Settlements
Oswald Jacoby and John R. Crawford, 1970
From The Backgammon Book, Chapter 13 (pp 181–186)

There is no rule that requires a backgammon game to be completed. The players may settle a game any time an agreement can be reached. Sometimes veteran players will call a game off entirely after the first couple of rolls because it looks as if it will be a long game and neither player has any advantage worth capitalizing on.

Or suppose the game starts with each player rolling 6-5 twice. Each man has brought his two back men to safety, and the man whose roll it is has a very slight advantage. He proceeds to dissipate this by rolling 3-1. This leaves him 4 points ahead, but it is the other player’s rolls. Since the average roll is 81/6 , the game is a tossup, and will depend entirely on who rolls the better dice. There will be no excitement or chance to use skill — so, unless the players start rolling the doubling cube over and over, they might as well just set up a new game.

A more frequent opportunity for settlement occurs when the game has been doubled several times and finally depends on the results of one roll. As an example, suppose the cube is at 8, and you have one man left on your three point and one man on your five point, while you opponent has two men left on his one point. It is your roll; to win the game you must bear off both men.

White to roll.

If you count up, you’ll find that you have fourteen possible winning and twenty-two possible losing rolls, and thus have somewhat the worst of the game. Your opponent now says, “I’ll take two points,” meaning he’ll settle for two points instead of playing it out and possibly winning eight points.

 11 12 13 14 15 16 21 22 23 24 25 26 31 32 33 34 35 36 41 42 43 44 45 46 51 52 53 54 55 56 61 62 63 64 65 66
14 rolls win
for White

It looks like a generous offer, but it isn’t anything of the kind. As a matter of fact, in almost all instances the correct settlement turns out to be mathematically less than you would expect.

Let’s work this out. Suppose you reached this position thirty-six times and played each game out. You would expect to win fourteen games and lose twenty-two for a net loss of eight games. Since the cube is at 8, your total expected loss in those thirty-six games would be eight times eight or sixty-four points, an average of one and seven-ninths points per game.

Thus, in this game a two-unit settlement is a trifle too much, and a one-and-one-half-unit settlement is a trifle too little.

When we settle games of this sort, we round off to a figure in dollars and cents rather than get involved in fractions of a point. In rounding off, we let the man who is ahead on the ledger take the worst of the settlement. Suppose that you were playing this game for a dollar a point. If you were ahead overall, you should pay two dollars; if you were behind, you should pay only a dollar and a half.

You may not want to get involved with settling games at all. In fact, we recommend that you leave settlements entirely alone unless you can be sure that you know exactly what you are doing.

Here is another example. The cube is at 32 on your side of the table. It is your opponent’s roll and his last two men are on the four and one points, while you have one man on your one point.

Black to roll.

He has twenty-nine winning rolls and seven losing rolls (check Table 4, or note that he loses if he rolls 3-2, 3-1, 2-1, or double 1).

 11 12 13 14 15 16 21 22 23 24 25 26 31 32 33 34 35 36 41 42 43 44 45 46 51 52 53 54 55 56 61 62 63 64 65 66
7 rolls lose
for Black

Thus, your net loss in thirty-six games will be twenty-two points. Since the cube is at 32, you multiply twenty-two by thirty-two to see how you would stand at the end of thirty-six games; this comes to 704 points. You now divide that total point loss of 704 by the total games involved, which is of course thirty-six. This will give you your expected loss per game, which comes to nineteen and five-ninths units.

If you were to offer to pay nineteen units in this position, the chances are that your opponent would refuse indignantly — not because the correct settlement would be a trifle more, but rather because he would feel that he was entitled to something like twenty-four to twenty-five units. In other words, he might also refuse a generous offer of twenty or twenty-one units.

Reverse the position, so that you become the man about to roll with that nineteen-and-five-ninths advantage. You might offer to take eighteen, seventeen, or even sixteen units, and your opponent would know that you were more than fair. If he did refuse your kind offer, it would probably be accompanied by some remark like “Shoot! I might as well be broke as the way I am.”

We saw an example of this recently. Black had doubled to 8 in a running game, and white had rolled very well and reached a position in which two men were left on his two point while black had two men left on his one point. It was white’s roll.

White to roll.

White was in a redoubling position. In fact, his position was so good that black barely had a proper basis for acceptance.

 11 12 13 14 15 16 21 22 23 24 25 26 31 32 33 34 35 36 41 42 43 44 45 46 51 52 53 54 55 56 61 62 63 64 65 66
10 rolls lose

The odds in white’s favor were twenty-six to ten, or just under three to one, and the correct settlement would have been seven and one-ninth units (twenty-six less ten, or sixteen times sixteen — the number of points the cube would be at — which comes to 256, divided by the number of games, thirty-six, equals seven and one-ninth.

White was fully aware of this, but he was well ahead on the score sheet and generously offered to take just six. Black had also figured out the proper settlement and knew that white’s offer was generous; nevertheless he replied, “I’d rather gamble the game out. Go ahead and redouble.”

White redoubled. Then he rolled 5-2 and collected the full sixteen units.

Other apparently simple one-roll situations may have complications. As an example, look at the following position

White to roll.

It is white’s roll, and he has a shot at the blot that black has been forced to leave on his five point. The odds against a hit are twenty-five to eleven, which means that in thirty-six games white would expect a net loss of fourteen.

 11 12 13 14 15 16 21 22 23 24 25 26 31 32 33 34 35 36 41 42 43 44 45 46 51 52 53 54 55 56 61 62 63 64 65 66
11 rolls hit

White has some slight chance to miss this blot and still win. He may stay off the board entirely this time and perhaps several more times, only to hit this or some other blot when he finally does come in. And he has a very slight chance to win in a running game: a roll of double 6 will put him almost even; so will coming in on any 6 if they are following by several big doubles. White also has some chance to lose a gammon. Roughly speaking, the gammon chance balanced against the other winning chances offset one another, and the proper settlement should be based on this one shot.

Assume the cube is at 4. In thirty-six games, white would expect to lose four times fourteen, of fifty-six points, an average loss of one and five-ninths points. At a dollar a point, a fair settlement would thus be a dollar and a half!

Change the black position a trifle by eliminating the man on the five point and two of the three men on the four point, leaving black’s blot on his four point instead.

White to roll.

Now the chance that white can win a running game has practically disappeared, and the chance that he will lose a gammon has considerably increased.

He will still win if he hits that blot, but the fair settlement at this point must take the gammon chance into consideration, so white should pay two dollars or even two dollars and a half to get out of his troubles.

Change the position gain by moving the two white men back from the white one point to black’s outer board.

White to roll.

White will still win if he hits black’s blot on his four point. He will almost surely be gammoned if he misses. His plus side is 11/36 times four (where the cube is at), or 44/36. His minus side is 25/36 times eight (the gammon certainty doubles the stake) or 200/36 and his net minus expectation is thus 156/36 or 4.33. Since there is some chance to miss hitting black’s blot and still avoid getting gammoned, a fair settlement is probably four units.

The following position arose in a five-handed chouette.

White to roll.

White was in the box and at one stage had redoubled to 16. Later on the captain wanted to double to 32. His three partners all objected, and, while the captain could have doubled without his partner’s consent, he offered to buy them out. They all sold their games to him at sixteen points each, and he doubled to 32. White took the double, and the game was actually for 128 since black represented four players. When the diagrammed position was reached, white doubled to 64 (representing a total now of 256), and black took the double.

 11 12 13 14 15 16 21 22 23 24 25 26 31 32 33 34 35 36 41 42 43 44 45 46 51 52 53 54 55 56 61 62 63 64 65 66
24 rolls hit

The take was correct. Actually, white should not have doubled, since he was only a two-to-one favorite to hit black’s man on the white twelve point. After the double, failure to hit black would cost him the game, since black would then control the cube and white could not afford to accept black’s double to 128; while if he had not doubled, he might miss and still win. Anyway, white did double, and so they were not playing for a total of 256 units. At this point, the other players insisted that there be a settlement.

Mathematically, white was entitled to one-third of 256, or eighty-five and one-third points. Black refused to pay that much, point out that he might still win even if white hit his blot. Finally, white accepted a very fair settlement of eighty.

## Midgame Settlements

You don’t have to settle an entire game. Sometimes when the cube has reached 16 or 32 the players may want to settle or perhaps call off part of a game. As an example, suppose that you have doubled to 32 in a complicated position, and then even more complications develop. You foresee all sorts of excitement ahead, including the possibility of a double to 64 being hurled at you. You offer to withdraw your last double.

This will have the effect of putting the cube back on your side of the table. Your opponent refuses this, but makes a counterproposition to let you turn the cube back to 16 provided it stays on his side of the table. This, of course, is a better proposition for him.

You now make a third proposal. Perhaps you suggest that the cube be turned to 16 and placed in the middle, so that either of you will have the right to double next. Or you might propose allowing the cube to stay on his side, provided that he will agree not to double for some specified number of rolls.

We have seen discussions of this sort go on for quite a while — and occasionally one player has said, “Let’s call the whole game off,” and had his offer accepted.

## Increasing the Stake

It is possible to turn the cube up by agreement. A typical situation would be when player is obviously thinking of doubling. His opponent says, “If you double, I’ll take it.”

The first player now says, “How about moving the cube to 2 and leaving it where it is?” — i.e., leaving it in the center of the board, and either player may double again later.

If his opponent accepts, it is the same as an automatic double.

## Beavers

Some backgammon players like to play that when a man doubles, his opponent has the right to “beaver,” or have the cube turned one extra notch. Beavers have no real part in backgammon, but they give desperate gamblers a chance to turn the cube over faster than otherwise.