Playing for Gammon

 Gammon Price Douglas Zare, 2001

 This article originally appeared in GammonVillage, November 22, 2001. Thank you to Douglas Zare and GammonVillagefor their kind permission to reproduce it here.

 I’m a professional mathematician and an expert backgammon player (my nick is “zare” on GamesGrid, FIBS, and GameSite 2000). In this column, I will discuss theoretical aspects of backgammon, but not just for their own sake. Better understanding of the theory of the game can help us to play better and to appreciate the game more. I will never recommend that you perform elaborate mental arithmentic at the table. I will also not require you to be comfortable with higher mathematics or to be able to plow through complicated calculations in order to follow this column. I’m writing for fellow backgammon players, not mathematicians.

 Contents Standard Definition of Gammon Price Gammon Prices in a Match to 7 Applications to Positions Cubeless Equity and the Gammon Price for Money Earlier Work, References, and Loose Ends Summary

Let’s consider the gammon price. We’ll look at the standard definition of the gammon price, how you can use the gammon price, and why the standard definition is wrong.

The gammon price is a useful idea. It tells you how valuable it is to win a gammon at the current level of the cube. Of course, if you can win a gammon rather than a single win, you would always choose to do so. The question is how valuable it is relative to the value of winning a single game rather than losing. Sometimes one move is clearly best because it wins more and wins more gammons, but often one has the chance to risk some wins in exchange for some gammons. The gammon price suggests how many wins you can accept converting to losses in exchange for each win that becomes a gammon.

### Standard Definition of Gammon Price

Suppose one play wins 100% of the time with no gammons, and another wins 90% of the time, including 15% gammons. Which play should you make in money play? What about if you are trailing 3–1 in a 5-point match, and you have already doubled to 2?

The standard definition of the gammon price is the difference between the payoff when you win a gammon and when you win a single game, divided by the difference between a single win and a single loss.

 gammon price  = gammon win − single win single win − single loss

For money, if the cube is at 2, when you win a gammon you win 4 points, when you win a single game you win 2 points, and when you lose a single game you lose 2. So the gammon price is

 4 − 22 − (−2) = 24 =  0.5.

This suggests that in order to get 15% extra gammons, you should be willing to give up only 15% × 0.5 = 7.5% wins. You should make the safe play.

Trailing 3–1 in a 5-point match,

• if you win a gammon, you win the match (100% match winning chances, or “mwc”),
• if you win a single game, the match is tied (50% mwc against an equally skilled opponent), and
• if you lose, you lose the match (0% mwc).
So the gammon price is

 100 − 5050 − 0 = 5050 =  1.

In order to get 15% extra gammons, you should be willing to give up 15% × 1 = 15% wins. You should go for the gammon.

### Gammon Prices in a Match to 7 According to Snowie

 1-Cube Opponent Needs 1C 2 3 4 5 6 7 You Need 1 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2 1.00 0.85 1.02 0.80 0.88 0.70 0.80 3 0.03 0.75 0.82 0.94 0.66 0.68 0.60 4 0.54 0.42 0.57 0.65 0.62 0.63 0.66 5 0.02 0.39 0.44 0.49 0.49 0.50 0.51 6 0.65 0.49 0.50 0.55 0.53 0.56 0.57 7 0.03 0.48 0.49 0.55 0.52 0.54 0.53

Here the first column is the Crawford game. It’s worth nothing extra to win a gammon if you need only one point to win the match. It’s valuable to win a gammon in the Crawford game if this decreases the number of games you have to win afterwards assuming that you double immediately, Crawford even-away, and not particularly valuable at Crawford odd-away.

I wouldn’t memorize this table, but it might be good to note that the gammon price on a 1-cube is often higher for the leader than the trailer when the leader needs 2, 3 or 4 points. Towards the start of the match, the gammon price on a 1-cube is close to that for money. It’s good to be aware of exceptional match scores such as 3-away 4-away, and to recognize the general pattern for the other scores.

 2-Cube Opponent Needs 1PC 2 3 4 5 6 7 You Need 1 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2 0.00 0.00 0.00 0.00 0.00 0.00 0.00 3 1.00 0.46 0.51 0.44 0.46 0.34 0.35 4 1.06 1.00 0.98 1.02 0.87 0.72 0.65 5 0.59 0.69 0.72 0.81 0.69 0.66 0.60 6 0.58 0.49 0.52 0.56 0.58 0.56 0.55 7 0.69 0.53 0.55 0.56 0.57 0.56 0.55

Here the first column is post-Crawford. The rest of this table is particularly important when one decides whether to accept an initial double. The gammon prices appear to be roughly equal to that for money when you have 3 points to go (except for post-Crawford). If you have 4 points to go, then the gammon price on a 2-cube is significantly greater than for money.

In general, the gammon price on a 2-cube is slightly higher than for money even at the start of a match to 7. While the gammon price is high while trailing 2-away 4-away, at 2-away high-away, the gammon price is not far different from the gammon price for money.

 4-Cube Opponent Needs 2 3 4 5 6 7 You Need 5 0.46 0.34 0.22 0.23 0.17 0.17 6 1.00 0.68 0.51 0.42 0.36 0.31 7 1.46 1.00 0.75 0.64 0.55 0.48

There should be no 4-cube post-Crawford. With 4 or fewer points needed to win the match, the gammon price is 0. The gammon price is high when winning 4 points will not make you a strong favorite in the match, but if you are already ahead in the match but losing 4 points would make you trail, then the gammon price is low.

As an example of how you might use the gammon price in match play, let’s suppose that you lead 3–1 in a 5 point match. You are 2-away, and your opponent is 4-away from winning the match. Your opponent doubles to 2. For you, the gammon price on a 2-cube is 0. If you accept the double, gammons are worthless for you. However, your opponent’s gammon price on a 2-cube is 1. Gammons are very valuable for your opponent, and very damaging to you. So, if you guess that your opponent is likely to win 65% of the time including 20% gammons, that is equivalent to winning 65% + 1 × 20% = 85% of the time, and you should pass.

This is an alternative to remembering the match equities to 3 digits of accuracy and performing computations on them: You can remember the take points and the gammon prices instead. A position with these statistics might be a clear take for money, and might not be a double, but it is a pass at this match score because of the high gammon price. Even if you can’t associate precise numbers to a position, you can still be more wary of accepting a gammonish double if the gammon price will be high after the cube is accepted.

### Applications to Positions

The positions considered in this article are towards the end of the game. The gammon price is applicable much earlier, but it simplifies the discussion if we only have to focus on one of gammon losses or gammon wins. In early positions we would have to consider both at the same time.

 13 14 15 16 17 18 19 20 21 22 23 24 12 11 10 9 8 7 6 5 4 3 2 1
Black, trailing 135–72 in the race, has a 5-2 to play.
Black is hopelessly behind in the race. The natural alternatives are 23/16, giving up on winning but almost certainly saving the gammon, and 14/9, 13/11, which is often attacked, closed out, and gammoned, but produces immediate direct shots on 6-3 and 6-5, and perhaps later winning shots.

We can ask Snowie what it believes to be the right play. Snowie evaluations are good, but not as good as Snowie’s rollouts. There are some difficulties in deciding which rollouts to perform and which to believe, and I’ll return to that issue in another column, but let’s assume that the following values from rollouts are accurate at all match scores.

 Win Lose BG G Game Game G BG 23/16 0.0 0.0 0.3 99.7 6.4 0.0

This means that after 23/16, you win no gammons or backgammons, and win 0.3% of the games, while losing 99.7% of the time including 6.4% gammons.

 Win Lose BG G Game Game G BG 14/9, 13/11 0.0 0.8 11.0 89.0 41.4 0.1

After 14/9, 13/11, rollouts indicate that you win 11% of the time including 0.8% gammons. You lose 89% of the time, including 41.4% gammons or backgammons, including 0.1% backgammons.

The difference between the plays is

 Win Lose BG G Game Game G BG Difference 0.0 −0.8 −10.7 10.7 −35.0 −0.1

Ignoring gammon wins and backgammon losses, the effect of staying back is to lose 35.0% extra gammons in order to win 10.7% extra games. This should be worthwhile if your opponent’s gammon price is at most 10.7/35.0 = 0.31.

Suppose the cube is at 2. Then by the second table above, your action should be to run at every match score where gammons matter, since all of the entries are at least 0.34. It might be that trailing 3-away 6-away and 3-away 7-away would be exceptions if you have access to the doubing cube.

If the cube is at 4, then you can only stay when gammon losses matter at n-away 5-away for n at least 4, and possibly at 7-away 6-away. At other scores, either gammons don’t matter and it is obvious to stay, or else it is too costly to stay, and you should run to save the gammon.

For money play, you should run.

 13 14 15 16 17 18 19 20 21 22 23 24 12 11 10 9 8 7 6 5 4 3 2 1
Black to play 1-1.
Here, you can play very safely with 5/2, 1/off, or try to get a few more gammons with 5/4, 1/off, 2/off. Snowie rollouts indicate that the latter play risks 3.6% wins to gain 5.1% extra gammons. So 5/2, 1/off would be appropriate any time the gammon price is less than 3.6/5.1 = 0.71. For money, you should play safely. If you trail 4-away 3-away, then you should go for the gammon.

Suppose in this position the score is 4-away 5-away. Should you go for the gammon? The gammon price, whether you are leading or trailing, is over 0.8, so surely it is correct to play for the gammon, right? Not neccessarily. There is a problem with the standard definition of the gammon price when the cube is not dead, and this problem shows up even in money play.

### Cubeless Equity and the Gammon Price for Money

The cubeless equity of a position is

 −3 × backgammon losses −2 × gammon losses −1 × normal losses +1 × normal wins +2 × gammon wins +3 × backgammon wins.

The cubeless equity is often mentioned, but what is important in real backgammon is the cubeful equity, the value of the position with the doubling cube in play. Sometimes people say that a cubeless equity of about 0.550 represents a borderline pass. Is there a way to convert a cubeless equity to a cubeful equity, so that 0.550 cubeless corresponds to 1.000 cubeful if you have access to the doubling cube?

No. There are ways of converting the estimated number of single, gammon, and backgammon wins and losses into a cubeful equity. One of the best is Janowski’s formula, a version of which is used by Snowie. However, there is no way to convert a cubeless number to a cubeful number because the gammon price is not 0.5!

Consider the following possibilities:
 13 14 15 16 17 18 19 20 21 22 23 24 12 11 10 9 8 7 6 5 4 3 2 1
Black to roll.
In this simple position, the bottom player always loses a single game. The cubeless equity (normalized) is −1. The cubeful equity is −1. If you play this position 100 times, you will lose 100 times the value of the cube.
 13 14 15 16 17 18 19 20 21 22 23 24 12 11 10 9 8 7 6 5 4 3 2 1
Black to roll.
In this position, the bottom player is an underdog, losing a lot of gammons, but winning a few games. The cubeless equity, according to Snowie evaluations, is −1.001. However, the same evaluations indicate that the cubeful equity is −0.898. If you play this position 100 times, you will lose only about 90 times the value of the cube. (Rollouts give different numbers, but at any rate the actual value of the position depends on how well you can play the resulting positions.) What is the difference between this position and the previous one?

The value of the position is about −1 times the current value of the cube if the bottom player promises never to double. However, if the bottom player turns the game around, he or she gets some value from owning the doubling cube. In fact, over 30% of the time, the bottom player is eventually able to redouble, either after hitting the initial shot or a subsequent one.

Is there a way to fix this problem, that one cannot convert the cubeless equity to a cubeful equity? Yes and no. A gammon price of 0.5 corresponds to no value from the doubling cube. We could assume that any redoubles are perfectly efficient: One never wins the game while holding the cube, and one always is able to offer a double that is precisely on the borderline of taking and passing. In that case, after some additional simplifying assumptions the gammon price is 2/5 = 0.4.

In actual backgammon, we are between these cases. The cube has some value, but doubles are inefficient, and sometimes we overshoot, or become too good to double, or can’t double on the last roll of the game. The real answer should be between 0.5 and 0.4. It depends on the position, and we can’t calculate the gammon price abstractly. A reasonable guess in this and many similar situations is to assume that the answer is about 70% of the way from the cubeless value to the perfectly efficient value. So the gammon price for money in most situations is about 0.43. More complicated adjustments need to be made to the gammon prices in match play at every match score and cube value in which the cube is not dead (a player might double).

When you woke up this morning, you might have thought the gammon price for money was 0.5. Now suppose you believe the gammon price is 0.43. How does this make you a better player? After all, most players don’t produce a numerical estimate of gammon wins for each alternative, and certainly not at the level of accuracy where the difference between 0.43 and 0.5 matters.

I think most backgammon players start with money play, and make their decisions in match play relative to how they would play for money. You might hear/calculate that the gammon price trailing 2-away 6-away on a 2-cube is 0.49 by Snowie’s table (winning a single game gets one to 2-away 4-away, which is surprisingly good for the trailer). If the gammon price is less than for money, you might want to go for the gammon less. However, 0.49 is greater than the correct gammon price for money, 0.43, so you should play more aggressively for the gammon than for money play. In particular, in the following position, Snowie rollouts say that you should play safely for money, but aggressively after doubling at 2-away 6-away:
 13 14 15 16 17 18 19 20 21 22 23 24 12 11 10 9 8 7 6 5 4 3 2 1
Black to play 6-4.
There are 2 natural candidates: 8/2, 6/2, which makes a full prime, and 7/1*, 5/1, which tries for an immediate closeout, but leaves 4 lethal return shots.

If you close out the checker immediately, then you should win a bit more than 50% gammons since your opponent has 8 extra crossovers to bear in. If you wait a turn or two before attacking and closing out the checker, then you will win fewer gammons, as more outfield checkers will have been brought home.

Here are the results of rollouts by Snowie:

 Win Lose BG G Game Game G BG 8/2, 6/2 0.1 26.7 93.0 7.0 0.2 0.0

That is, you should win 0.1% backgammons, 26.7% gammons and backgammons, and 93.0% wins of all types. You should lose 7.0% of the time, including 0.2% gammons.

 Win Lose BG G Game Game G BG 7/1*, 5/1 0.1 43.4 85.6 14.4 0.5 0.0

As expected, hitting risks losing more, but wins more gammons.

So the difference between the two plays is

 Win Lose BG G Game Game G BG Difference 0.0 −16.7 7.4 −7.4 −0.3 0.0

Snowie rollouts indicate that playing safely wins 7.4% more games, but wins 16.7% fewer gammons. 8/2, 6/2 also loses 0.3% fewer gammons. Let’s ignore the difference in gammon losses, and the fact that in the rollout, Snowie used some assumption of how to play for gammons in the future. At which match scores should one make the risky play, and at which match scores should one make the safe play? What about for money? The first approximation is that one should make the risky play if the gammon price is greater than 7.4/16.7 ≅ 0.443.

Snowie rollouts indicate that for money, one should play safely, and that 8/2, 6/2 is better than 7/1*, 5/1 by 0.014. (Incidentally, this is true despite the fact that the cubeless equity is higher for 7/1*, 5/1.) On the other hand, at 2-away 6-away, you should attack, and this is better by 0.017. If you assumed that because the gammon price of 0.49 is less than 0.5 that you should play even more conservatively, you would be wrong, though not horribly so.

A more common match score in which this arises is 2-away 3-away after the trailer doubles. According to Snowie’s match equity table, the gammon price is 0.51, but according to other tables the gammon price is as low as 0.43. It would be an error to play more conservatively for the gammon at this match score than for money.

### Earlier Work, References, and Loose Ends

That the gammon price for money is not 0.5 is contained in Janowski’s formula, and also implemented in Jellyfish. That the gammon price is less than 0.5 is why one can take (correctly) blitzing cubes in which the cubeless equity is below −0.6, even though −0.55 is a reasonable threshold for races, and it is almost never correct to take in a race where the cubeless equity is below -0.6.

More complicated adjustments can be made to the gammon prices at match scores and cube values where the cube is not dead. Snowie makes some adjustments, but I believe Jellyfish does not, and that this is one of the reasons Snowie is better at match play than Jellyfish. On the other hand, when one performs a cubeless rollout with Snowie, I’m not sure whether it uses a gammon price of 0.43 or 0.5, and neither of these might be appropriate for a position from a match.

Additional adjustments need to be made when you are too good to double, and when large swings result in overkill, merely increasing the size of a pass. In each of these cases, a large change in the distribution of gammons and wins may result in a small change in the equity, and it could be that you should go for the gammon more as a result, or less.

In the bearoff position with a 1-1 to play, if Black owns a live cube but is too good to double, then Black can play more aggressively than the usual gammon price indicates. The reason is that the down side of getting hit is less than normal: It hurts to drop from being too good to double to being on the borderline of being too good to double; it doesn’t hurt to drop from borderline too good to double to a borderline take/pass decision, but then it hurts to drop more. So the large drop from getting hit is not as large as one might expect because the middle part of the drop doesn’t affect the equity.

You can think of the cubeless distribution after 5 turns as a payoff in tokens, which you redeem later, but not 1-to-1 for prizes. Suppose if you have between 50 and 80 tokens, the prize is the same as for 50. In that case, if you have 80 tokens, you would happily trade that for a 50% chance of having 90 tokens, and a 50% chance of having 50 tokens, or possibly even a 50–50 chance of having 90 versus 45 tokens, even though that looks like you are risking 35 to gain 10. It is inefficient to go from not good enough to double to too good to double, and similarly inefficient to regain your market from a position which is too good to double.

In the bearoff position, if you own the cube in money play, the aggressive play is very slightly wrong, but not nearly by as much as the gammon price indicates. When you are too good to double, there may be no gammon price which indicates the best strategic play, but usually you should play at least as aggressively as if you did not have access to the cube.

### Summary

The gammon price is a useful idea. I recommend learning not just the take points at important match scores, but also the gammon price. This can help in both cube and checker play decisions. However, the gammon price needs to be adjusted to take into account the possibility of future doubles. The gammon price for money when you are not too good to double is about 0.43. That cubeless equity is based on a gammon price of 0.5 means that it is inaccurate for comparing plays which lead to different numbers of gammons, even for money play.

© 2001 by Douglas Zare and GammonVillage.

Douglas Zare is a mathematician and backgammon theorist. He writes a monthly column at GammonVillage on the theoretical aspects of backgammon. His web site is douglaszare.com.

Backgammon
Sets & Boards

 More articles by Douglas Zare More articles on playing for gammon Return to:  Backgammon Galore