Basic endgame theory
In the endgame the life and death status of large groups has been settled, and the board can be divided up into a number of independent or nearly independent regions. The focus of the game shifts mostly to the borders of territory.
Per Tomoko Ogawa, By the endgame the board has been more or less divided up into separate territories, and most of the fighting tends to affect only two of them, occurring at a mutual boundary. The opening and middle game are much like a single large battle between two armies; the endgame is like a number of smaller battles going on in different places simultaneously.
That doesn't mean that attack and defense are not important anymore, but rather that they are used more for immediate gain than for long term strategic goals. The life and death of smaller groups may be unsettled, particularly if they depend upon ko.
In this article we first give a short introduction to basic endgame strategy, focusing mainly on the importance of sente and gote. We will then go into more detail on precise evaluation of endgame moves, introducing a new concept of sente and gote in the context of the endgame and their effect on the value of moves. We will explain swing values, count, local tally and average gain values, and their application in evaluating endgame moves.
After reading this page you should have a pretty good grasp of the value of any single move, which makes choosing a good one quite a bit easier. If endgame theory does not simplify things for you, one can fall back to basic reading with the aim of optimising the score and imagined scoring at the end of every move sequence.
Table of contents |
Principles
Beginners often struggle with the endgame, especially against stronger players. They feel like they are being pushed around, points they thought secure vaporize before their eyes, and what they thought was a comfortable lead turns out to be a hopelessly lost position.
Struggle for Initiative
Initiative (or sente) is one of the most important concepts in go. Gaining and keeping it is as important in the endgame as it is elsewhere. To keep the initiative you have to play moves with a follow-up, that is open-ended moves. To gain it you have to ignore those of your opponent's moves which lack such a follow-up.
Beginners are often frightened about their territory and tend to follow their opponent around, because they overestimate the loss of not responding to moves that are seemingly sente. They can improve their endgame tremendously by simply considering alternatives to responding to every move the opponent makes.
Instead of letting yourself be pushed around, you can push back.
One prime example of this principle is the concept of mutual damage.
Mutual damage
Here's a simple example of the concept of mutual damage. In the initial position, on a 7x7 board, both players have claimed part of the board. Black can now choose an endgame move.
The second diagram shows the typical way in which beginners get pushed around. threatens white's corner. White instinctively defends the corner with
and
, then the same process repeats at the top.
But White can do better. Instead of defending against , White launches a counterattack with
. Now, Black has a choice. He can continue with his attack in the lower right, scooping out White's corner. But if he does that, White will counter by scooping out Black's upper left corner. Both players receive damage, hence the term mutual damage. In diagram three we can see that Black would actually take a larger loss than white. Black's corner is bigger and thus more vulnerable.
Instead of the third diagram, Black should settle for the fourth diagram, respecting White's counter-attack. In this diagram, both players have received small damage symmetrically, and the exchange is fair.
Defending where necessary is as important as not defending where unnecessary.[1]
What is Sente?
The first approach is to simply consider the local sequence of moves you expect, and then if your opponent needs to make the last move we call it sente, other wise we call it gote. Although correct in theory, it depends upon whether your opponent will indeed play the sequence you've envisioned. This depends on the value of plays elsewhere. See Value Of Sente.
Here's a sequence that seems to be sente for white. White plays first, black plays last, so sente for white.
But what happens if Black, the opponent, deviates from our planned line?
Here's a possible point where Black can deviate. Black decides to play elsewhere with instead of taking gote to defend the cut at a.
White cuts and devastates black's territory, the end result is quite drastically different from the earlier line we showed.
Compared to the original diagram, all the marked points are gone for black, and White captures 1 black stone.
The above devastation of Black's territory is painful, but Black got one move elsewhere (move ). If the gain made by that move was large enough, then Black may well have been right in playing elsewhere with
.
From the above, we can learn an important basic principle:
sente depends on the size of the follow-up.
Often, the size of the follow-up is unclear. The above example is relatively straightforward, but what if we remove one stone, like this:
With that stone gone, the size of the follow-up becomes a lot harder to calculate. After in the earlier follow-up diagram, the position was settled. In this position, white would be able to push up further.
This is a genuinely hard question, to which there is no easy answer. Even the most experienced players will, at some point, have to make an informed guess about whether a given follow-up is big enough to make a move sente. Here, experience is key.
Sente in the endgame
If we take another look at this diagram, we can identify that was sente. The follow-up, shown earlier in the mutual damage diagram, is large enough that black must respond.
But there is another concept of sente and gote in the endgame, which is important in evaluating moves.
A move or sequence in the endgame is said to be sente if its follow-up is larger than the value of preventing it. If you play such a move when the follow-up is bigger than the other moves on the board, then your opponent will want to respond immediately, and you will be able to keep sente, which means that you get to be the first to play elsewhere too.
We will ignore complex issues like ko for now. The above is sufficient to allow us to start tackling the question of what the value of endgame moves is, which we will examine next.
Other principles
Other principles about avoiding losing points, neutral intersections, premature endgame; taking more than the opponent; best global and local choices; reading; other topics avoid more than half of the practically ocurring endgame mistakes and can be found in the literature.
Swing counting
There are several ways to estimate the value of a move. We will first explain the basic principle of swing counting before we move onto more advanced concepts, such as the effect of sente and gote on the value of moves.
Swing counting, also known as deiri counting, estimates the value of an area of play by considering the swing, which is the difference between White playing first and Black playing first.
Examples
Lets have a look at some simple example endgames positions, and see what their swing values are.
First Example
From the given initial position, we consider the resulting position after either Black or White plays first. The difference between these positions is:
- If Black plays first, Black gets the
marked point, White does not get the
marked point
- If White plays first, Black does not get the
marked point, White gets the
marked point
So the difference between Black playing first and white playing first is 2 points. This is what we call the swing value.
Second Example
Again we consider the resulting position after either Black or White plays first. The difference between these positions is:
- If Black plays first, Black gets the
marked points, White does not get the
marked point
- If White plays first, Black does not get the
marked points, White gets the
marked point
So the difference between Black playing first and White playing first, the swing value, is 3 points.
Third Example
Again we consider the resulting position after either Black or White plays first. The difference between these positions is:
- If Black plays first, Black gets the
marked point, White does not get the
marked point and stone
- If White plays first, Black does not get the
marked points, White gets the
marked point and stone
So the difference between Black playing first and White playing first, the swing value, is 4 points.
This method of putting a value on moves is reasonably straightforward.
Exercise: In the earlier section on sente we gave an example of not defending a cut. Revisit that example and try to work out the swing value between Black defending and White cutting and devastating Black's territory. Answer
Is the Swing Value Sufficient?
There are a couple of problems with swing values. First, a naive approach would be to simply calculate the swing value of every move on the board and then play the biggest one. This approach fails because it fails to take into account the very important aspect of sente and gote that we introduced earlier. Second, as we shall see, swing values do not always add up correctly.
Suppose we take the naive approach with the above three examples. Suppose all three are on the board in different corners, and there is no other endgame.
White plays first and takes the 4 point move with -
-
, then Black takes the 3 point move with
-
-
, then White takes the 2 point move with
-
-
. Using these values we might expect that White got 6 of the points in play, while Black got 3, for a net result of 5 points for White. But when we count the marked points White has 3 + 1 = 4 points, and Black has 2 points, for a net of only 2 points to White instead of 3. The naive method gives us the wrong result.
But that is not what swing values measure. They measure the difference between the results of a Black play and a White play. To make that comparison, what do we get if Black plays first?