Wealth Growth in Guild War 2, part 2

Today I would like to continue my discussion of wealth growth in Guild Wars 2. This discussion was inspired by a post made by John Smith on the forums. 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)

In part one of the series I mainly answered the questions, what does it mean for something to grow exponentially? and what does it mean for something to grow logarithmically? Today I would like to tackle the final question I presented in the first part, why might wealth grow logarithmically in GW2?

Exponential vs Logarithmic

This graph shows the opposite nature of these two functions.
The blue line is exponential growth and the purple logarithmic

To quickly recap, in the last article we found that if something is growing exponentially, than the value of that "something", x(t), at some time in future, t, is given by the formula;

x(t) = a * b^t

where, a is the initial value, and b is the rate of growth. On the other hand we discussed that logarithmic growth is the exact opposite of exponential growth in the way that division or subtraction is the opposite of multiplication and addition. If something is growing logarithmically than its growth is described by the formula,

x(t) = a * log_b(t)

where x(t) is the value of the formula as some time, t, the initial value is a, and b is a base of the logarithm and governs the rate of growth much like it does in the exponential function.

The primary difference between the two can been seen by looking at their respective derivatives. Without getting side tracked into calculus, all you need to know is that the derivative of a formula merely describes that formula's rate of change from one time to the next. The derivative of the exponential formula is;

    x'(t) = a * b^t * ln(b) 
          = a * b^t * C

and the derivative of the logarithmic formula is,

    x'(t) = a / (ln(b) * t) 
          = a * / (C * t)
          = a * 1/C * 1/t

You can see the two formula are drastically different. Firstly, let me explain the ln(b) term that appears in both expressions. Really the only thing you need to know is that ln(b) will always be the same for any value of the functions main variable time, t. That is why I made the substitution C = ln(b). This substitution just makes the formula easier to read because ln(b) is a constant term. For the specifics, ln is a special type of logarithm where the base is equal to the number, e. This number, e, is a constant like pi. For more details see the natural logarithm. Since this logarithm is being taken on the number b, which is also constant, the logarithm of this number will always be the same as well.

Quickly examining the simplified derivative of the exponential formula and you can see that as the value of t gets bigger so does the value of the derivative. This means that as time goes on our exponential formula increases the rate at which it grows. The rate of change itself is actually increasing exponentially. Look at the function and see that the only term that depends on t, is b^t. You can see that this term will clearly get larger as t gets larger.

On the other hand if you look at the derivative of the logarithmic formula you will see that as t gets larger the value of the derivative actually gets smaller. This means that as time goes on our logarithmic function slows down in growth. You can see that by looking in the equation for the term that depends on t. In this equation that is 1/t. As t gets bigger this clearly gets smaller as any fraction would when you increase the bottom term.

So, exponential functions grow at an ever increasing rate, and logarithmic functions grow at an ever slowing rate. That is the main difference between the two and a difference between the GW2 economy and the US economy. This is also why it is interesting! So, why would the two be different?

Why does the Guild Wars 2 Economy grow Logarithmically?

Now that I have described what exponential growth and logarithmic growth are and what the differences between them are we can start to talk about why the Guild Wars 2 economy grows logarithmically versus the US economy which grows exponentially. The main things I am thinking about are what would cause the rate of growth of the economy as a whole to have the brakes applied as time goes on?

The Guild Wars 2 economy as a whole has been fabricated by individuals using gold sinks and gold taps. A gold sink is a way for the developers of a game to constantly remove quantities of gold from the economy, and a gold tap is the exact opposite, it is a way for developers to add gold to the gold supply of the game.

Believe it or not, this kind of thing happens in real economies as well. The federal bank adds and removes money from the general money supply through the buying a selling of bonds to help regulate economic growth. This is sometimes referred to as "printing money", but because no money is actually printed the correct term is "Quantitative Easing". There are other ways for wealth to be generated (a mine creates wealth from work done by people removing materials from the earth), and for wealth to be destroyed (a war destroys wealth every time something of value is blown up).

A comic showing the situation the central bank finds itself
in when attempting Quantitative Easing.

The whole point of controlling the money supply is to attempt to control inflation. Inflation is a general rise in the cost of goods and services as time goes on. Inflation is always going to happen and it is the reason why your Grandpa can tell you about the hundreds of things he could buy for a nickle when he was a kid, and now a days you can not buy anything. A steady amount of inflation is healthy, but uncontrolled inflation can lead to all sorts of problems, such as people starving in the real world, or new players not being able to afford basic items, in GW2.

In general, since the GW2 economy has been designed from the ground up, we can imagine that there are plenty of gold sinks in the economy to prevent crazy amounts of inflation. Some examples of gold sinks in the game include, the 15% TP fee, anything purchased from a vendor, soul binding, or account binding items (they can still be sold to an NPC, but the difference in the old sale price on the TP and the new price to a vendor is a gold sink). The more you think about it they more you will see that a large amount of wealth is pulled out of the economy, enough to prevent it from growing on an exponential scale, and instead limit it.

This comic depicts the amount of fun to be had
fighting against inflation.

I have played MMOs before where the economy is so inflated that a new player does not have a chance at purchasing any of the items in the game. The MMO's solution to this is to start new server and start the inflation cycle all over again. The designers of Guild Wars 2 are smart people. We can trust that they have specifically designed the game in a way that money will be drawn out of the system with each cycle. Think about the places in GW2 that you can purchase things with your gold, then think about where that gold is going. When gold is "sunk" in GW2 it isn't just moved somewhere, it is GONE. It is not coming back. This is utter wealth destruction. This prevents the uncontrolled inflation seen in other MMOs.

If you compare this to the real world, the scale of the wealth being removed through the gold sinks does not even compare to what the federal bank does to control the money supply in our real world economies. In order to have the kind of wealth destruction seen in GW2 economy in the real world there would need to be something actively undoing human production, like a massive disaster or a massive war.

The best case for economic growth in GW2.

The only way to create wealth is to play the game and farm gold from the gold taps, such as dungeons, or materials from creeps, etc. There is a fixed limit to how much a player could earn from this and if every player was doing this and not spending their money it would ensure that the GW2 economy would grow at a constant rate (aka linearly). Playing the trading post does not add wealth into the economy, it actually only destroys it through the 15% transaction fee. Even without the fee it would be a zero sum game with nothing produced in the economy (gold and items just change hands nothing is produced).

In the real world you can innovate and create new technologies which can create wealth at ever increasing rates. Then you can use the wealth you have earned from that innovation to fund a new innovation that help generate even more wealth. For example, think of the industrial revolution and its impact on the number of useful things that can be produced. There is nothing like this in Guild Wars 2. There is no way to take wealth and use it to generate more wealth in game.

Conclusions

Everything in GW2 leads towards some sort of gold sink. If you are making money in GW2 it is usually because you want to buy something big. If you buy it from a vendor that gold is instantly gone. If you buy a large item and equip it, most likely that will soul bind or account bind the item, then effectively that wealth is gone. This leads to the conclusion that most player's goal in GW2 is to destroy wealth. This is the exact opposite to the real world where most peoples goal is to create wealth. When put that way, does it surprise you that the two economies grow in exactly the opposite manner as well?

So why does the GW2 economy grow logarithmically? It grows logarithmically because that is what makes the game fun for the most number of people. This is done through the control of gold sinks that utterly destroy wealth and gold taps which ensure that at best the economy could grow linearly (at the constant rate). Playing the game requires players to destroy wealth. Players are essentially earning wealth in order to destroy it.

I hope this two part series on the growth of the GW2 economy has taught you something, or at least given you something fun to think about. Economics is an interesting topic that I hope to discuss more here on my blog. Let me know what you thought of this discussion in the comments. If you think I have it wrong, I might I am no economics expert, leave me a comment below. If you want more details on something or clarification also leave a comment. I would love to talk about this more.

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.