Hello,
I've been thinking about the value of ko threats.

I've created a position where there is one ko, and several ko threats. I scored every line until the end of the game, in order to see what's the real value of each move (not the estimated value), and see what ko threat should be answered, and what ko threat should not.

This situation is very simplified. Here are its important characteristics :

• The ko is gote-gote. It means that winning the ko is gote for white, and gote for black.
• All ko threats are gote. It means that if the opponent doesn't answer the threat, the continuation is gote.
• Winning the ko leads to a stable sitation for both players. There is no continuation.
• The follow-up of a ko threat (when the opponent doesn't answer the threat) has no continuation.
• There is nothing else to play than the ko and the continuation of the threats. Experts say that the "temperature is zero". It means that having the sente at the end of the fight is worth nothing.

In the game below, the numbers in the comments give the final score of black minus the final score of white after perfect play.

The "swing value" of the ko is 15 points gote-gote (difference between the local score after white wins the ko and the local score after black wins the ko).
The biggest ko threat's swing value is 10 points gote-gote (difference between the local score after the defender answers the threat and the local score after the threatener's gote continuation). The next ko threat is 8 points, then 6 points gote-gote.

I found that for this 15 points ko, the 10 points ko threats must be answered.
The first 8 points ko threat must be answered. The second might be answered or ignored.
Both 6 points ko threats must be ignored.

The loss for ignoring the first 8 points threat is very small.
The loss for answering the first false 6-point threat is very small.
the loss for answering the second false 6-point threat instead of winning the ko is very big, because there are no more ko threats, therefore when Black wins the ko, White has nothing in compensation.

Maybe compare with this?
https://senseis.xmp.net/?KoThreatRuleOfThumb

In non-ko endgames, such as one local endgame with one or both players' simple follow-ups in an environment of simple gotes, decision-making differs for the early versus late endgame. Nevertheless, there would be some approximative relation for every class of positions. I suspect the same for ko endgames. Currently, you study the late endgame. Expect any findings to differ somewhat for the early endgame. In general, if you do not start with an ideal environment of constant drops, you might study arbitrary values as variables. IMX, ideal environments are already difficult enough. Partly Bill and more so I have solved them (approximately for the early endgame). It would be surprising to find that ko would ease environements with arbitrary drops. If you are a dreamer, study them immediately. If you are realistic, start with ideal environments.

Thank you !
It works perfectly in the position that I posted.

The following conditions for Bill's rule of thumb are met:

• we have a simple, ordinary ko with 3 plays between a Black win and a White win,
• we can ignore the local plays left after a win.
• when a ko threat is played it leaves a (relatively large) gote and we can ignore the local plays left after the gote is played

In the position posted, K=15, T=0.

After move 2, H0=10 and M0=10.
Is K superior to H0+M0-T ? No, 15 < 20. The rule say that we must answer the threat. That's the correct answer.

After move 5,
15 < 10+8-0. White must answer. -> correct.

After move 8,
15 < 8+8-0. Black must answer. -> correct.

After move 11,
15 > 8+6-0. White must win the ko. -> here, it doesn't make any difference.

After move 14,
15 > 6+6-0. Black must win the ko. -> correct

After move 17,
15 >> 6+0 (that's the last threat). White must really win the ko. -> correct, it really makes a big difference.



The case studied here is not very difficult. It is the case with T=0, and all threats are neutral threats played in the opponent territory (i.e. they are not endgame sente sequences, and the local score doesn't change before and after they are played and the opponent answers them).

It seems that Bill's formula works well. At least in this position, and within the approximation that the territory score is nearly equal to the aera score.

The case of the ideal environment seems harder to me. The modern counting method obviously works well as long as any ko threat leaves an endgame position (with the choice left to the opponent to answer or not) of the exact same temperature as the environment. But does it still work if the threats are bigger ? I mean, the best move would it still be the one with the highest value ?

Your position may be simpler but you do not want to only study it, don't you? I think you seek some general insight.

Most ko fights are exchanges. Presumably the "theory" of exchanges, if we can call it that, is less confusing. In this case each move in the ko is 10 2/3 gote or 5 1/3 miai value, if we are using area scoring. When one player gets to make two moves that make 5 1/3 each then the other player needs to make up 10 2/3 points or the exchange is a gain for the first player. We need lot of assumptions to call one value the correct value for the ko threat if we are not speaking about a specific position. Fortunately, there is the good advice from Rob van Zeijst's on the SL page addressing a common case.

Let's say White made a 6-point ko threat. The correct answer for Black ignore it and fihish the ko. How to apply the formula in this case?

Assume each player has a 10t, a 8pt and a 6 pt ko threat, black takes the ko and White makes a 6-point ko threat.
Here, H0=M0=10, H1=M1=8, H2=M2=6.
Since K=15 > H2+M2=12, Black must take the ko.

The tree exploration confirms that taking the ko is the best move : Black wins +17.5.
Answering the 6 points threat, Black wins +15.5 only.

I'll try and see what happens if Black has two large threats and White two small threats.

I have setup positions where Black has two large ko threats and White has two small ko threats.
Here are the conclusions of the tree exploration based on real scores, chinese scoring.

The ko is still the same : 15 points of swing value (japanese counting), no follow-ups, nothing left to play after the ko fight.
The threats are gote.

Black threats : 10, 10
White threats : 6, 6
a) White plays the first 6 points threat : Black must answer
b) Black plays the first 10 points threat : no difference if White answers or takes the ko
c) White plays the second 6 points threat : Black must answer
d) Black plays the second 10 points threat : White must ignore and take the ko

The explanation of d) is that there is nothing left to play. Therefore it doesn't matter if it takes one or two moves to win the ko, the opponent will get nothing while we play the second move. Thus, in the classical way of counting endgame moves, the values of the ko (15) and of the threat (10) should be directly compared. In the modern way of counting, it is as if the tally of the ko had fallen from 3 to 2.

The explanation of c) is that if White wins the ko, Black has prevented the 6 point threat and executed another 10 point threat in exchange. Thus 10+6 is a good exchange for a 15 point ko.

The explanation of b) is too complicated for me.

The explanation of a) is that if Black has some reasons to believe that he may win the ko whatever happens, then it is better to do so without leaving anything in compensation for White, than to do it and leave 6 points in compensation for White. Why should White get 6 points if Black can prevent her from having them, and still win the ko ?
Of course this is only valid if there is nothing to play elsewhere. At some point, White should choose to play elsewhere if this play is worth more, in itself, than a tiny ko threat.


Black threats : 12, 12
White threats : 6, 6
White plays the first 6 points threat : Black must answer
Black plays the first 12 points threat : no difference if White answers or takes the ko
White plays the second 6 points threat : Black must answer
Black plays the second 12 points threat : White must ignore and take the ko

Black threats : 14, 14
White threats : 6, 6
White plays the first 6 points threat : Black must answer
Black plays the first 14 points threat : no difference if White answers or takes the ko
White plays the second 6 points threat : Black must answer
Black plays the second 14 points threat : White must ignore and take the ko

I have no explanation for the fact that in situation b), a ko threat of 10, 12 or 14 points are worth exactly the same in actual score.

Black threats : 16, 16
White threats : 6, 6
White plays the first 6 points threat : Black must answer
Black plays the first 16 points threat : no difference if White answers or takes the ko
White plays the second 6 points threat : Black must answer
Black plays the second 16 points threat : no difference if White answers or takes the ko

Black threats : 18, 18
White threats : 6, 6
White plays the first 6 points threat : Black must answer
Black plays the first 18 points threat : White must answer
White plays the second 6 points threat : Black must answer
Black plays the second 18 points threat : White must answer

At last the value of Black ko threat is enough to overcome the value of the ko.

Applied to the 20 situations above, Bill's rule of thumb never gives a bad advice.

I think I know.
In this situation, if white takes the ko, Black executes the threat.
If White prevents the threat, Black will execute another identical threat anyway. So it changes nothing in the end.

The working of a ko is incredibly complicated. I think that there are some very important basic principles embedded in Bill's rule of thumb.
So far, the most important concept I'm discovering is the importance of the question "what will the other player get in exchange for the ko" ?

In the kind of simple situation you describe, Bill's rule of thumb is not just a rule of thumb but a theorem. More precisely, suppose that locally, if Black wins the ko then Black gets K points and if Black loses the ko, Black gets 0 point. Let B1 ≥ B2 ≥ B3... be Black's ko threats and W1 ≥ W2 ≥ W3... be White's ko threats. Let k be the largest integer such that Bk ≥ K-Wk, and suppose it's White's turn to play a ko threat. Then

• Under optimal play from both players, the final score (for Black) is min(Bk, K-W(k+1)).
• Any ko threat Wj for 1 ≦ j ≦ k is optimal. Responding to the threat is optimal for Black.
• If White plays a ko threat Wj for some j>k, then ignoring the threat is optimal for Black and the final score is K-Wj.

The preceding statement can be proven by induction on the total number of ko threats of both players.