
Backgammon Articles by Douglas Zare


My Backgammon Articles

I find backgammon to be one of the most mathematical games. Most of it is
not the sort of mathematics that one would initially think about
associating to the game: the probability of accomplishing some goal in the
next roll. Instead, I view it as the archetypical game of skill and
chance. The depth of the game is substantial, though frequently
underestimated by weaker players. The doubling cube requires one to
understand not just the relative strengths of positions, but their
absolute strengths as well. One must learn despite the the
nondeterministic feedback.
The following are articles I have written, primarily on the mathematics of
backgammon, and had published in GammonVillage, an online backgammon magazine. I'm now a monthly columnist for GammonVillage.


Backgammon Ends
I filled in the gaps of a folklore proof attributed to Curt McMullen that backgammon ends with probability 1, no matter what the strategies of the players are.
 It suffices to show that if someone chooses the dice (for 9020 rolls) they can end the game.
 One can choose the dice so that not both players are on the bar within 20 rolls.
 9000 consecutive 24's will then end the game. I introduce a modified pip count which decreases on each exchange to establish this.
I enjoy playing backgammon misere, in which one tries to lose (ignoring gammons and backgammons). Wellplayed games take much longer than normal backgammon games, and usually I get to mention this theorem.


The HalfCrossover Pipcount
Backgammon is essentially a race with obstacles, and knowing whether one is ahead in the race or not is vital for correct playing strategies. Here I introduce a new, simple method for establishing the exact pipcount. As a bonus, one first gets an approximate pipcount, which usually suffices.
 Count the halfcrossovers to bear in.
 Multiply by 3 and add to 75.
 Adjust by −1, 0 or +1 for each checker according to where it is within its halfboard
triplet.
Most of what one does is count, and the arithmetic is relatively simple. I think this is the easiest practicable pipcount for most people to learn.


A Measure of Luck
I discuss criteria a measure of luck should satisfy and a mathematically correct measure of luck which has some surprising consequences. Luck is the equity gained through the roll of the dice.
 The luck involved in any match won is about the same as the luck involved in any other, barring serious errors.
 From any nongin position in a match, you need good luck to win and bad luck to lose.
 Strong players are lucky more often than weak players.
 Forced rolls might be good luck and they might be bad, but you play them perfectly.
 Someone who is manipulating the dice to their advantage will show up as "luckier."


Hedging Toward Skill
In this article I discuss a method of variance reduction to remove most of
the luck from the game. This follows up on an idea of Fredrik Dahl, the
author of Jellyfish in a recent
preprint. It provides a way of a backgammon player or program to analyze
games between stronger players without bias, and I applied this to a game
between Jellyfish and Snowie. I introduce "hedged backgammon" in which one
makes sidebets cancelling most of the luck of the game.


Ratings: A Mathematical Study
This was coauthored with Adam Stocks. We studied the properties of the
ratings systems in place on various internet backgammon servers. Our
results include the following:
 There is a halflife of ratings variations that is approximately 600
on FIBS and GamesGrid and 500 on GameSite 2000.
 We determined the stable distributions of ratings (of a player of
fixed strength). The standard deviation on FIBS for someone who plays only
5point matches is 41.7.
 The maximum rating ever achieved by a player has a tighter
distribution than the current rating for a large experience level.
For players on GamesGrid of lower playing strengths and experience levels
below 10,000, the maximum rating was often achieved during the ramped
period.
 We explained some discrepancies between our results and articles in
the rec.games.backgammon archive, and further developed some of their
ideas.
Here is a Mathematica 4 notebook which shows how some of the computations in the article were done, and allows one to vary the parameters.


Burying vs. Bearing In
This is a tutorial for players who don't know the theory of bearing in yet.
It illustrates why it is bad to move checkers deep into your board while
bearing in by simultaneously displaying the results of correct and incorrect
bear in technique for a sequence of rolls. It introduces one to the 753
target position.


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