How does bga calculate multiplayer ELO?

[traincopter]

I am quite new on this platform, and I recently looked up how the Elo system works. During my search I learned that there is no standard way to calculate the multiplayer Elo score but multiple possibilities exist.

I am curious on how BGA does it.

[FrankJones]

I know that for win percentage, it counts wins like this:

If you player a 3-player game, and you win, you will have one game played but two wins. It counts as a win against each player.

So, you would have a win % of 200%.

Elo? Who knows? It is not done in a way that makes much sense to me. I have seen multiple instances of winning a 3-player game, but getting the exact same Elo as though it were only a win against one of the players. Which seems odd to me.

I have not been able to get a clear answer on this from anyone.

[Meeplelowda]

I know that for win percentage, it counts wins like this:

If you player a 3-player game, and you win, you will have one game played but two wins. It counts as a win against each player.

So, you would have a win % of 200%.

Elo? Who knows? It is not done in a way that makes much sense to me. I have seen multiple instances of winning a 3-player game, but getting the exact same Elo as though it were only a win against one of the players. Which seems odd to me.

I have not been able to get a clear answer on this from anyone.

The Elo change is calculated as a win against each other player but gets scaled down for a win in multiplayer games in a way that isn't transparent.


As the FAQ says:

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

[Jellby]

I know that for win percentage, it counts wins like this:

If you player a 3-player game, and you win, you will have one game played but two wins. It counts as a win against each player.

So, you would have a win % of 200%.

Well, you got it wrong (unless it has changed from the last time I checked). I believe it counts as number_of_players*50%. So a win in a 2-player game is 100%, in a 3-player game 150%, in a 4-player game 200%, etc.

[FrankJones]

I know that for win percentage, it counts wins like this:

If you player a 3-player game, and you win, you will have one game played but two wins. It counts as a win against each player.

So, you would have a win % of 200%.

Well, you got it wrong (unless it has changed from the last time I checked). I believe it counts as number_of_players*50%. So a win in a 2-player game is 100%, in a 3-player game 150%, in a 4-player game 200%, etc.

I doubt it has changed, and I know you know the mechanics of this sit as well as anyone.

My statement was almost certainly incorrect, in that case.

How does it work then with losses? What happens if a person plays a 3-player game and finishes 2nd?

Overall, is it calculated in a manner that results in a player's win % showing as 50% if that player performs at probability expectation?

Meaning, a player who plays 3 3-player games and finishes 1st, 2nd, and 3rd, would show as having a 50% win rate?
And likewise for a player who plays four 4-player games and finished each game in one of the four possible positions?

Somehow, with all my games played, this still eludes me.

Is there any reason that BGA keeps this so opaque?

[Jellby]

How does it work then with losses? What happens if a person plays a 3-player game and finishes 2nd?

It's not a win, so it counts as 0%. I believe a tie counts as half a win, regardless of the number of players tied

Overall, is it calculated in a manner that results in a player's win % showing as 50% if that player performs at probability expectation?

Yes, I think that's the intent. If you play n-player games, you're "expected" to win 1 out of n games, and that would show as 50%.

[FrankJones]

How does it work then with losses? What happens if a person plays a 3-player game and finishes 2nd?

It's not a win, so it counts as 0%. I believe a tie counts as half a win, regardless of the number of players tied

Overall, is it calculated in a manner that results in a player's win % showing as 50% if that player performs at probability expectation?

Yes, I think that's the intent. If you play n-player games, you're "expected" to win 1 out of n games, and that would show as 50%.

That makes sense. Thanks.

You're very knowledgeable about BGA workings. Ever consider becoming an admin? Maybe they could use a volunteer whose tak it is to help explain site functionality and such.

The Elo calculation is still a bit mysterious.

But maybe winning a 3 player game gives the Elo one would gain by beating a single opponent with the average of the opponents' Elos?

That would be consistent with a game I saw in which all 3 players had the same ELo, and the winner gained only 10 Elo.

[Jellby]

The Elo calculation is still a bit mysterious.

In multiplayer games you just win or lose some elo based on each and every match-up with the other players. Each of these match-ups works exactly the same as with 2 players, except that the k-factor may be adjusted.

[Meeplelowda]

But maybe winning a 3 player game gives the Elo one would gain by beating a single opponent with the average of the opponents' Elos?

That would be consistent with a game I saw in which all 3 players had the same ELo, and the winner gained only 10 Elo.

It's not averaging the Elos. There are some things that aren't clear from the FAQ but this isn't one of them:

A player's ELO change is determined by comparing their finishing position against each of their opponents. For example, finishing second means a loss against the player in first place but a win against the player in third.

Based on these individual outcomes and considering the current ELO ratings of the compared players, a single ELO change is calculated for each comparison. Finally, all these individual ELO changes are summed to determine the player's total ELO adjustment for that game.

That last sentence is true or false depending on how you look at it. It's true if you look at it from the perspective that the K-factor modifier for multi-player games is applied before the "calculation," so it is a sum of the Elo changes if each opponent was considered a single match-up with a modified K-factor. But it's not the sum of the Elo adjustments that would result from N 1v1 games against N opponents.

[FrankJones]

I'm still confused, so let's use a concrete example.

Let's say 3 players play a game.

John, Sam, and Keith.

Each have an Elo of 200, and are past 21 games played for purposed of determining their individual k-factors.

John wins.
Sam finishes second.
Keith finishes 3rd.

What will each player's Elo be afterward?