
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

Every backgammon player has seen outrageous or ridiculous sequences of die
rolls. Most of us have been in matches where every risk we take is
punished, and every risk our opponent takes is rewarded. Is it worth
quantifying the amount of luck in a match, and if so, how should one
measure it? In this article, we will look at a correct measure of the luck
in a match. There are many interesting consequences of measuring luck
correctly, and they aren't necessarily what one might expect.
First, some basic properties of luck:
 The important quantity is the net luck.
The net luck of a player exactly cancels out with the net luck of the
opponent. Just as the total number of heterosexual partners of women must
equal the total number of heterosexual partners of men, so luck must
balance to 0, no matter what surveys say. An individual player
might have been unlucky, but players overall have not. (On no backgammon
server are the players usually lucky or unlucky, since they play each
other.) Though it seems less rude to cheer after your good rolls than
after your opponent's bad rolls, these have the same effect.
 The right units are money for money play, and match winning chances
(or equity) for match play.
Being lucky when the cube is higher
counts for more than when the cube is lower. If one doesn't adjust
for the cube level, one might as well include when the cube is
at 0—luck in other activities than backgammon.
 Your expected luck, the average luck you should expect in the
future, is 0, regardless of your play.
If a position will definitely look bad after the next roll, it is already
bad. Some people think that they are definitely going to be unlucky. These
people are wrong.


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

Some consequences of this measure of luck:
 The luck involved in any match won is about the same as the luck
involved in any other, barring serious errors.
If neither side makes errors, then if you lose a 5 point match 54 by
losing a close race at double match point you have had exactly as much net bad
luck as if you had lost the match 240 by being backgammoned with the cube
at 8 or if you lost 5 single games in a row (within the match). Your luck
was 1 in each case. In the first case the last few rolls had tremendous
quantities of luck, and it is easy to underestimate their importance.
If someone exclaims about how much luck you had after you win, they are
rudely asserting that they played much better than you did. Of
course, it is much more obnoxious if you lost.
 From any nongin position in a match, you need good luck to win and bad
luck to lose.
Example: Suppose you are down 100 in an 11 point match. You have about a
3% chance of winning the match if you and your opponent play
perfectly. Your opponent still needs +3% good luck in order
to win: if you come back to tie the match 1010 and then lose, then you
have either had bad luck or made errors. Though the score looks more
respectable than the "median" result of losing 111, if neither side makes
errors you have had just as much net bad luck as you would by losing
110. To go from 100 to 1010 with perfect play requires +47%
luck, and losing at 1010 requires 50% (bad) luck, for a total of
3%.
 Strong players are lucky more often than weak players.
Strong players still have to be lucky to win, just less so. Because they don't
need to be as lucky, their good luck is spread out to win many matches and
their bad luck is concentrated in a few losses. Someone who bears in and
bears off efficiently doesn't roll larger numbers than someone who aims
for the ace point. He/she just uses the pips received more
efficiently. The same thing happens with luck.
A strong player throwing away .2 less equity than I do will still have to
be lucky by +.8 to beat me (in a match). Of course, this will happen 60%
of the time.
 Forced rolls might be good luck and they might be bad, but you play
them perfectly.
Some measures of a player's errors average the errors over the number of
unforced moves. Having a proper measure of luck allows one to see that the
total errors of the two players are comparable, rather than the errors
averaged over unforced moves.
Suppose you have a choice between making an error of size .03, and having
your next three moves forced rather than no error this play but errors of
size .02 for the next three moves due to your unfamiliarity with the type
of position. The first way has a higher error per unforced move (.03 rather
than .015) but afterwards it requires .03 less luck to win.
 Someone who is manipulating the dice to their advantage will show up as
"luckier."
A cheater will not necessarily have a higher average number of pips
rolled, and there will not necessarily be any particular pattern (e.g.,
more 44's than normal) but their luck will inevitably be
better, assuming all of their moves are legal (and that they are not
cheating through getting unfair advice). Over the long run, they will
probably appear unreasonably lucky.
A stronger player will probably not show up as unreasonably lucky over the
long run. To distinguish between the two, one needs to measure luck
carefully, more carefully than most human players can do by
hand. Many people claim that the free versions of Jellyfish cheat
despite documented methods for checking that they do not, but unbiased
estimates indicate that Jellyfish is not particularly lucky.


Conclusion

The measure of luck described in this article is mathematically correct,
but it does not capture the emotional experience of the game. It does not
represent the human perception of luck. Winning a close race with the cube
at 16 will not make as good a story as a "sure loss" turning into a won
backgammon with the cube at 4. You might look for another measure
of luck that would describe preceived luck. However, the
mathematical measure of luck is consistent and useful. Backgammon is a
game of luck and skill, and an appreciation of the luck involved may allow
one to focus on the skill.
How much of backgammon is luck? Almost all, if you and your opponent play
it very well, and play relatively short matches. Both players are vying
to be luckier than the other. If not, luck could be relatively
insignificant.
That's why I don't say "good luck" at the start of my matches.


© 2000 by Douglas Zare.
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.
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