The Mathematics behind the "True Skill" Ranking System
"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. There is one caveat to consider before we get under weigh. Bungie is not likely 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:
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.
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.
"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.
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.
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.
For actual formulae and graphs, please visit this page by Microsoft."
In Lamest Terms...
Ok, that was a little complicated and you probably did not catch all that. In other words, to rank up, you need to win more than you lose. Consistently winning is what allows you to rank up. If you lose every other game, it will be very difficult to rank up. Your level is determined by your win/loss ratio and by the total games you have played in that particular playlist. Please note that your rank in one playlist has no effect on other ranks. If you have a 24 in Team Slayer, you will not rank up faster/slower if you are a level 1 in Team Doubles. Also, your total or global experience has no effect on your ability to rank up. A common misconception is that Force Colonels or Brigadier Generals have a difficult time ranking up because they have a lot of EXP. This is NOT true. What must be pointed out is that if you play hundreds of games in one playlist, it can be come difficult for you to rank up in that particular playlist. As pointed out by Microsoft, the ranking system requires many wins to detect a difference in skill. If you have played 300 games in one playlist, and you win 55% of them, you probably need to win 80 out of the next 100 for you to rank up several levels. Doing so will improve your win/loss ratio - which the higher that is, the higher your rank will be (most of the time). The difference between a 50 and any other rank is that the 50 has won more than half of their games overall or that they win consistently.
Ranking up in Parties
In this section I will explain how the ranking system works when you play in parties because this seems very confusing.
What should be stressed first is that you will rank up much slower in a full party (2/2, 4/4, 5/5 or 6/6). The ranking system EXPECTS you to win when playing in parties. When you play in a party, the ranking system has difficulty determining YOUR rank. The True Skill ranking system is based on the individual's stats and not the "team's stats." When I say stats, I mean the mathematical model behind YOUR rank as explained above. If you play in a mixed party (meaning a party with a level difference greater than five) your rank becomes even more skewed. As a side note: playing in a full party can either be really difficult or very easy. You either play a very skilled team or a team of newbs.
For some reason, each win with a party is worth less than winning a game in half a party or alone. I would estimate that three to five wins with a party is equivalent to one win when playing alone or in half a party. Each win is worth less (in contrast to playing alone or in half a party). Because of this, I would suggest only to play with really good people and in parties when you reach a higher rank. Some people have performed interesting experiments with new accounts. They play with a full party from level 1 and try to get a high level. Surprisingly, some people have won 100 straight games, but only reach level 24. Again, this is because the ranking system can not properly determine the person's rank since they have never played alone.
The People in Your Party vs. Other Parties
The type of people you play with can dramatically affect the speech at which you rank up. For now, let's assume you are playing with a full party. If the players in your party have a very good win/loss ratio (i.e. a hundred wins on their 50), it will be even more difficult for you to rank up. The ranking system expects you to win, because everyone in your party has a good win/loss ratio; however, if you are in a party with people who have a poor win/loss ratio you will rank up much faster, especially if you beat a team that HAS a good win/loss ratio. At the same time, if you are a 6 (playing your friends who are 30s) and you lose to a bunch of 30s, you have the possibility of ranking down because you have played very few ranked games.
Half a Party
A half a party is having half of the maximum party size allowed in a playlist. In other words, playing in a party of two in Team Slayer (which has a maximum party size of four) or a party of three (which has a maximum party size of six) in Squad Battle. Note that a half a party in Team Doubles would be playing alone. The great news about a half party, is that you rank up just as fast as if you are playing alone. There is little effect your partner can have on you (unless you play with a booster). This is the best method to ranking for every playlist (except the higher ranks in Squad). You usually match against less challenging opponents - because they do not know one another - and you rank up faster. The only downside is that you can have one or players who bring down your team.
"I'm a level 36 and my friend's a 10; why does he rank up eight levels and I go up one rank?"
The reason why the level 10 ranks up faster is that he is not expected to beat people 20+ levels above him. Also, the opponents are 36 or LOWER. The ranking system EXPECTS the 36 to beat people lower than him, so it sees no reason to rank him up; a higher rank will most likely be awarded if he beats people 37 or higher. When playing against low ranks the 36 will still rank up, but it will take many more games for that to happen.
Ties - Who Really Wins?
One of the most frustrating and confusing aspect of the ranking system in Halo 3 is how it treats games that end in a tie. Most of the following information is based on theories and deducted logic from the information provided thus-far. Sometimes the opposite results occurs when you end in a tie.
I would like to once again stress that kills, medals, flag captures, MVP, headshots, et cetera, have absolutely no correlation to the ranking system. Also, EXP does not mean you get win. EXP is awarded only if you do not “lose.” For instance, the top half in FFA get EXP, even though players did not necessarily win. In team games, if both teams tie, neither “technically” lost so both teams are awarded EXP. If the game ends in a tie, such as 44-44 (Slayer) or 2-2 (flag) the only thing that matters is how do the ranks compare between the players on each team.
Let’s examine the following situation. Team Blue and Team Red tie a 4v4 slayer game. It does not make a difference whether or not they tie at 20-20 or at 49-49. Team Blue has a level 49, 48, 50, and 35 while Team Red has 48 48 45 47. One theory is that, Team Blue will win because their average rank is 45.5 ([49+48+50+35]/4=45.5) while Team Red’s average rank is 47 ([48+48+47+45]/4=47). As mentioned earlier, if the lower “rank” ties the higher “rank,” the lower rank gets the win. This situation is probably the case if both teams were in a party of four.Another theory is that the ranking systems examines each player by matching the ranks one to one. So, 50-48 (the 48 wins), 49-48 the (48 wins), 48-47 (47 wins) and 35-45 (35 wins). So if we go case by case, the only person on Team Blue that would win/rank up would be the 35 while the other three players on Team Blue would lose/rank down. The only thing that can really change this is if one team plays with a boosting account.
Boosting accounts have a horrible win/loss ratio, and if you play with one, the ranking system expects you to lose. So if a team has a booster, they can potentially get the win because the ranking system assumes the team will lose with a player whose account is so terrible. The last theory is that all players are examined independently (so you disregard parties); so the rank of each player is compared to the ranks of others in the game. Overall, the general rule of thumb, theoretically, is that the lower ranks will always get the win while the higher ranks get the loss should two teams tie.
Ranking up in FFA
Lone Wolves, Legendary Brawl, and Mythic Brawl have been known to be notoriously difficult to level up. People "lose" a game and rank up while others tie for second place and rank down. The best way to ensure that you will rank up is to always finish in 1st place. You will NEVER rank down if you come in first (when I say first, I mean first place with no ties). Obviously, it is very difficult to win every game.
The most significant factor in FFA ranking are the ranks of each player. Your opponents will more or less determine your ability to rank up or down. Beating people that are a equal OR higher level than you is what allows you to rank up. Just because you earned EXP after a match does not mean you should rank up or down. In FFA, Bungie's EXP system awards EXP to the top half finishers (1st-3rd out of 6 players, 1st-4th out of 7, 1st-6th out of 12). EXP has no correlation or influence in your ranking ability.
How (I think) it works in FFA
The best way to explain how the ranking system works is through the following example. Observe the table the table below. Now let's answer the question: Who will rank up and who will rank down?
My guess and reasoning
Summary
Ranking up in Halo 3 is largely determined by the type of people you play with and the opponents you match against. Playing in full parties makes ranking up difficult since the ranking system has trouble determining your specific rank. Winning consistently is what allows you to rank up. Having a poor win/loss ratio will not only prevent you from ranking up easily, it can make very difficult for you rank up at all. Playing in half a party or alone is the best way to play to rank up. In FFA, you receive credit for beating everyone that is above you and you are penalized for losing to everyone below you.
Best advice: win as many games in a row as possible in a half party and come in first place every time for FFA.
Introduction
The purpose of this guide is to clarify the ranking system of Halo 3. I will explain how it works and best method to rank up (without cheating of course). There is a mathematical model used to determine your rank. Do not worry if you can not understand the variables or formulae. It has been incorporated to provide the mathematical facts for skeptics and for people who really wish to understand the math behind it. For those who have not visited the Halo 3 Forums at Bungie.net while it was active, there was a "sticky" (originally found here but the link has been removed by Bungie) explaining the mathematics behind the ranking system. It is difficult to change the original text because it is the best explanation for the technicalities of the ranking system. Unfortunately, the user's name was deleted so am unable to cite the author. If anyone knows, send me a message!
In addition to the explanation, I will also explain the ranking system in my own simpler terms to clarify all the technical jargon. I will also explain FFA ranking and parties as that is not strictly covered below.
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