Taking Risks

Kit Woolsey Many readers will remember a television commercial for oil filters a few years ago. The punch line of the commercial was, "You can pay me now, or pay me later," which meant that the consumer had a choice of paying a small amount for an oil filter now, or a large amount for a ruined engine later.

It would not be surprising if the writer of that commercial were a backgammon player. The problem of whether to pay now or pay later — that is, whether to take a chance now or play safe for the roll at the possible cost of taking a greater risk later — recurs constantly in backgammon.

Television Ad

"You can pay me now ...            
      ... or you can pay me later." In the early stages of the game, when the opponent hasn't built up his board, the cost of being hit is small, so it is usually correct to "pay me now." For example: on an opening 2-1 roll most experts will slot the five point, since the cost of being hit now is much smaller than the cost of not owning the five point at a later stage of the game.

It is with bearing in positions, when getting hit will cost the game, that the most crucial pay-me-now or pay-me-later decisions arise. A player must often decide whether to take a risk on the current roll, with relative safety remaining if he gets away with it, or to play safer for now but leave a more awkward position with the likelihood of leaving later shots. Many good players misevaluate this type of position. They forget that the potential later shot still has to be hit to cost, so they tend to pay now when it is safer to pay later.

If we assume that getting hit loses and escaping wins, we must calculate for a given play the probability of getting hit either now or later. This figure is: probability of being hit on this roll + probability of being missed on this roll × probability of leaving shots(s) on future rolls × probability of future shot(s) being hit.

Example 1. Black to play 5-4

Look at Example 1. Many good players will clear the midpoint with 5-4, figuring that they would rather leave eleven shots now than seventeen shots later. However, a careful analysis shows that they are wrong. If we leave eleven shots now we will be hit with a probability of 11/36 or .306. (I assume, for simplicity, that surviving this roll is complete safety.)

Suppose we play safe, playing the number in our inner board. Let us assume we will have, on average, two rolls before the eight point must be cleared (this assumption is on the low side), and during this time we might roll double 2, 3, 4, or 5 for safety. (I realize that 2-2 and 3-3 are not completely safe, but I ignore 1-1, which is helpful as a compensating factor. I'm just trying to get an approximation, so I'm keeping the assumptions simple.)

So for each of the two rolls we have 8/9 probability of not safetying, or 64/81 chance of still being on the midpoint when the eight point is cleared. Once the eight point is cleared we have ten bad rolls and four good rolls, so the probability of leaving a shot is 10/14, and it is a seventeen-number shot, so the probability of being hit is 17/36. Consequently if we "pay me later," the probability of being hit is approximately 64/81 × 10/14 × 17/36 = .267, so this is the safer play. I realize that there are several things I haven't taken into account, but I tried to make my simplifying assumptions equally favorable to both sides to make a rough calculation possible.

Example 2. Black to play 5-4

Example 2 comes from Backgammon for Profit by Joe Dwek. Dwek suggests paying now, clearing the eight point and slotting three point, leaving a direct shot. The alternative is to move two checkers to the one point, leaving an extremely awkward position to bear in from. Dwek suggests the reader roll it out a few times to convince himself. I did just that (100 times each play) and became quite convinced — paying later was much safer! Even though Dwek's play is considerably safer if he gets away with it, Black is far from home free and is still quite likely to leave one or more shots in the subsequent bear-in. Since the "pay me later" play has to average almost one extra future shot to compensate for the immediate shot left by "paying me now," it is better to play safe for the roll.

Example 3. Black to play 6-5.

The problem also occurs when there is a question of how many shots to leave. Consider Example 3, which is from Magriel vs. Eisenberg in Backgammon with the Champions by Kent Goulding. Magriel played 13/7, 6/1*, getting Eisenberg off the one point by leaving fifteen return shots, while 13/2 leaves only eleven shots. Since Eisenberg will probably have a game-winning double if he hits any shot, we can assume that a hit is a loss. Is it worth the four extra shots?

I don't think so. Even if we assume that if Magriel doesn't ever get hit he will win, which is clearly not a valid assumption in this position, the safer play (for now) would have to leave a shot and have it hit at least 4/25 more often than Magriel's play to compensate for the immediate danger. This would involve leaving a direct shot almost 40% more often, which seems pretty high to me.

Look at it as a proposition in the following form: Would you take Magriel's play and play it play-vs.-play twenty-five times each way on the assumption that the initial shot was missed, but spot your opponent four wins? This is essentially what Magriel's play does.

Now, when is it correct to "pay me now"? If your opponent hasn't built up his board, it may be correct in any of the previous positions. If his board is built up it is almost never correct to leave an unnecessary direct shot, unless doing so otherwise improves your winning or gammon chances (i.e., there is more involved than just safety). However, it may be correct to leave a few extra shots for future safety, if playing safer now is likely to lead to later problems.

Example 4. Black to play 6-2.

Consider Example 4. Black can either clear his eleven point, leaving sixteen shots now but virtually no later problems, or clear his bar point leaving eleven shots now and the eleven point to be reckoned with later. If he clears the eleven point he gets hit 16/36 = .444. If he clears the bar point he is immediately hit 11/36. Of the remaining 25/36 he will have five "good" numbers (double 1, 2, 3, 4, 5) and eight "bad" numbers (6-1, 6-2, 6-3, 6-4). Therefore he will leave a shot 8/13 of the time, and it will be a seventeen-number shot. (I realize that there are other variations possible, but this calculation takes the major factors into account.) Therefore, his probability of being hit is: 11/36 + (25/36 × 8/13 × 17/36) = .507, so it is proper to "pay me now." Experts are familiar with this type of position, but they would probably be surprised that the decision is that close. In fact, if the opponent's board were about to crunch in a couple of rolls, it would be better to "pay me later."

Example 5. Black to play 4-2.

Another "pay me now" position comes up in Example 5. Black could play safe by bringing in the man on the eight point, but this would probably lead to an awkward bear-in later on. It would be impossible to prove without a computer, but experience has taught us that it is better to clear the bar point and pay off to the 17-to-1 shot to achieve the smoother position.

The concept may apply to potential shots as well as immediate shots. Here, it is often correct to pay now. Consider Example 6.

Example 6. Black to play 6-2.

Black can take two men off, paying off only to 6-5, or he can come down to two men on the six point, leaving a shot on both 6-1 and 5-1. It is correct to "pay me now" by coming down to two men on the six point. The reason is that, unless Black rolls doubles, he will eventually have to come down to two men on the six point anyway. Consequently, the "safer" play is likely to have to pay off to both 6-5 now and 6-1 and 5-1 later, so to avoid this double jeopardy Black should get it over with quickly.

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