What about reverse sente?
We can write a simple potential reverse sente, r, for White as
r = {z | 0 || -s}, z > 0, s ≥ 0
If z > 2s, r is a reverse sente for White. It has a count of 0. We can add or subtract a constant to all results and change only the count. So
{z+m | m || m-s} = {z | 0 || -s} + m
The average gain for White for playing the reverse sente is s.
If 2s > z, then r is a gote. It has a count of (z-2s)/4. The average gain for either player is (z+2s)/4.
Such a value does not appear in our little theory, for a very simple reason. The theory is concerned with exact results, not averages. (I think that making this clear to people would nip in the bud the idea that it is always right to take the largest play. Taking the largest play is simply playing the averages. It does not guarantee correct play.
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Sente gains nothing
The saying goes that sente gains nothing. That is because, if a player makes a play with sente, he gains nothing more than he ends up with. And, because the play is sente, that is what he is entitled to when it is his turn to play.
Global sente
Now, the above test for whether r = {z | 0 || -s} is a sente for Black is that z > 2s, It makes no reference to other plays and it defines a local or intrinsic characteristic of r. It determines the local count and helps us to play the averages. But our little theory is not concerned with playing the averages, it wants exact results. r is global sente when, if Black plays to {z | 0}, White has nothing better, in terms of the score, than to reply to 0.
On this board with r and some number of simple gote of the form {a | 0}, {b | 0}, . . . , r is global sente for Black when z ≥ a. Our question is whether White should play in r or in a. If White plays in a, that leaves a board with r and zero or more simple gote. With no gote left, r is of course a global sente for Black. In our example, r = {5 | -1 || -7} is local gote with a count of -2.5. But if r is a global sente we may regard it as having a count of -1, and White's reverse sente gains 6 pts. instead of the average gain of 4.5.
Suppose that z ≥ b, which makes r a global Black sente at that point. Then when White takes a, Black will play sente and gain nothing, and then the rest of the board will be played out. The total gain for White will be this:
1) T1 = a'/2 ; Remember that in {g | 0} each player gains g/2
OTOH, if White plays in r to -s, the total gain for White will be
2) T2 = s - a'/2
White should play in r if
2) T2 > T1 <=> s - a'/2 > a'/2 <=> s > a'
This test appears when we have two or more gote. (Note: It applies in all of them if we let a' apply to a single gote. OC, with only one gote we may regard r as a global sente if White takes a.
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For 2 gote the rule is this:
Play in r if ((s > a') and (t > a)).
Let's do some rewrites.
Play in r if ((s > a - b) and (s + z > a)).
Play in r if ((s > a - b) and (s > a - z)).
If z ≥ b then a - b ≥ a - z and the rule reduces to
Play in r if ((s > a').
And that rule applies to all cases with two or more gote.
Well, if b > z then r is a global gote. However, we can still apply the idea of comparing z with the values of local gote. Doing so gives the theory is this form:
Play in r if
((z ≥ b) and (s > a')) OR
((z ≥ c) and (s + z > a + c')) OR
((z ≥ d) and (s > a - d')) OR
* * * *
This form is simpler and easier to comprehend. In fact, it was the first form of the theory that I came up with in the 1970s. But back then I was not constrained to avoid the concepts of gote and reverse sente.
N.B. When considering whether to play r as a reverse global sente, we do not worry about whether it is so now, but whether it will be on Black's next turn.
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Estimation
Suppose that r is not a global reverse sente and there are so many gote that we do not know how many there are and z is so small that we do not know which comparison to make. In that case we can estimate our comparison as this:
s >? a - z'
That is some help, but we still have to estimate z'. It has a maximum value of z and a minimum value of 0. The estimate that minimizes its maximum error is z/2. That gives us this comparison.
s + z/2 >? a
I.e, we are comparing the swing value of r as a gote with the swing value of the largest simple gote. Back to playing the averages.