This isn't, strictly speaking, an RI. But I wrote it, and I want people to say nice things about me, so I'm posting it anyway. If the mods would prefer this in another subreddit or in the stickies instead, delete it and let me know.

Why we use math in economics

I run into a lot of comments along the lines of "just because you have a mathy model written down doesn't mean its right". Some of them seem to argue that by using math the way we do in economics, we're implicitly assuming people are rational. It's also very common to see comments that point to explicit assumptions of rationality and say "well, we know people aren't actually rational, so this whole thing is worthless".

All of this represents a fundamental misunderstanding of (i) what we're trying to do as economists, and (ii) why we use math the way we do. So I'm going to try to take a step back and explain what we're trying to do, and why we use math, in a way that's clearly laid out and easy to understand (now that I look at this, I'm pretty sure I failed, but I don't know how to do better).

Part I: What are we trying to do?

Economists are trying to understand economic systems. Wikipedia says "Economics focuses on the behaviour and interactions of economic agents and how economies work." "Economic agents" is a bit of jargon, and basically means "things that make decisions". Some economists might argue with me about whether or not agents make decisions, but they're invariably wrong or missing the point. We're interested in understanding how individuals make decisions (this is an oversimplified description of microeconomics) and what happens when a bunch of individuals make decisions together (this is an even more oversimplified description of macroeconomics).

In general, we follow a general sort of algorithm when we're trying to understand economic systems or agents. We look outside, or we look at some data, or we think about our own decisions, and we collect "Stylized Facts" - then, we try to create an economic model that produces those same facts.

For instance, when I was doing my Masters degree, I had a 100mbit/s download speed - which was a big change from what I'd had in my last house, a mere 18mbit/s speed. This changed the amount of time it took to torrent movies and video games. I didn't collect data, because I'm not that big a nerd, but I downloaded a lot more movies and video games during my Masters degree than I did before. So what I would argue is that the "stylized fact" is that when the cost of doing something goes down, and the benefits stay the same, people will do more of that something. If I go and write down an economic model - whether I write it in English, French, or math - and it doesn't produce this kind of behavior, I will rightly be skeptical.

This is why economists don't really understand comments along the lines of "economic models don't reflect reality" - because we make them specifically so that they do, in terms of the behavior of the model. I think most people who make these kinds of comments would say "but I don't do math in my head when I'm making decisions," and they'd (probably) be right. It's actually impossible for us to determine objectively if they're telling the truth or not, but since I don't think I do math in my head when making decisions either, I'm inclined to believe them.

The key point, though, is that whether I do math in my head or not doesn't matter. To explain why, lets turn to an example I'm familiar with - matchmaking ratings, or more generally ELO-based rating systems. This is a pretty big departure, and I don't think its absolutely essential to understanding what I'm saying here, but I like it. Feel free to skip ahead and come back if you still want to.

The question that rating systems are designed to answer is one that goes "is Bob better at (insert game of skill here) than John?" Think about trying to decide whether Kobe Bryant is a better basketball player than Michael Jordan, or LeBron James. Think about the arguments you've seen on this topic, or similar ones. It's not an easy question to answer - people talk about all sorts of things, like win/loss records, points per game over a career, number of championships, the rules of the game when X was playing, and hundreds of other things too. We fundamentally struggle to turn "skill at basketball" into a number, but if we want to say "X is better than Y at basketball", we have to be able to turn it into a number that we can rank. So, if we wanted to create a model that reflected reality, we'd want to list all of the little pieces that go into being a good basketball player, like height, reach, top speed, acceleration, "game sense", dribbling ability, and so on and so forth.

Lets say we had a list like that, and we all agreed that every single skill that matters to basketball was in the list. We still can't directly compare one player to another - what if one guy was 2 inches taller than another, but had a lower top speed, and everything else was the same? We need some way to turn a list of numbers (a vector) into one number. We could average all the numbers, but people would complain - they'd say that height is more important than the size of your hands, for instance, and they'd probably be right. So we need a weighted average, where the weights reflect how important each little skill is for "total" basketball skill. I think it's pretty obvious that we're never going to get everyone to agree on this. Creating a model that's realistic in this sense is impossible.

However, as it happens, we can go at the entire question from a different direction. What do we mean when we say "X is better than Y at basketball"? It seems reasonable to say that if X is better than Y, if they played each other a bunch of times, with equal teams around them, that X's team would win more than Y's. So, when we're talking about skill at basketball, we're actually talking about the likelihood of winning if they players faced each other, and everything else that matters was equal.

What if we could make a system that could tell us, accurately, how many times X would win against Y (everything else held equal)? We'd have a system that let us put players in a list, where the player on top was better than all of the other players. We'd have a system that told us how to answer "is X better than Y?" for every pair of players.

The key point here is that it doesn't matter how that system works. If it tells us accurately how many times X would win against Y, it's telling us about what we call skill. If the system worked by having cats shit into a blender, and looking at the color of what came out, I still wouldn't care, as long as it worked. Obviously, the color of blended cat poops has nothing to do with basketball skill. The system probably wouldn't work. But if it did, I wouldn't care that it was doing something completely unrelated to what we think happens in reality, because the results would be realistic.

This is why economists try to make models that show the same outcomes as we observe in reality, without being worried that we're making up agents that do perfect math constantly and never make mistakes.

Part II: Why we use math

Now, you might come out of what I just wrote thinking "yeah, ok, but there's no way blending cat poops would ever work". You might go from that to thinking "the only way your system could produce the right results on the back end is if it was realistic at the front end". I will now try to show you that this is not true.

Lets talk about video games. Midair is a FPS that is doing serious damage to my career prospects as we speak. I play it a lot, and I'm pretty good at it. Since it's an FPS, one of the primary things that I need to do as a player is aim my weapons at other players, with the goal of hitting them. So lets talk about what's going on in that one task.

When I play, I definitely don't do math in my head. I don't draw vectors in my head, or anything. I look at the game, I aim at a place that "feels right", and I fire. So I really have no idea what a "realistic" front end for my model would look like.

Lets pretend, for a second, that I never miss. I hit every shot I take. I can write down a mathematical model that produces that behavior, easily. I know what the projectile speed is, I can see the game, so I know what my target's trajectory is. Lets imagine a supercomputer that can perform infinite math in no time at all. If I program this computer to minimize the distance between where its shots explode and where its targets are, it's never going to miss. It's going to hit every shot it takes. I have produced a model with the correct back-end results, despite knowing for a fact that the front end behavior is nothing like what's actually going on.

Of course, I miss some shots. But this is pretty easy to deal with, too. Lets say I miss shots at random. Well, to get the same behavior out of the model, I just need to stick a random number generator somewhere in it, so that the computer aims at the wrong spot every once in a while.

Of course, I don't miss at random. I miss more often when targets are far away, or when they're going really fast, or when they're moving at close to right angles relative to me. But this is also pretty easy to deal with. Instead of adding random misses, we add a random number generator that generates bigger numbers when the target is going fast, or is far away, or is on a bad angle. Now the computer misses more in the same situations that I miss more.

Now, I hope I haven't caused anyone to go "wait a minute, this is impossible" - because it isn't, not if I have a good enough supercomputer. But I wrote all that out in English, and this is supposed to be about why we use math. Look at this statement:

program this computer to minimize the distance between where its shots explode and where its targets are

This is the same as writing "the agent solves min_(aim point) { || loc_(explode) - loc_(target) || }". I can write similar, but more complicated, mathematical statements for all of the modifications I just proposed.

Fundamentally, we use math because its easier. We can be more precise about the assumptions we make, and about the behaviors we specify. We can write all of them out much faster. But the key point, the point that I think people don't stress enough, is that it doesn't matter if the model reflects how people actually make decisions - what we care about is that the model produces the same kind of behavior.

Part III: Trying to tie it all together

Lets talk about how we make decisions. When I go to the grocery store, I'm not doing math in my head (I mean, sometimes I am, but thats because I'm an enormous nerd). I'm definitely not doing the kind of optimizations that basic microeconomic theory says agents do when they make choices. But what I am doing is trying to choose the best option of those that are available to me. I probably don't actually get it right very often, but I think I do better than just walking around at random, flipping a coin, and buying whatever is closest when it comes up heads.

The key point is that I'm trying to make the best decision I can. Have you ever made a decision that you thought was a bad decision the very instant you made it? If you said yes, I don't believe you. I don't believe anyone is even capable of doing it. Sure, half a second later you might think "oh fuck this is a disaster" - but you got new data. You're evaluating the decision with the benefit of hindsight. If you actually believe people make bad decisions knowing that the decision they're making is bad, I don't know what to tell you. You're departing from assumptions that are so fundamental that not even economists really realize we're making them.

If you believe that people make decisions believing that the decision they're making is the best one they could make, then you shouldn't have a problem with using math. It doesn't matter that people don't actually do math - I can model someone who's trying to make the best decision they can with a maximization problem. I just need to get the objective function and the choice variables right. How do I know I got them right? I check if my model is producing the right behaviors. Thats what economists do.

We use math because it makes things easier - and we ignore the fact that people don't actually do math in real life because it doesn't actually matter if they do or don't.