In depth explanation of the skill system

Posted by: El Kafungus
I am NOT an employee of Bungie nor one of Microsoft, and my views and opinions do not necessarily reflect those of either of the aforementioned entities.
Also, thanks for sticking this Bungie, I hope it is useful.

There is one caveat to consider before we get under weigh. Bungie is not likely, nor or ever, to release the true inner workings of the Halo 3 ranking system, but as it is based somewhat on the Microsoft Trueskill system, this is a fair approximation of the way skill ranking works in Halo 3.

What is the Trueskill system?

The Trueskill system is a player skill rating system for Xbox Live. Halo 3 uses a version of the Trueskill system in order to make the matches fairly even.

How Does Trueskill Work?

Trueskill assigns skills on a normal distribution (bell curve, Gaussian). The Trueskill system has two main variables to consider. They are your player skill level (Mu) and your uncertainty factor (Sigma).

Mu is an approximation of your skill level based upon past performance. Win games to raise Mu, and lose games to lower Mu.

Sigma is the numerical representation of the range in which your true skill could lie. Play really consistently, and you have a low Sigma. Play streakily, and you have a higher Sigma.

After a win, the Trueskill system adjusts your skill ranking based upon the Mu and Sigma of all the players in the game. We'll go deeper into how much it is adjusted later.

Trueskill systems use a conservative ranking Ranking = mu - (K * Sigma) so your skill is likely to be higher than the actual number represented in your Halo 3 highest skill level. K is a constant assigned an arbitrary value by the developers of the game.

Why don't I level up after winning X games in a row? / Why does my friend go up three levels a game and I don't?

Mu increases after a win. Always. The increase is proportional to the winner's Sigma and the Mu difference between the winner and the loser. So, if your Sigma is high, you will proceed faster through the ranking system (in BOTH directions). If your Sigma is low, you will both gain and lose rank more slowly.

So, I want a high Sigma value?

While it sounds as though a high Sigma value is desirable, it CAN be a double edged sword. A high Sigma can mean you increase by one level for every win. It also means you decrease by one level for every loss. In addition, a high Sigma means that your skill ranking will be significantly lower than your actual skill owing to the conservative nature of the Ranking = mu - (K * Sigma) ranking equation.

Also, Sigma value starts out VERY high. So, if this is your friend's first time in a particular ranked hopper, and he wins consistently, his rank will rise meteorically due to his high Sigma.

Myth Busting

Trueskill DOES NOT take into evaluation performances inside the game. The ONLY statistics that Trueskill takes from a game are the Mu and Sigma values of all the players, and how you placed.

This means that out of all of these statistics:
Getting the MVP
Having a High K/D ratio
Getting the most medals
Skill levels of those you killed / killed you
Weapons you used
Headshots
Captures, detonations, stops, ousts, etc.

NONE of them matter when calculating the Mu increase/decrease. Any link between any of these and the skill ratings are purely correlative and not causative.

For example. When I play Team Slayer with my buddies on my alternate gamertag, much of the time I end up carrying the team. I created this tag to play with my buddies, and as such, have only ever played on that party with that tag. I ranked up to the level of my friends within 20 or so games, while it had taken them 50 to 100 games to reach their level. My friends thought that it was because of my frequent MVP status and my high K/D ratio that I leveled up so fast.

Wrong. I leveled up fast because A) I initially had a very high Sigma, making my Mu more variable after each win, B) Starting out at a 1 and playing people in the mid teens gave me a considerable boost every time I won, and C) We won more frequently than they had in the past.

Summation

If you're worried that you're not getting your fair shake in the ranking system, look at the last 50 games that you've played in that hopper. (If you haven't played 50 games, play more. The system needs more data.)

Take your win/loss average. If it's 65% or above, and you've been in that playlist for a while, you've probably increased in level 2 or more times over the last 50 games (or you will in the next 20 or so). Otherwise you are going to be hovering around the same level or dropping in rank.

Addendum, Opinion, and Technical Notes

In strict Trueskill systems, Sigma only decreases. I believe the "momentum" system in Halo 3 means that Sigma can be increased through consistent winning or losing. This is total conjecture, but I think that Sigma in Halo 3 matchmaking hoppers is proportional to the absolute value of the win/loss ratio of your last X number of games. The formula probably looks a little like: Sigma = C ( | wins - losses | ) where C is an arbitrary constant. This would explain why when some people go on a win streak with two or more different parties, their skill goes up faster than their teammates' skill.

I also believe that the system is weighted so that the peak of the normal distribution is around level 20, not level 25. That means it will take more wins to level through 1-25 than it will 25-50. Which makes a bit of sense, if you don't want the highest levels too terribly crowded.

The higher the Sigma of your opponents, the less Mu you gain from beating them.

Predictable outcomes (e.g. win against a lower ranked opponent) are treated as statistically less significant. Upsets are given more weight. So, winning against lower ranked opponents doesn't do much for your Mu rating. Neither does losing against higher ranked ones. But beat a team that's 6 or more ranks above you and at least someone on your team will rank up.

Trueskill ratings have no discernable correlation from hopper to hopper. When you play on a hopper you've never played on before, you are starting with a fresh Mu and a high Sigma.

Team Mu and Team Sigma are the summation of the Mu and Sigma values of all the players on the team. When calculating the skill of an opponent in a team game, the system uses the Team Mu and Team Sigma values in calculation wherever rational.

*Removed equations for space. They can be found in the link below.*

The rest of the equations and a less Halo 3 centric analysis of the Trueskill system can be found here.

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Important additions, Appendices, and Member Questions Answered
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In-depth response to a post LoveNub made later in this thread. Should clear things up for some.

More data analyzed, and a request for help with research

Kudos to Jay120171 for finding the post detailing the Mixed Party Bug in matchmaking.

"Why do I win 10 games and not level up, then delevel when I lose 2? Help!"

There are several possible explanations for this. The easiest one, and the one that fits the Trueskill system EXACTLY as explained by Microsoft, is that losing against a team that is X (I believe 6 or more, but experimentation is needed to determine the true value) skill points below you can cause you to lose as much as 5 times the Mu value that winning against even level teams gains you. Of course, winning against a team that has X more skill levels than you can net you as much as 5 times the Mu gain. The reverse is true as well. Winning against a weak team can count as little as 1/5th of an even skill win.

Second, and this is enlightened conjecture, your Sigma value was low at the beginning of your streak, and since your win streak is raising it, the conservative ranking system (Remember, Rank = Mu MINUS (K times Sigma)) is kicking in to reduce your apparent skill level. Your real Mu is higher, but your higher Sigma is having a negative impact.

mo0vaf00 writes: (edited to clarify intent)
So is there some kind of value that requires you to be inconsistent to rank up? If I win 1 game going even and win the next going +10 do I have more of a chance of ranking up than just winning 2 games getting the same score?

This conclusion is popping up fairly frequently. The answer is no. The system does not look at "In Match" statistics. At all. These stats are all included in a list above. The game only looks at your wins and your losses, and the strength of the opponents you play against.

Also, the system does not "reward" you for having a high Sigma value. There is ONE positive associated with high Sigma, and there are TWO negatives.
Positive: Your level can move up faster through winning.
Negative: Your level can move down faster through losing.
Negative: The conservative ranking system ranks you lower when your Sigma is higher.

Higher Sigma simply means your level is more fluid. Sigma is good to have high when you are still reaching your appropriate skill level, but good to have low once you've reached it and are working on improving.