The Price Of Gammons
Danny Kleinman, 1980
Vision Laughs at Counting, Vol 2, © 1980 Danny Kleinman

We need to know the price of gammons in four situations.

In money games, the price of gammons is easy to compute. Because of the Jacoby rule eliminating gammons with centered cubes, the price of gammons is zero prior to any cube turn. Then it becomes exactly 12, for the swing of gammon versus plain game is the size of the cube, while converting a loss into a win gains twice the size of the cube.

We can apply this factor in all four situations. When we elect to risk getting gammoned, we need at least half as great a chance to win the game to justify this. When we get greedy and try to gammon our opponents, the extra gammon chances we create should be at least twice as great as our extra risk of losing the game.

When our opponent turns the cube, we should subtract from our winning chances half of the probability that our opponent will gammon us, and add half of the probability of any gammon threat that we have.

Finally, when we have been too good to double because we are playing for the gammon, we should give up on the gammon and {double our opponent out=double out} as soon as the chance of losing the game looms as much as half as large as our gammon chances.

Complications arise inn tournament matches. Unlike in money games, each point won or lost at each score has a different value. Therefore our unit of measure cannot be merely the points put at stake, but must be percentage increments in the chances to win the match as a whole, the match-winning equity.

Computations of the price of gammons depend on estimates for match-winning probabilities at various scores. And the price of gammons not only differs between the two opponents, but from one turn of the cube to the next. Only when a match still has a long way to go, and at relatively low cube sizes, can we assume the the price of a gammon is its regular money price of one half.

Since the Jacoby rule does not apply to matches, a centered cube does not make the price of a gammon zero any more. But there are free gammons. When winning the game at the current cube size suffices to win the match, then the price of a gammon is zero. These gammons are free to the trailer in the match.

The leader in the match as a different kind of free gammon to give: During his Crawford game with a lead of more than one point, or during the Crawford game with a lead of an odd number of points, the trailer’s gammons have a very high price, very nearly as high as 1. For the gammon essentially doubles the trailer’s chances of winning the match, reducing the number of future games he must win by 1.

Gammons which suffice exactly to win a match are extremely valuable. But gammons which win the match with a surplus of points are less valuable.

Suppose that the match is to 9, and that one player has 6 points. In one instance we may suppose that the player trails 7–6 and has doubled. His gammons are worth less than in money play. In another instance we may suppose that the player leads 6–5 and has been doubled. His gammons are again worth less than in money play.

Another consideration affecting gammon prices is the odd-even factor. In a 9-point match, for example, an 8–5 lead entering the Crawford game is far more commanding than an 8–6 lead. This means that it is somewhat more important to gammon your opponent at the 1-level with a 6–5 lead than at a 6–6 tie score. Conversely, with a 7–6 lead it is more important to win the match by gammon your opponent than with a 7–5 lead.

If winning a plain game brings you to the Crawford game with an even number of points in your lead, this inflates the price of a gammon. If winning a plain games brings you to the Crawford game with an odd number of points in your lead, this depresses the price of a gammon.

Once we have estimated gammon prices in match situations, we can use them in deciding how to move, whether in gammoning our opponent or in averting his gammon threats. We can use the same gammon prices when — at the last possible moment, of course — we must choose between doubling our opponent out or playing on for the gammon.

But in our pass-or-take decisions, we must be careful to use the gammon prices for the new level to which the cube is being turned, not for its prior level. Thus the price of a gammon in a 2-point match is very high — .750 — but a cube turn is gammicidal, eliminating the value of gammons entirely.

In contrast, at the start of a 4-point match, gammons have a fairly ordinary price of .529, but once the cube is turned to 2, gammons become match winners and should be priced at .927.

To illustrate the method of determining gammon prices, we may examine the 1–0 score in a 3-point match. We start with some estimates: A tie score provides a 12 chance of winning the match. A 2–1 lead entering the Crawford game provides a 710 chance. And a 2–0 lead entering the crawford game provides a 34 chance.

With the cube centered, let us assume that the player leading 1–0 is about to win a plain game but can risk losing in order to try for the gammon. Instead of a 34 chance by playing safe, he can either win the match outright or reduce his chances to 12. He risks 14 match equity to gain 14. This makes the price of a gammon 1.

Now let us chance this by assuming it is the player trailing 1–0 who can try for the gammon. Instead of a 12 chance by playing safe to tie the score, he can either win a gammon and take a 2–1 lead for a 710 chance or lose the game and fall behind 2–0 for a 1/4 chance. He risks 14 match equity to gain 15. This makes his gammon price .800.

Finally, if the cube has already been turned to 2 by the player trailing 1–0, then the leader’s gammons are reduced to meaninglessness. But the price of the trailer’s gammon is reduced somewhat too. Sacrificing the win of a plain game to try for a gammon can win or lose the match, playing safe to win the game provides 710 match equity. Thus trying for the gammon risks 710 equity to gain only 310. The price of a gammon at the 2-level for the player trailing 1–0 in a 3-point match is thus 37, or .429.

We can use this method to find the gammon price for any score and any level of the cube.

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