Elo and Trickle-Up Economies

Preface: I realized after writing it that this post is pretty long and could use an abstract. It's an overview of the two scoring systems used in Warlocks viewed as economies. The objective is simply to note what kind of properties will emerge from having an economy with these particular constraints. What does emerge is that the resources in both systems are only acquired from other players, not independently generated, which results in a pyramid of sorts, where a large number of players lower down in the hierarchy provide a source of points for the higher ranked players. Larger economies will result in a higher peak, but otherwise do not effect the average player in these systems.

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As I've mentioned before, I'm an avid player of the game Warlocks, based on Waving Hands. This game uses two different ranking systems for competitive players, and they have some interesting differences between them.

The first, and simpler ranking system is ladder points, and they work as follows: every time a player wins a ladder match, they gain one ladder point; every time a player loses a ladder match, they lose one ladder point; every time a player dies during a ladder match, their ladder points reduce to zero (note that in this game, most matches end with one player surrendering, not dying). Every player starts with a ladder score of zero, and you cannot have negative ladder points. This means that every time a player with no ladder points loses a match, a ladder point is created from the ether, and every time a player with ladder points loses a match, their point is effectively transferred to the winner. There's one more feature of ladder matches that's worth mentioning, and that is that you cannot challenge a player to a ladder match if your relative ladder scores are more than 5 points apart.

The result of these features is that ladder scores rarely get very high. Since your ladder score will get reduced to zero by a single death it takes a lot of skill (or luck) in order to continually grow your score. Moreover, since you cannot challenge an opponent who is more than 5 points apart from you, the high possible ladder score for any player is 7 points higher than the second-highest score (assuming they began 5 points apart and that the higher-ranked player won). This means that in order for me to have a ladder score higher than 20, there need to be other players with a ladder score of at least 15 I can challenge. This means that the upper limit of ladder scores depends on the presence of a population of successful ladder players who collaboratively create ladder point (by playing those with 0 ladder points) and then transferring them up the ladder to the best players.

We'll see a similar dynamic with the second ranking system: elo. Like in chess rankings, elo is a system in which the change in a player's score is weighted depending on their expected likelihood of winning (which is, in turn, based on the competing players' relative elo scores). Each player who registers begins with an elo score of 1500, which defines that score as the expected skill level of an average new player. Each match results in one player gaining a number of points and the other losing an equal number of points - in other words, once again, matches effectively cause a "transfer" of points from one player to another. If a player with a lower score beats a player with a higher score, they earn more points from the win, and if a player with a higher score wins, they earn fewer points. The difference in points earned corresponds to a player's expected likelihood of winning - meaning that if I'm expected to have a 75% chance of defeating an opponent, I will earn 1/3 as many point for winning as he will if he wins, so that over the course of many games, elo scores will stabilize if players tend to win as often as they are expected to given their elo score. Since all starting players start with 1500 points, they begin ranked as equals even though some may be stronger players than others. However, the differences in skill level will fairly rapidly be reflected in their score once they begin playing ranked games.

Let's look at an example. I register a new account and start with 1500 elo. If I play and beat another new player, I will gain 12 points, to have a score of 1512, and their score will go down to 1488. Now the difference in our scores is 24, so if I play that same player again and win, I will gain slightly fewer points than I did the first time. Once the elo difference is over 100 points, I will gain 8 points from a win and my opponent will gain 16 points if he wins - as long as I win approximately twice as often as I lose, the elo difference will remain stable, but if I win more often, it will continue to go up, and if I lose more often, it will go down.

Notably, if the winning player is ahead by enough elo, they effectively gain no points from victory, so many high-ranked players will simply refuse to play ranked matches with much lower-ranked players (since they have nothing to gain and much to lose if they make a mistake). In practice, the maximum effective difference between players who can fairly compete in ranked matches is a little over 200 points. Any more of a difference and fluke wins by inexperienced players will unduly throw off the scores of high-ranked players.

All of this together suggests some interesting features of the elo economy - since a winning player gains as much as their opponent loses from a match, the sum elo score of the player population cannot grow except by the addition of new players, and that the existence of players with more than 1500 elo requires the existence of players with less than 1500 elo. Moreover, a player can only effectively grow their elo by playing opponents with an elo score within 200 points of their own, which suggests that growing your elo depends on a population of players with elos near your own, so the highest possible elo in the system depends on the number of successful players, which is in turn limited by the number of total players. That is, a population of new players is needed in order to support the elo growth of players with elos between 1500-1700, and a population of players with elos of at least 1700 is needed to support the elo growth of players with elos between 1700-1900.

As of this writing, there are 1577 players who have registered to play Warlocks, about 200 of which never played a ranked duel. Of the players who have played ranked games, 281 have an elo higher than 1500, and 419 players have an elo lower than 1500. The lowest elo in the system is 1298 (202 points lower than the average) and the highest elo is 2106 (606 points higher than the average). This suggests that in practice, a large population of weak players is needed to support the heightened elo scores of a relative few. There are two reasons for this: first, players who repeatedly lose will likely stop competing at some point, and players who repeatedly lose will have their elos fall to the point where they no longer effectively feed the elo growth of stronger players.

Since the value of a win is weighted by the likelihood of the win, players who perform as well as expected will have stable elos - if you are about twice as good as the average new player (meaning twice as likely to win), your elo should stabilize around elo 1600. However, once a player enters the higher echelons of play, the relative dirth of other high-ranked players makes it harder to play enough balanced games to maintain a representative elo. In a population of players with elos from 1400 to 1600, it is unlikely for me to grow my elo above 1800, no matter how good I become at the game.

So the grand result is this: The total size of the elo economy of the game is determined by the number of players in the system, and the larger the total elo economy is, the higher the elo ratings of the best players can be, but that for the vast majority of players, the size of the elo economy will have no impact on their personal elo scores. That is, as a resource, the total quantity of elo in the system will only effect the players at the top.

Now there are obvious disanalogies between the elo economy in Warlocks and market economies in the real world, but it nonetheless serves as an interesting model of a competition driven economy. This is also not meant in any way to be some kind of moral statement about how "just" the elo system is - the numbers simply represent the fact that some players win more often than others, and it is the explicit goal of the elo system to represent this. I simply believed that the unintended emergent features of the system are noteworthy, since they result from the interactions of thousands of players.

What is a Bubble? (Philosopher's Economics)

I've been thinking about economics a bit recently (I doubt I'm the only one), and a few things regularly creep back up into my mind. One of those is the question, "What is a bubble?" Now I'm not asking what people mean when they use the word - I know that much - I'm asking what it represents. In theory it's a disconnect between the perceived state of the market and its actual state, but I want to see what it is one level deeper than that. Where did the money lost go? Why is it that bad accounting results in our society being poorer?

Before we begin, full disclosure: I'm not an economist, and I'm not an expert on the intricacies of the market. In fact, I'm historically the kind of person that doesn't even like thinking about economics. In my defense, however, recent history has shown that many supposed experts on the market are nothing of the kind, and it's also forced many of us who would prefer to spend our time on other topics to think a bit harder about the economic system we live with. I'm also not going to try and explore the intricacies of the market here; we're going to be looking at fairly high-level concepts instead.

What I propose is a mental exercise: imagine, for the time being, that you do not know what "money" is, or quite what an "economy" is, and you are presented with our society ("our" meaning American society, here, but much of this will apply to other developed countries). You do know that the people in this society adhere to a complex codified system of behavior in order to allocate resources, which they refer to as "the economy".

So how does this system work? It seems to have a number of conditions and goals:

1) It seeks to produce value for the system as a whole. Unfortunately (very unfortunately, as we shall see), what qualifies as "value" is vague and isn't universally agreed upon.

2) The system needs to allocate resources so that they are spent on the projects best capable of adding value to the system (in order to fulfill the goal above). Note that just because a project contributes value today doesn't mean it will contribute value tomorrow, so this must be constantly re-evaluated.

3) The system motivates citizens to be productive by allowing those who create the most value to consume more of the value produced.

4) The system requires citizens to consume its products. This is for two reasons: first, the act of consumption is an expression of faith in and acceptance of the system. Second, and more importantly, it is a means of measuring the perceived value the system is producing.

So (4) ties back into (1) - it's how the system deals with the ambiguity of "value." If a citizen consumes or makes use of the products of the society, they are assumed to be getting value out of it (or else they would choose not to consume those products). Now you also have a way of measuring the productivity of projects - if their products are consumed, they are productive projects.

We have a basic idea of how this system likely works, now. When citizens produce value, they are rewarded by being able to claim or use up some of the value produced, and their consumption is an indicator that the producers of whatevever they consumed are contributing value, so these producers are also allowed to consume, etc. This is how it works once you "get the ball rolling", but how does the consumption chain start? Who is the prime mover, if you will?

In this case, it's financial institutions like banks that have the authority to declare that a citizen deserves to consume before they've proven that they're creating value. In theory this is because the bank has determined that the likelihood of this citizen producing future value is very high, and because the consumption chain needs to get started to keep the society as a whole productive. Since the objective is to have the rate at which value is created constantly increase, the rate at which value is consumed must also increase, so the purpose of these institutions is to bring the rate of consumption closer to the rate of production, so that the system gets more feedback.

At this point, things are starting to look pretty familiar, and we can see where the current crisis fits into this chain: bad predictions about how much value citizens are likely to contribute. However, let's step back and consider what that means.

In the case of a real estate bubble, we're talking about building houses. The financial institution is deciding that a citizen should have a house, but it turns out, eventually, that the citizen didn't create enough value to justify the consumption. In other words, the society has decided that the current house owner does not deserve the house. That's what the bubble means. The bubble popping is the large-scale recognition that house owners have not earned their houses, and the subsequent punishment enacted by the system.

Now you may not remember getting together with your fellow countrymen and having a vote where you decided to punish house owners. That's because there was no such vote - instead we have a tool in this system that automates the decision for us, called "money" (along with contracts and credit scores and such). If you remember, however, we agreed not to talk about money when describing the system, and there was a point: by giving ourselves a little distance we get a new view of the system. It is the same core system whether we vote to punish or not - money is just a means of implementation.

So back to the situation at hand: the system is punishing people, both house owners and financial institutions, but on a large enough scale that it has destructive consequences. When the financial institution loses sufficient authority in the system, it is no longer able to delegate resources, and the rate of consumption falls. This has a chain effect, or viewed another way, it interrupts the chain effect the system so carefully created. If you do not have consumption, you don't know where to allocate your resources (because you don't know who's contributing value), so resources will simply sit unallocated, and projects grind to a halt. In our society, this manifests in a rise in the rate of unemployment.

Now here's the weird part: at no point in this story was a problem caused by a decrease in the rate of value production. That's the result of the problem. The problem was actually too much value production, or rather, "unjustified" value production. Now we have nice houses, which carry a lot of value, just sitting around unused because we can't decide who should get to live in them.

From this perspective, our current situation is rather obviously absurd. As a society, we have the resources and infrastructure we need to be prosperous, but we're stuck in an extended period of indecision about how to allocate these resources. In order to fix the situation, you simply need to start allocating resources again (in fact, to a large extent it doesn't matter where you allocate them, as long as you enable consumption and get the chain effect restarted).

This makes recent proposals seem much more reasonable than they appear at face value to many Americans. Bailing out banks is, in this little model of ours, synonymous with "not punishing" those banks, so that they can continue to allocate resources. If (and this is a big if) you add well balanced regulations to the mix, you can force the bank to do a better job of allocating resources without interrupting its operation, and getting it to do a better job allocating resources was the whole point of the punishment feedback loop in the first place.

A stimulus package is an alternative of the same idea - but in this case you skip the institutions that performed poorly and have the government allocate resources directly. Again, as long as the policy enables consumption, we start getting feedback about value production, which allows the system to adjust resource allocation and slide back into an efficient state of operation.

Note that this doesn't say anything about how these policies are actually being implemented. Bailing out banks without adding regulation might reinforce their poor performance, and result in the chain effect starting back up, but at low efficiency. A stimulus might not allocate resources fast enough to effectively jumpstart the chain effect. Either way, the principles behind the policies clearly have some grounding.

We could go further, but I have to stop at some point, and going further into policy risks getting us too far away from the core concepts. So what's the takeaway? Well if you've gotten to this point and think that what I've said makes some kind of sense, I consider the post a success, but if there is one idea I want people to come away with, it's this: money is a means, not an end.

We were able to tell a perfectly coherent story about our economy without the concept of money being involved, and we can see where it fits in - it's the common unit of measure for the value of any product. Its purpose is to distribute the evaluation mechanism among all citizens in the system, which is important because, as mentioned before, our products need to be constantly evaluated to have efficiency in the system. In other words, dollar bills are evaluation tokens, or a mini-vote for a the value of things we consume. This is clearly not how citizens tend to think of them, however; since these tokens can be exchanged for items of value, we think of them as having inherent value, and treat them as property.

The first step to a more sane system might be recognizing what money actually is.

-Silent Ellipsis