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I'm a total Go novice, and barely even understand the rules. From what I understand, at the end of the game each player has an amount of "points" which is equal to the amount of stones they captured (or something like that). The player with most points wins.

If two players with the same Elo play against each other, it should be that the expected difference in their amount of points is 0. Yet Go also has a handicap system, where someone with Elo 3500 can play against 4000 and to compensate starts with a set of initial points.

My question is: What are the statistics on how many handicap points one needs to give a player of rank X in order to still have even winning chances against one of rank Y?

For example, it might be that for a 4000 Elo player to have even odds against a 3000 one, they need a handicap of 8 stones (random guess). I'm interested in what these numbers are for the whole range of Elo.

Note:

  • I'm not sure if this is actually equal to "the amount of handicap points actually given based on different ranks", because they might not adjust the latter to match the former.

  • It probably also isn't equal to "the amount of excess points actually obtained in games without handicap", because maybe the better player achieves fewer excess points than they could, by taking risk-averse strategies.

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  • There was a paper describing exactly what you're asking, but I can't find it immediately. The gist of it was that it depends on a) the Elo-like pattern in use - the win rate between players 100 points apart varies from 1:1.5 to 1:4 or so - and b) the strength - at high levels, a single stone of handicap is worth more than 100 Elo points, at low levels a handicap stone may be worth only a fraction of that.
    – mafu
    Jun 17, 2022 at 7:55
  • Check out Sensei’s , Library and you will find a lot of information on this and many other useful topics in Go.
    – PJTraill
    Jun 17, 2022 at 14:09
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    If you are playing against a skilled player, they'll be able to explain it to you. If instead, like many beginners, you are playing against other beginners, you'll quickly work out your own relative handicaps based on past games. (E.g. whenever you beat someone three times in a row, you increase the handicap by 1 (and reset the count to zero), where the handicap is the number of stones they can play on their first move.) This handicap system works far better with Go than with many other games (e.g. chess), and allows skilled players to play challenging games against beginners.
    – Ray Butterworth
    Jun 19, 2022 at 1:40
  • My answer gave a source, but not much detail; I have extended it to address more of your questions.
    – PJTraill
    Jun 22, 2022 at 10:41

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Clarifications

Before answering, I would like to clarify some things;

  • Your score is not how many stones you have captured¹, but how much of the board you control. This means areas where your opponent cannot play without being captured. Details: Scoring
  • A stronger player does not generally give a weaker player a handicap of so many points of score (though that is possible), but rather gives them free moves for the difference in their ranks. Details: Handicap
  • In most events, only a win or loss matters, so players do not try to maximise their score: if they seem clearly behind, they may risk losing a lot more even for a small chance to win; if this fails they are likely to resign. This means that statistics about winning margins in points (which are often not recorded) would not allow reliable conclusions.

European statistics

Source

There are different ranking systems in different countries and on different Go servers, as described at Rating Systems. In particular, the European Go Database (EGD) operates an ELO system, and provides statistics here on win probability against rank difference and handicap given.

Examples

For example, if I select the period 2015-01-01 – 2022-06-22, it churns for quite a while (perhaps 20 seconds), and then shows several tables of statistics. There one can see, for example, that in games between players 2 ranks apart with 2 stones handicap (i.e. the standard handicap), Black (the weaker player) won 39% of the time (34% for kyu – weaker – players, and 39% for dan – stronger – players). For a 5 rank difference Black won 35% (in all cases) with 5 stones but only 16% with 1 stone (17% for kyu players, 7% for dan players).

Conclusions

I have not studied the statistics in detail, but one thing that stands out is that the standard handicap (1 stone per rank) only gives Black about a 35% chance of winning in most cases, except for a difference of 1 rank, where it is about 45%. Further, to answer your bolded question, it looks as though the weaker player needs 1 or 2 stones more than the rank difference for an even chance, but such a handicap is almost never given, so that is an extrapolation.

(You actually seems to be asking about the points needed to make a game even, but the score difference is often not recorded, and I think that these (EGD) statistics do not allow a conclusion about this. Also the approach I mentioned of playing for a win rather than to minimise your loss would confuse the issue further.)

Secondly, weaker players get better results than stronger against opponents a given number of ranks stronger (also in games without handicap); perhaps this means that weaker players vary more, or that their ranks are less accurately estimated by the system.

Notes

¹ This (score = captives) may possibly have been the case in the origins of the game, though the situation seems not really clear to me, and is now definitely not the case anywhere I know of. See Ancient Chinese Rules and Philosophy and Prisoner Scoring for some more details.

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    Your first point isn't strictly true, while Prisoner Scoring isn't commonly used there is some evidence it was used for early Wei-Ch'i games.
    – KMR
    Jun 21, 2022 at 4:05
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    @KMR: Interesting, but that article is very short and the discussion makes it sound quite controversial. And in the Ancient Chinese Rules and Philosophy, Bill Spight remarks that “the extant game records indicate that territory was counted, not just captured stones”.
    – PJTraill
    Jun 22, 2022 at 9:47

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