Backgammon Articles

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 non-deterministic 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 2-4'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). Well-played games take much longer than normal backgammon games, and usually I get to mention this theorem.
The Half-Crossover 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 half-crossovers 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 half-board 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 non-gin 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 side-bets 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 half-life 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 5-point 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 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 7-5-3 target position.

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