Re-Volting Backgammon

With a population of seven billion, and more and more of them using computers, smartphones and even kettles and toasters with their own microchips, the world will run out of electrons in six years’ time.  That’s the worrying conclusion of Dr. Mustafa Shok of the University of San Serif in Thailand.

Fortunately the so-called “Shock Doc of Bangkok” has come up with a solution based on the science of Chirality.  Using his idea of mirror images we can reduce our need for electrons by up to 50%.  He has dubbed this process “Re-Volting” because of the electrons freed up.

Following on from his pioneering work I have applied it to the backgammon world.

Diagram 1 & 2 – The Jacoby Paradox

Cube Action?
1. Playing Right to Left
2. Playing Left to Right

Bearing off to the left White should redouble but if bearing off to the right it’s correct to hold the cube.

With the opponent’s checker on the one point White is 19/17 favourite. His positive equity is (19-17) in 36 games, multiplied by the cube value, so best to have the cube on 4. His equity in 36 games will be (2*4) = 8.  Now with the opponent’s checker on the 6 point we need two calculations.

Suppose we redouble…

Danny Kleinman came up with a way to calculate the recube value for Black, when we fail to bear off immediately, called “Excess 18 arithmetic”.  Take the number of wins above 18 and add this back to the number of wins to get the effective cubeless wins. Here Black wins 27 in 36, 9 more than 18. Add 9 to 27 and we see Black wins 27+9 = 36 in 36.  The recube means he effectively wins all the games in which we fail to bear off immediately.  White’s equity will therefore be 8 in 36 games – the same as when the opponent’s checker is on the one point.

If we keep the cube…

Now we win a quarter of the 17 games we don’t get off on the first roll when Black also misses.  Approximating, we win 19+4 games, leaving 13 out of 36 which the opponent wins.  (23-13) * the cube value gives us a net of 20 in 36 games, 150% more than when we redouble.

Sometimes it’s obviously not possible to save electrons by including two positions in one diagram.  The alternative is to use one analysis for two diagrams.

Diagram 3

Diagram 4

I first saw diagram 3 in an article by Kit Woolsey in the Las Vegas Backgammon magazine in the 1980s.

If White wins with a centred cube he goes to -1-4 Crawford for approx 81.5% Match Winning Chances (MWC).  If he has already doubled he wins the match for a gain of 18.5% MWC. Black will be trailing -3-2 if he wins an undoubled game and -2-2 if the cube has been turned: his MWC will increase from 40% to 50%.

So White is receiving odds of 18.5 to 10 on MWC when he is only 19 to 17 underdog; therefore he should double. Diagram 4 is the reverse position where the trailer is on roll.  Since the leader wants the game to finish with the cube on two the trailer should not double.

Diagram 5 & 6 (consider both directions)

White on roll, cube centred
White leads 2-away, 4-away

You should try this problem bearing off to the left and then bearing off to the right.

Read the analysis below twice; once for each direction.

The point of the Jacoby paradox is that you should take care giving your opponent cube access when the position is likely to evolve into one where he can recube close to your dropping point.  The same is true in Diagram 5 & 6.  Although Black always has access to the centred cube, Diagram 4 shows he shouldn’t double if White fails to get off with say 61.  However, if White has foolishly turned the cube, his take point on the recube is 50% so White must pass even though he wins 47%.

The gain for Black when he cashes these recubes far outweighs White’s gain when he wins on roll one.  It is a double blunder to cube either Diagram 5 or Diagram 6 costing about 2% MWC.