Wealth Growth in Guild Wars 2

Today's article was inspired by a post made by John Smith on the forums. The quote has to do with Wealth Growth inside Guild War 2 versus a real world economy. He stated

"There are several good reasons why wealth distributions in game don’t mimic the US Economy. Another reason is that the growth of wealth is exponential in many developed economies, where it’s closer to logarithmic in GW2. The entire set of reasons would be a very interesting discussion…Well for me at least." - John Smith (from forums)

These discussions are interesting to me as well, thus the reason I thought I would write an article on the topic. What does it mean for something to grow exponentially? and what does it mean for something to grow logarithmically? And finally, why might wealth grow logarithmically in GW2?

Exponential Growth

What is exponential growth? It is the growth rate of a lot of things around you. Already mentioned, is the US economy, other things include populations of humans, cultured microbes, compound interest, computing power (see Moore's Law) and internet traffic. There are lots of other examples, these are just some of the things you might encounter in your day-to-day lives. Let us look at the math a little more closely by examining the example of compounded interest. I will try to keep the math as simple as possible.

Firstly, what is compound interest? Well firstly, interest is a fixed payment paid on a principle amount over a given period of time. The payment is usually a percentage of the principle amount. For example, if your principle amount was $100.00 and you were to receive interest of 10% per month, than every month you would receive a payment of 10% of $100.00 or $10.00. These interest payments would not change amd  month to month you would keep receiving $10.00. With compounding interest, the interest payments get added to the principle amount each month and the interest is calculated with the new principle amount in mind. Using the same example, in the first month you would receive $10.00 making the principle amount $100.00 + $10.00 or $110.00. In the second month you would receive 10% of $110.00 or $11 making the principle now $121.00. In the next month you would receive $12.10, and so on.

So, lets try to generalize this. Let our portfolio's value be defined by the variable x. The value of our portfolio at any given time will be given by the function x defined at time t, or x(t). The initial value of the portfolio, the principle amount, will be given by the variable a. Thus the initial value of our portfolio would be,

x(0) = a

In other words, the value of our portfolio, x, at time t = 0 is a. There is no reason for this other than these are the variables and values we are selecting to setup the equations that will describe compound interest growth. The only other variable we need to define is a rate of growth for our portfolio, or the percent interest to be paid each term, t. Let, i, be the rate of growth of the portfolio. Thus, at t = 1 we would have,

x(1) = a * i + a  = a * (i + 1)

This is simply, the new portfolio value at after the first payment (t = 1) is the value of the interest payment, a * i, plus the initial value of portfolio, a. We can simplify this equation further by letting b = i + 1. That way we have,

x(1) = a * b

For compounding interest, we have to add in the interest from the last payment when calculating the interest to pay for the second period. Thus, the interest we pay at t = 2 is based off the value of the portfolio at t = 1. So, if we write that out we have,

x(2) = x(1) * b  = (a * b) * b = a * b²

And we can continue this,

x(3) = x(2) * b = ((a * b) *b) * b = a * b³

x(4) = x(3) * b = (((a * b) * b) * b) * b = a * b⁴

You should notice a pattern by now. The power b is raised to is equal to t. Thus we can generalize the above into one formula to describe the value of our portfolio, x, at any given time, t, given compounding interest i, such that b = 1 + i.

x(t) = a * b^t

This is the form that all exponential growth takes, whether it be compounding interest, human population growth, or internet traffic.

Logarithmic Growth

Another beast completely. Most people have not worked with logarithms, so I will not go into as much detail as I did with exponential growth. The main thing you need to know is that a logarithm is the inverse of the exponential function, as division is the inverse of multiplication, and subtraction is the inverse of addition. The logarithm is defined in the following manner. If you have,

x = b^t

Then,

log_b(x) = t

That is, the logarithm, with base b, of x equals t. Basically the logarithm returns the power that the base of the logarithm would need to be raised to in order to equal x. Here are some examples,

log₁₀(10) = 1

log₁₀(100) = 2 

log₁₀(1000) = 3 

log₂(4) = 2 

log₂(8) = 3 

log_x(1) = 0

It is hard to put logarithmic growth into a context that most people can understand as not a lot of day-to-day things grow logarithmically. It is an important part of computer science though. A lot of mathematical problems are broken down and solved in ways that as the complexity of the problem increases the amount of time it will take to solve the problem grows logarithmically with respect to the complexity of the problem. This makes the problems compute faster when put into a computer. A population that is growing and has its growth limited by resources will often grow logarithmically. If you do not understand logarithms, it doesn't really matter. What does matter is what the graphs looks like when we compare them. 

Exponential vs Logarithmic

The differences between the two can wait until tomorrow when I will continue this discussion. The graph to the right show them both graphed together on the same chart. You can see that one is the inverse of the other. In fact, when I made the image above for exponential growth I merely rotated and flipped the logarithmic graph. Start thinking about why wealth in GW2 would grow logarithmically and not exponentially. Remember wealth growth in the real world is exponential, ie the exact opposite  Leave the reasons you think there is a difference in the comments below. I find this topic really interesting and I will be posting my thoughts tomorrow.