Questions about Points in League Ranking

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Wriggler
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Questions about Points in League Ranking

Post by Wriggler »

What is the formula again?

It would be nice if when you click on to see individual statistics, there would be a display for points against each player in a game to get an idea of where the points come from. I think the old system showed this.

Also, do you get zero points if you beat an opponent with zero wins? For example, look at Dark, who beat SPW 2-0, yet has zero points to show for it. That doesn't seem right. Didn't the old system give 100 points in such a case?

Thanks for clearing that stuff up. I'm just curious. Thanks. :D
Dario
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Post by Dario »

The shortest version of the explanation:

The points you have are not really "points", it's the addition of the expected winning ratios against each player in the league pondered by the points they have. Where the expected winning ratio is expressed in this way:"at least x% or more, and I am y% sure that it won't be lower".

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From the sticky in this part of the forum, the summarized and simplified version:

The points you get from an opponent depends on the total number of rounds you have played against him, the total number of rounds you've won and the points of that particular opponent at the moment you are watching the ranking (doesn't matter how many points that player had when you played the game). In this way if you win against someone who has very few points, at that moment you won't get many points from him but if some time later he goes up into the ranking and ends up being one of the best, then now you will be getting more points for that victory.
This implies that the ranking is very dynamic and you can't accurately predict your score because it depends on how all the scores of the people you've played change, then if you want to climb in the ranking all you have to do is win rounds against the largest variety of opponents possible.

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From the private forum, the long and detailed explanation:

Let's start a league, players: Wormer1, wormer2, wormer3

There are no games played, we have nothing to make a ranking, so we give them all 1000 points to start.

After some time you have these results:
wormer1 2-2 wormer2; wormer1 4-2 wormer3.
wormer2 2-2 wormer1; wormer2 2-1 wormer3.
wormer3 2-4 wormer1; wormer3 1-2 wormer1.

And we want to know what is the ranking at that precise moment.

The points of wormer1 will be a proportion of the points of wormer2 + a proportion of the points of wormer3. The proportion of the points he gets will be the proportion of games he is expected to win against that same player if they played a very large number of games.

Now I ask you, given the 2-2 result, what will be the minimum proportion of games wormer1 will win after infinite games between those 2 players?. you could say 0,5 , sounds logical. And maybe you are right about this, but what if I give you another 2-2 result from different players? you should be fair and, again, say 0,5, but maybe this time you are wrong, and after infinite games it turns out that wormer1 won only 30% of the games.

If I give you 100 results of 2-2 and you make the same >0,5 prediction for them all (you have to, it is the fair thing to do since you don't know another thing about them) I can say that you will be wrong 50% of the times. About 50 the results will end up with a proportion of won games equal to 0,5 or higher, and about 50 lower than 0,5.

Now I give you the 4-2 result, the fair thing to do is make a prediction so that, just like the one you made before, there is a 50% chance of being right.

In other words, if c=0,75 and p=0,5 that means 75% of the predictions you make will end up with a result higher than 0,5 , while 25% of them will be lower than 0,5.

After calculating the "p" scores for each of the pairs, this is what we get:

wormer1 2-2 wormer2; P= 0,3599 (since this is the P of wormer1 to wormer 2, you can call it P(1-2)
wormer1 4-2 wormer3. P(1-3)= 0,5138

So wormer1 gets 0,3599 the points of wormer2 + 0,5138 of the points of wormer3. Remember that in our first ranking we gave 1000 points to them all, so for the next ranking wormer1 will have 359,9 points from his games against wormer2 and 513,8 points from his games against wormer3. Points for wormer1 for the next ranking = 873,7

wormer2 2-2 wormer1; P(2-1)= 0,3599
wormer2 2-1 wormer3. P(2-3)= 0,4565

So wormer2 gets 0,3599 the points of wormer1 + 0,4565 of the points of wormer3. Remember that in our first ranking we gave 1000 points to them all, so for the next ranking wormer2 will have 359,9 points from his games against wormer1 and 456,5 points from his games against wormer2. Points for wormer2 for the next ranking = 816,4

wormer3 2-4 wormer1; P(3-1)= 0,2531
wormer3 1-2 wormer2. P(3-2)= 0,2431

So wormer3 gets 0,2531 the points of wormer1 + 0,2431 of the points of wormer2. Remember that in our first ranking we gave 1000 points to them all, so for the next ranking wormer2 will have 359,9 points from his games against wormer1 and 456,5 points from his games against wormer2. Points for wormer3 for the next ranking = 496,2

Then our new ranking is
Wormer1: 873,7
Wormer2: 816,4
Wormer3: 496,2

But to calculate those you were considering them all with 1000 points. Now we know that wormer 1 is the top player, so a victory against him should be worth more than a victory against wormer3. We will get a third ranking by repeating what we did to get the second ranking, but this time instead of using 1000 as the points of wormer1, wormer2 and wormer3 we will do the calculations, using the same "P"s obtained before.

Then for the second ranking, the points of wormer1 will be = [P(1-2)*(points wormer2 in the previous ranking)] + [P(1-3)*(points wormer3 in the previous ranking)] .
Then points wormer1 = [0,3599*816,4] + [0,5138*496,2] = 548,76
points wormer2 = [0,3599*873,7] + [0,4565*496,2] = 540,9
points wormer3= [0,2531*873,7] + [0,2431*816,4] = 419,6

Do the same thing again and you will get another ranking. Again and a new one, and again, and again until the scores are proportionally stable.
After doing that about 10 times, these are the scores:

Wormer1: 11,78
Wormer2: 11,34
Wormer3: 8,166

For a real league, the number of times you have to re-calculate the ranking to get it stable is 3, or 4 to be a bit more sure. Also, for a real league the scores go from decimals when almost no games have been played yet to really, really high numbers (talking about millions of points) near the end of the season, so there is one last adjustment before showing the points to the public:
Calculate the average score = (11,78+11,34+8,166)/3 = 31.286
Divide the score of each player by that average, and then multiply it by 1000, and you get the final ranking:

Wormer1: 376
Wormer2: 362
Wormer3: 261

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The accurate formula is something so complex to write as a formula that not even my maths professor couldn't give me a clear answer when I asked him how to write "that" down in one equation, but the logic behind it is clearly explained in the post I just pasted above, with the addition of an indirect way of estimating the probability of beating an opponent (if you have been beaten by 5 players and I win against those 5 players, then we can say that there is a good chance that I will also beat you).

Dark and SPW don't really have 0 points. They just have less than 0,5 and the rounding turns it into 0. And they will keep on having almost no points until one of them plays against someone else of the league.

The logic behind what is happening there could be expressed like:

1-We have a ranking with over 30 active players.

2-we have one player (Dark) and we know absolutely nothing about him.

3-We have another player (SPW) and we know absolutely nothing about him either.

4-Dark and SPW play a game (Dark vs SPW) and Dark wins.

Knowing only that, where would you put Dark and SPW in the ranking?.

Dark won a game against someone who the system has absolutely no idea how good or bad is. So far, being there no proof of how good SPW is, there is also no proof of how good Dark is and the system will tag them as "newbs" until they prove the opposite ;).

Therefore, if in a future SPW played and beated the top 3 (Dario-Koras-Jakka) there will be proof that SPW is good. Knowing that SPW is good and that Dark won against him, then Dark will also climb up in the ranking (even if he doesn't play another game) because we could safely say that there is a good chance Dark is a very good player (although not too high because there is also a chance that Dark won that game only because SPW was in an awful day and had some awful luck).

I will now report a fake game for SPW, where he beats me 2-0, and you can check the ranking to see the effect.
Joschi

Re: Questions about Points in League Ranking

Post by Joschi »

Is the old explanation still correct for our current ranking system?
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