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This is a (shall we say muddy) attempt to address Kirby's question of how to play correctly in the endgame, and at the same time to clarify endgame terminology for non-specialists (me).
At the beginning of a go game, neither player has any points. At the end of the game, each player has made moves that have secured points. A game is won by making more points than your opponent. Many moves do not secure points. Instead, they create possibilities to do so. With each move, we attempt to increase our potential to make more points than our opponent.
As the game progresses, opportunities arise for positions to be finalized and points to be secured. In order to decide where to play in this phase, we need to know how much a move gains from what we already have. But if the postion is not finalized, how do we know what we have? For this, endgame specialists have devised a way of calculating the points each player can expect from an incomplete position. This calculation, called the count, depends on whether or not the position is sente for one player. If it is, then the count reflects the fact that we expect the sente move to be played. If the move is gote for both players, the count is an average of the locally available points.The count should be distinguished from the score, which is the points of a finalized position.
As play progresses, the players alternate making moves which can secure points and/or narrow the options for securing points. The point value of these moves can be calculated by adding the secure points made to the average of the remaining potential (as Mitsun demonstrated above). This however is different from what a move gains. This is important because the gain is relative to the expectation (the count), and also comes at a cost, namely the cost of the stones necessary to make that gain. This cost is expressed by endgame specialists using the local tally, which shows whether one side would spend more in a local exchange.
The local tally is the difference between the number of stones each side would play if white went first and the number of stones each side would play if black went first. If an exchange is sente for one side, but not the other, then the cost of the move is greater for the side needing gote, and the local tally expresses that cost numerically. This allows the the cost of the stones can be included in the calculation of the move's value. The local tally will be 2 if the position is equally advantageous for both sides, and 1 if it is sente for one and not the other.
This value is called miai value, and is not to be confused with the miai value of the count. Both are determined by averaging the remaining possibilities, but in the first case we are calculating the value of a move, and in the second case we are calculating what each player can expect on the average from a position. The calculation of a move's miai value is as follows:
M(iai value) = C(ount)/T(ally)
In simpler positions, it is possible (and perhaps even preferable) to calculate what a move gains - its miai value - without the help of the above equation. The simplest such position is one in which the move is the last move to be made in that exchange, thus finalizing both the local position and score. In this case, the miai value is the point difference between a white move and a black move divided by 2. Its divided by 2 because what it gains is not the points made, but rather the difference to the count - which as we recall was the average of what either side could make locally.
The second simplest position is the one explained by Mitsun, in which a move clarifies the score and also leaves two possible further outcomes. In this case we add the score change of the first move to the average of the two outcomes of a follow-up move. This again gives us points, but not miai value, which again is half of the points.
So now that we have a way of determining the value of moves, the question remains, when to play them. Common sense would dictate that larger moves be played first, and in most cases, this is correct. There is however an interesting exception to that rule. As players alternate plays, we can assume that the miai values roughly cancel each other out. The last move however cannot be answered and as such gives the player who can make it an inherent advantage. This fact can sometimes be exploited due to the existence of asymmetrical positions such as the one in the first post. Asymmetric refers to the fact that if one player gets the move, a follow-up move is available, whereas if the other player plays it, no follow-up exists. By calculating the values of the remaining moves, and jockeying to get that last move, a predicted outcome can (at least theoretically) be upended.
I'd like to apologize for this post in advance, because I am sure it contains numerous inaccuracies and could easily be construed as an example of those who can't do trying to teach. Its just the result of a day's struggled thought while digging in SL, and a night's deep glass (which accounts for the disorderly stones above). I'm ready for my test now. Please make it easy.
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Patience, grasshopper.
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daal
if none of the local moves affect it in any way?
Been staring at a screen too long today. So the local tally is calculated from one point of view (black's) by comparing the difference in each scenario thusly?: