Settlements and Fracture Points
Danny Kleinman, 1980
Vision Laughs at Counting, Vol 2, © 1980 Danny Kleinman

Many backgammon players disdain settlements. To an extent, as backgammon players they are also gamblers, willing to entrust their financial fate to the fall of the dice. I do not wish to dissuade backgammon players from gambling. Some willingness to gamble is a necessary ingredient for winning at backgammon.

If you are such a gambler, you may ask: “Why should I try to guess the value of my game? The dice will let me know if I just roll it out!” Despite this protestation, you do place a value on your game and you do accept settlements every time you play backgammon. When you take a redouble to 4, in effect you decline the settlement of 2 which a pass would have brought you. You also imply that you are at least somewhat of an underdog in the game — else you would have beavered.

Some cube turns occur in complicated, unresolved positions. The future course of the game is very much in doubt, and you must rely on judgment and guesswork in deciding how to respond. Other cube turns come in simple, resolved positions. For example, when you have just been hit and closed out on the bar (an easy pass). Or during a pure race or a holding game you may count to see just how close the race is before deciding whether to pass or take.

But often your toughest opponents will turn the cube at a fracture point. At a fracture point, though the position is unresolved, the very next shake will determine which of two or three divergent paths the game may take. The technique of fracture-point analysis provides fair settlement values for the game as well as sound cube decisions.

In this technique, you assume a cross section of 36 games played, 1 for each permutation of the dice. Then you divide the net total of points won in these 36 games by 36 to determine the fair settlement value.

Figure 1 illustrates a fracture point. For the sake of simplicity, we may assume that the player who hits a shot here wins the game. The occasional losses after hitting figure to be compensated by the occasional gammons to be won.

Figure 1
Black on shake

Figure 1 will resolved itself along three different lines. Black can roll a 1 and hit white’s blot. Black can miss this shot and fail to escape. Or black can escape with his laggard and convert to a pure race.

Now let’s make some rough estimates. Of the 36 games in the cross section, black figures to win 11 by rolling an ace. There are 7 shakes which keep black from escaping: 42, 32, 66, 33, and 22. Let’s estimate that black will win 2 of these 7 games. Finally, by virtue of owning the cube, black will be a slight favorite in the 18 remaining games where he escapes. Let us credit him with winning 10 of these. Thus black wins 23 and loses 13 games. His net (with the cube at 2) is 20 points in 36 games, so a fair settlement becomes 59 of a point to black.

What if black redoubles to 4? He’ll still win the 11 games where he hits with an ace, gaining 44 points. But in the 7 games where he fails to escape, he’ll have to take the cube back at 8. Now his 5 losses versus only 2 wins at the higher cube value cost him 24 points. Finally, with white owning the cube, black becomes a slight underdog in the remaining 18 games, winning only 8 but losing 10 and losing an additional 8 points. Netting all 36 games, we see that black wins 12 points in 36 games by redoubling. Since a cube turn would thus reduce black’s fair settlement value to 13 (from 59), the fracture-point analysis indicates that black must keep the cube.

In general, this is the proper mathematical approach to determine whether a given position is a double or a redouble. Compare the settlement value without the cube turn to the settlement value after a cube turn. This purely mathematical technique ignores a crucial factor, however: white’s cube psychology. Surely black should double if there is a real chance that white may pass.

In the actual game from which Figure 1 was taken, black doubled. A mistake? No, for white beavered. This made black’s game worth 23 of a point to him — better than the 59 it was worth before the cube turn.

Figure 1 poses a more difficult problem than most real-life settlements, in which the game can take only two different paths, not three. Most settlements are made on the very last shake of the game. Classically, settlements come at the end of the bear-off, and can be determined by a simple formula.

Suppose that with the cube at Q, you have W winning shakes and L losing shakes among the 36 permutations of the dice. Then your game is worth

 Q(W − L) 36
points.
Since L = 36 − W, we can also express this as
 QW 18
Q,
which reveals that you need to have 18 winning numbers to get an even game, and that each winning shake above or below 18 is worth exactly 118 of the cube.

Another common settlement situation arises from a back game. You own the cube, and have built and preserved your board sufficiently to win by doubling your opponent out if you hit a shot in time. Finally, you get the very last possible shot: If you hit you win, but if you miss, you surely get gammoned (losing a double game). Now your game is worth

 Q(W − 2L) 36
points. Again substituting 36 − W for L we arrive at the alternate expression
 QW 12
2Q.
Thus you need 24 winning numbers to get an even game, and each winning shake above or below 24 is worth exactly 112 of the cube.

On the other hand, you may be the player defending against the back game. Now the formula becomes

 Q(2W − L) 36
,  or
 QW 12
Q.
Your break-even comes at 12 winning numbers; again, with each winning shake above or below 12 being worth 112 of the cube.

Finally, on double-gammon swings, where whoever wins the game will also win a gammon, the formula is

 Q(2W − 2L) 36
,  or
 QW 9
2Q.
The break-even again comes at 18 winning numbers but each shake above or below 18 produces 19 of the value of the cube.

To apply any of these formulae, you must be sure that all the numbers the player on shake can roll truly guarantee the outcomes assumed. If not, you must fall back upon detailed fracture-point analysis in which you substitute your estimated net for the definite outcomes of any imperfectly decisive group of shakes.

Two warnings about settlements. First, you shouldn’t accept an opponent offer of a settlement if you feel there is something about the backgammon position you don’t understand. You should know enough about the particular position to feel sure that you are getting at least your fair settlement value. Second, you must be careful to include cube turns in your calculations.

Figure 2
Black on shake

Just as white picked up his dice from the last shake, white proposed that they settle this game. White offered to pay 6 points in Figure 2.

Black knew the settlement formula and proceeded to do his calculations. He had 25 winning numbers and only 11 losing numbers. His game was worth 718 of the cube. Seeing that 16 times 718 was somewhere between 6 and 7, black made a counteroffer to take 7 points.

White accepted cheerfully. For what white realized, and black overlooked, was that black had a redouble to 32. Black’s game was thus worth 32 times 718 points, or almost 12.5 points. And white, knowing the mathematics of it, would gladly have paid 12. White thus saved more than 5 points in the negotiations.

Did white cheat black? Not at all. It was black’s own ignorance or lack of alertness that cost him. Black was no more cheated than if he had simply rolled the game out without redoubling to 32. Black cheated himself out of the 5 points he lost in the “unfair” settlement. White was under no obligation to offer a fair settlement, only to bargain in good faith and without lying.

You can protect yourself against such unfair settlements by adopting a simple rule. If it is your shake at the time when a settlement is being proposed, take any cube turn you are contemplating first and only discuss the settlement after your opponent takes your double.