In order, Competitive Skill Rank, MatchMaking Rank, Expected Skill Rank, and Performance Skill Rank. The first two are terms created by the developers of Halo, and there is a lot of official developer literature on what they are and how they work. The latter are partly creations of my own.

TL;DR Version: CSR is what you see in game, MMR is what the game uses to match you with other players and is invisible to us, ESR is a reasonable approximation of MMR, and PSR is for approximating how well you did in a single match.

Competitive Skill Rank (CSR)

CSR is the most visible of all of the statistics; it's the one you see in game. When you win, it goes up, when you lose it goes down. The formula for how much it moves for a given win/loss is not publicly available, but based on my observations, two things influence it: your in-game performance and the difference between your MMR and your CSR. While the first is pretty self-explanatory, the second is by far the most impactful and requires a basic understanding of...

MatchMaking Rank (MMR)

This is a statistic maintained by the game that is mostly invisible to us*. It represents how good the game thinks you are, and (at the time of my writing this), appears to be what Halo is using to ensure fair matches. Regardless of if you win or lose, your in game performance will be judged and factored into this metric. In what is most likely an effort to prevent gaming the system, Halo has declined to expose the exact nature of how MMR is calculated.

However, there are some hints. On the official Halo Waypoint website, there is a graph featuring your expected kills and deaths for every match you play. If you outperform your expectations, your MMR appears to increase. Miss your expectations, and you can expect it to dip accordingly. It's likely that kills and deaths are not the only factors that influence your MMR, but they are the only ones that we can see.

Expected Skill Rank (ESR)

That expected kill/death graph on halowaypoint.com has a lot more data available to it than what they've chosen to display. Here's a sample of the data from a single match of my own:

{
  "SelfCounterfactuals": {
    "Kills": 13.02902159740169,
    "Deaths": 11.779710508499637
  },
  "TierCounterfactuals": {
    "Bronze": {
      "Kills": 0.7181542783576873,
      "Deaths": 17.6524283446301
    },
    "Silver": {
      "Kills": 2.220069373790745,
      "Deaths": 16.31645277818232
    },
    "Gold": {
      "Kills": 4.78171550865068,
      "Deaths": 14.99185741714148
    },
    "Platinum": {
      "Kills": 8.085594967544454,
      "Deaths": 13.64572850930582
    },
    "Diamond": {
      "Kills": 11.907774926653307,
      "Deaths": 12.19888568969912
    },
    "Onyx": {
      "Kills": 16.05313653186301,
      "Deaths": 10.65299514545297
    }
  }
}

As you can see, in this particular match, with these particular opponents/teammates, it was expecting me to put up about 13 kills and die 12 times. However, it also included a TierCounterfactuals property, containing expected values for other ranks. At the time of that match, I was ranked as a low Diamond player, and you can see that the game expects me to slightly beat the Diamond benchmark. It seems safe to conclude that those counterfactuals represent the average player at the base MMR for that division (Bronze starts at 0, Silver at 300, etc.).

If we take these values and linearly interpolate them, we can create a graph of what a player at any given skill level would be expected to perform for kills and deaths in that match. In this example, the counterfactuals map like so:

If we find the SelfCounterfactuals values on this line, which purportedly represent the kills and deaths that the game expects us to achieve given our MMR and the MMRs of everyone else in the lobby, we can match the point on the line to a skill ranking.

Each game variant has its own ESR rating (including one for 3 flag CTF and another for 5 flag CTF), and they can move independently of one another.

Performance Skill Rank (PSR)

Same thing as ESR, except we use the actual kills and deaths that you recorded in the match instead of the ones the game expected of you. This may be useful for determining how well you did in a single match, but it does result in some strange values (like negative skill) if you have a particularly good or bad game.

A few disclaimers...

• MMR likely has more than just kills and deaths going into its computation, so ESR will always be merely a small window into that stat.
• Skill for kills and deaths can be different, even for the same player. ESR/PSR averages those two values together. For ESR, they're usually very close, but for PSR the -D and -K values can skew wildly.
• The counterfactuals stop at 1500 skill. Beyond that, I can only linearly extrapolate the Diamond/Onyx line. While this looks accurate-ish for pros I have checked, I have less confidence in those values.

Your in-game performance has very little to do with the amount of CSR you are rewarded. In fact, before the match even starts, the game has a good general idea of what to award you if you win, and what to penalize you if you lose. That's because the primary factor in CSR rewards/penalities is the difference between your pre-match CSR and your pre-match MMR.

If your CSR is much lower than your MMR, you will receive large rewards for wins and small penalties for losses, and vice versa. A CSR that's matching your MMR will generally result in rewards/penalities of +/-8 points.

A really good or bad game performance can adjust this payout slightly, however it'll usually be by no more than a single point in either direction. A truly uncharacteristic performance is required to move the payout by even 2 CSR.

For an official overview from Halo on how CSR rewards work, see this article.

To put it simply, if you want to raise your ESR, kill more, die less, and win your matches. Those are the three metrics that are used to compute your skill ranking. Of the three, kills per minute seems to have more of an impact than the other two, though exactly how much likely varies from one game type to the next.

Objective points in the role graphs are a weighted aggregation of various statistics that we feel accurately show that a player is "playing the objective", rather than the traditional Slayer or Support roles. Usually the stats selected are those where excelling in the stat will actively hurt the amount of kills or assists a player will accumulate.

These are the current statistics and their weights:

Strongholds

Secures - 33.33%

Captures - 66.67%

King of the Hill

Points - 100%

Capture the Flag

Captures - 70.59%

Capture Assists - 23.53%

Steals - 5.88%

Extraction

Initiations Completed - 50%

Conversions Completed - 50%

Oddball

Points - 100%

Minigame

Score - 100%