Komi is compensation for white having the first move. The Japanese justify their 5 points komi by saying it is half the value of the first move. How many points is it really worth? Why half the first move value?
The value to black of having the first move can only be estimated by probabilistic means. This assumes that different games have different numbers of moves of each particular size. The first move has a size, which should be the same for each game, which has the value n.
Assume the size of the first move available decreases monotonically from n to 0. (That means that the first move is worth n points and the size of a move is never greater than the move before it). The game ends when only moves worth 0 points are left. For each size of move there are either an even or odd number of moves of that size. If each size has an odd or even occurence with equal probability the komi should be n/2. (Note that with Chinese counting the size n is 1 greater than with Japanese so the komi should be 1/2 point greater.)
This 50% probability may not be correct for the first few moves. For example with the current habit of playing first in 4 empty corners there are an even number of these and the size of the komi should be half the size of the 5th move. This analysis could be extended if we could say exactly how many moves there are with size n-1.With perfect play the numbers of moves having a particular size is fixed and the correct komi could be anywhere between 0 and n.
Note that if the best first move were at tengen and then there are an even number of moves of each size after that then komi should be n (ie the value of the first move). However if black plays at tengen and then mimics white's subsequent moves he should win by 1 point on the board. This is because white's move after black has played on the centre point are actually bigger than the same moves for black. They have the effect of nullifying black's thickness.
How big is the first move?
The size is not the same as the size counted for yose moves. In estimating yose we look at the difference between our move and our opponents so the size calculated is actually twice the actual size. The same is true of capturing stones where the actual size is just the number of captured stones plus 1.
I can think of 2 ways for estimating the size of the first moves:
It seems that the correct komi calculated as half the size of the first move is at least 9 points and probably 10. the only way to check this is to get the top professionals to play on larger komis and see what the percentage wins for black are. The results of Ing's tournament, played with 8 points komi seemed not to favour black. A further analysis of pro-pro handicap games could indicate the true value of the different moves. Of particular interest would be the value of the centre point which should be the largest of all.
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