*by Douglas Zare*

13 April 2015

Douglas Zare

*"Judge a man by his questions rather than by his answers."
-- Voltaire*

For 13 years, this has been a monthly column on the mathematics of backgammon. When GammonVillage first asked me to write this column, I was worried about running out of things to say, and I wrote down a year's worth of column ideas before I agreed. More than a decade later, I have a backlog of ideas and unfinished series, from covering Janowski's formula when too good to double to using roll variance to supplement the effective pip count in the bearoff. Due to some exciting life events (including impending fatherhood), this column will become more sporadic. I hope that I will have the time to write several articles each year.

Normally, I take a topic in backgammon, and try to analyze it mathematically, and report the results. Or, I take a topic in mathematics, and look for places in backgammon where it turns up. Instead of answers, in this column (and in Part 2), I want to share some of the problems in the mathematical theory of backgammon where I think significant future work can be done.

The Gammon Price

The first column I wrote for GammonVillage was on the gammon price. It's easy to say what the gammon price is for cubeless money play: Promoting a normal win to a gammon gains one point, while changing a win to a loss loses two points, so gammons are half as valuable as losses are costly (the opposite of the values if the base point were breaking even instead of a normal win). However, few people play cubeless money games. With the doubling cube, gammons are usually less valuable since they don't activate the doubling cube. In match play, the different values of the points affect the gammon price.

I have several questions about the gammon price with the doubling cube.

How can we estimate the gammon price when we are or might become too good to double? When we are too good, the gammon price may become not just greater than 1/2, but in some cases it is greater than 1. This should affect the gammon price before we become too good to double, too.

What is a simpler way to see how the doubling cube affects the gammon price, or other trades involving backgammons? If we really understand the cubeful gammon price well, we should be able to say what trades are worth if there are other payoffs than just losses, normal wins, and gammon wins. You can try to use something like Janowski's formula, but this takes in a cube efficiency estimate. If the cube efficiency should be 70% (on some scale) when you are considering risking a win for a gammon, is the cube efficiency also 70% when you risk a win to scoop up extra blots for possible backgammons, or does the cube efficiency depend on the payoffs?

How can you determine which gammon price to use in match play? For example, suppose you lead 2-away 4-away, and the cube is centered. If the cube stays centered, gaining 2 points instead of 1 is worth a lot for the leader, but not so much for the trailer. However, we know the trailer doesn't lose much by doubling immediately, particularly if the position is gammonish. Once the cube is held by the leader on 2, it is gammon-go for the trailer while gammon wins are the same as normal wins for the leader. So, even while the cube is centered, gammons are not that valuable for the leader, while steering for gammonish positions is good for the trailer.

A simplification we often make is to assume that positions are worth a fixed number of backgammons, gammons, and wins, and that the main difficulty is determining this distribution. In all of these situations, the gammon price might depend on more than just the bg/g/w distribution in cubeless money play. We have to look ahead to the efficiency of future doubles, and two positions with the same bg/g/w distributions might lead to different doubling scenarios and different gammon prices.

Clustering

We all know backgammon positions can be put into categories. There are races, blitzes, prime-versus-prime positions, holding games, mutual holding games, deep anchor holding games, backgames, containment positions, etc. These labels are useful because we can associate strategies and priorities with each type of position. In a blitz, making your home board points in order is not important. We can remember that certain assets have different values in different types of positions. A big racing lead may be valuable against a high anchor holding game, but not valuable against a backgame. However, when you try to categorize actual positions, or checker play decisions, it is much messier.

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Positions from real games are not easy to classify. There are often tactical components such as checkers on the bar, blots, builders in good or bad positions, or duplicated numbers. Does the bar point count as a high anchor? Does the 3 point? Sometimes yes, but sometimes no.

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