This article originally appeared in the January 2001 issue of GammOnLine.|
Thank you to Kit Woolsey for his kind permission to reproduce it here.
The following position is a prototype of a very common backgammon problem --
whether one should leave a few extra shots in order to gain a moderately
small advantage when the shot is missed. Since most backgammon decisions
are a matter of balancing risk vs. reward, one could say that it is a
prototype for backgammon problems in general.
The astute reader will notice that there a few checkers missing from the diagram. This is intentional, as the position is meant as a prototype. Imagine the rest of Blue's board filled in with perhaps a few checkers borne off, and imagine White having a reasonable board of his own but perhaps a few checkers in the outfield so a gammon is a possibility. We will look at specific examples later, but for now we will just consider the general concept.
We have all been in this sort of situation countless times. We are confidently bringing the position home, and all we have to do is clear that last hurdle. Naturally we roll a 5-1. Not only are we forced to leave a shot, but we have to think about how to play the roll.
If gammons didn't matter, there would be no question about the play. 6/1, 1/0 would be best. This both takes a checker off and leaves only 11 shot numbers. However in real life gammons do matter most of the time. For now we will assume that there is some decent chance of winning a gammon if we hit, and that the gammon is close enough so hitting is clearly correct. Thus, for now the two plays we will examine are 6/5*, 5/0 and 6/5*, 6/1.
The advantage of 6/5*, 5/0 is very clear -- Blue gets an extra checker off. The disadvantage is also very clear -- Blue leaves 13 shot numbers instead of 11. We have to decide whether the advantages outweigh the disadvantages.
Both plays leave 11 direct shot numbers. In addition, 6/5*, 5/0 leaves two combination shots. These aren't the same direct shot numbers, since they are fives after 6/5*, 6/1 and sixes after 6/5*, 5/0. For counting purposes, however, we can assume they are the same rolls, since that will simplify our counting. Thus, we have:
11 rolls which hit with either play.
Taking the checker off is potentially helpful both when the shot is hit and when the shot is missed. If the shot is hit, that checker off will improve the chances of winning the race after later scrambling around the board. If the shot is missed, the checker off will improve the gammon chances. Of course, both of these gains are mild gains, while the cost when White rolls one of the two numbers which hit after 6/5*, 5/0 but miss after 6/5*, 6/1 is huge.
To put in another way, White will be mildly happy after playing 6/5*, 6/0 34 times out of 36. However he will be very unhappy with that choice 2 times out of 36. What we have to estimate is how big the very is compared to the mildly in order to choose our play.
Let's examine some concrete examples:
In this position, Blue's gammon chances are pretty small whether he takes a checker off or not. He increases his gammon chances, but since they are small to begin with the increase is smaller still. In addition, since White's board is so strong getting hit is a sure loss for Blue since White will be able to double Blue out. Thus Blue doesn't even get his mildly happy on all 34 rolls -- only on 23 of them. So, Blue has to estimate whether the gain from taking the checker off (when he is not hit) is 2/23 of the loss when the checker is hit. Intuitively it seems pretty clear that it is not, so Blue should play 6/5*, 6/1.
White's checkers have been moved back so now a gammon is a distinct possibility. This means that the extra checker off is more likely to swing the gammon than before. Now we are talking about a much closer decision. The safer hitting play of 6/1*, 6/5 is still probably correct since those 2 rolls turn a win into a loss, but it may be a close call.
Same men outside for White, but this time we have done some damage to his board. Now we have other factors to consider. For those 11 rolls which hit whichever play is made, the extra checker off can make a difference. No longer does White have a claim with the cube -- he has a lot of work to do. Additionally, getting hit those 2 extra rolls isn't necessarily swinging the game from a win to a loss -- Blue will still win some of the time. Thus the very part of very unhappy of the 2 extra hitting rolls goes down a bit, while the mildly part of mildly happy when the shot is missed goes up a bit. It still looks close, but this time my judgment would be to leave the two extra shots and take the checker off.
This time we have improved Blue's position by taking a few more men off. Now Blue is a distinct favorite to win a gammon with either play, but taking the extra checker off still helps the gammon chances. More important is that Blue will be the favorite after being hit, so the cost from being hit isn't as great and the checker off in the variations where Blue is hit is more likely to matter. In this position it feels very clear to me to play 6/5*, 5/0 and leave the extra shot numbers.
The same concept can be used when deciding whether or not to leave a few indirect shots with mild positional improvement if the shots are missed.
Blue must choose between 13/6 and 13/10, 13/9. Bringing the builders down is a moderate improvement for Blue if White doesn't hit, since these builders will increase Blue's chances of making key points and playing more flexibly in the future. However White does have six indirect shots after 13/10, 13/9, and if White hits one of them Blue is quite unhappy. Since White has 30 misses and 6 hits, the question is whether Blue is 5 times as unhappy when the shot is hit as he is happy when the shot is missed. In this position, my judgment is that he is not 5 times as unhappy, so he should bring the builders down. The main reason is that getting hit is not fatal, since White only has a moderately strong board. White's board will improve in the future, making it more important for Blue to build key points so he can play safely later. But change the position to:
Now White's board is so strong that Blue probably won't have any life after death if he is hit. While the motivation to build new points is still there, the downside from the 6 hitting numbers has gone up greatly. In this position I believe Blue will be more than 5 times unhappy when he is hit than he will be happy when he is missed, so I think he should play 13/6.
Here White's position is the original one, but Blue's flexibility has improved. Now the gains from 13/10, 13/9 aren't nearly as great as they were before because Blue already has decent builder distribution and flexibility, while the losses from getting hit are still pretty severe. I believe these diminished gains are sufficient to make the unhappiness from being hit more than 5 times the happiness from not being hit, so here I would play the safe 13/6.
Yet another improvement for Blue, and a big one. Now 13/10, 13/9 leaves only 4 shot numbers, so we are talking about an 8 to 1 ratio of misses vs. hits. However, the gain from the extra builders is no longer as great. Blue has already made one of the key points and he has an extra landing place on the bar point for safety. Thus 13/10, 13/9 gains relatively little when Blue isn't hit, while when Blue is hit the loss is still very severe. In this position I think that Blue will be more than 8 times unhappy to be hit as he will be happy to be missed, so I would definitely play the safe 13/6.
The same type of reasoning can be applied when examining the risk of leaving future shots. This is particularly true when gammons are at stake. Most players are far too conservative in these sorts of positions. They forget that it takes a big parlay for the shot to be left and hit in the first place.
Blue can play very safe with 5/1, 4/3, or he can take a checker off with 5/0. How dangerous is 5/0? For it to cost, the following parlay has to occur:
1) White has to flunk.
White flunks about 2/3 of the time. Blue rolls big doubles to leave that shot only 1/12 of the time. If that happens, White hits the shot about 1/3 of the time. If White hit he won't be able to claim immediately, but due to his recube vig he will be a very good favorite -- let's say White will win 4/5 of the time. So, the probabality of 5/0 costing is about:
2/3 X 1/12 X 1/3 X 4/5
which comes to about 1 in 68. This is not very likely. In the actual position the chance of winning a gammon is very small anyway, and while taking the checker off will improve that chance it won't improve it very much. This gain looks so small to me that it wouldn't make me 68 times as happy to take the checker off and be missed as it would to be hit and lose the game, so I would play the safer 5/1, 4/3. However if the gammon chances were slightly larger, that would swing the issue.
White now has two checkers outside, so Blue's gammon chances have gone up. Not by much, but in my opinion enough to make it worth the risk of playing 5/0. Now I do feel more than 1/68 happy when I don't get hit than I feel unhappy when I get hit and lose.
The above examples only scratch the surface of this sort of problem. As we have seen, any backgammon problem is essentially a balance of risk vs. reward, so this sort of analysis can be applied to all positions. Of course it generally can't be broken down so mathematically, since there may be too many variables to include. Still, just knowing the concept of deciding how happy you are when things go right after a risky play vs. how unhappy you are when things go wrong will very often lead you to the right conclusion. Our instincts about the game are usually pretty good. If we can use these instincts properly, we can find more winning moves.