Komi on smaller boards

[shapenaji]

In the limit of infinitely many moves where the value of moves decreases slowly and steadily the komi is exactly half the value of the first move. But of course that disregards cases with sudden drops in temperature and parity related effects. But I think it should still be around half the move value.

Well, if we plan on using a single komi, then temperature and parity effects don't matter. If there exists a feasible game without these effects, then its komi should be the same as a game WITH those effects

[Bill Spight]

In the limit of infinitely many moves where the value of moves decreases slowly and steadily the komi is exactly half the value of the first move. But of course that disregards cases with sudden drops in temperature and parity related effects. But I think it should still be around half the move value.

Well, if we plan on using a single komi, then temperature and parity effects don't matter. If there exists a feasible game without these effects, then its komi should be the same as a game WITH those effects

Under territory scoring the temperature of the empty 3x3 board is 8. That is also proper komi, although the expected komi for a board with temperature 8 (knowing nothing else) is 4. The difference is the result of the huge temperature drop after the first move.

We do not know the temperature of the empty 19x19 board, but, based on a komi of 7, we can guess that it is around 14. Perhaps it is higher, because the four corners can be treated as miai.

[Li Kao]

Under territory scoring the temperature of the empty 3x3 board is 8. That is also proper komi, although the expected komi for a board with temperature 8 (knowing nothing else) is 4. The difference is the result of the huge temperature drop after the first move.

Can you explain why the temperature is 8, is temperature different from the value of a move? If I calculate the value of the move as score-with-that-move minus score-when-passing I get 16. And the same if I use coupons.

When thinking a bit more about the problem I came to the conclusion that since the initial board is invariant under color swapping the correct komi is always half the value of the first move. The sudden temperature drops affect the value of the first move and the komi in the same way.

[Bill Spight]

Under territory scoring the temperature of the empty 3x3 board is 8. That is also proper komi, although the expected komi for a board with temperature 8 (knowing nothing else) is 4. The difference is the result of the huge temperature drop after the first move.

Can you explain why the temperature is 8, is temperature different from the value of a move? If I calculate the value of the move as score-with-that-move minus score-when-passing I get 16. And the same if I use coupons.

When thinking a bit more about the problem I came to the conclusion that since the initial board is invariant under color swapping the correct komi is always half the value of the first move. The sudden temperature drops affect the value of the first move and the komi in the same way.

The value of the empty board is 0, by symmetry. On the 3x3 B1 gains 8 points. That is the temperature, which is how much the largest move gains.

Suppose that there is a board such that the first move gains 8 points, leaving a play that gains 2 points. The game tree looks like this, with / indicating a Black play and \ indicating a White play.

Code: Select all

                       A
                      / \
                     /   \
                    /     \
                   B       C
                  / \     / \
                 9   7  -7  -9

The temperature of the board is 8, the komi is 7. After the first move the temperature drops by 7. The size of the temperature drop does not affect the size of the first play.

[RobertJasiek]

We do not know the temperature of the empty 19x19 board, but, based on a komi of 7, we can guess that it is around 14.

Why twice the komi value? Are you speaking of local temperature here? This would express the per move value (miai value). I thought that it would be one time the komi value.

Suppose the game would be over after just one play. Since the komi is supposed to be fair, the score must be 0 then. 7 komi points for White versus 7 points for Black gained by the one play. That is, the per move value was 7 points - not 14.

What do I not understand?

[Bill Spight]

We do not know the temperature of the empty 19x19 board, but, based on a komi of 7, we can guess that it is around 14.

Why twice the komi value? Are you speaking of local temperature here? This would express the per move value (miai value). I thought that it would be one time the komi value.

Suppose the game would be over after just one play. Since the komi is supposed to be fair, the score must be 0 then. 7 komi points for White versus 7 points for Black gained by the one play. That is, the per move value was 7 points - not 14.

What do I not understand?

Why twice the komi value? There are two extremes. One extreme is that the first play is the last effective play (as with the 3x3 and 5x5 boards). In that case the komi (starting from an empty board) is how much the first play gains (the temperature). The other extreme is that the game is miai, with a score of 0. That leads to an estimate of komi as half the temperature.

Edit:

The error, of course, is rather large. On the 19x19, however, the estimate of the value of sente as one half the temperature is quite good. The main reason is that the temperature drop between plays is normally small. Let us assume that komi is a proportion of the temperature, and that the temperature drop is small. I. e.,

k(t) = s(t) = a*t

The komi when the temperature is t equals the value of sente when the temperature is t, which equals a times t.

k(t0) = t0 - s(t1)

The komi when the temperature is t0 equals t0 minus the value of sente for the new temperature, t1.

a*t0 = t0 - a*t1

By substitution

a*t1 = (1-a)*t0

a/(1-a) = t0/t1

If the drop in temperature is small, then t0 approximately equals t1, and so a/(1-a) approximately equals 1, and so a approximately equals 1/2.

[palapiku]

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