Ko Threat - Rule of Thumb

 Table of contentsBill's rule of thumb Whether to answer a ko threat or win the ko Book advice Derivation of formulas Example 1 Example 2 Credits and Other Links Table of diagramsKo Evaluation Example

Bill's rule of thumb

Bill Spight discovered the following rule of thumb in the early 1970s.

Say that

• 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.
• we let the point difference between a Black win and a White win be K.

Say that

• 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
• The point difference between answering the threat and completing the threat is indicated by M for my threats and H for the opponent's threats.
• Let M0 indicate my largest threat, M1 the next largest, etc., and similarly for H0, H1, etc.
• the other plays besides the ko and threats are such that the the gain for the player with the move after the ko fight is settled is approximately half the miai value of the largest of those plays. (This is normally the case.)
• Let T be the miai value of the largest play.

Whether to answer a ko threat or win the ko

Case 1. The opponent has played his largest threat, leaving the gote, H0, on the board.

• If K > H0 + M0 - T, win the ko; if the opposite, answer the threat.
• If you have no ko threat, then win the ko if K > H0 + T.

Case 2. The opponent has played his second largest threat.

• If K > H1 + M1 - T, win the ko.
• If you have only one ko threat, win the ko if K > H1 + T.

Etc., etc.

The books, e.g. Rob van Zeijst's All About Ko, tell you to compare the ko with the threat, 2K/3 vs. H. That is not very good advice, unfortunately.

Note that the values of your own threats, as well as the value of other plays on the board, are relevant. In particular, you can afford to reply to a small threat of the opponent if your corresponding threat is large enough. :)

This formula works when you make two key simplifying assumptions:

1. The ko is only fought when the ko is the same size as the ambient temperature (miai values are equal: K/3 = T). This will be true for many kos, because if the ko is smaller than the ambient temperature, players will simply ignore it until the temperature drops to the same value as the ko, and only at that moment fight the ko. However, sometimes a ko arises from local fighting, and the size of the ko is larger than the ambient temperature (K/3 > T).
2. They ignore the size of the threats that you still have in reserve. (TODO: Add some discussion about why ignoring the threats you have in reserve is bad)

After making these assumptions, you can substitute K/3=T into the simple case formula:

```K vs H + T
K vs H + K/3
2K/3 vs H
```

Derivation of formulas

First the values K and H used in the formulas above are deiri values, but to do calculations it's easier to use miai values. Each play in the ko is K/3, each play in the threat is H/2.

The ambient temperature value T is already a miai value. So playing the largest "normal" move on the board (which will be the largest gote or reverse sente) is worth T. The value of sente is T/2.

Ignore threat:

• You win the ko (+K/3)
• He finishes his threat (-H/2)
• You get sente (+T/2).

• You answer the threat (+H/2)
• He takes the ko (-K/3)
• You play a normal everyday move on the board (+T)
• He finishes the ko (-K/3)
• You get sente (+T/2).

So to compare if Ignore vs Answer:

• K/3 - H/2 + T/2 vs H/2 - K/3 + T - K/3 + T/2

Simplifies to:

• K vs H + T

Example 1

Ko Evaluation Example

• 'a' - ko worth 21 points (K=21, or in miai values 7).
• 'b' - Black's ko threat worth 12 points (H0/M0=12, or in miai values 6)
• 'c' - White's ko threat worth 12 points (M0/H0=12, or in miai values 6)
• 'd' - Largest play on the board, worth 2 points (miai value). T=2.
• 'e', 'f', 'g' - Other plays

Assumptions:

• "Outside" framing stones are alive.

Correct play:

• Black takes ko: Ko is 7 points miai, ambient temperature is 2, so play ko.
• White makes a threat: The ko is larger than this threat, but White can't retake the ko by rule. So this threat is the best.
• Black answers the threat: K vs H0 + M0 - T ==> 21 vs 12+12-2 ==> 21 vs 22.
• White takes the ko
• Black makes a threat
• White ignores and finishes ko: K vs H + T ==> 21 vs 12+2 ==> 21 vs 14.
• Black follows up on the threat
• White gets sente, worth T/2 = 2/2 = 1 point.

TODO: Add some more examples to illustrate various key tipping points. Smallest valid ko threat, etc etc.

Iago isnt K worth 23 in this exemple ? if white connects there is 2 points here + black 21 points for the kill... edit... I think I see my mistake here, black gets 19 points for the kill : 8 prisonners worth 2 points each + 1 prisonner worth 1 point, + 2 empty spaces...

Example 2

Example for a large ko, let K=99. Let T=6, a typical middle game value(?).

The table below shows conditions for answering or ignoring threats of various sizes:

```For K=99, T=6:
H
99+  always answer since T >= 0
98   always answer since T >= 1
...
93   always answer since T >= 6
92   answer if M >= 13    ( M >= K-H+T )
...
67   answer if M >= 38
66   answer if M >= 39
...
12   answer if M >= 93
11-  Opponent made a mistake by playing a threat smaller than 2T=12.
Assume he will realize his mistake on the next turn, so refer back to the H=12 line
```
• For H=99+, answer since the threat is bigger than the ko.
• For 99<=H<=93, answer since the threat is almost as big as the ko, and the board plays (T) would compensate your opponent for the shortfall.
• For H<93, even though you could win the ko now, it may pay to wait if you have adequate threats in reserve.
• For H slightly smaller than 93, you can typically answer as long as you have adequate threats slightly larger than 12.
• The smaller H gets, it becomes more likely to be correct to win the ko and allow the opponent to complete his threat H. For these smaller values of H, you need larger threats of your own to delay winning the ko.
• Also be careful that the value of H and M you are comparing are correct. As H gets smaller, answering the threat based on mistaken counting, calculating, and comparing threats gets more expensive.