OK, here it is, the most complicated part of the book. I’ll try to explain this **urban mathematics** thing as clearly as I can, but I guess I’ll need to re-write it few times. This article would probably be tricky to follow but fortunately, there are pictures.

So, deep breath…

My goal here is to find the mathematical equivalent of (don’t panic) Nash equilibrium on a large scale with many players. Just as in the example of the ice cream vendors, I am looking for the easiest way to explain why businesses are clustered in the city, using the simplest possible algorithm.

Hmm, sounds reasonable so far. Are you with me? let’s go.

The ice-cream vendors example uses a simple one-mile-long beach and starts with the steady state of one vendor in the middle of it interrupted by a second vendor. The goal of each ice-cream vendor is to earn as much as possible. The rules of the game allow him to move each round in one step until the system stabilizes. Things that are missing from this simple story are – a situation in which customers are willing to walk only a few meters to get the goods and a situation where the relocation of the vendors is costly.

And therefore, that’s what I’m trying to do with my algorithm.

Let’s start with the one-dimensional situation, like the beach of the ice-cream vendors, it’s easier and simpler.

This time we have a long, endless beach. We have customers who are distributed in a uniform manner, every 10 meters there are 10 customers and one vendor. Each vendor has 10 popsicles to sell each day. Each customer buys one popsicle a day. We are in a very stable situation where all suppliers manage to sell all the popsicles to all of their customers every day.

However!

We allow customers to move around. In fact, our customers don’t really care where they buy their popsicles. We assign a 10% hassle-distance to each 10-meter cell. That is if they buy in their original location, their hassle is only 10%. If they move to the next cell – the hassle is 20%, the third – 30% and so on.

The distribution of the demand for popsicles will be relative to the hassle-distance. In other words, customers will buy more from vendors that are closer and buy less from vendors that are farther away. Since our customers are not willing to bother when hassle-distance exceeds 100%, they will not purchase anything from over 100 meters away from their ‘home base’. That’s the maximum hassle they’re willing to “pay” to buy a popsicle.

I need some sketches to explain what’s going on here but at the base it’s very simple … are you with me?

Let’s phrase it in a formal way:

- The supply in each cell is 10
- The demand in each cell is 10
- The hassle-distance of each cell in the row is 10%

Looking at it, the situation is stable. Every vendor sells 10 popsicles, although he sells some of it to people that came from his left, some to those who come from his right and some to those who are located in the same cell as him. No one have an interest in moving, reducing demand or increasing supply.

We have to set another rule before we start playing with the system: The vendors don’t want to produce or hold too many popsicles, if he’ll have more customers, he will increase the supply and if he had more popsicles than customers he would reduce it the day after. We’ll limit the daily change a vendor can do in to 10% of its current supply. Each day represented by a single round of calculation.

Now we can start playing with the system. You’re welcome to join me, I’ve built it for you in excel and you can feel free to follow my steps. I’ve added a button to recalculate each round and you are invited to download it. The objective of the game is to perform rounds of calculation, to see the evolution of the supply though time, and check what happens until we reach a steady state.

Since buyers are not allowed to change their base cell location and will not change the demand (for now at least), there is no need to display the demand on graph. We will focus on only two data sets – the Supply in each cell, shown in blue bars, and the pressure to change the supply, shown on the red line.

We start from a completely stable situation as described:

You can see that the red line stands at zero along its entire length of the dataset. Zero is – no pressure to increase or decrease supply. If we will run a calculation round, since the need for change is zero, nothing will change.

### Let’s get to it and play

Let’s say that in one of the cells (Point A, marked in orange) we now have an additional vendor. The supply was immediately increased to 20, and the graph will look like this:

Let’s take a second to examine it. Supply was increased only at point A, but since customers are willing to go and buy out of their home cell, the effect of over supplying hurts vendors on neighboring cells. You can see the pressure on the supply in the red line. The maximum pressure to decrease supply is at the point A. The farther away from it, the effect goes down till the hassle-distance is over 100%.

Let’s run one more calculation round and see what happens … Now we’re with this result:

**Huh? what?!?**

That was exactly my reaction … What just happened here? Why is the red graph so different?

Let’s try to figure it out, If we look at point A, the pressure on it is negative and will still reduce the supply BUT – much less than on the first round. The reason is that not only the vendor at this point reduced supply but also vendors that are at the nearby cells. Still, since the supply in point A is still large, there is still pressure to reduce it.

**But then why do we get positive pressure on both sides?**

To understand it, we need to examine what the vendors there just experienced. A vendor at point B is out of range for the influence of point A, meaning that he has no notion of excess in supply there. On the other hand, it is certainly affected by the decline of the supply in cells between A and B. In fact, consumers located in B feel that someone has reduced the supply. There is now more demand than supply, and sellers get “positive” pressure to increase supply accordingly.

For the next few more rounds it seems that the situation will be stabilizing back to the starting point:

The orange line is approaching balance, the pressure to reduce supply at point A still exists, and it seems that the system will balance itself in few more rounds of calculation.

So, let’s run a few more rounds …

Well, it seems almost balanced but … wait … at point B there is a supply higher than 10 for sure and yet positive pressure continues to rise. That is because on the range between A and B, competition is coming from both directions, A and B, and continues to pressure it down to decrease supply.

Few more rounds and … surprise. The system is out of balance and aspires to balance in a totally different place:

More rounds and the system is almost stable:

You can see where it goes. All the supply is now located at defined cells that aspire to grow further while between those cells there is almost no supply. The new equilibrium will be when all the supply will accumulate to specific points and drain the areas between them.

This mechanism of accumulation of supply to defined points in space (as in the simple case of the ice-cream vendors) has many implications. I will go into more details about the formula and its implications later on.

Till then here is a small an example that you can relate to:

**Imagine a commercial street with three shoe stores one next to the other, and you want to open your own shoe store**.

These will be location options you will consider:

- Locating your store right next to the existing stores. If you are close, you create a block of 4 stores that will attract supply from the entire area. People prefer to get to a place where they possibly have more selection and competition. Not only that your store will not reduce the turnover of other stores, but it may actually increase it.
- Locating your store 5 minutes away from the existing stores. That will be a bad option, even someone that lives right next to you will prefer to go to the center. It is true that you are closer but if all the difference is a five minutes drive most customers will prefer the larger center.
- Locating your store far from the existing shops, let’s say on the city outskirts. Most of the residents in the nearby neighborhoods will prefer your store. True, the center may have more variety but you are much closer. Thus, you will not be competing for the same customers.

In other words, just as the mathematical graph demonstrated, a small accumulation of supply creates positive economic pressure, which calls for a more accumulation of supply in that same place while drying, creates negative economic pressure, on its immediate periphery.

**Ugh! That was tough.**

Hope you’ve managed to chew through that. I’ve tried to make it as simple as possible. I’ll assume that you understood everything, and I’ll go on to the broader meanings of this economic model before I’ll come back to the hard math behind it.

If you have any comments or suggestions about improving this article, drop me a note please, I can use all the help I can get.