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 Post subject: Math ko
Post #1 Posted: Fri May 14, 2021 6:54 am 
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I was tinkering around with the idea of creating a basic rating system.

It may be that the weight of each game should be reduced the more times that person has or will play with that person, it being reduced more the closer those games are to it in the past or future . . . (<-edit: forgot to write that last part)

If A's score against B is 3/4, we imply A has three times more exp than B, and if B scores 3/4 against C we assume the same. However, what if C scores the same to A? It's the classic style triangle. Well apparently one way of solving this implies something interesting about dividing by zero . . .

What if the reason top professionals overall between those affiliated between the Nihon and Kansai is because goratings uses a distribution which overestimates the chance of a lower-rated player winning, whereas mamumau0413's distribution is more accurate?

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Last edited by ElomKW on Sat May 15, 2021 2:34 am, edited 2 times in total.
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 Post subject: Re: Math ko
Post #2 Posted: Fri May 14, 2021 7:01 am 
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In the whole-history rating experiment, first I ascertain the the number of games between A, B and C. For the purpose of simplification I assume they've played the same number of games. Solving from B's perspective, which is taking B's exp as the unit exp value 1, B's direct game with A implies A's exp is 3. B's direct game with C implies C's exp is 1/3. B's indirect game with A implies that A's exp is 1/9, but since it's an indirect game it has half the weight of a direct game.

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 Post subject: Re: Math ko
Post #3 Posted: Fri May 14, 2021 7:09 am 
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Intransitivity is easily explained by the fact that there are several different skills at go. I.e., a player's "correct" rating is a vector, not a single number. Reducing the rating to a single number loses information. The single number rating calculated by a rating system is an average. In addition, if the player's opponents in different systems have significantly different skills, then the player's single number ratings in the different systems will almost surely differ.

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 Post subject: Re: Math ko
Post #4 Posted: Fri May 14, 2021 5:15 pm 
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Bill Spight wrote:
Intransitivity is easily explained by the fact that there are several different skills at go. I.e., a player's "correct" rating is a vector, not a single number. Reducing the rating to a single number loses information. The single number rating calculated by a rating system is an average. In addition, if the player's opponents in different systems have significantly different skills, then the player's single number ratings in the different systems will almost surely differ.


Yes! And from what I've seen it seems to be some players naturally have a wide range, especially if they're prone to blunders (Fujisawa Shuko) of play particularly well when inspired Yuki Satoshi). Whereas others are the opposite, consistent (Mok Jinseok). The best I've seen a rating system account for this is with a sort reliability factor which doubles as a way to handle the wide range of beginners and their faster improve ment . . .

If B's exp is 1, B's impression of A's exp is 3^(1)*1/9^(1/2), putting =3*1/3=1, which we know should be the correct outcome as A, B and C score equally to each other in a style loop. This so far should be easy to follow for someone with even the most rudimentary math level, although this next part is were things get slightly interesting, but the concept is still basic (how not to be wrong quadrant).

Imagine if the score for everyone in the style loop was not 3/4, but 1! Yes, a rating system should have a way to handle players with only wins, but we'll keep it pure for now and carry on as normal and deal with the mathematical ridiculousoties like the brave souls who discovered imaginary numbers. Again, everyone has played the same number of games with each other, so any weighting is based purely on directness. With B's rating as 1, from B's perspective, A's perfect score over B means A's exp is an idiotic-sounding 'infinite' a pretend number. Likewise with B's rating of 1 B's perfect score over C means C's exp has to be 0. Now, ignoring temporarily the silliness of the next phase as those handling imaginary numbers did, if C's exp is 0, and C scores perfectly over A, it implies A has an exp that makes C's 0 look infinite. What could this mystery value be?

A exp is the 'infinite', exp multiplied by the square root of the mystery exp, which we know should equal 1.

It's possible to solve by, instead of rooting the 1/9, squaring the 9. So we have 9*9*1/9=0.

Here infinity times infinity--bare the nonsense--times the mystery is 1.

This all seems to imply that infinity time 0 does indeed equal 1. The mystery number must be 0^2, which turns infinity squared to 1.

Now it's best here not to think of infinity and 0 as numbers, but speeds . This is described in the mathematical concept of different sizes of infinites. 2 times infinity means getting bigger twice as fast. A half, twice as slow. For 0 it reverses; 2*0 means getting smaller twice as slow and 1/2*0 means getting smaller twice as fast. So ∞ and 0 are like pretend numbers. ∞ & 0.

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 Post subject: Re: Math ko
Post #5 Posted: Sat May 15, 2021 5:10 am 
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ElomKW wrote:
With B's rating as 1, from B's perspective, A's perfect score over B means A's exp is an idiotic-sounding 'infinite' a pretend number. Likewise with B's rating of 1 B's perfect score over C means C's exp has to be 0. Now, ignoring temporarily the silliness of the next phase as those handling imaginary numbers did, if C's exp is 0, and C scores perfectly over A, it implies A has an exp that makes C's 0 look infinite. What could this mystery value be?


Statisticians are way ahead with regard to this question. Going back to Laplace, if A and B play N games and A wins N games, then Laplace estimated the probability that A would beat B on their next game as (N+1)/(N+2), not 1. There are problems with Laplace's estimate, which were well known by the 20th century. But an estimate of 1 is, without other evidence, ridiculous. That is obviously so with cases of intransitivity.

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 Post subject: Re: Math ko
Post #6 Posted: Sat May 15, 2021 7:05 am 
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Bill Spight wrote:
ElomKW wrote:
With B's rating as 1, from B's perspective, A's perfect score over B means A's exp is an idiotic-sounding 'infinite' a pretend number. Likewise with B's rating of 1 B's perfect score over C means C's exp has to be 0. Now, ignoring temporarily the silliness of the next phase as those handling imaginary numbers did, if C's exp is 0, and C scores perfectly over A, it implies A has an exp that makes C's 0 look infinite. What could this mystery value be?


Statisticians are way ahead with regard to this question. Going back to Laplace, if A and B play N games and A wins N games, then Laplace estimated the probability that A would beat B on their next game as (N+1)/(N+2), not 1. There are problems with Laplace's estimate, which were well known by the 20th century. But an estimate of 1 is, without other evidence, ridiculous. That is obviously so with cases of intransitivity.


I did think of assuming there was a 50 percent chance that the number of games A and B had played so far was 1 games less than the minimum needed to represent the true score fraction, (N+(1/2))/(N+1)≡(N+1)/(N+2) . . . And then right after that I thought it would make sense to add the chances for being 2 games less, 3, games less, and so on (assuming each chance is half as likely as the previous one) . . . (N+((1/2)+((3/4)^(1/2))+((7/8)^(1/4)) . . . +((∞-1/∞)^1/(∞/2))/∞))/(N+1)≡(N+1)/(N/(1/((1/2)+((3/4)^(1/2))+((7/8)^(1/4)) . . . +((∞-1/∞)^1/(∞/2))/∞)) :).

(for others reading who might not immediately get the concept, because it can be said for certain A's rating is not infinite, it can be said that the number of games they've played is to small to represent their true score. in other words I thought the most basic form of correction would be for the rating system to recognise that the number of games played is too small a bandwidth to properly express the true difference between A and B. If it's 2/3 you need 3 games, 3/4 you need 4 games, 4/5 you need 5 games, and so on, for example if the 'true' score is 3/4, but they'd need to play four games at least to express that properly, and also if the total number of games isn't divisible by the required number of games, the fewer the games, the less the true score can be expressed, and even after all that, the laws of probability mean it's still very possible for one player to outperform their usual self to the point were A could still get a perfect score over the required number of games, which is another reason why the more games played the better for the rating system).

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 Post subject: Re: Math ko
Post #7 Posted: Mon May 17, 2021 12:26 pm 
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A simpler way to show the strange infinities concept might be to have:

A* 1-0 A 1-0 B 1-0 C 1-0 A*

Using B's exp as the unit exp 1 A's exp is ∞, and (A*'s exp/∞) = ∞/1 ≡ A*'s exp = ∞^2.

Using B's exp as the unit exp 1 C"s exp is 0 and 0/1 = (A*'s exp/0) ≡ A*'s exp = 0^2.

Averaging between the two impressions B has of A*, (∞^2)*(0^2)=1 by necessity as we know beforehand the implied exp of all players must be 1 from their equal results against each other.

So making a basic rating system I've accidentally found out about notation to represent different sized infinities (for those who haven't heard of it, the infinite series of even integers being half the size of the sum of all infinities is the kind of thing they mean by this) that implies 0 and ∞ is the same kind of symbol.

0 and ∞ seem to force meta-mathematics, for example if you multiply a number by 0, it's important to know if that 0 was multiplied by another number beforehand, such as 2*0 or 0^2, from which we can infer it's better to represent zero as 1/∞ and x multiples of zero with x/∞. Now it seems hard for me to see how this is inherently bad even though it goes against my previous beliefs :lol:

Fortunately (N+((1/2)+((3/4)^(1/2))+((7/8)^(1/4)) . . . +((∞-1/∞)^(1/(∞/2)))/(N+1)≡(N+1)/(N/(1/((1/2)+((3/4)^(1/2))+((7/8)^(1/4)) . . . +((∞-1/∞)^(1/(∞/2))) and better removes the need to handle infinite exp's and infinities of exp. Although all this is to apply the principal of weighting games less if a game between two players is in the near past or future, so best of all would be for goratings.org to weight games less if two players have played a lot of games and those games are near in the past or the future of that point. It would be extremely interesting to see as I'm very sure, above the degree I have right to be so it is faith, that it would do a lot to balance out the inflation and deflation that can occur within any player pools.

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Post #8 Posted: Wed May 19, 2021 5:52 pm 
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rating.0

Of course halving the weight of games between players of the same country (in terms of their affiliated pro association) would also be useful (I remember during the hilarious Iyama Ranking Obsession era that in reply to someone saying that Iyama Yuta was overated on goratings, he essentially claimed that the problem was the Japanese go environment and not the elo-based rating system, but the entire discussion was partly based on the idea that maybe rating systems should be immune to this if they were to be considered complete so I don't understand that response . . . In any case I can't imagine why he didn't just halve the weight of games between players of the same country from then. Indeed I don't know of any rating system that can handle these sorts of player bubbles without an adjustment like that. But I can't seem to be able to reach Rémi Coulum . . .

point zero səoʊl impact

An interesting consequence of concluding that zero and infinity force meta-mathematics is a possible hint of an explanation as to why there is something rather than nothing. Before the sphaleron process and the like. Although I think it's a bad habit to separate the question of why there is consciousness and why there is matter. They probably relate to each other in the same way electricity relates to magnetism (a touch of egg-order theory).

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