What is the formula of rank difference to handicap stones?

Question

When the difference in rank is one (examples: 1 dan vs 1 kyu, or 1 kyu vs 2 kyu), the weaker player takes black and the komi is 0.5 for white (sometimes called "no komi" even though that's not technically correct).

From then on add one handicap stone per rank difference (black playing first with white getting 0.5 komi is equivalent to one handicap stone if free placement is used).

However, this formula doesn't work after about a 5-rank difference, because handicap stones start reinforcing each other. With some discussion I was able to find others who seem to know this is true. I can find no authority however, nor anybody who seems to have any idea of the correct formulation.

Answer 1

Regarding your question: To my knowledge, handicap is intended to be linear.

The observation that large handicap games tend to end with a large score difference is not necessarily true. A large difference in score is usually a group dying involuntarily, which happens in both high handicap and even games on a regular basis.

However, it is far less common to resign in handicap games. Regular games in comparison often feature a high score difference as well, but those games are usually just won by resignation.

Another thing to keep in mind is that the rank system in use may very well be wrong for larger rank differences. You can see this very clearly by just comparing the rank difference of various players in different rank systems. For instance, KGS ranks tend to be a little bit more spread out than EGF ranks.

This means that two players with a difference of 8 stones on KGS may be only 7 stones apart in EGF rank. Which makes you wonder which is correct - and unless you got a huge database of high handicap game results, you won't know which of the systems is closer to the statistical truth.

This leads to my final thought: The EGF rank system (along the European Go Database) clearly specifies the intended win percentage for certain rank differences, also taking into account handicap. According to their formula, it is linear. I believe that AGA uses a similar system, too.

Edit (2018): I have recently worked more closely with rank computations than I hoped, and there is one thing I want to add, described here (I should note that some other information on that page appears to be outdated or wrong, in particular regarding AGA rating).

The skill difference that is supposed to be balanced by handicap stones depends on the skill of the players. That is to say, a single handicap stone in a high level match has a lot more impact than it does in a game between beginners.

This can in fact be quantified: A handicap stone in a pro game equals around 300 elo difference, around 150 for high dan amateurs and only around 50 (!) for kyu players. This adds another dimension in how rank differences and stone differences between players interact.

By the way, these data also show that adding more handicap stones is linear at least up to 4.

Answer 2

The formula I use for the point differential for handicap stones is 2/3 (X**2) +12x-6.

That means 6 2/3 points for one stone, 21 points for two stones, 36 points for three stones, 53 points for four stones, 71 points for five stones, 90 points for six stones, 111 points for 7 stones, 133 points for eight stones, 156 for nine.

Each stone is worth more on an exponential basis, while handicaps are linear. A nine stone handicap worth 10-11 levels rather than nine.

Answer 3

Black going first without Komi is equivalent with 0.5 stones. 6.5 points of Komi is also equivalent with 0.5 stones (they even out). 3 stones handicap is equivalent with 2.5 stones. E.g.: on IGS, players with a rank difference of 3 will be auto matched with 3 stones of handicap and reverse Komi. Rating systems also use this rule to predict the outcome of a game.

Back to your question, you are basically asking if handicap is linear: if A and B require a handicap of X and B and C require a handicap of Y, does that mean that A and C require a handicap of X+Y ?

In my experience this is pretty much true. However, a handicap game requires somewhat different tactics (or at least a shift in balance). So some players are more experienced at receiving handicap while others are better at giving handicap. That can influence the odds.

Answer 4

I don't think there is a "correct" formulation that has been in any way validated. It is supposed to be approximately linear because we essentially define it that way: a 1 dan is defined relative to a 2 dan by being able to win around 50% of the games with a komi of 0.5. It is fairly traditional, going back to old game series (pre-komi) where they would progressively change the handicap to indicate relative strength ("forced X to take black" etc).

Part of why you generally should think of ratings as relative to others in the same system–and possibly the same area–rather than absolutes. Don't get me started on Korean "1 kup" players… -.-

The challenge (beyond all of this) is that handicap is essentially an imperfect approximation, and there is a skillset that is particular to handicap games. Due to playstyle/psychological differences, handicapping is also not strictly transitive, to quote David Mechner:

Most people consider the handicap system very elegant because it's simple and it works well. However, it's not perfect. For one thing, the value of a stone is not really necessarily constant. For example, against an opponent of constant strength going from 4 stones to 3 stones is a bigger jump than going from 5 stones to 4. Also, there are style interactions that cause intransitivities. Player A might be able to give B at 2 stones, and B might be able to beat C even, but because of the specific strengths and weakness of the players, C might be able to give A two stones.

Combined with that DDK (and even some higher SDK) ratings tend to fluctuate wildly game-to-game since they haven't "stablized" yet, and what you are essentially looking at is a system that is considered a "good enough approximation." Straight-komi systems also tend to break down after a certain point, because the difference in skill isn't linear no matter how much we try to shoehorn it into that model.

Answer 5

A Kris noted the first handicap stone is actually just half a stone. Sensei's has a related page: Proper Handicap

The main part of the question seems to be if scaling the handicap linearly in the rank difference works.

In my experience it is a good approximation, at least up to 6 stones. But this depends a lot on style. I'm a relatively aggressive fighting oriented player, and I tend to do pretty well in handicap games as white. But there are other players whose strength is a proper opening game in an even game. I assume they will have a harder time in high handicap games.

In my experience high handicap games tend to be very unstable. Wins by a 100 points in both directions are very common for me.

Answer 6

I am very much a novice, so I claim little experience and no expertise. But I think handicap stones are supposed to be linear. I know I have taken a 9 stone handicap and lost before, and even as a novice I have given 6 stones to complete beginners and won.

Naturally the handicap stones work together (even with just 2, though its less obvious, and they reinforce each other much more above 5), but they are supposed to. If they are refinforcing each other too much for White to have any chance, perhaps that is a sign that one of the two players is being misranked.

If you want to fine tune handicap a bit without giving more stones, you could use a reverse komi or permit free placement of the handicap stones, but give fewer of them, rather than enforcing traditional placement.