How Elo ratings work

jasonrohrer · 2014-12-29 02:10:34

Updated formula

Here are the details of the Elo rating system as implemented on the Cordial Minuet server. Elo in general is described here: http://en.wikipedia.org/wiki/Elo_rating_system

New players in the game start out with a rating of 1, but their ratings are provisional until they've played more than 20 games. Provisional players ratings are adjusted by the usual formula after each game that they play, BUT, their opponents' ratings are not adjusted when they play against non-provisional opponents. This prevents new players with inaccurate ratings from unfairly dragging the ratings of established players up or down (all new players are assumed to be over or under rated, but we don't know which).

Ratings can never drop below 1---that's the rating floor.

Score outcomes from a game are 1 if you leave with more chips than you started with, 0 if your leave with less, and 0.5 if you leave even.

Note that because of the house tribute, unlike Chess, score outcomes for the two players do not always sum to 1. It's possible for both players to lose money, for example, or one to break even while the other loses. In a recent survey of CORDIAL MINUET history, this happened in less than 2% of games.

When you play a game, the Elo rating of you and your opponent allow us to compute what we expect your score should be, according to the E_a formula here: http://en.wikipedia.org/wiki/Elo_rating … al_details

This number will be between 0 and 1. If your opponent's rating is much higher than yours, we expect your score to be closer to 0. If your rating is much higher, we expect your score to be closer to 1. If you and your opponent have equal ratings, we expect your score to be 0.5 (in other words, we expect you and your opponent to tie).

What actually happens in your game determines how your Elo will change, post-game. If you leave with more chips than you started with, your Elo will go up. If you leave with fewer, your Elo will go down. If you break even, your Elo may go up if we expected you to lose, but it will go down if we expected you to win.

How much your rating changes depends on how many chips were taken. More chips taken means a bigger rating change for both players.

Your rating changes by

( chipsTaken / totalChips ) * K * ( actual_score - expected_score )

totalChips are how many chips were at the table at the start. chipsTaken is the difference between chips of the two players at the end plus the house tribute. Thus, if you take half of your opponent's chips, your Elo will go up by half the maximum amount. If you take all of your opponent's chips, your Elo will go up by the full amount. If you and your opponent walk away even, your Elo change will be weighted by the house tribute (in other words, roughly how long the game lasted).

To summarize, games where very few chips change hands by the end don't affect Elo very much. Long grinds that come out as an even loss for both players affect Elo the longer they last as the total house tribute grows.

Then there's also the K factor, which is the overall speed that Elo changes.

Provisional players have K of 64, while established players (more than 20 games) have K of 32. Thus, provisional players have Elo ratings that change faster and then settle down somewhat when the player becomes established.

As an established player playing against a player of equal rating, the most your rating can change by after a single game is 16 points (in other words, we expect you to walk away even, and you walk away with everything, so your score is 0.5 higher than we expect it to be, and we multiply that by K = 32 and 100% chips taken).

Last edited by jasonrohrer (2014-12-29 18:07:49)

jere · 2014-12-29 02:48:30

Wow, cool! It's neat how you rolled through the logs to generate this.

Here's something you might find interesting. I convinced a friend to get into the game just for the tournament (Inducement Vanity). We played a couple dozen practice ($0.01) games beforehand and the only other time he played was the tournament. Somehow he's at #3 Elo. I'd like to think it was due to my superb coaching, but I have another hypothesis. I was losing a fair number of games against him because instead of playing conservatively, I was seeing games out to the finish. I wanted to first explain the game fully and then see if his bets were actually reasonable. I wonder if all those practice games gave him a high Elo because I would have had a high Elo? Anyway, what a way to introduce a game to a friend: "hey, all you had to do is play for a few hours and you won $10 and are now the #3 player!"

There's one more problem: Elo ratings tend to encourage strong players to stop playing to protect their high rating. That's a problem for another day, though.

I suppose this is why League of Legends has Elo decay.

computermouth · 2014-12-29 02:52:50

Ahhhhhhh. And now I'm glad the leaderboards are anonymous

Edit: StarCraft 2 utilizes a pool of points where users are subject to rank decay, but upon returning to play, earn a raised increase in rating until they climb from that pool back to the standard ranking system. That said, that's going to be a decent bit more sql actions/data for a small neat feature.

Last edited by computermouth (2014-12-29 02:56:37)

jasonrohrer · 2014-12-29 03:02:44

Yeah, I though about decay... But how would that work? Daily? I guess it could just be a nightly sweep, and if you haven't played at all today, your Elo goes down X points.

jere · 2014-12-29 03:13:32

Prior to the Season 2 rating system remake, Elo decayed over time when you were above 1400 Elo:[6]
Elo decayed at a rate of 50 Elo for Diamonds, 35 Elo for Platinums, 25 Elo for Golds, 10 Elo for Silver, and 0 Elo for Bronze for every 4 consecutive weeks of inactivity and every 7 days thereafter.

http://leagueoflegends.wikia.com/wiki/Elo_rating_system

Not sure if the logic can be pulled directly, but daily sounds a bit too harsh.

Last edited by jere (2014-12-29 03:17:13)

computermouth · 2014-12-29 03:37:27

Perhaps decay could be proportional to standing? Higher spot, higher decay?

jasonrohrer · 2014-12-29 04:53:06

Yeah, so I'll table this for now. We're not close to the "four consecutive weeks of inactivity" yet for the top players. It would not be hard to add Elo decay later, when the need arises.

Pox · 2014-12-29 06:26:33

Doesn't this system easily fall victim to one of the issues we discussed in the other thread? If you walk away from a table with 99 chips then this is interpreted as (almost) tying with the opponent, and thus if you do this enough against good players your Elo will rise without ever actually playing well.

jasonrohrer · 2014-12-29 16:44:21

Yeah. Working on fixing this. I think I misread your suggestion for weighting based on chips taken. In the current Elo, chips taken is your score. I think I should still have binary win/loss as score, but weight the effect on Elo based on chips taken. Then, your Elo can only go down when you lose chips, and only go up when you win chips, but the amount you go down or up will depend on how many chips.

jasonrohrer · 2014-12-29 18:05:55

Oh, I fixed it.

Now it treats each player in isolation to determine their score for the game. If you lose money, it counts as a loss for you, regardless of whether you have more chips than your opponent by the end. If you gain money, it counts as a win for you. It's only a "draw" if you leave even, regardless of your opponent.

Thus, "doing well" in terms of Elo is the same as making money at the game.

Finally, your Elo change post-game is simply weighted by how many chips moved by the end, including the house tribute.

Thus, if you walk away 1 chip up after one round, neither Elo will change much. But if you walk away 1 chip up after 100 rounds (and your opponent is down 25 chips, 24 taken by the house tribute), then both Elos will change more.

So, this balances the two factors.

Whether you won or not depends on whether you walked away with a profit or not.

How much this will affect your Elo depends on how much you trounced your opponent (or how much you were trounced).

And happily, you will NEVER win money and have your Elo go down as a result. You will NEVER lose money and have your Elo go up as a result.

(One unfortunate side-effect is that Elo changes are no longer zero sum, which means the Elo average can change over time... when both players lose money, both Elos go down by the same amount---though this does kinda affect the nature of the game, so slight Elo deflation is probably okay).

Elo ratings have been recomputed using full history, with some interesting results.

CORDIAL MINUET ENSEMBLE

#1 2014-12-29 02:10:34

How Elo ratings work

#2 2014-12-29 02:48:30

Re: How Elo ratings work

#3 2014-12-29 02:52:50

Re: How Elo ratings work

#4 2014-12-29 03:02:44

Re: How Elo ratings work

#5 2014-12-29 03:13:32

Re: How Elo ratings work

#6 2014-12-29 03:37:27

Re: How Elo ratings work

#7 2014-12-29 04:53:06

Re: How Elo ratings work

#8 2014-12-29 06:26:33

Re: How Elo ratings work

#9 2014-12-29 16:44:21

Re: How Elo ratings work

#10 2014-12-29 18:05:55

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