When I first went through games played by TD-Gammon (the first neuralnetwork backgammon playing program), I looked for unusual plays which weredifferent from what I would have done. One recurring theme I noticed wasthat TD-Gammon would often break up its board in a holding game for noapparent reason. For example:
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110 | 
        
 
         
| White
money game
Blue |
What could be more obvious than 13/8, 6/5, right? Yet TD-Gammon played theseemingly inexplicable 13/8, 3/2! Initially I assumed it was one of thosequirks with the neural network programs. I knew that in simple positionsthe programs often make strange-looking (and wrong) moves because theirtraining doesn't give them enough information to work out the proper principles.Human experts will usually outplay the bots in these sort of positions whenit comes to small technical differences. I joked about this with GerryTesauro, the inventor of TD-Gammon, and called it the TD weird play.
A little later when Gerry had programmed TD-Gammon to roll out positions,I had him roll a couple of these types of plays out just to make sure. Muchto my surprise, the TD weird plays consistently outperformed the normal playsin the rollout. This was something else. While the program might havesome strange ideas, it generally played well and it was hard to see how itcould badly screw up the rollouts for a position like this. Even if therewere some kind of bias in the program's playing of holding games, thepositions after the two plays in question are so similar that the bias wouldcertainly apply to either play so the rollout results would be accurate.The absolute value might not be correct if the bot is misplaying one sideor the other, but the relative results between the two plays almost has tobe accurate.
Offhand, it doesn't appear as though it could make much difference even ifone play were superior. Rollouts show that the difference can be moresignificant than we might think. I had Snowie roll the two plays out(1-ply, cubeless, 3888 times each with the same dice), and the resultswere: 13/8, 3/2 +.41013/8, 6/5 +.362 This is quite a significant difference for what seems to be an almostmeaningless ace. Granted the sample size isn't huge, but with the duplicatedice and the similarity of the positions it appears quite clear thatthe TD weird play is noticeably superior. This is not an isolated case.I found this sort of play consistently won in the rollouts. What is going on?
Let's look ahead and see how the game is likely to continue. It is clear thatBlue isn't going to be hitting any blots for quite some time. White isbehind in the race, so even if White had a free ride to clear the bar pointhe would rather remain there to keep contact. Only if White rolls largedoubles will White even consider leaving the anchor. Therefore, there isno urgent necessity to have the inner board points closed for the nextfew rolls.
If Blue rolls doubles, Blue will of course clear the midpoint and probablywin the race. It won't matter much which ace Blue has played, but splittingto the two point does diversify Blue's checkers for the bearoff so that can'tbe bad. Similarly if White rolls large doubles White can run off the anchorand turn the game into a straight race. Again, splitting to the two point isfine. The more interesting scenario occurs when neither side rolls doublesfor several rolls. Blue will have to bring his checkers in as best he can.His first priority will be to play safe, but he also wants to waste as littleas possible. One large doubles by White will turn the game into a straightrace, and if Blue has wasted too much his pip lead will go down the drain.Blue's goal is to fill his inner board smoothly, avoiding gaps on the lowerpoints. This will be best both for the race and if White gets squeezed offthe anchor and Blue gets a shot. What Blue wants to avoid is piling abunch of checkers on one point while leaving another point empty.
Now that we see what Blue is trying to accomplish, it becomes clearer why13/8, 3/2 is superior to 13/8, 6/5. First of all, having the spare on thesix point is a definite plus, since that increases Blue's flexibility. Moreimportant, the split to the two point improves Blue's chances of smoothingout his position. After the split, if a Blue checker lands on either thethree point or the two point it looks fine. If Blue plays 13/8, 6/5, thenhe needs to have a checker land precisely on the two point for the smoothingprocess to work out.
As an illustration, suppose White rolls 6-4 playing 13/7, 13/9, and Blue's next roll is 6-4. Naturally Blue will bringa checker in from the midpoint. Blue wants to clear the midpoint as soonas possible, and he wants to keeps spares on the eight point to handleawkward numbers. If Blue had played 13/8, 6/5 the position after henow plays 13/3 will look like this:
It's starting to look a bit ugly. Three checkers on the three point, andthe ace and two points still bare. That third checker on the three pointbelongs on the two point, but it won't get there unless Blue rolls an ace.The key is the Blue did roll that ace his previous turn, and that is whenthe checker should have been moved to the two point. Had Blue playedproperly, the position would instead look like:
Isn't that much nicer? Now if Blue's next roll is something like 5-3, hecan play it decently without tying himself in knots. In the positionwith three checkers on the three point and the six point stripped, a 5-3roll plays awfully.
I'll grant that I chose a particularly awkward sequence of rolls for Blue toillustrate my point. However, these awkward sequences are the ones whichmatter. If Blue rolls a bunch of 2-1's it won't make any difference how heplayed previous rolls, since he will have a chance to reshuffle his checkersto where they belong. It is the large rolls such as 6-4 which create theawkwardness if you have failed to plan ahead and place your checkers properly.
Blue should play 8/1, not 8/2 5/4. This is a fairly important play. Theproblem is that if Blue rolls a six next, he will have to clear his eightpoint in order to play safe. If Blue has played 8/2, 5/4 then this sixwill force him to put a third checker on the two point, which is awful fromflexibility, racing, and board building considerations. After playing 8/1Blue will be putting a second checker on the two point, which is fine.
The main thing you have to watch out for when considering this sort ofboard breaking play is that you won't be getting a shot in the next coupleof rolls, so you will have Time to put the board back together. If youare ahead in the race this is almost always the case, since your opponent willbe hanging back in order to retain as much contact as possible. If youare behind in the race, however, breaking your board may give your opponenta chance to pay now and make a run for it. For example:
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| White
money game
Blue |
Since White has three checkers on the midpoint, there is no way Blue willbe getting a shot next roll. Even if Blue messes up his board, White willnot be moving two checkers from the midpoint and leaving multiple shots.Therefore, Blue can and should play 8/6, 4/3. This will make it easier forhim to complete both the four and three points in his next couple of rolls,which is what he is trying to do. There is little danger in temporarilyunmaking the four point, since Blue has so many rolls to remake it. IfBlue intead plays 9/6, he will have to roll more perfectly to make thethree point. To see the difference, imagine Blue's next roll being 6-1and look how the roll plays from both positions.
A small modification of the position can change things. Consider the following:
Same position except that now White has only two checkers on the midpoint.The difference is that Blue may be getting a shot next roll. If White rollssomething like 5-3, White may choose to pay now and play 13/8, 13/10. Thispayment is obviously more attractive if Blue's board is a mess. ThereforeBlue should play the normal 9/6 rather than the fancy board breaking play.
Most readers are familiar with point switching plays in blitz situationswhere you switch to make a lower point in order to have more buildersaimed at the slotted point. If you can't Move the builders to the point,move the point to the builders. The same principle can be applied inholding games, using board breaking plays.
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| White
money game
Blue |
I believe Blue's best play is the odd-looking 13/9, 5/3. White definitelywon't be leaving a shot next roll, so Blue can afford to break his boardtemporarily. The loss of the five point is meaningless, since Blue getsit back with any roll but 5-2. The key is that Blue slots the three point,which is the next point he wants to make. If Blue makes another play thethree point remains unslotted, and while Blue will usually be able to slotand make it before White leaves a shot occasionally the dice won't cooperate.In essence by playing 5/3 Blue activates the builder on the 11 point withouthaving to move the checker. Naturally if White had only two checkers on themidpoint Blue could not afford the luxury of breaking his board, since Whitemight volunteer a shot and clear the midpoint.
Granted the gains from these board breaking plays is small, but every littlebit helps. Our goal is to maximize our equity all the time. Also this sortof position is very common, so the cumulative gains over many games can besignificant. The next time you are building your board while playing aholding game and it seems as though your plays are awkward and you are tiedup in knots, go back and see if you missed an opportunity for a profitableboard breaking play. It is likely you will find those awkward rolls wouldhave played much smoother had you prepared for them appropriately.
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