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
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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.
Money
Gammon Go
Gammon Save
By Julian Fetterlein. Jigsaw photo courtesy of droetker0912.
Thanks for your kind comments Jesper, good to know one’s efforts are appreciated.
Perry, I thought of going further but there’s some problems made me stop here.
The data is from GNU which I know nothing about and was obtained by Mochi, so all I have seen is a summary.
I think we can make an educated guess at some of your questions.
In races we will typically make an initial double at something like 70/30 favourite. I would expect the trailer in the race to be able to recube slightly over 20% of the time. Since the data shows only 17% of games finishing above 2 points this number must be depressed by the Holding/High Anchor games where the cube efficiency will be less.
Things are harder with the exciting games since we don’t know if four point games are gammons at 2 or recubes.
However if you take the rollout data for a typical blitz (say 54 split, 55 attack, fan) there’s a total of 40% gammons so the magnitude would be 2 + 40% of 2 or 2.8. If it’s 3.4 for this subset then the 0.6 difference must come from recubes.
Colin
There’s obviously arguments both ways but my strategy is to avoid the exciting games.
The dice and the opponent also have a say on the type of game and I generally aim to double my opponent out on the exciting games.
The justification is the gammon price is lower for the stronger player and higher for the weaker one.
As an example, if you are 54:46 favourite at DMP; with the cube at 2 at a score of -3 -3, the gammon prices become 0.4 for the stronger player and 0.6 for the weaker.
very nice article
Where does this leave a significantly stronger player at a reasonably level score? He wants more complexity, which he’ll tend to get in ‘exciting’ games, but conversely wants less magnitude in order that the winner of the match will require to win more games.
Well I never knew that!
A wonderful contribution to our knowledge base. Now that you have quantified the magnitude of the value of differing Game Plans, I wonder if you can refine the data even more.
In addition to revealing the number of points exchanging hands at the end of a game can you extract the number of initial cubes given by the losing and winning side in Boring & Exciting games, as well as when the score is tied? The number of re-cubes from this perspective could also turn out to be an interesting and valuable statistic.
Great article, well written, well explained and such great input..
Thanks for sharing !