ELO calculation change

[FrankJones]

It says in the FAQ, linked to at the bottom of nearly every page in the site:

I can't recall exactly how the other somewhat recent discussion about this went, but -

Didn't we figure out that the Elo change is modified with some unknown formula in such situations?

For example, when 3 equal Elo players play, the winner gets credited with 2 wins (explaining how some people can have a win% higher than 100%), but is not warded the Elo that would be expected from beating both players individually.

In fact, I recall two instances in which the winner of a 3-player game received exactly as much Elo as though it had been just a win in a 2-player game.

It is modified (reduced), but when it works out to be exactly what it would be in a two-player game I think it's just coincidence.

I cannot recall - were we able to figure out or reverse engineer the formula?

I have seen that when 3 beginners play, the winner receives 30 Elo - exactly what would be received if the game has just been a 2-player game.

The only non-team multiplayer game I have played in non-friendly mode is hearts. I could check some of my results there to see if I can figure out how it works.

[Meeplelowda]

I can't recall exactly how the other somewhat recent discussion about this went, but -

Didn't we figure out that the Elo change is modified with some unknown formula in such situations?

For example, when 3 equal Elo players play, the winner gets credited with 2 wins (explaining how some people can have a win% higher than 100%), but is not warded the Elo that would be expected from beating both players individually.

In fact, I recall two instances in which the winner of a 3-player game received exactly as much Elo as though it had been just a win in a 2-player game.

It is modified (reduced), but when it works out to be exactly what it would be in a two-player game I think it's just coincidence.

I cannot recall - were we able to figure out or reverse engineer the formula?

I have seen that when 3 beginners play, the winner receives 30 Elo - exactly what would be received if the game has just been a 2-player game.

The only non-team multiplayer game I have played in non-friendly mode is hearts. I could check some of my results there to see if I can figure out how it works.

Here is a concrete example with three players with established ratings (i.e., k-factor = 20) from https://boardgamearena.com/table?table=746815265:

Expected Elo change for Flo37rm vs. flwchin in a two-player game: +6.8
Actual vs. flwchin: +5.13

Expected Elo change for Flo37rm vs. Meeplelowda in a two-player game: +9.7
Actual vs. Meeplelowda: +7.25

The actual Elo Flo37rm was awarded was 25% less than they would have gotten in two 1 v. 1 matches against the other two players.

I just checked a four-player game, and the winner won around 32-33% less that they would have in three 1 v. 1 matches against the other 3 players. So it looks like the more opponents, the more of a reduction in the "raw" 1 v. 1 Elo adjustment.

So it could be that beginners are scaled at 100% instead of 75% for whatever reason. I'm not sure what would happen with a mixture of non-beginners and beginners.

[FrankJones]

I don't understand why there is a lack of transparency in the system.

My most recent game of Hearts:

https://boardgamearena.com/table?table=677268991

I calculate an expected Elo change of +26.5. I actually gained 18.

The FAQ states that a win in a multiplayer game is considered a win against each opponent, but does not say anything about modification to the expected ELo. At least, I don't think it does. I'll take another look.

[Meeplelowda]

I don't understand why there is a lack of transparency in the system.

My most recent game of Hearts:

https://boardgamearena.com/table?table=677268991

I calculate an expected Elo change of +26.5. I actually gained 18.

The FAQ states that a win in a multiplayer game is considered a win against each opponent, but does not say anything about modification to the expected ELo. At least, I don't think it does. I'll take another look.

It doesn't say it, but we can observe that it's true. And it's consistent. Your Hearts win was about 67% of what it "should" have been, which is the same 33% reduction I saw in a completely different four-player game (Tokaido).

[FrankJones]

I don't understand why there is a lack of transparency in the system.

My most recent game of Hearts:

https://boardgamearena.com/table?table=677268991

I calculate an expected Elo change of +26.5. I actually gained 18.

The FAQ states that a win in a multiplayer game is considered a win against each opponent, but does not say anything about modification to the expected ELo. At least, I don't think it does. I'll take another look.

It doesn't say it, but we can observe that it's true. And it's consistent. Your Hearts win was about 67% of what it "should" have been, which is the same 33% reduction I saw in a completely different four-player game (Tokaido).

Interesting. So, in a 4-player game, the actual Elo gained is 2/3 of what is expected. And in a 3 player game, from what I've seen, the gain is 1/2 of what is expected.

Maybe the formula is (n-2) / (n-1) where n is the number of players.




Here is what the FAQ says:


How does the ELO change calculation work?
...

While you can find general information about the Elo formula online, be aware that Board Game Arena incorporates its own modifications. Primarily these are reflected in modifiers that together determine the magnitude of the ELO change (called the K-factor in the standard Elo formula):

* Game Experience: players with fewer games played on BGA have a higher ELO adjustment to facilitate reaching the appropriate level within the ranking system.
* Consecutive wins: beating the same player repeatedly makes the outcome less outstanding.
* Teammates: for some team games, teammates are ignored.
* Game abandonment: when a player quits or is expelled, the ELO gain for the opponent(s) is proportional to the game's progress at that point.
* Player count



So, there it is in the 5th bullet point. A vague mention of "player count" affecting the expected Elo, with no further explanation.

[Meeplelowda]

I don't understand why there is a lack of transparency in the system.

Come on, we have a multitude of examples of things for which there's no good reason to hide the ball (i.e., it doesn't involve internal finances or site security) yet there's no official statement anywhere of how things are done. Take how ties after someone is expelled from a tournament are handled.

[Meeplelowda]

And in a 3 player game, from what I've seen, the gain is 1/2 of what is expected.

No. I just showed above that in a three-player game it's 3/4 of what is expected.

[FrankJones]

True.

Okay, I viewed a 3-player game among 3 established players.

The winner received 9.78 Elo when the standard calculation would have awarded that player 13.1. Thart's just under 75% of the expected.

I am officially baffled. That's higher than the 67% that occurs in a 4-player game, even though it's easier to win a 3-player game than it is to win a 4-player game.

[FrankJones]

And in a 3 player game, from what I've seen, the gain is 1/2 of what is expected.

No. I just showed above that in a three-player game it's 3/4 of what is expected.

Whoops. I am getting mixed up because some of my replies were written before I saw your intervening reply.

ok, so, 3 players = 3/4 of expected Elo gain.
4 players = 2/3 of expected ELo gain.

And of course, 2 players means 1/1 of expected Elo gain.

2 possibilities:
1) There is a mathematical formula that can be expressed in closed form, such as the incorrect [n-2]/[n-1].
or
2) It's not a formula but just an arbitrary fraction that decreases in value as the player count increases. Maybe in a 5-player game it is 60%, and in a 6-player game, 50%.

And obviously whatever method is used, it is applied differently for beginners.

[Meeplelowda]

True.

Okay, I viewed a 3-player game among 3 established players.

The winner received 9.78 Elo when the standard calculation would have awarded that player 13.1. Thart's just under 75% of the expected.

I am officially baffled. That's higher than the 67% that occurs in a 4-player game, even though it's easier to win a 3-player game than it is to win a 4-player game.

I don't think the goal is to scale it by "difficulty." What the actual rationale is....who knows? If they won't even publish the formula they sure is heck aren't going to disclose the thinking behind the formula. And that's assuming there's still someone left in the organization who was around when the decision was made who even knows what the thinking behind the formula is.