its forms. May they guide your perspectives on evolution, government, and the wider world.