C0nfuseki/FindingCorrectKomi

In the game of Go, players take turns to place their stones on the board. Firstly, we need to ask, "is there an advantage of playing first?". Apart from the obvious strategy to pass if playing first is problematic, it seems logical that being the first to play does indeed assist in winning the game.

What are the advantages/disadvantages of the first move?

+ The first move gains an unchallenged board position.

+ The starting player is always either equal to the number of moves played or one ahead. (Therefore having marginally more choice during the entire game)

? It presents the start of the game for the second player.

- The first player reveals his/her goals and strategy first.

Solution to Komi?

Working out the correct Komi is a very tricky problem. To be absolutely certain that Komi is correct, the problem solver needs to work out the handicap White has on a perfect game. That is, a game where neither side made any errors. Please note that we are using the perfect game idea as a guide and not as an actual algorithm of playing go. (If an actual perfect algorithm solving go were to be made public, then the existence of komi would be useless because the maximum point advanatge would already be known... and as such, setting the Komi would instantly result in who would win the game.)

Analysis of Gameplay

In an ordinary game, Black's first move gains the advantage on the board. We can see that there are a few choices available which have varied levels of usefulness for the very first move. Remembering that we are only thinking about a perfect game for mathematical modelling purposes, we give this Black move a hypothetical value +(x).
When White responds, it appears that she regains equality of influence on the board. There are now an equal number of stones back oin the board again... yet evrything isn't exactly equal. For example, if Black played on the Tengen, then White couldn't play there. But since we are thinking about the perfect game then we can't _ on a specific move that might not be a perfect move. We can however, understand that the White stone is indeed subjected to the influence that the Black stone exerts on the board.^[1] We can give this White move a value of -(y).
The Black reply to the White move gives Black two stones on the board to White's one stone. Black combines the usefulness of the stones together to use the situation on the board so far to perpare for White's next move. (from an equavalent position to start off with) ^[2] A value of +(z) can be given to this move.

[1] Luckily for White, the White stone also exerts influence on the Black stone once played... but the nature of that influence may have already been pre-meditated by Black as the first move was played.

[2] White also had to think about the second Black move (Move 3). Therefore, White's very first stone had to do two things (counter Black's 1st and perpare for the next Black) while Black's very first move only needed to prepare for White's move.

Diminishing Influence Value

The usefulness of a move is generally observable when there are other stones on the board. So the importance of placing the initial stones, directs the game in both flexibility and dominance. Once the positions are layed out, the moves that follow, utilise the situation already on the board. The number of choices in the early middle game are astonishingly high. However, opportunities to change the influence flow on the board, are now limited to complicated tesuji which may be bad inefficent moves.
As the game continues and more moves are made, the combinations of moves get simpler and the variation of choices get smaller. The actual point differences on certain plays can be micromanaged at this point, with complicated computation designating only a single possible line of best play. The moves which have given these values have already been played. The players needing to now only play the stones on the board for the sake of them being on the board (rather than through being played in the mind).
In progressing this way, the role of a single move has started from that of being influential in planning for the game... to resulting in moves being played to simply make counting the endagame points clearer.

How much do stones influence the Board?

If we return to our notation of having the stones on the board defining an influence value, we can (in theory) provide a set mathimatical value of how much advantage of a certain colour has over the other one. (Seeing that Black goes first, I would assume that a positive value would indicate a measure of how well Black is doing. A negative number should likely be a value where White would be doing well.)
When looking at the Go board, a value could be given on the position of stones and dominance on the board. But such a method alone would lack taking into account the potential flexibility and moves that haven't been made yet. A game of Go isn't a stagnant diagram, but a growing situation that slowly changes. Thus, although the meanings of shape evaluation do matter, the dynamics of go playing is a more thorough analysis.

Black's 1st move +(x)

White's 1st move -(y)

Black's 2nd move +(z)

White's 2nd move -(a)

Black's 3rd move +(b)

White's 3rd move -(c)

Black's 4th move +(d)

White's 4th move -(e)

I believe that x > z > b > d
as well as y > a > c > e

( - 1 ). Black's next move plays yet another stone on the board and gets an advantage of 2 stones to 1... and thus the influence is less than it was than the first move made ( + 0.5 ) and White responds yet again canceling out the Black advantage ( - 0.5 ). Black's move after ( + 0.3333 ) and White's response ( - 0.3333 )... etc...

We will eventually come to a point in the game where the difference of one move is so small, that it causes no real difference in the endgame. (Besides the one extra point advantage if using the Chinese counting method). But what about the fact that the game constantly see-sawed in Black's favour? (As shown below:)

Move#:    || Board Value:
Black 1   || + 1
White 2   || - 1
Black 3   || + 1/2
White 4   || - 1/2
Black 5   || + 1/3
White 6   || - 1/3
Black 7   || + 1/4
White 8   || - 1/4
Black 9   || + 1/5
White 10  || - 1/5
...
Black N-3 || + 2/(N-2)
White N-2 || - 2/(N-2)
Black N-1 || + 2/N
White N   || - 2/N

The resulting sum of Board Values up to move N has a total of 0. Which can be interpreted as White continually nullifying Black's advantage. However, Black has the advantage in half the situations during the game. To correctly assess an evaluation of how much advantage Black gets, we need to take a snapshot of the game after each move. (As shown below)

Move || Black Advantage
 1   || + 1
 2   || - 0
 3   || + 1/2
 4   || - 0
 5   || + 1/3
 6   || - 0
 7   || + 1/4
 8   || - 0
 9   || + 1/5
 10  || - 0
...

If the last (White) move "N" is move 100, Black has an advantage of 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 1/10 + 1/11 + 1/12 + 1/13 + 1/14 + 1/15 + 1/16 + ... + 1/49 + 1/50
Which is equal to 4.4992053383294250575604717929648 units.^[1]

What does this calculating tell us? When both Black and White continue to play equally important moves, Black will gain an advantage on the board simply by being the first player to continually break the equalibrium.

Note: This makes no sense when we think about moves generating specific territory. (The outcome of the game relies on the difference of the territory and prisoners at the end of the game) These point valuations are based on the ability of making influence on the board. Unlike solid countable points, influence is vague and relies on stones elsewhere on the board to give it meaning. (Which is why there is an advantage by directing the game in a certain direction with no immediate benefit).

Alternative Go

If simply playing first gives an advantage, and since attributing this advantage into a hard-coded score penalty is an uncertain balancing act, it may be possible to handicap the player with an extra move on the board by penalizing them for a move at the same time.

I propose a system where the first player plays one stone on the board and then two stones are played by each alternating player from then on. This has the advantage of creating a genuine see-saw between the players. What would we expect the board values to be? (See the table below)

Move#:    || Board Value:
Black 1   || + 1
White 2   || - 1
White 3   || - 1/2
Black 4   || + 1/2
Black 5   || + 1/3
White 6   || - 1/3
White 7   || - 1/4
Black 8   || + 1/4
Black 9   || + 1/5
White 10  || - 1/5
White 11  || - 1/6
Black 12  || + 1/6
Black 13  || + 1/7
White 14  || - 1/7
White 15  || - 1/8
Black 16  || + 1/8
Black 17  || + 1/9
White 18  || - 1/9
White 19  || - 1/10
Black 20  || + 1/10
...

Using the similar move valuation system as before, it can be seen that Black has less benefit than previously modelled.

Move#:|| Black Advantage
  1   || + 1
  2   || - 0
  3   || - 1/2
  4   || + 0
  5   || + 1/3
  6   || - 0
  7   || - 1/4
  8   || + 0
  9   || + 1/5
  10  || - 0
  11  || - 1/6
  12  || + 0

The total sum of all moves up to 100 is equal to 0.68324716057591818842565811649003 units. This valuation is much closer to zero advantage -- but still isn't zero.
Which means that either:

• The first move still carries significant value in dictating the type of game to be played.

  -OR-

• The model by which moves are weighted is totally wrong.

I cannot be sure of the rate of attrition. The 1 to 1/2 to 1/3 to 1/4 ...etc seems to work roughly well for communicating the decreased usefulness of a move as the board fills up with stones. However, I have only guessed at this weighting method and have not analyzed the flexibility of Fuseki to come up with an accurate description of what is going on.