Basic endgame theory
The endgame is the phase of the game where the life & death status of many groups has been settled. The focus of the game shifts mostly to the borders of territory.
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.
In this article we first give a basic introduction into 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. Be aware though, that no endgame theory can beat actually reading all the possible combinations to the end of the game.
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.
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 out 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.
Swing counting
There are several ways to estimate the value and urgency 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 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 stones
- If White plays first, Black does not get the marked points, White gets the marked point and stones
So the difference between Black playing first and White playing first, the swing value, is 6 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?
Now that we have established an easy way to evaluate moves, it would be tempting to use it indiscriminately to choose moves. 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.
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 6 point move with --, then Black takes the 3 point move with --, then White takes the 2 point move with --. So White got 8 points of endgame, while Black got 3, for a net result of 5 points for White.
Can we do better? Assuming Black doesn't ignore any moves of White that contain a follow-up, then yes! Here's another order of play.
White takes the 3 point move with ---, then White takes the 6 point move with --, then Black takes the 2 point move with --. Now, white got 9 points, while Black got 2, for a net result of 7 points to white. So despite choosing a smaller move first, white made 2 points more.
Why is that? Because the 3 point move is sente for white. So white can play it, and then immediately choose another endgame play. So the two point gain was the result of playing sente and gote moves in the right order.
Sente and Gote
Lets look at sente and gote in a little more detail. With the above three examples, we can say that:
- Example 1 is gote for White, and gote for Black
- Example 2 is sente for White, and gote for Black
- Example 3 is gote for White, and gote for Black
Instead of using the specific colors, it is more interesting to look at it from the point of view of one of the players. After all, you are playing only one of those colors. In general, we can differentiate four types of endgame moves:
- Moves that are sente for us, and sente for the opponent
- Moves that are sente for us, and gote for the opponent
- Moves that are gote for us, and sente for the opponent
- Moves that are gote for us, and gote for the opponent
Those four cases are generally dubbed "Double sente", "Sente", "Reverse Sente" and "Gote" respectively.
Play sente first
A fairly effective approach, is to play moves in the order given below:
- If there are endgame moves that are sente for you, play those. Those with the largest follow-up first.
- Otherwise, all moves are gote for you so play the largest swing value first.
This approach is a lot more effective than the first approach of playing the largest swing value regardless of sente or gote.
Note that it is very important here to realize, again, that sente depends on the size of the follow-up. If the follow-up to a sente move is smaller than the biggest gote move, then it is not truly sente!
Moves that are sente only when the follow-up is large enough are not sente yet when there is a larger gote move. This means that we can keep moves that are not sente yet in reserve for when they become sente. So we play moves that are currently sente first, but if there is a gote move which is worth more than the follow-up of an ignored sente move, then that comes first.
Next we will look in more detail at the effect of sente, gote and follow-ups on the value of moves.
Average gain (miai counting)
Note: it is highly recommended to watch the Kyle Blocher's lecture on Miai counting before reading any of this. Link can be found at Miai counting page.
Since it may be difficult to judge if a move is sente or gote - see "What is Sente?" - to adjust the swing value, we could also take a look at the difference in the number of stones played to achieve the swing value. If a move is gote for us, we will have to invest 1 stone more than the opponent in order to reach the end position. This means we invest 1 stone to get our points. If the move is also gote for the opponent, then they would also invest one stone to get the endgame points. So the difference is two stones. This total, of black investment and white investment combined, is called the local tally. So in simple double gote, the tally is two, in sente or reverse sente it is one, and in double sente it is zero.
If we divide the swing value by the local tally, we get a value which indicates how much was gained, on average, per stone. Instead of trying to judge the "sente-ness" of the move, we can calculate scores for different variations, including playing elsewhere, each time dividing by the local tally and find a maximum among the variations. This way of counting is known as miai counting.
Value of a position (the count)
To be able to use miai counting, we also need to know the value of a position, which is an average value of local endgame outcomes.
Lets take a look at a simple example:
Here, we have a simple double gote endgame. The swing value is exactly 1 point.
What is the count in this position? The count is the average value of the position. If black plays first, black makes one point locally, marked . If white player first, no point is made there. Since there is 1 point in 2 positions, the count is 0.5 (Black gets half a point, on average).
The miai value of moves
In this corner, there are four endgame moves left, a through d. Each of them is exactly the same.
Since the count (remember: average value) of each of these is 0.5 points for black, this means that with four of them, the count is 2.
The local tally for each of them is two (one more black play if black plays first, plus one more white play if white plays first), and since the swing value is 1, this means that the average gain, per stone is 0.5 points.
Lets see if that is correct.
Here Black has made four moves locally, and has thereby scored 4 points (the marked intersections). Since the initial count was 2, and each play was said to gain 0.5, the result is consistent with our calculated miai value for these moves.
Here's the same position, but this time Black played three times and White played once. From the initial 2 points, Black has gained 1.5 and White 0.5, for a net gain of 1 to Black. The result is three points, which is what we observe (the three marked points).
What this means for the order of moves
Armed with this new knowledge, we can take another step at determining the correct order for endgame moves. Given the average gain (miai value) of all moves, we could play them in order from largest to smallest.
What does this mean for our earlier approximation of playing in the order double sente, sente, reverse sente, gote?
We can immediately notice is that this order has a declining local tally. Double sente moves have a local tally of zero, sente and reverse sente have a local tally of one, and double gote has a local tally of two.
Since determining the miai value of moves involves dividing by the local tally, what does that give us?
Well, for double sente moves we're dividing by zero, which is a strange thing to do, according to mathematics. Effectively we can say that because of the division by zero, the value of a double sente move is infinite. So always play double sente moves immediately!
We can also see that both sente and reverse sente have a local tally of one. This means that playing a sente move or preventing one of the opponent are equally valuable.
Finally, since double gote moves have a local tally of two, we can give the rule of thumb: sente moves are worth twice as much as gote moves.
Note that this is still only a rule of thumb, because we need to take the value of the follow-up of sente moves into account. Lets see if our new knowledge of count, local tally and miai values can give us fresh ideas on this concept.
Evaluating branches
Earlier, we showed that this position has a count of 0.5 (it is worth 0.5 points to Black, on average), and playing in it has a miai value of 0.5, since it gains 0.5, making the value of the position either 0 or 1.
Here's three ways in which the position can play out. Black gets either two, one, or zero points here. But not every one of these variations is equally likely, so how do we calculate the average value?
Here, our knowledge of the earlier position comes to the rescue. Lets consider two positions:
If black plays first, black gets 2 points. This is easy and straightforward. But what if white plays first? Well, the position after is equivalent to the position we showed earlier. So when white plays first, the resulting position is worth 0.5 for Black on average.
So now we have two positions with exact values. One is worth 2 points, the other is worth 0.5 points. That means that the initial position is worth, on average 1.25 points. The count is 1.25 points.
So what is the miai value of a move here? Well, given that the count is 1.25, and Black can move to a position with a count of 2 using one move, that move is worth 0.75 points. Similarly, white can move from a position worth 1.25 points to a position worth 0.5 points, which also gains White 0.75 points.
Exercise:
Given the above knowledge, can you show what the count (average value) of this position is? Can you show how much a move by black or white is worth? (miai value). Answer
See also
[1] From Toshiro Kageyama: Lessons in the Fundamentals of Go
[2] Answer to the exercise: The swing value is 9 points (eight points lost plus one prisoner).
[3] Answer to the exercise: Black first gives a position worth 3 points, White first is the position from above worth 1.25 on average. So the position is worth 2 1/8 (=4.25/2) on average. A move by black or white is worth 7/8 of a point.
Question:
I do not understand any of this. Where does the 7/8 come from? Should not it be 2.125/2? And where does 0.75 come from? Impossible to understand.
Answer:
The average gain for a move is the difference between the value of original position and the value of the resulting position. That gives us 3 - 2 1/8 = 7/8. Or equivalently, 2 1/8 - 1 1/4 = 7/8. For more, see miai value.