Backgammon sometimes seems like a giant jigsaw puzzle: there are multiple pieces of seemingly unrelated information but, very rarely, we find a way to join the pieces together and the big picture becomes clearer.
The First Piece…
Suppose you are four points behind in a long match. If you have to win four games at one point to equal the score, the chance of this happening is 1/16. If each game is worth two points you only need to win two games and your chance is one quarter. If each game is worth four points your chance is a half. This is to show what may be obvious to some – that the trailer wants each game to be worth more points.
The Second Piece…
A few years ago I played a match in Las Vegas against Japanese master Michi. One game started with me rolling 31 making the 5 point; then Michi rolled 11 and made the 5 and 7 points. Usually I wouldn’t bat an eyelid but Michi is a renowned opening expert and I knew that all the rollouts since Jellyfish favoured making the 5 point and splitting to the 22 point. So I asked him if the rollouts had changed and he replied “More gammons”. At the time I was leading 7-4 in a match to 15.
The Third Piece…
Suppose you are 52% – 48% favourite in a race. Your cubeless equity is 0.04, being the difference in percentage wins.
Compare this to a position where all the wins are gammons but you are only 51% – 49% favourite. Your equity is again 0.04, being the difference in percentages multiplied by 2 for the gammon wins.
So we have two very different positions with identical equities. For some time I have been searching for some measure to distinguish the difference in positions and came up with the term “magnitude” defined as the average number of points that will change hands at the end of the game.
The Final Piece…
The final piece of this puzzle was a seminar given by Mochy on 5 point matches. He divided games into two types: type A and type B. I have given these more descriptive names, ‘Boring’ games and ‘Exciting’ games.
Type A, Boring games, includes races, holding games and high anchor games.
Type B, Exciting games, includes blitzes, priming games and low anchor games.
Mochy used a facility unique to the GNU program of analysing the number of points that changed hands at the end of each game, with the following results:
Boring games produced 83% of 2 point games, 15% of 4 point games and 2% of games above that level.
Exciting games produced 46% of 2 point games, 45% of 4 point games and 9% of games above that level.
I then took this a step further by calculating the average number of points for each type of game, which results in 2.4 points for Boring games and 3.4 points for Exciting games. So the magnitude of Exciting games is a full point or almost 50% more than for Boring games.
Michi’s mysterious move steers the game towards a priming battle while the splitting play is more likely to lead to a racing or holding game where both players have high anchors. If his plan succeeds, my match lead of three points is now less than one game at 3.4 points, rather than more than one game at 2.4 points. The XG rollout for this position at this score shows that Michi’s move is superior to the splitting play by about 20 millipoints.
Michi’s Mysterious Move
Having found the idea how can we utilise it in our own games?
I am going to look at the opening moves at the two extreme cases of Gammon Go and Gammon Save, where the players’ incentive to maximise or minimise the magnitude will be at its highest.
Five of the opening rolls have one play that is obviously best at all scores, 65, 31, 42, 53 and 61. Of the remaining ten possible rolls, 90% of the best plays involve moving one of the back checkers at Gammon Save and not moving the back checkers at Gammon Go.
Put simply, we increase the magnitude by playing offensively on our side of the board and decrease the magnitude by playing defensively on the other side of the board.
Finally here’s a second roll example which shows the idea. Our opponent opens with 42 making the four point and we roll 22 in reply.
Three XG rollouts are shown below, for Money, at Gammon Go and at Gammon Save.
By Julian Fetterlein. Jigsaw photo courtesy of droetker0912.