Match score sensitive endgame cube action

[Johan]

Both players have 2 checkers on their deuce point. What is the correct cube action when leading 2-away 3-away? The double is clear, but what about the take/pass decision? This situation arose between Patrick and Paul in the Brugge-Leuven 4 Cubes encounter. According to Michel, taking would be correct for money but a blunder at the match score. Why?

[Toon Van Herreweghen]

I know you try to entice me to comment this one…

Match play:
If you take, this game is played for the match.
So you only take if your winning chance is at least 25% (= your match winning chance at 3-away, 1-away crawford)
You win when your opponent misses (10 rolls) and you bear off (26 rolls or around 3/4 of the time)
So out of 36 games, you win around 10 x 3/4 or 7.5 which is less than the 9 needed.
Pass.

Money play:
Start by classifying the situation based on the cube actions after you take and your opponent misses.
You have 26 winning rolls. You will redouble (26 > 18) and you opponent will take the 4-cube (26 < 27).
Realise that you will effectively quadruple the expected win compared to a cubeless game.
I like to express this expected win as R-18 = 26-18 = 8 points for 18 games.
So – in 18 games – you expect to win 4 x 8 = 32 points, 4 short of 36, thus 2 – 4/18 (or 2 – 1/5 for easiness)

Back to the original position. If you take:
You loose (-) 26 x 2
You win (+) 10 x (2 – 1/5) = 20 – 2 = 18
-52 + 18 = -34 which is slightly better than the -36 loss if you pass.
Take.

[Toon]

My apologies: in match play, if you take as the trailer this game is NOT played for the match.
Yes, you would redouble automatically, but no, the leader will not take the 4-cube.
(He needs something better than the 30% he would get after passing with the cube on 2, and he only gets your 10/36 missers)
Therefore, a miss by your opponent means a mere 2 points win for you, and leaves you still with only 70%.
The resulting decision is the same, though: you get 0% (26/36) or 70% (10/36)
Nearly 20% MWC by taking.
You are happy to grab the 25% by passing.

[Johan]

Toon’s analysis also shows that for money, the position (each player having 2 checkers on the deuce point) is indeed an initial double. But is it also a redouble? If the player on roll decides not to double, then in case he misses, he takes away his opponent’s powerful redouble.

[Toon]

I have to confess that I don’t torture my brain anymore with these calculations.
I’ll share a first-class approximation with you.

Situation: 2-roll bear off, money play, you ponder about doubling.
You roll out in Ra, you opponent in Rb (and let b = Rb-18).
And in the case: you know you will have to take a recube if you miss (18<Rb 2b
Redouble if Ra > 3b

I may get it slightly wrong in 5% of the marginal cases. But I won’t keep me from sleeping.

[Toon]

The previous text was garbled.
Another try

Case: 18 < Rb 2b
Reouble if: Ra > 3b

[Toon]

Johan, all,
there seems to be a problem with publishing texts including less-than and greater-than signs.
These are probably taken as delimiters of html elements.

Last time, in words:
Case: you will have to take a recube if you miss (Ra between 18 and 27)
Double if Ra is bigger than 2 times b
Redouble if Ra is bigger than 3 times b

[Author]

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