What's the miai value here?

Here's my analysis

Black plays first

(B)  

After White can play a or Black can play b

White first (B-W)  

The count is 4-2=2

Black first (B-B)  

The count is 5-1=4

The tally is 2: each of the plays is gote The average of the two counts is (4+2)/2=3 The value of playing here first is 1, moving the count from 3 to 4 or to 2.

So the count of the position after Black plays first in the original diagram is 3. This is the easy part.

White plays first

(W)  

This is more complicated. We'll figure out the count of this position by exploring the branches.

(WBW)  

If both continue playing here, the count is 2-3 = -1

(WBBWB)  

When Black plays two times here, is sente, because if Black responds, the count is 3-2 = 1 if Black doesn't respond, then White captures to make the count -2 and this is worse for Black than if he hadn't played in the first place

So the count is 1 here

(W-B)  

The tally of the previous two diagrams is 2 Hence the count of this position is the average of the counts of those positions: (1-1)/2=0

(W-W)  

Now for two successive White plays, which is the most difficult position to assess (IMO).

(WWW) - 1  

The chance of getting a is 1/2 so the count here is -3.5

(WWW) - 2  

If White plays here instead, then we get

(WWWW)  

a count of -4 OR

(WWWBW)  

if here, then White has the right to take (2/3) and the count is -3-2/3 assessing further

(WWWBBW)  

After Black can't cover, so there will be a capture - recapture; later White can block for a 1/3 ko the count is -3-1/3

(WWWBBB)  

playing here allows White to block, capture recapture and make another 1+1/3 which gives a count of -4-1/3 this means Black won't play and this means White has the right to play in the previous diagram

(WWWB)  

Hence the count here is -3.5 and the value of a is 1/6

(WWW) - 2  

So White should play here the count of WWW is -3.75 The value is 0.25

(WWBW)  

If -, the count is -2-5/6 being the sum of 1/2 for Black a -3 for White's points and -1/3 for White to take the ko.

(WWBBWB)  

If answers then is sente and the count is -2

(WWBBWWB)  

If Black doesn't answer then we get a capture recapture (earning White a bonus point) and the next diagram

(WWBBWB)  

If Black takes a then we end with a count of 1-3 = -2 If White takes a she gets the right to play the ko later and the count is 0-3-1/3 = -3-1/3 The average count is -8/3 = -2-2/3

So, in retrospect it is better for Black to answer then not, in other words

(WWBB...)  

If --- is a sente sequence and the count is -2

(WWB)  

Hence the count here is the average of WWBW and WWBB, which is (-2-2-5/6)/2 = -2-5/12. A move is worth 5/12.

(W-W)  

Finally the count here is (-3-3/4 - 2-5/12)/2 = -3-1/12 and the miai value is 8/12

(W)  

Since the count of () was 0, the count of (W) is -1-13/24 and the miai value is 1+13/24

Result and verification

 

Wrapping it all up, (B) has count 3 and (W) has count -1-13/24 Hence the count of this position is 35/48 and the miai value of a is 2+13/48

In the case where Black plays first the miai value of the moves are 2+13/48 > 1

When White plays first 2+13/48 > 1+13/24 > max(1/4, 5/12)

So the explored moves are all gote.