Why is glicko->dan/kyu non-linear?

Is it possible to explain to someone who does not have deep knowledge of glicko and ranking systems why it is that glicko->dan/kyu is non-linear?

I’d be curious for insight, but I certainly don’t have a mathematical gasp of glicko.

1 Like

i believe it reflects how much harder it is to overcome 50 points of strength at the elite of the game compared to 50 points of strength amongst beginners. But that is just my suspicion.


The first step to a non-mathematical explanation would be a non-mathematical question.


I think it’s always hard to explain why things aren’t a certain way. Why isn’t the World flat? Whoever is going to answer your question well is going to need to understand why your expectation is that it should be linear.

For me it makes sense that it isn’t linear because there is an upper bound (theoretically perfect play) and a lower bound (random play) so with asymptotes involved I would expect some curviness in the relationship.


Here’s what started this question.

I was wondering why a particular Dan was showing as [?]. I thought “Oh, maybe their rank uncertainty is high”. So I looked at it, and it was +/- 1.9d . And I thought “that’s odd, my own uncertaintly is +/- 2.1k, and I am not [?]”

Then I found that my uncertaintly is smaller in glicko than the Dan’s. In glicko points, their uncertainty was much higher than mine, even though in d/k it is not.

This means that 1d is a bigger jump than 1k at 10k, and that is a bigger jump, in glicko points, than 1k at 20k.

It is not obvious why I should have to earn more glicko points to go from 11k to 10k than from 20k to 19k

I was wondering if there is an intuition that would help understand this?

1 Like

Specifically, it seems the conversion formula is an exponential function:

1 Like

Yeah - that’s the fact. I wonder why it is like that? It’s nice when the maths has an intuitive explanation to help feel like it “makes sense” (though I appreciate that not all math and not all real world phenomena do make sense).

If I were to guess, it’s because the devs looked at the statistics for handicap games (as that is supposedly the basis for kyu/dan rankings), and figured that it was a good statistical predictor of game results.


The difference in Glicko points gives you the probability to win,
the difference in rank what handicap is needed to get an even game.

Now I’ve to speculate a little, why the relationship between the two is not linear:
One additional stone changes the probability to win depending on the players strength.
A Dan can make more use of the additional stones than a DDK. For DDK an additional stone is more or less an additional stone the opponent capturs eventually later. For a Dan the whole game will more or less evolve around the additional stones.

Why exp:
Coincidentally a good fit for the nonlinearity.


What’s a piece of maths that doesn’t make sense? I thought the whole point of maths is that if something doesn’t make sense, then you must have made a mistake.

For what it’s worth, Glicko is in fact linear with regard to the rating of the player.

We could think of two analogous situations, where all players in the first situation have exactly the same rating deviations and history and such as the players in the second situation, but the only difference is that each player in the second situation has a fixed amount X of rating more than their corresponding player in the first situation.

Looking at the computation as described on wikipedia, we’re only interested in the effect of this higher rating that the player in the second situation has compared to player in the first situation. Hence everything that does not depend on rating can be assumed to be constant. This means everything except for the term E(s | r, ri, RDi).

Now in this term there is only one thing that is being used that depends on rating, and that is the difference (r - ri) between the rank of the player and the rank of the i-th opponent. This is equal in both situations. Hence both situations should lead to the same new rating, as never in the whole computation the absolute value of r is used; only the relative difference between ratings is used.

The nonlinearity that you describe comes purely from the way we translate the Glicko points into kyu/dan ranks.

This makes the most sense to me. Thankyou flovo. Now let me test my understanding:

If a 25k gives a 29k four handicap and a 9d gives a 5d four handicap then these two games should be more or less even.

However if the games are played ‘no handicap’ then the probability of the 9d defeating the 5d is higher than the probability of the 25k defeating the 29k simply because less skilled players make the result less certain.

This makes good sense too and is in line with complaints that DDK tend to think komi is too high while dans might prefer to have Black against the same komi.


For me is very obvious that improving from 11k to 10k is way harder than from 20k to 19k.
So I would say that the learning curve for Go is nonlinear if we compare strength and effort.

So maybe the intuition could be: rank just measures strength while glicko involves effort.

Actually glicko is just a mathematical model that tries to simulate a phenomenon, so it doesn’t actually measure any effort. But as a metaphor it could make sense.


An intuitive explanation, hm.

Think of running. For an average person of average height and ability, “running” 1mph is trivial. It’s also fairly trivial to run at 1mph+1. Out of all the people who can run 1mph, almost all of them will manage 2mph.

It is less trivial to run 20mph, and inconceivably hard to run 20mph+1. Out of all the people who can run 20mph, almost no one will manage 21mph.

As for an actual explanation, albeit a cheap one, I would say there are simply few factors limiting people to 1mph, whereas there are many factors limiting people to 20mph.


As I said above, Glicko is linear. It’s our translation from Glicko to kyu / dan that is not linear. So Glicko doesn’t really have anything to do with this discussion…

1 Like

And now to add in the handicap stones, just imagine playing with N handicap stones is like running a race against an opponent that runs on a (very long) treadmill going N mph in the opposite direction.

1 Like

I left a word out - makes sense intuitively. Quantum mechanics was what I had in mind, which is both maths and real world phenomena that are hard to intuit. Even the mathematics (tensors) and physics (what you see) of flipping a ruler is a little unintuitive, or at least to me.

I mostly made that comment to guard against the inevitable “why do you think maths should make sense intuitively?” if I didn’t say that :smiley: )



In my opinion the maths is solid, it’s the real world that should be intuitive (as it’s the real world), so my intuition must make no sense, then. :slight_smile:

Correct :slight_smile:


Yes. That’s exactly what the OP asked and what we are discussing here: why the relationship
isn’t linear.

I confess that I don’t understand what you mean by “glicko is linear” .
You need a relationship or a function to say if that’s linear or not.

y = f(x) is linear if you can write it as
y = ax + b

Nothing is linear in itself (or maybe we could say that everything is linear in rapport with itself, y = y, but it’s meaningless).
Moreover glicko is a recursive algorithm, so it seems to me very hard to call it linear.

1 Like

It’s linear in the following way: if we have rank r and we play against an opponent of rank ri, then there is a difference of dr = r-ri between our ranks. How many times to I need to win against an opponent of this rank difference to get to rank R? It turns out that this relation does not depend on the value of your rank r, but only on the difference between r and R.

If we make some reasonable assumptions, such as that you on average play against opponents of the same rank as you have, and opponents have on average the same rating deviation (on a large scale this will be the case), then the question above basically asks “How many games (against opponents of the same rating and constant RD) do I need to win to gain (let’s say) 200 points of rating?”

This is one of the questions at the heart of the OP’s question: why does it take more games to go from 1d to 2d than it takes from 10k to 9k. There are two possible factors that could account for this:

  • it could be that the glicko system does not scale up (this is not the case), so you need games to go from 2100 to 2200 rating than you need to go from 1200 to 1300.
  • it could be (and is the case) that the difference in rating between 1d and 2d is a larger number of rating points than the difference between 10k and 9k.

If the translation rating > kyu/dan is exponential, but glicko would somehow counteract this by being logarithmic, then in total it would take the same number of wins from 1d to 2d as it would take from 10k to 9k. So checking if the first point is indeed not true is necessary to answer the question sufficiently.

If you want to express the linearity as a function, it would be something like f( r ) with r being your rating, and f telling you how many games against an equal (average) opponent you have to play to gain 200 rating points. It turns out this function is linear. Actually, it’s constant. The number of games against equal opponents you need to play to gain 200 points does not depend on the value of r. Even when we change “opponents of equal rank” into “opponents an X amount of rating stronger / weaker”, this function stays constant.

Of course glicko is not linear in any way if we ask more complex questions. But regarding this discrepancy between going from 1d > 2d or 10k > 9k (in the amount of effort one needs to put into it), if we put all other circumstances to be equal, it turns out the answer is that Glicko doesn’t care about if you go from 2100 > 2200 rating or if you go from 1200 > 1300.

1 Like