Keywords: Question

This is a Discussion of a Ko and its Value.

Starting Position

Diagram 0  

Major Local Strategies

1) If Black starts and wants a seki, then it is a seki in sente for Black.

2) If Black starts, wants a "0-ko capture-first White", and wins it, then the white semeai string is dead.

3) If White starts and both want a seki, then it is a seki in sente for White.

4) If White starts, plays the "1-ko with White approaching, flower-viewing for White, capture-first White, 1 local threat for Black", and wins it, then the black semeai string is dead.

5) There are some more variations with unusual tenukis.

First Questions

Does it make sense to assess the size of the ko with a single number or should each case provide a separate number? What is the size of a move in each case? Does it make sense to compare local move sizes with numbers of tenukis (incl. the first sente play, if any)? Which methods of counting are useful to assess which kinds of moves here? Should one introduce an abstract environment or calculate temparatures? How to compare the ko with other kos or with "ordinary tenukis"?

I see too many possibilities to introduce numbers here, but which are really useful in practice?

Counts of Stable Positions

The area counts[1] of some (rather) stable positions are determined. These are the most interesting positions that can occur after some sequence starting from the starting position. Later the area counts can be used to characterize it.

Diagram 1  

area count = 13 - 13 = 0

Diagram 2  

area count = 20 - 9 = 11

Diagram 3  

At some point, Diagram 3 will be converted to Diagram 4 when the exchange will be made. So both have the same area count.

Diagram 4  

area count outside the ko = 8 - 20 = -12

area count inside the ko[3] = 2 - 4/3 = 2/3

area count = 8 2/3 -20 = -11 1/3

Typical Ko Threat Environments

To evaluate kos we have to take the ko threat situation into account. It would be possible to assess a ko purely locally without considering the global context, but in practice it is much more interesting to know how a ko behaves in a global environment of ko threats. Therefore the global view is studied here. Calculated values have a meaning in the global context of a ko threat environment on the rest of the board.

Neither Player Has a Ko Threat

The position will become seki as in Diagram 1 and then the count is 0.

White is Komaster and Black to Play

Black can fill the ko with sente[2]. The position will become seki as in Diagram 1 and then the count is 0.

It is sente because Black's threat to capture White's stones is larger than the ko. Filling the ko is Black's privilege.

White is Komaster and White to Play

White to play can win the ko following the local strategy (4). This results in Diagram 3 (or the equivalent) in 3 net plays.

Diagram 3  

(repeated for convenience)

Since the area count of Diagram 3 is -11 1/3, each play gains on average (11 1/3)/3 = 3 7/9, which is the miai value of the play.

"net plays" is the "absolute difference of number of black stones played locally minus number of white stones played locally in this particular move-sequence".

Diagram 4  

(repeated for convenience)

The area counts of Diagram 3 and Diagram 4 are the same because White's capture of Black's stones and Black's taking the ko are miai.

Black is Komaster and Black to Play

Using the local strategy (2), in 2 net plays Black can reach Diagram 2 or its equivalent, for an area count of +11.

Black is Komaster and White to Play

White can play W1 in Diagram 5.

Diagram 5  

Now the result will be seki[4], which White can force as in Diagram 6.

After W1 in Diagram 5 Black no longer threatens to capture White's stones. The best Black can get is seki. White's threat is still to capture Black's stones for an area count of -11 1/3 (as in Diagram 4). As we already know, there are 3 net plays between these results, and a play in the ko is worth 3 7/9 points.[5]

Since we are 1 Black play away from 0, the area count at the start of Diagram 6 is -3 7/9.

Diagram 6  

B4 = ko threat, W5 = reply, B6 takes ko, W7 elsewhere, B8 fills ko.
White gains something from W7 elsewhere, so White has the incentive to play the ko, even though Black wins it. Black is quite happy to wait until W7 is worth nothing.

Diagram 7  

What does it mean for further calculations of values if Black plays first after Diagram 5, i.e. B1 in Diagram 7, to make a gote seki with the area count 0? Since just after Diagram 5 the area count is -3 7/9 and after Diagram 7 it is 0, in one net play Black achieves an improvement of 3 7/9.

Thus regardless of whether White or Black moves first just after Diagram 5, the result seki and the miai value of a play are the same.[6]

So far we have determined the area count of the position just after Diagram 5. Now, in the current case of Black being komaster and White to play, what is the area count of the original position (Diagram 0)?

In 2 net plays from the original position Black reaches an area count of +11 (Diagram 2, until which Black has spent 3 plays while White has spent 1 play). In 1 net play from the original position White reaches an area count of -3 7/9 (just after Diagram 5, i.e. just before Diagram 6; until that moment Black has spent 0 plays while White has spent 1 play). Each play is then worth (+11 - (-3 7/9)) / (2 - (-1)) = (14 7/9)/3 = 4 25/27, about 4.93.

For the currently discussed case, we now know the miai value of a play: 4 25/27. We still do not know the area count of the original position. However, finally we can get it: In 1 net play[7] worth 4 25/27 by White a position worth -3 7/9 is reached. Thus -3 7/9 + 4 25/27 = 1 4/27, about 1.15, is the area count of the original position.

Furthermore we can observe that the value of a local play is lowered from 4 25/27 to 3 7/9.[8]

Interpretation and Application

Things work like for non-ko values. The calculated numbers are a guide (not infallible) to when to play the ko. Normally it is right to make the largest play.

If the position is associated with

  • no ko threats, then the position becomes seki,
  • Black komaster and starting, then when the ambient temperature is about 4 25/27, Black captures White's stones,
  • Black komaster and White starting, then when the ambient temperature is about 4 25/27, White fills an outside liberty to save the White stones,
  • White komaster and Black starting, then Black makes seki with sente before the ambient temperature drops to about 3 7/9,
  • White komaster and starting, then when the ambient temperature is about 3 7/9, White takes and wins the ko.

Ko with threat

Diagram 10  

Bill: Komaster analysis is abstract, so I thought I would take a look at an example with a concrete ko threat.

Black has a six point ko threat, which is hotter than the ko. Is that enough to make Black komaster? Not even close!

Diagram 11  

B9 takes ko. W10 elsewhere. B11 fills ko. W12 elsewhere.

If Black plays first White ignores the ko threat, but then Black can take and win the small ko. The final local area score is -2.

Diagram 12  

If White plays first W1 forces seki. The final local area score is -4.

Black gains 2 points in 2 net plays. The miai value of a play in this ko ensemble is 1 by area scoring (the same as a Japanese dame).

Since White ignores Black's threat, Black is not komaster. OC, neither is White.

Additional Notes

RobertJasiek: What exactly is a "ko threat"? Presumably it is not the same as "any tenuki or any local ko threat". I guess it depends on sizes of potential ko threat moves for them to be threats, but which sizes?

Bill: As I pointed out in my multiple kos paper, the evaluation of plays in go, including kos, may be derived from play in an idealized environment (the rest of the board) consisting of simple gote. In such an environment, simple sente may serve as ko threats. You may also just give a player the right to take a ko once (or some finite number) of times.

In real life things are more complex. But we need the theory to deal with the complexity.

Do we calculate the miai value as count/#net_plays and is this so in general, provided the number of net plays is not 0? Can we compare kos with 0 net plays with kos with non-0 net plays at all?

It's the difference in count between stable followers. If the number of net plays is 0, the position has a score, and the miai value is -1 (by territory scoring, 0 by area scoring).

Final Questions

RobertJasiek: Now, how can we apply and interpret all these calculated figures in practical play?

Bill: The values of non-ko plays provide a very good guide to play. As a rule the biggest play is best. Kos are different. If they are placid, their value is a good guide. But if they are hyperactive, as this one is, things are less clear. Still these calculations give some guidance.

Seki appears to be a common result for this ko. When White is komaster Black can make seki with sente. When Black is komaster, White to play can keep Black from doing better than seki. Also, the fact that White's play when Black is komaster reduces the size of a local play may be important information about how to play in real life.

RobertJasiek: Inhowfar should the miai values and area counts for the five major cases of global ko threat environments be treated separately or may be compared with each other?

Bill: I am not sure what you are asking. Perhaps you are dissatisfied with the abstractness of komaster analysis. It does make the results hard to apply to real games. Anyway the different cases apply to different whole board situations. Just what they are is not specified in detail.

RobertJasiek: I am not worried about abstraction at all. My question aims at something else. E.g., in particular we have the case "White is Komaster and Black to Play" and "White is Komaster and White to Play". For both the ambient temperature is 3 7/9 where some player should act locally. Can we draw such connections in general is it just here so that regardless of the player to move the relavent ambient temperature is the same? Is there any advantage or disadvantage of Black or White, respectively, playing first locally when they both should do so at the same temperature?

E.g., "White is Komaster and White to Play" and "Black is Komaster and White to Play". We have different miai values; 4 25/27 in the latter case. Does it make sense to compare the two cases? How? Or should we treat everything separately because there would be no useful relation? Of course not, but then what kinds of comparisons are useful between the two cases and what does this mean for global play until the temperature drops to 4 25/27 and then to 3 7/9, if not even simultaneously so.

Bill: Traditional evaluation of go plays does not depend upon who plays first, and that holds for the extension of the theory to ko situations. This is different from minimax evaluation, where who has the move is significant.

But there is a close connection between the two. For any position there is an ideal environment of simple gote in which the optimal minimax results differ by the miai value of the largest gote in the environment (the ambient temperature). You can use that fact to define the miai value of non-ko plays, and I did so in my multiple kos paper. For kos things are not so simple, but the value of the play does not depend upon who has the move.

RobertJasiek: Since there can be situations with some ko threats but neither player being komaster, is the case analysis sufficiently detailed or should we add more general cases?

Bill: The question is one of generality. There is an approach using a Neutral Threat Environment (NTE), for instance, in which each player has equivalent ko threats. It has produced some promising results, but is impractical because it requires algebra.


For an exact definition of "count" see

  • Berlekamp's "Economist's view of combinatorial games" chapter in Games of No Chance
  • Bill Spight: paper on multiple kos (exact title?)
  • Mueller, Berlekamp, Spight: paper on generalized thermography (exact title?)



Bill: I have adjusted the area count to include all stones and territory. Since we started with an equal number of stones on each side, it corresponds to the territory count, as well, for the original position. :-)


Bill: The question may arise, even if White is komaster, should Black settle for seki when playing first? Maybe it is better to try to take White's stones, and then get compensation elsewhere in exchange for the ko.

The answer to that is that with W2 in the next diagram White can prevent Black's threat, leaving seki as Black's best option. It yields a curious ko fight, where each player is trying to lose the ko. ;-)

White forces seki  

After W2 Black has nothing better than to win the ko with B3 to make seki.

RobertJasiek: Is there any theoretical chance where White would correctly play 2 at 3 while Black 1 was correct?


Each move in this ko is worth 4/3 points because there are 3 net plays between the stable follower 2 points (Black has connected the ko) and the stable follower -2 points (White has captured and then connected the ko). In one play (worth 4/3) Black can get a score of 2. This is one of the values we know since it is of a stable follower position. From both the value of this stable follower and the value per move it follows that the current count is 2 - 4/3 = 2/3.


Although Black is the komaster, we do not speak of "seki" yet, but we are one black move away from seki. After 8 in Diagram 6 we may speak of the position being "seki".


These two results are seki after one net black play more than white plays (Diagram 1 shows the result, the number of net plays is counted starting just after Diagram 5) or removal of the black semeai string after two net white plays more than black plays (Diagram 4 shows the result, the number of net plays is counted starting just after Diagram 5). The seki of Diagram 1 has the area count 0; the removal leading to Diagram 4 has the area count -11 1/3. The difference in counts is 0 - (-11 1/3) = 11 1/3. The number of net plays from just after Diagram 5 to Diagram 1 are +1 and from just after Diagram 5 to Diagram 4 are -2. The difference in net plays between both is +1 - (-2) = 3. Therefore we can calculate the value of a play in the ko just after Diagram 5 as (11 1/3)/3 = 3 7/9.


Why would Black play B1 in Diagram 7 at all? He might want to defend against the ko when White might later become the komaster. This possibility cannot be ignored entirely.

As for such shifting of komaster status, an analysis would require explicitly modeling how that shift would come about. The great advantage of komaster analysis is that we can evaluate kos without having to know the details of the ko threat situation. But not knowing the details means that we do not know the answers to such questions. The question of a change of komaster was debated among Berlekamp, Bill, and others in the mid-90s. The main problem comes when there is a discontinuity in the value of a play at the point when who is komaster shifts.

Bill's reformulation of thermography in '98 allows us to evaluate kos in ko threat situations where komaster might change. But even then there may be no single answer to the question of how to model a change of komaster status because there may be different ways that that could happen.


White fills an outside dame to save the stones and reduce Black's threat to making seki.


Bill: So local play stops, as a rule.

RobertJasiek: Why, exactly when?

Bill: After White fills one outside liberty, saving the White stones. Now that Black no longer threatens those stones, the value of a local play has dropped to 3 7/9, and normally the will be larger plays elsewhere.

RobertJasiek: So a "stop" is just temporary. The players wait until the ambient temperature will have dropped a little further again.

Bill: Yes, maybe "pause" is a better word when there is still play left in the position.

Page contents: Bill, RobertJasiek.

BQMRJ000 last edited by Dieter on July 5, 2008 - 12:49
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