Despite knowing next to nothing about the Elo system, I am strangely drawn to this discussion, with the hope that I might have something meaningful to say.
I may have something to add about fractional errors.
moha wrote:
The 1 point example was just an example, in reality "rounding" your mistakes to integers cannot affect (or affect differently) your class. If you only make average 0.1 pt worth of mistakes per game, you will be rated much closer than a class from perfect play once tested widely and seriously - exactly because 0.1 pt mistakes has, on average, much less effect on the game than one point mistakes.
lightvector wrote:
{That} might simply not be true.
Note: Below I am going to use octal notation. I know that the discussants are familiar with it, but not all of the readers may be. In octal notation, 0.7 = ⅞, 0.6 = ¾, 0.4 = ½, etc. Outside of ko positions, fractional go values will be in terms of powers of ½. A mistake of 1/10 is very unlikely to occur.
There are at least two possible meanings to an error of some number of points in go. One sense is that the player makes a play that is dominated by a play that gains more points, on average. Another is that the player makes a play that, after correct play, gets a worse score. I don't know if the players are talking about either one of these, or if they have something else in mind.
Here is an example of domination. Suppose that there are two plays left on the board, one that gains 0.6 pts. and one that gains 0.4 pts. by territory scoring. Then the play that gains 0.6 dominates the play that gains 0.4. IOW, the play that gains 0.6 pts. will never score worse than the play that gains 0.4 pts., but the play that gains 0.4 pts. may score worse than the play that gains 0,6 pts., and we consider the play that gains only 0.4 pts. to be a mistake. Using | notation, let the plays be these: {1|0} and {2|1||0}. The local scores are pts. for Black. If you don't know this notation, I think it will be clear after I go through this example.
Suppose that Black plays first. Then if she makes the 0.4 pt. play she will move to the left in {1|0} to get a local score of 1 pt. and {2|1||0} will remain on the board. White will next take the remaining, 0.6 pt. play, moving in {2|1||0} to a local score of 0, for a combined local score of 1 + 0 = 1. Now suppose instead that Black makes the 0.6 pt. play first. She will move in {2|1||0} to {2|1}. That will leave two 0.4 pt. plays, {1|0} and {2|1}. They are miai, so the combined local score will be 2 + 0 when White moves to 0 in one and Black moves to 2 in the other, or 1 + 1 when White moves to 1 in one and Black moves to 1 in the other. The local scores add to 2 in either case. Black does 1 pt. better by making the 0.6 pt. play.
Now suppose that White plays first. If he makes the 0.4 pt. play there will be a local score of 0 and {2|1||0} will remain on the board. Black will next take the remaining, 0.6 pt. play, leaving the {2|1} play on the board, which White will take, for a combined local score of 0 + 1 = 1. Now suppose instead that White makes the 0.6 pt. play first, leaving a local score of 0 and the {1|0} play. Black will take that play, for a combined local score of 0 + 1 = 1. In this case it does not matter which play White makes first, the score is the same. Still, we may consider the 0.4 pt. play to be a mistake in the first sense.
OC, bots, as currently written, do not calculate the value of plays this way, so they don't make mistakes of this sort, except as viewed by humans.
As I have been writing this, I have come up with a third possible meaning of a fractional mistake. It involves the concepts of an environment and its temperature. What the environment is for any board depends upon the player. The environment consists of background plays, and there at least one play in the foreground. The player knows or assumes the following. There is a play in the environment that gains a certain number of points, T, on average, along with other plays that gain T or less, such that we may estimate that playing in the environment will add T/2 pts. to the current value of the board. T is the temperature of the environment, or ambient temperature. T/2 plays a similar role in the concept of the environment as komi does at the start of the game.
Now suppose that we have one gote in the foreground that gains T + 0.1, octal, on average, for each player. We may then estimate the result of making that play and then letting our opponent play in the environment as T + 0.1 - T/2 = T/2 + 0.1. But if instead we play in the environment and let our opponent take that gote, we make estimate the result as T - T - 0.1 + T/2 = T/2 - 0.1. That is 0.2 pts. worse, on average, than if we take the gote. So we may think that not taking the gote is a 0.2 pt. mistake. OC, we are talking about estimates and average scores, so it may not be a mistake at all. Also, note that the error associated with the estimation of the final score is associated with the environment, not with the small difference. That error is on the order of T/2. Yes, if we have a lot of small errors, our estimation of their sum should be approximately normal and dependent on the size and number of the errors, but our uncertainty about the environment trumps that of the size of any particular small error. Besides, bots do not think this way, either. But humans might.
Another sense of the size of an error is the eventual difference in the score, given the following assumptions. Suppose that if you make play A. then Black can play to guarantee that the final score will be at least S for Black and White can play to guarantee that the final score will be at most S for Black, we may consider S to be the theoretical value of the game after you make play A. Following Berlekamp, I refer to such play as canonical. Similarly, if you make play B, the theoretical value is T. If S > T and you are Black, then play B is a mistake that costs at least S - T, and if you are White, then play A is mistake that costs at least S - T. OC, this is not what our discussants have in mind, because by both territory and area scoring S - T is an integer, not a fraction. But there is a form of go that has fractional scores, Chilled Go. (Environment, temperature, and chilled go all have explanations on SL, if not here.
)
In chilled go each move costs 1 pt. by comparison with territory scoring. In territory scoring each move costs 1 pt. by comparison with area scoring. So we may think of territory scoring as chilled go for area scoring.
If there are no complications by ko, correct play in chilled go is also correct play in territory go, which is also correct play in area go. It is quite possible, then, that the theoretical value of a go game is 7 pts. for Black in all three forms of go. For simplicity and convenience I will make that assumption. A perfect player playing Black can guarantee a final board score of at least 7 pts. for Black and a perfect White can guarantee a final board score of at most 7 pts. for Black.
Let me show how fractional scores can arise in chilled go. Here are the two positions we looked at before, {1|0} and {2|1||0}, using territory scores. In chilled go we subtract a point for each move, so {1|0} becomes {0|1} and {2|1||0} becomes {0|1||1}. {0|1} = 0.4, octal, and {0|1||1} = 0.6. Together they add up to 1.2. That's the same as their average value by territory scoring and by area scoring, as well. It's just that neither player wishes to lose a fraction of a point in chilled go by playing in either position. (This is also discussed on SL and here.)
Now let's suppose that 7 pts. for Black is the game theoretical value of chilled go and that we have a perfect player for it. Suppose that this player is Black, and White is almost as good. Then if White makes a 0.2 pt. mistake and otherwise each player plays canonically, the final chilled go score on the board will be 7.2. We do not have to rely upon estimates, we can take the final score to evaluate fractional errors. In this case, what would the final territory score be? That depends on who got the last play in chilled go. If it was Black, then White can round the 7.2 score down to 7, and if it was White, then Black can round the 7.2 score up to 8, which will usually round up to 9 by area scoring. That is true for any fractional value, with no kos. 7.2 is just as good or bad under territory scoring and area scoring as 7.8.
Now, I am not suggesting that bots switch to playing chilled go. If ko makes headaches for territory scoring, think of the headaches it will make for chilled go.
But if we are talking about near perfect play and fractional errors, chilled go offers a way of thinking about them.