Bill Spight: The traditional scheme gives an advantage to White. In an even game Black gives komi. If Black is 1 stone weaker, the proper handicap is for Black to put 2 stones and give komi. Alternatively, Black could play first and White could give komi, since komi is worth 1/2 stone. Simply having Black play first is inadequate compensation for the difference in strength.
Also, the equivalence between one stone and one rank is inexact. Is the fifth handicap stone worth the same as the fourth? However, through 9 stones the deviation appears to be quite small. This could be the topic of a good statistical study. :-)
Jasonred Gotta agree with this one. Not only that, but the size of the board has gotta make a difference too. IMHO, the strength of the players also makes a difference, as even 3 stones on a 19x19 is not much help to a 30 kyu, but is a HUGE difference to a 9 Dan... Especially if you're following French handicap, or whatever you call the "free placement of handicap stones" thing.
Instead of statistics though, I would ask a top Dan pro how much komi he'd be willing to give an equally strong opponent for a 1 stone handicap, for a 2 stone, and so forth.
I DO know that the 26th stone isn't worth as much as, say, the 5th stone... and on a 9x9, anything over 5 is probably a liability. More than 10 stones is just silly on a 9x9, you're just wasting your own territory, so I can safely say that the 10th stone should have a NEGATIVE value.
Flower: Why would placement of more than 10 handicap stones be negative? They might waste n moku of your territory (assuming territory scoring and Handicap n) Yet at the same time they create a very strong moyo that cannot be invaded easily. Thus more than 10 Handicap stones still serve to give Black a considerable advantage. For example you could place 40 handicap stones ins uch a manner that each stone still has 4 liberties left and ample space to connect to the adjacent stones which are just one space apart. Of course this would be a ridicoulous handicap. But it should only serve to show that a large amount of handicap still benefits black and does not become a liability unless it reaches amounts where it is impossible to form two eyes. (around Handicap 77 on a 9x9 I would assume) --Flower, 2007-01-30
Warfreak2: The only way the 10th stone could have negative value, is if it is worth -1 point; that is, if it just fills in one point of territory for black with no other effect. This assumes that the whole board is black's territory with nine stones, that is, that white cannot live. If white cannot live against this weaker player against nine stones, then the handicap should be lower!
Flower: Yes indeed =]. Yet Jasonred's comment regarding Handicaps larger 9 implicates that there is a reason for awarding these large handicaps. Thus Black's skill would be abysmal to an almost unimaginable degree. And as such we can neither assume that the nine handicap stones cause the board to be blacks territory nor that White will fail to live with at least one group. (In fact I played two 9x9 games where I gave 9 handicap stones to my (first timer) opponent and still managed to live. One of them against my dear Mom :) --Flower, 2007-01-30
Herman Hiddema: Note that the above refers to handicap on a 9x9 board. 9 stones in quite enough (actually too much) to claim the 9x9 board entirely. So under territory scoring your score will be lower with 10 stones than it would be with 9, hence the negative value for the tenth stone.
Flower: My above reply assumes 9x9 (as Jasonred to whom I replied did). Jasonred said, that it is likely that any stone beyond the first nine Handicap stones would have a 'negative value'. To me this claim translates to: "Any Handicap stones beyond 9 actually decrease Black's chance of winning the game" As explained above I find this claim quite suprising and would not agree. (Of course I agree that the territory is reduced and as such the maximum margin with whom black can win is reduced. Yet as the amplitude of the winning margin is irrelevant I fail to see why it is of any importance) --Flower, 2007-01-30
Flower: I think it is worthwhile to add that a 9+ Handicap in 9x9 would translate to 40+ Handicap in 19x19. So in order to interpret above situation correctly one needs to assume players with equally large rankdifferences. (Like a child who plays Go for the first time playing against a strong dan player who for some reason does not want a teaching game but a challenge including proper handicap. (did I mention ridicoulous earlier ;-) --Flower, 2007-01-30
Dieter Verhofstadt: Let me get this straight:
A and B are of equal strength. A takes Black and B White. B gets 5,5 points of komi. Second game: B takes Black and A White. A gets 5,5 points of komi. We expect the same result. This means that the difference between having the first move or not is equivalent to 11 points. Removing komi is only worth 5.5 points. The next step we add a stone to Black's handicap. If one stone is worth 11 points, then this is a bigger leap than the first one, and so White consistently receives 5.5 points more than he deserves. If one stone is worth 5.5 points, then this system is OK.
So, Bill, your assumptions are:
Bill: Each handicap stone (up to 9) has approximately the same value. Playing White (sans komi) is equivalent to having a handicap of 0 stones (i.e., of playing first but passing on the first play). Receiving (proper) komi makes the game even. So (proper) komi is worth approximately half a stone.
About 25 years ago I did some statistics on pro-pro handicap games and concluded that each handicap stone was worth around 13.5 points (territory scoring). (Of course, the estimate was not very precise. I was surprised to find, however, that the relationship was nearly linear. My initial assumption had been that it was not.) Since that implied a proper komi of 6.75, I predicted that the Japanese komi would increase to 6.5 by the turn of the century. Wrong again! ;-) (Nearly: In October 2002, the Nihon Ki-in announced it would start using 6.5 komi).
Somebody else published an article in the AGA journal (Summer, '77, I think) based on over 2,000 Japanese pro games, that indicated a proper komi of around 7.
In the late '70s in New Mexico I ran 4 tournaments a year (around 20 players showed up for each tournament). We divided our ranks into two half ranks. If there was a 1/2 rank difference, the lower ranked player took Black (White winning jigo); if there was a 1 rank difference, he took Black and White gave komi; etc. Komi was 6.5.
After a few years game statistics indicated that White had a slight advantage. However, most games were decided by such large margins that a small change in the komi would not have mattered. Besides, in our small pool of players, the better players (who were more often White) tended to advance more than the weaker players. They were probably more interested in learning the game. :-)
How were handicap games invented?
Frs: A theory: Ancient gobans were smaller than 19x19. One could easily claim territory by starting at tengen, making games rather uninteresting. To prevent this, handicap stones placed at the corners were invented.
Jan: Just a quick question, for now: has any SL deshi ever played a game with reverse komi instead of normal handicap stones? So, for instance, when by rank difference a four stone handicap game should be played, has anyone played with forty points of komi for Black?
The reason I'm asking is that while I like going up in rank by winning handicap games, this doesn't help my 'normal' play - joseki, direction of play and so on. I'd like to get started on that aspect of the game, because I don't want to be without a clue when I finally reach the level to play even games (at tournaments for example). Also I think I'd like to get better at estimating the score.
WillerZ: How many tournaments have you been to? The first I attended was Britain's Furze Platt in Maidenhead; I entered as my club rank of 18kyu (having only played games where _I_ got 9 stones), and my first opponent had entered the tournament at 30kyu. I had to give him 9 stones and 30 reverse komi, which is not what you want to face as your first game with the white stones.
That aside, I read somewhere on SL that a handicap stone was worth 13.5 points when used correctly -- your 4-stone game should therefore be 54-points komi to black, rather than 40.
The other point worth considering is that while you only get to play one corner style with fixed handicap, you can get a lot more variety with a free handicap 4 stones is enough for 2 corner enclosures for instance.
Finally, have pity on white -- remembering you get 6.5 bonus points can be hard enough, but giving 40 (or 54) would be even harder to deal with. At least handicap stones are on the board where you can see them.