position that we can draw conclusions as
to his designs and will, therefore,
— Karl von Clausewitz
Settlements are an important strategy in all money games, and though they are used in European tournaments, they are not as yet allowed in the United States.
A settlement is a compromise negotiated between the opposing sides, wherein one army agrees to give up a portion of the disputed stake in return for an immediate end to the hostilities. It is a complicated form of bargaining in which one faction attempts to buy off the other by saying, in effect, “Surrender this much now or ultimately I may take it all.” Settlements are the game’s politics.
Given the esoteric mathematical hagglings such negotiations involve, many players of backgammon have chosen the simplest course: they never settle. One man of our acquaintance plays regularly in a running chouette. He is a better than average player but knows nothing about this facet of the game and therefore has adopted the tactic of refusing all settlements, regardless of how attractive they may seem. By refusing settlements he reasons that he will at least break even. It is sound strategy, since he is incapable of determining how much he should give or take in specific situations.
|‘||Many players of backgammon have chosen the simplest course: they never settle.||’|
But the art of settlements is not as difficult as it may at first appear, and since these proposals to compromise usually occur at crucial moments in the game, a rudimentary understanding of how they work can be beneficial.
Settlements may crop up in the middle of the game, but normally they are made toward the end, when the game suddenly takes, or seems about to take, an unexpected turn from certain victory to defeat, or from defeat to possible victory. During these crucial moments, when, for example, the player in the stronger position sees that his superiority may be undermined, or the player in the weaker position can turn the game around by rolling one specific number, players may attempt to reach an immediate compromise rather than risk the whole stake. If a player can determine the precise odds against his winning or losing, in situations of this kind it is not difficult to compute what percentage of the stake he should either take or give.
Calculating a Fair Settlement
In all settlements there is a basic formula to use. Take the recurring position in which black is bearing off and leaves a blot which if hit will give white the game. It is a 25 to 11 shot in favor of black. The doubler is at 32. Since white is the underdog, how much should white give?
On an average of 36 games, given the odds, black will win 25 and lose 11, for a net gain of 14.
|Black wins 25 times:||25||games|
|Black loses 11 times:||−11||games|
If you multiply 14 (his net gain) by 32 (the stake), you get a figure of 448. Dividing this figure by 36 (the number of games played), you get an exact settlement of 12.44.
|Total points gained over 36 games = 14 × 32 = 448 points|
|Average points gained per game = 448 ÷ 36 = 12.44 points|
Since 12 is the nearest whole number, 12 is the correct settlement that white should give to black. You will note that the underdog gets a tiny edge here, since he should give 12.44, but this kind of fraction is generally overlooked.
But just because you know the fair settlement is 12 in no way commits you to negotiate that figure. Though it helps if you at least know what the fair figure is, it is not considered unethical to ask for more if you are black, or to offer less if you are white. Both sides are bargaining. Sometimes unfair settlements are intentionally offered; since the less skillful player is either ignorant or confused by settlements, he may accept a lesser amount or surrender a greater amount than the correct figure.
|‘||Just because you know the fair settlement in no way commits you to negotiate that figure.||’|
There have even been cases in which the less skillful settler, having been offered more than he deserved, has demanded even more than that. In a money game played a few years ago, a position arose in which black had four men remaining on his 1 point and white had two.
This position is illustrated below. Black needed any double in order to win the game. The doubler was at 64 on black’s side and white offered a settlement. What would the correct offer be?
The cube is at 64.|
How much should black pay to settle?
Thirty shots lose for black and six (the six doubles) win, so he was exactly a 5 to 1 underdog. Again, using the formula — in 36 games, white will win 30 and lose 6 for a net gain of 24. As before, multiply 24 by 64 (the stake), which is 1,536. Divide by 36 (the number of games played) and we get 422/3 , or, to the nearest whole number, 43.
But this was the last game, and for other reasons white was not prepared to court disaster. Thus, in a burst of generosity, he offered to take only 32, 11 points less than he was technically entitled to.
Black was behind and steaming. He considered the proposal, looked up and said, “You want 32? I’m not giving you anything. As a matter of fact, I’m going to double you to 128.” White was amazed, but he shrugged and accepted the double. Black then rolled double 4’s to win the game, and white, for the hundredth time, wondered why he had ever become involved in this cruelest game.
Settlements in Tournaments
To date, settlements are not permitted in American tournaments, which is unfortunate, for they are a definite part of the game. Should beginners find them difficult, they have only to refuse. In Europe, however, they are allowed and contribute greatly to the game.
Recently, in an important London tournament, black, an experienced and wily player, was pitted against a comparative beginner. They were playing a 15-point match with no Crawford Rule, and white was ahead 13–12. The doubler was at 2 on white’s side, and the match had progressed to a critical position in a critical game.
MATCH TO 15
||Black to roll.|
At this juncture, black, whose turn it was, said, “I’ll take 1 point.” The offer looked and seemed reasonably fair, since white had to roll a double to win. But white was uncertain, and after thinking about it declined the settlement.
The game continued and after four rolls by black and three by white, neither black nor white having rolled a double, black halted play before white’s roll and again made the same offer.
MATCH TO 15
||White to roll.|
“Perhaps I’m foolish,” he said, “since it’s three rolls later and I’m in much better shape than before. I’m probably getting the worst of it, but I’ll still take 1 point.” On this occasion white, seeing that black would be off in two rolls and that he still needed a double to win, relented; he agreed to the offer of the point and the score became 13–13.
Before reading on, try to determine which of the two got the better of the deal. Was it unfair or fair? It is, in fact, probably the most outlandish swindle imaginable. But why?
Consider the situation. There is no Crawford Rule, so there are no restrictions on the doubler. They are playing to 15 and white is ahead 13–12. If the doubler had been on black’s side in any money game in this position, it would be a clear-cut drop. Also in any money game, an offer to take 1 by black should be promptly accepted by white.
But in this special instance, white has virtually nothing to gain and everything to lose by conceding a point. What he has failed to comprehend is that this is the game that can win the match. Had white played out the game and lost, the score would have been 14–13 against him. This is no worse than being 13–13, which the score became when the offer was accepted.
|‘||White has virtually nothing to gain and everything to lose by conceding a point.||’|
Behind 14–13, white will certainly double on the first roll of the next game. (You will remember that to be ahead 14–13 is slightly better than being at 13–13, because when your opponent doubles on the first roll of the next game, which he will do, you have the choice of dropping if you wish.) Except for this slight compensation, however, the loss of 1 or 2 points had no bearing on the outcome of the match, and thus for white to concede 1 point and give up the possibility of rolling a double to win the match was lunacy. It is hard to conceive of a more lopsided settlement in favor of black.
Incidentally, there was nothing unethical in black’s proposal. It was little more than a political gesture which white was at liberty to accept or refuse. This is what bargaining is all about. The point to remember here is that regardless of how persuasive your opponent is, if you know you are receiving the best of it, accept; if not, decline.
To be able to settle well is a form of money management, and the judicious control of one’s money is essential in all games of chance. Oddly, correct money management often goes against the odds — and rightly so.
For instance, assume that you have $10,000 to your name and no other assets of any kind. You are approached by a man with $20,000 who requires an additional $10,000. He offers to bet his money against yours on the flip of a coin. It is an even-money bet. Would you accept? It is the sort of situation bookmakers and professional gamblers dream about, for you are getting 2 to 1 on an even-money bet.
|‘||There is another axiom which states: when you are down to your case money, never bet it all.||’|
But though the bet adheres to the normally excellent axiom which states that when you have the best of a proposition, bet all you can, it would be folly to accept the challenge. There is another axiom which states that when you are down to your case money, never bet it all on one roll, regardless of the odds you get. Today is important in the world of chance, but not when it eliminates tomorrow.
This is an example of wise money management ignoring the odds. This sound principle is applied particularly in places like Las Vegas, where casinos not only have the best of the odds, but limit the size of your stake as well. It is a sensible practice, since it would be possible for someone on a streak actually to break the house. The limit which the casinos impose on their clientele is an essential form of money management.
In backgammon the same principle should be applied. Be wary of games that become so high and wild that in order to manage your money properly you must take the worst of the odds. Suppose, for example, that you are in a five-handed chouette at $5 a point, a stake somewhat higher than you usually play for, and that the doubler has somehow been turned and turned to 64. You are in the box and are feeling distinctly uncomfortable. While bearing off, you leave a blot. Your opponents have a single shot; it is 25 to 11 against their hitting you, but if they do, they win the game.
||White to roll.|
At this point they offer you a settlement. The correct settlement in this example is 25, but you would take much less; in fact, you would take almost anything you could get. By allowing this kind of situation to develop, you have squandered valuable equity. Avoid these predicaments and play only in games where settling depends exclusively on the position and not on the financial catastrophe that would occur should your opponents obtain the right roll.
In backgammon there are certain “insurance” situations in which players will intentionally take the worst of the odds. This is most frequent in the final matches of a tournament when one player reaches a position whose odds can be calculated exactly.
An example of this occurred in the finals of a recent championship match. Each player had 16 points in a 17-point match, and victory would be decided on the final roll of the final game. In that final position (see Diagram 78) black, an Englishman, had two men left on his 1 point. White, an excellent player from the Far East, had four men left, one of them on his 6 point. It was white’s roll, and so he needed double 6’s to win the first prize of $6,500.
MATCH TO 17
||White to roll.|
At this point, black stopped the game in order to take insurance against the possibility of white rolling double 6’s — the odds of which, you will recall, are 35 to 1 against.
Black extracted a $100 bill from his pocket, turned to the crowd of spectators and asked what odds he could get. One spectator said he would give black 30 to 1 against the double 6’s, and black agreed. Another offered black 31 to 1 against for a further outlay of $100, and again black agreed. Of course black knew he was getting the worst of the odds, but in this case it was worth it as insurance against the potential disaster of double 6’s.
It was white’s roll, and he promptly threw double 6’s to win the match. By losing, black collected not only the $2,500 second prize, but also a total of $6,100 from the two spectators — $2,100 more than he would have received had he won the tournament. As the crowd surged round the table congratulating the players, a woman turned to another spectator and asked, “Aren’t the true odds against double 6’s 35 to 1?” “No,” the man said, “not when a Chinaman’s rolling.”