Backgammon Articles
 A Measure of Luck by Douglas Zare

 This article originally appeared in GammonVillage in 2000. Thank you to Douglas Zare for his kind permission to reproduce it here.

 Ne soyez pas superstitieux, cela porte malheur. —Tristan Bernard (It's bad luck to be superstitious.)

Basic Properties of Luck

Measuring Luck
Counting jokers and calculating how unlikely a sequence of rolls would have been if it had been called in advance do not work. For example, it is possible to alter the number of jokers you get by changing your playing style, e.g., by taking more cubes when your opponent's possible improvements will be in small increments and yours will be in large ones. If you want to try to factor out the effect of luck on the outcome of a series of games or rollouts (see David Montgomery's article on variance reduction in the February 2000 issue of GammOnLine), your method for estimating luck should satisfy these properties or else you may introduce biases for or against certain playing styles.

The mathematical measure of luck gained on a roll of the dice is your equity after the roll minus your equity before the roll, i.e., luck is the equity you gain through the roll of the dice. It is the equity of your best possible play minus your equity before the roll, or your equity after your opponent's best play minus your equity before the roll. (For an introduction to equity, see Gary Wong's article on equity in the rec.games.backgammon archive.)

Note that there is no luck involved in using the doubling cube, though it might affect which subsequent rolls are lucky and by how much. Further, there is no luck involved in how one plays a roll. When you make imperfect plays or imperfect cube decisions, you lose equity, but that is not due to luck. If your equity is +0.3 and you roll well so that the best play has an equity of +0.5 but you blunder, only getting +0.4, then your luck was +0.2 and you erred by 0.1.

We have a reasonable definition of the luck on a roll, but it is hard to compute—that takes some understanding of backgammon. Programs such as Snowie, Jellyfish, MonteCarlo, Motif, and Gnu have guesses about what the equity is in each position, but they disagree about the equities, and even about what the correct plays are, so they can't all be correct. Nevertheless, they are better than nothing, and their guesses are extremely useful for the purpose of variance reduction. Most beginning and intermediate players would do well to assume that Snowie 3 is correct. Snowie's analysis can reassure you that your opponent was very lucky, or point out that you blundered away your equity rather than losing it with the dice.

We generally don't care that much about the luck on a particular roll, but rather the whole game or whole match. If one adds up the luck from each roll, it turns out that this is much more interesting.

 Theorem: (folklore?) The outcome of a match or money game equals the net luck plus the net skill difference.

We would like to estimate the sum of the changes in the equities after making the best possible plays (whether or not those were made). What we can see is the sum of the changes in equity, since we know what the equity is at the start and at the end. At each step, the equity lost or gained through your or your opponent's technically incorrect plays is the difference between the equity of the best possible play and the equity of the actual play. So the difference between the total luck and the outcome is the sum of the net errors.

Example: Suppose you are playing a money game. If you give up 0.2 due to incorrect play and your opponent gives back 0.5, then if you win you have been lucky by an amount of +0.7 and if you lose your total luck has been -1.3. If you win a gammon with the cube on 4, your luck has been +7.7.

It doesn't matter how you won or lost the match, what its score was, or what your strategy was: If you know the outcome and the relative skill displayed, you can determine the amount of net luck for each player.

Consequences of This Measure of Luck