- Article 7 -
Very often the last strategic decision made by either player in a game of backgammon is whether to finally break contact, leaving the outcome to be determined mainly by the pip count and the dice, or to leave an anchor or a blot in enemy territory, hoping to hit a shot that will decide the game.
Sometimes the decision is trivial: a player with a big lead in the race will almost always want to break contact if he can. Conversely the player losing the race may be able to hold an anchor at no cost, in which case he usually should.
Many of the interesting plays in this category leave a blot behind. The position of such a blot, alone, exposed, and cut off from friendly forces, has always led me to think of him as a “sniper.” The analogy is imperfect. Unlike his military counterpart, a backgammon sniper-blot must save himself, whether or not he carries out his special mission, or the whole war is lost.
Problem 1: Red to play 5-2.
Problem 1 illustrates a standard, easy decision. Red is so far down in the race that he has almost no chance at all (something like one tenth of a percent) unless he hits a blot. Red wastes no pips – either for saving the gammon or for preserving the ultra-slim chance of winning the race – by playing 20/13. He has almost nothing to fear. If White rolls 1-1 he will hit (6/5*(2), 4/3) but the decision is a close one which increases the risk of losing the game for additional gammons.
The upside is that when normal things happen and White does not clear his 6 point with doubles he will have a few blot numbers and Red will win between two and three percent of the time, which is better than nothing.
Problem 2: Red to play 4-4.
As Red’s deficit in the race is reduced it eventually becomes right to just leave and try to win the race. In Problem 2 the decision is almost a coin-toss.
Problem 3: Red to play 6-4.
Sniping opportunities can arise without a massive race deficit. In Problem 3 the count will be even after Red plays his 6-4. White’s position is a bit inferior, so being on roll will give him only a small edge. But White’s two home board blots give Red a chance (easily overlooked!) to hang back and hope for a shot. Note that Red will always be getting a direct shot unless White rolls more than five pips, and 9 rolls fail to do so. White hits with all his 5-x rolls, but only 5-2 cleans up both his home board blots to leave him as better than a 2-to-1 favorite. White’s 5-5, which would otherwise be a great racing roll, turns into a nightmare with the essentially forced play 13/8*, 7/2(2), 6/1 which leaves a direct shot and two blots, meanwhile burying three checkers on deep points.
Problem 4: Red to play 6-5.
Following Problem 3 I’m sure you had no trouble spotting Red’s chance to leave a blot behind. But can it possibly be right to risk those damaging hit/cover rolls by White just to maintain contact between blots at a distance of three pips?
Let’s begin by roughly estimating Red’s chance of winning if he breaks contact and tries to race. After the 6-5 is played, White will be on roll with pip count 62 to 76. Racing formulas predict a bare take of a money double at 62-70, with a cubeless winning probability near 22%. White will be six pips worse than that. A linear extrapolation of the even-money and last-take points would actually put White down near 8%, but the change is not really linear (the leader gets diminishing returns as his lead increases), and White’s position, with the gap on the four point, is worse than Red’s. Without resorting to more advanced methods I would put White’s chances in the 10%-15% range.
If White neither leaves a shot nor hits, then Red’s play has no effect on the outcome, since his play does not waste pips. The rolls where the play helps Red are 2-1 and 3-3 (did you notice that one?) The rolls that hurt are 3-1, 3-2, 3-4 and 1-1, which hit Red’s blot and cover White’s. White will not hit with 3-5 or 3-6, of course. On 2-1 and 3-3 we can put Red at least 35%, since he will have additional (though small) chances beyond the immediate direct shot. When White rolls a hit-and-cover Red will have the immediate 4-4 return (2.8%) plus chances of hitting later when White bears off from his gapped board. Even winning the race without hitting is not completely out of the question. So it should not be too surprising that altogether Red wins 9% of the time when White hits and covers.
If we peg Red at 15% in the race, which was the high end of our crudely estimated range, we can now see that the sniping expedition rates to work out favorably on average. We incur a 6% loss (15% - 9%) on White’s seven favorable hits but we rake in a 20% gain on any of his three bad rolls, so our profit is 3*20% - 7*6% = 60% - 42% = 18%. Of course this actually gets spread over 36 rolls so we are only ½% more likely to win by sniping than by racing.
Problem 4 was not contrived, by the way. It came up in an on-line match of mine. Nor does it represent the minimal amount of contact worth preserving. It is not hard to construct a position similar to this one where all you have to hope for is the opponent’s 2-1 roll. That’s not the minimal snipe, though...
Problem 5: Red to play 6-5.
This rather silly position is contrived. Obviously Red should leave a blot on the 17 point and hope for White to roll 1-1. It gains when he does and it costs nothing if he doesn’t.
Problem 6: Red to play 4-4.
Problem 6 is too complex to try to solve quantitatively in practical play. The best approach is to estimate the racing chances if contact is broken, look for the favorable and unfavorable rolls by the opponent if contact is kept, and then apply a little judgment and intuition.
Red will be down six pips and the roll after the 4-4. White’s position is ugly but it will improve some if he can bear in unhindered. It looks like Red would be about a 2-to-1 underdog in a pure race. As a 2-to-1 dog, Red would not mind trading in all his racing chances for a winning direct shot plus a little extra “slime vie” in the form of small racing chances plus possible repeat shots.
At first glance we see, after Red’s 20/4, only 6-2 as a blotting roll for White, with 1-1, 1-2, 2-2, and 5-2 as rolls that make Red sorry he stuck around. But looking a little deeper we can find quite a few rolls for White that have to be played awkwardly, increasing wastage for the bear-off, just because the Red blot is in the way. These include 6-1, 6-5, 3-2, 5-4, 5-3, and to a lesser degree (because White would like to fill the gap on the five point) 4-1. 4-2 and 3-1 are puny racing rolls for which the safe plays are quite wasteful, and White should probably hit, leaving a direct return, with these rolls even if the shot is not forced. White’s 5-5 would be a great roll if the blot were not in the way, but with Red’s sniper in place the only safe play would be the awful 7/2(2), 6/1(2). And here too White does better to hit, making the five point.
Being able to compel your opponent to make his fifth best bear-in play can be worth quite a bit. Imagine that Red runs with the 4-4 and White rolls 6-5, for example. What would White be losing if he played off the seven point instead of the ten? About 6% of his chance of winning the game, as it turns out. That is quite a significant difference for Red, who starts with a wining probability somewhere in the 30% range. The large number of gains of this type turn out to make the sniping play 20/4 right for Red.
Problem 7: Red to play 4-4.
Problem 7 is quite a different case, though the roll is the same and the pip counts are similar to the situation of Problem 6. Once again, if Red plays 20/4, we have the single forced-blot roll 6-2. The Red blot gives White some strong hit-and-cover rolls – 4-1, 4-2, 2-2 – plus the crushing 1-1. The existence of these rolls might be tolerable if the sniper did enough to interfere with White’s bear-in on a lot of other rolls. Red does get some gains of this sort on White’ 5-2, 3-2, and 2-1 but they are less numerous and less significant than in the previous example. Red does best simply to run with 20/16, 20/8.