ARBITRAGE AND EQUILIBRIUM IN THE TEAM FORTRESS 2 ECONOMY

By yanis

ARBITRAGE AND EQUILIBRIUM IN THE TEAM FORTRESS 2 ECONOMY

Arbitrage is the holy grail of every trader. The dream of buying low and selling high (for this is what arbitrage is all about) is the driver of all commerce but also its own worst enemy: for as everyone is trying to pursue it, the potential for arbitrage disappears. And when it does disappear totally, we have equilibrium (the holy grail of the economists).

One of the problems of living in the analogue world is that these concepts (e.g. arbitrage and equilibrium) are extremely hard to pin down. No one has enough information on how actual prices diverge from their ‘equilibrium’ level so as to be able to measure, or to visualise, the potential for arbitrage. All we can do is guess what it might be. Not so in our digital economies. The following diagram, for instance, captures the room for arbitrage (what we call an Index of Arbitrage Potential for Steam trades of Team Fortress 2 items) as it arose during the period 13th November 2011 to 23rd May 2012.

The peaks represent moments when there was a great deal of room for arbitrage (i.e. for buying low and selling high), while the line’s thickness reflects the volume of actual trades. It is no great surprise that these peaks coincided with major new releases and sales (e.g. the Christmas sale) that the community required some time to price properly.  Where did this Index of ours come from? How come we could compile it for our economy when no economist can compile a similar index for the ‘real economy’ out there? The answer to the second question is simple: Unlike the analogue economy (where we can at best sample some prices at discrete points in time), at Valve we have all the real time data there is, courtesy of Steam: a peculiarly sophisticated barter economy. Let me explain why I am calling it ‘peculiarly sophisticated’ before explaining how we put together our graph above.

A peculiarly sophisticated barter economy

Steam enables Valve’s gamers to trade freely with one another, effectively to establish a substantial economy in which thousands of items, also imaginable as assets, are exchanged for one another. This is a typical barter economy, in that every exchange necessitates a double coincidence of wants (i.e. when Jack offers Jill some Team Fortress 2 hat in exchange for a couple of keys, the trade will go ahead if, at the same time, Jill also prefers that particular hat to her two keys).

Barter economies are cumbersome precisely because they require this double coincidence of wants before any bilateral trade proceeds. For this reason, throughout history, whenever the number of transactions (and ‘assets’) grew in number, one of those assets soon emerged as a numéraire – a basic form of money that is. Once the numéraire acquired currency, suddenly the prerequisite of some double coincidence of wants vanished and people could trade anything for the numéraire–asset which they could then use in order to buy whatever else tickled their fancy. In short, as economies grew in sophistication, they ‘monetised’ and ceased functioning on the basis of barter. This is why never in history have we witnessed truly sophisticated barter economies (for reasons similar to why we have not developed hugely sophisticated training wheels for professional cyclists).

Initially, I had expected that a similar pattern would be replicated in digital economies, like Valve’s. I was expecting to find that some item or asset would emerge as currency in the context of games such as Team Fortress 2. However, a close study of our Team Fortress 2 economy revealed a more complex picture; one in which barter still prevails even though the volume of trading is skyrocketing and the sophistication of the participants’ economic behavior is progressing in leaps and bounds. 

What is an economic equilibrium?

Those of you unlucky enough to have suffered the privilege of studying economics 101, will recall our fixation with something we call equilibrium. Why is equilibrium such a central concept in economics? The simple answer is that, without it, we economists stand no chance of explaining anything, let alone predicting (prices, quantities, etc.).

The idea of an equilibrium sprang up, like most scientific ideas, from physics. Suppose that you see a rock rolling down a mountain. Can you predict its path? Or, equivalently, can you predict its final resting point? If you can, then you have a pretty decent idea of its actual path. Well, this ‘resting point’ is what we mean by equilibrium: the point at which some ‘system’ will reach a resting place; a place in which there will be no tendency to carry on ‘moving’. Back to economics, suppose that we are witnessing the price of oil increasing, after (say) a period of relative stability. Will it stop at some level? Which level will that be? In other words, will it reach a new equilibrium, and if so what is the equilibrium price?

To complicate things a little, both in Nature and in some economy, an equilibrium can be either static or dynamic. A static equilibrium means no change. The ‘system’ under study is in complete standstill. Like the rock that stopped rolling. A dynamic equilibrium, by contrast, entails movement but of the sort that is eminently predictable, periodic. For example, the Earth’s orbit around the Sun (while neither the Earth or the Sun are stationary, the Earth’s orbit is). Or the demand for toys which, predictably, peaks before Christmas, every Christmas.

Equilibrium in a barter economy

Consider a small scale version of an economy like that of Team Fortress 2 (TF2) – the first Valve economy that I had the pleasure of studying. Suppose that there are four tradable items only: Some laser gun (G), one type of hat (H), earmuffs (E) and keys (K). Now suppose that we observe that players trade these items in the following proportions, or exchange rates (also known in the economics trade as ‘relative prices’):

  • 2 hats for 1 laser gun
  • 1 key for 2 laser guns
  • 3 earmuffs for 1 laser gun

Question:       Recalling that equilibrium is reached when there is no further room for arbitrage, how many hats should 1 ear muff buy if this economy is to be in equilibrium?

Answer:          From the ‘numbers’ above, we know that, in terms of exchange values:

Value of 1 laser gun = value of 2 hats = value of 3 ear muffs = ½ a key

OR

1 key buys 4 hats, or 6 ear muffs, or 2 laser guns.

Thus, the relative price of hats to ear muffs ought to be 4-to-6, which is the same as an exchange ratio of 2:3 hats to earmuffs, or 3:2 earmuffs to hats. In terms of a table, or matrix, the above reduces to the simple question: What should the missing entries be (i.e. the numbers in the red, blue and orange cells) if this tiny economy can be said to be in equilibrium?

We have already found the missing number in the case of the red cell: It is 3/2, i.e. the exchange ratio, or relative price, 1.5 earmuffs to one hat (or 2 hats for 3 ear muffs). Before moving to the missing numbers in the blue and orange cells, let’s first make sure we understand that unless the number in the red cell is 3/2 there will be room for arbitrage – which, in turn, means that some entrepreneurial person will try to exploit, with the result that the prices in the first row will end up changing.

Suppose that the number we actually observe (from our trading data) in the red cell was 3 (as opposed to 3/2). What would that mean? It would mean that a smart Jill, some fictitious player, could do the following starting with, say, 6 earmuffs:

  1. Jill could trade her 6 earmuffs for 2 laser guns (since, from the first row, we know that she needs 3 earmuffs to get 1 laser gun)
  2. Jill could then trade these 2 laser guns for 4 hats (since, again from the first row, we see that the exchange rate is 2 hats per laser gun)
  3. Finally, if the number in the red cell is 3, she could trade each of her 4 hats for 3 earmuffs – ending up with 12 earmuffs.

This is a typical case of arbitrage. Jill starts with 6 earmuffs and, just by bartering, she ends up with more earmuffs than she started off with. While this is great for Jill, economists believe that arbitrage opportunities of that sort cannot survive for long. Why? Because, if these opportunities for gain (without pain) are freely available, they will be exploited. So, others will emulate Jill, trying to trade earmuffs for laser guns and then laser guns for hats, before trading hats back to earmuffs. But all these offers of earmuffs (corresponding to what Jill did in the first step above) will exert downward pressure on the relative value of earmuffs. E.g. it will cause players selling their laser guns to demand more earmuffs for each of their laser guns. Thus, the price of earmuffs will fall (relative to laser guns) and, therefore, the opportunity to end up with more earmuffs (through selling them before buying them again) will disappear.

In short, as players take advantage of arbitrage opportunities, the economy tends to equilibrium and, when it gets there, there are no arbitrage opportunities left! What would our economy look like in such a state of equilibrium? The following tables answer the question: Each relative price in the oval shape must equal the relative price in the rectangle divided by the relative price in the triangle.

More generally speaking, in economies like (our TF2 one) with many, many items that are being traded, say N, there will be (N=1)N/2 relative prices; e.g. even if we only had 100 assets, we would end up with 4950 relative prices. Imagine the complexity of the TF2 economy where we have more than 35 thousand items. We are talking about more than 600 million relative prices at any one point in time. What would an equilibrium look like? The answer is less complex than you may think. This is what our N by N matrix of relative prices looks like when assuming that keys (k) are the Nth item:

Just like in the simple 4 item example above, our prices are in equilibrium when the following equalities hold – which are no more than an extension of the conclusion above that each relative price in the oval shape must equal the relative price in the rectangle divided by the relative price in the triangle:

Why care about equilibrium?

The simple answer is because it makes our lives (as economic analysts) much, much easier. Take again the simple 4-item example above, in a state of equilibrium:

The beauty of equilibrium is that it makes it possible for us to quote only one price per item. Indeed, the matrix above can be reduced to a simple menu list which gives us the price of laser guns, hats and earmuffs in terms of number of keys necessary to purchase one unit of each:

Suddenly, instead of 6 prices we have 3. And in the case of 99 items, the number of prices reduces drastically from 4950 to just… 98 (one per item, excluding the keys that have no price, as they are the effective currency).

Should we expect the economy to be in equilibrium?

We wish! But if we assume, as economists tend to, that equilibrium prevails most of the time we are inhabiting La La Land. Indeed, the first great economists (people like Adam Smith and David Ricardo) never thought that an economy would ‘equilibrate’; that it would ever reach equilibrium. They just hoped that competition will cause economies to tend to an equilibrium, which is quite different to saying that they will actually get there. So, to cut to the chase, we should never assume that our economy is in equilibrium.

Now, the beauty of digital economies is that we have all the data at our disposal. We can actually observe how far or how close to equilibrium our economies are. So, one of the first things I did at Valve was to put the TF2 economy under the microscope to see whether it is in equilibrium or not. Guess what: it is nowhere near it. So, folks, it is official: There is plenty of room for arbitrage. Moreover, this arbitrage window varies in size constantly – as we shall see below via the medium of some nice graphs.

But first, to measure the arbitrage potential, or window, we needed to develop a method for assigning one price per item even though our economies are hardly ever in equilibrium (a fact that, according to what I just wrote above, should rule out assigning one price per item). This is how we did it:

Measuring the relative prices/values of the TF2 economy

Having observed that the TF2 economy is never in equilibrium  long enough for us to be able to derive one price per item, we asked ourselves: Is it possible to measure the size of our economy? To offer a valuation of some hat or other item? To answer any of these questions, we had to find a way of pinpointing one price per item (e.g. some hat) when, in reality, there is not one but many (e.g. one in terms of laser guns, one in terms of keys, one in terms of earmuffs and, to boot, these prices are in conflict with one another – thus giving rise to arbitrage opportunities). How could we do this?

Suppose that, returning to our trusted 4-item example, we have the following disequilbrium prices (averaged out over a period of, say, a week):

Because we are in disequilibrium, it makes no sense to take the last column as a vector of our relative prices for G, H and E. We need to take into consideration the fact that the other numbers (in the other columns) are inconsistent with the last column and pack important information about the value of each item (that we should not throw out). Here is what we did in order to compute single price estimates for each item:

We began by noting that, if the economy were in equilibrium, the following equations would hold:

Then we observed how many times, during the same period of observation (week in our case), guns were traded for keys (say NGK), guns were swapped for hats (NGH), guns for earmuffs (NGE), hats for keys (NHK) etc. Beginning with the item that was traded most often, we computed the price of guns as its exchange ratio with that most heavily traded item. E.g. if keys were the most traded, we would define the price of guns as . Next to compute the price of hats in terms of keys we would use the equation (see above) =, which would have held in equilibrium. But because we are not in equilibrium, we augment it by the relative frequency of trades as follows:

Estimated price of hats in terms of keys =

In other words we weighed the two sides of the equation that would have held in equilibrium in a way that reflects which of these bilateral trades were relatively more frequent (and thus influential). In this manner, and after having dealt with a significant number of computational issues that I shall not bother you with here, we ended up with a method deriving a list (or vector) of prices per item. What were they expressed in? They were expressed in something I call notional keys; i.e. the ‘keys’ that would have been observed if our economy were in equilibrium. In other words, a made-up currency that converges to the actual keys of the TF2 economy when the latter approaches to equilibrium, as arbitrage opportunities vanish.

And the result? Here is a graph depicting our estimates of the (relative) price of three popular TF2 items: earbuds (in red), Bill’s hat (in blue) and refined metal (in purple). [Note that this diagram pertains to the period between November 2011 and 1st June 2012.]  

Arbitrage opportunities come and go in the TF2 economy

So, the time has come to explain the derivation of our Index of Arbitrage Potential as it appeared in the diagram with which this post began.

In the preceding analysis (as evidenced by the last diagram) we managed to compute one price per item in the TF2 economy. From these it was easy to derive our Index of Arbitrage Potential. Here is how we did it: We have already worked out that the following equations must hold in equilibrium:

By computing the extent to which these equations do not hold (by means of computing the least squares of the differences between the left hand and the right hand side of these equations, all added together), we can gauge how far our economy is from equilibrium at any point in time. And we can then graph these against real time, looking out for fluctuations in the presence of arbitrage opportunities as new items (ones not previously priced by the community) enter the game. The result is a series of rather cool graphs, one which you saw at the beginning of this post.

Conclusion

When trading on Steam, there is always potential for gaining by buying items cheaply and selling them expensively. By the same token, you may be short-changed, so to speak. In this post I tried to show you how these opportunities and dangers come and go; how they expand when new items or sales are released and then wane as the community learns to price things more consistently (i.e. as arbitrage gives way to equilibrium). I think this is fun to see. The tantalising thought also comes to mind that we could, perhaps, let you see this graph in real time, to guide your trading. Or perhaps not? What do you think?