I was searching for the Elo formula that's used in WoW but was unable to find one. The reason I'd like to know it is that I'm having a spread sheet with our arena games for this season and simply would like to add a cell with "suspected enemy arena rating" on it.

Unfortunately the search function on this board doesn't allow searching for terms shorter than 5 letters, so I'm kind of stuck (and I'm quite sure this thread wouldn't be found either when searching for this topic ). Looking at the offical forums (with already no hope really) resulted simply in "Text search temporarily disabled". Using Google's search I found this thread, unfortunately noone answered there.

So, has anyone already tried to figure out what formula Blizzard is using to deterime the gain/loss after a game?
If not, does anybody have any results where the ratings for both teams were known and one could derive the formula from this?


As for the formula itself, I looked at two sources for the Elo system, one on Wikipedia and one on WoWVaul.

They are using different approaches for calculating this. Both are seperated in two steps.
First there is calculated an expected outcome that lies between 0 and 1 (if you are rated higher than your opponent, this expected score is higher. Basically it's just the chance of winning the game) and then use this expected outcome and the outcome of the match (victory, draw, loss - where there's no draw possible in WoW right?) to determine the rating change.
Also there is a variable that defines the maximum possible change in rating.

Let's take these ratings as an example:
Team A: 1500 points
Team B: 1580 points


Wikipedia's formula is:
Expected Outcome = 1 / (1 + 10^((Enemy Rating - Your Rating)/400))
New Rating = Current Rating + Maximum Possible Change * (Outcome - Expected Outcome)
(Outcome is either 0 for a loss, 0.5 for a draw or 1 for a victory.)

So this becomes:
(note: I have found out that the windows calculator doesn't accept "^" for calculating the power. Instead, use an "y" if you want to do this via copy&paste. At least for the German version.)

Team A:
Expected Outcome:
1/(1+10y((1580-1500)/400))= 0.387 [38.7% chance of winning]

Loss:
1500+32*(0-0.387)= 1487 [-13]

Victory:
1500+32*(1-0.387)= 1519 [+19]


Team B:
Expected Outcome:
1/(1+10y((1500-1580)/400))= 0.613 [61.3% chance of winning]

Loss:
1580+32*(0-0.613)= 1560 [-20]

Victory:
1580+32*(1-0.613)= 1592 [+12]


The formula on WoWVault is a bit different:
Expected Score = Current Rating / (Current Rating + Opponent Rating)
New Rating = Current Rating + Maximum Possible Change * (Outcome - Expected Score)

Team A:
Expected Outcome:
1500/(1500+1580)= 0.487 [48.7% chance of winning]

Loss:
1500+32*(0-0.487)= 1484 [-16]

Victory:
1500+32*(1-0.487)= 1516 [+16]


Team B:
Expected Outcome:
1580/(1580+1500)= 0.513 [51.3% chance of winning]

Loss:
1580+32*(0-0.513)= 1563 [-17]

Victory:
1580+32*(1-0.513)= 1595 [-15]


So currently I'm having two formulae (and I'm sure there are a lot more out there) and have no clue which variation Blizzard uses. Albeit the second one doesn't look to be very likely, as with a 80 points difference in rating, you wouldn't gain as much points as you are loosing in WoW.
So Wikipedia's formula looks quite appealing, yet of course without data there's nothing you can do, really.

Any data could shed some light on the formula. Especially interesting is the maximum points you have gained/lost against another team (to determine the 'Maximum Possible Change' variable) and of course the ratings of your and the opposing team together with the point outcome.

Of course the easiest thing would be if somebody already knew the formula.

Jesta from Jesta.us had a calculator but it was taken down. I guess you can find more info on his WOM post. http://www.worldofming.com/?p=376

I think someone one these board recomended some addon which showed the teams rating on the score board but I am unable to find it now.

ArenaPointer shows you your Arena Rating on your PvP tab, but that's not what I was looking for.
I'm looking for the formula which determines the points gained / lost.


// Edit
Just looked at Ming's page, the calculator would have been what I was looking for. It is down, but Jesta wrote some comments on Ming's page, which seem interesting.

So the change in rating is 0 if a team wins with 545 or more points in arena rating than the other, and 32 points if the lower team wins.
That was from his calculator though, did anyone *ever* receive 0 / 32 points?

Zero point wins happen quite often once you get in to the 2300 range, but I don't think I've lost more than thirty points in a single game.

I think Arena system is supposed to choose equal rating teams if available.

Well, that's only possible if there are actually playing teams with equal or similar ratings.
We've been crushed sometimes for 2 arena points due to the system can only be as good as the participants.

In my experience this happens mostly because the other team has probably been queued for a while, and has broadened their search to eventually reach your team. This is why sometimes you'll get an instant game after queueing and only win/lose like 5 points, or 25 points, etc.

That wouldn't really explain why it went from always taking 60+ seconds to find a game and go up against a drastically higher/lower team from arena start too patch, then after that it usually would match people up much faster but wouldn't be based on time, this is over lots and lots of games not just one night of arena.

Ok, it seems Wikipedia's formula is (near) correct.
Of course without real data it is hard to validate, but these are the formulae to calculate the min/max opponent team rating, considering Blizzard is using the standard Elo formula:

MAXIMUM POINT CHANGE seems to be 32, but could be 30 as well. 32 means that you can gain 32 versus a very high ranking team when you are very low, or that you gain/lose 16 points against an even team.

RESULT is either 1 for a victory or 0 for a loss.

Minimum Opponent Rating = CEIL((LOG((1 / (RESULT - (POINTS/MAXIMUM POINT CHANGE))) - 1) * 400) + OWN RATING)
Maximum Opponent Rating

[top] FLOOR((LOG((1 / (RESULT - ((POINTS+0.9999)/MAXIMUM POINT CHANGE))) - 1) * 400) + OWN RATING)


So if you just began in season 2 and have a rating of 1500, and you won your first match for 15 points, the opponent's rating was:
MIN:
CEIL((LOG((1 / (1 - (15/32))) - 1) * 400) + 1500)

(((1 / (1 - (15/32))) - 1)l * 400) + 1500 [for the windows calculator, you have to replace LOG with l (small L) and place it *after* what has to be calculated]

[top] CEIL(1478.257)

1479

MAX:
FLOOR((LOG((1 / (1 - ((15+0.9999)/32))) - 1) * 400) + 1500)

[top] (((1 / (1 - ((15+0.9999)/32))) - 1)l * 400) + 1500

FLOOR(1499.998)
= 1499

So for 15 points and a 1500 rating, your opponent should have had between 1479 and 1499 rating.




If you have knowledge about the rating of the opponent team, please try to verify.

I'm fairly certain that 15 is the break even rating, and 30 is the max, not 16 and 32.

I can verify that you can win 32 points in a match.

When I was first starting arenas, I was q'd up with a very highly rated team, whose members must have dc'd, because as soon as the gates open, the victory sheet popped up saying that we had won and received +32 points

I've had the 32 point win against a no-show team happen before, I think it is just a bug when the system tries to match you against a team that you shouldn't be playing. I've never actually faced off against a team where I won more than 28 or lost less than 2 points, and the average seems to be around 15 for the most part. In fact the vast majority of our matchups during prime time are 15 point games, with a handful of 14/16 pointers scattered in.