Skill and Luck in Backgammon

In general, I want to make this blog about games and tournaments in general, not about backgammon tournaments, or any other particular kind of competition. But, as I often used to admonish colleagues when I worked, in a past life, developing information products for lawyers, you need to solve someone’s problem before you’re ready to solve everyone’s problem.

Backgammon is the only game I play competitively, these days, and it’s the one I know best. So it can, I hope, serve as a model for how to adapt insights about tournaments in general to help improve particular events by adjusting the general model to suit the specific game.

This is the first of three posts on Backgammon. It discusses the balance of skill and luck in the game, and how that affects tournament simulations. The second will explore what that means for how the game is learned and played. And the third will discuss computer players, and the curious fact that every decent computerized backgammon player is frequently accused of cheating.

 In More Bad Byes, I offered a rather critical analysis of Sean Garber’s bracket for running a 24-team tournament. Among his reasonable responses was that the parameters I use for my base case tournament yield results very different from those observed in backgammon tournaments, and it was for backgammon tournaments he designed his bracket.

Sean  noted that many of my experiments showed the best player in the tournament winning the whole event forty-something percent of the time. But even the best human backgammon player in the world these days, Mochy, wins only about 63% of his matches, and that means that the percentage of the time Mochy wins whole tournaments should be somewhere in the teens.

A rough and ready way to calculate this more precisely (if not more accurately) would be to assume that Mochy’s reported 63% win rate was constant from one round to another. Thus, Mochy should win the four games of an unseeded, 16-player single elimination tournament this often:

(0.63)^4 = 15.753%

Now, the whole premise of much of the analysis of what I’ve been doing is that this isn’t true – that some parts of a bracket are tougher than others, and that generally the competition gets stiffer the further one progresses. It seems likely that Mochy’s record in the early rounds of a tournament is at least a little better than his record in the later rounds where, on average, he’s up against somewhat better players. But it would be massive task (if it’s possible at all) to analyze Mochy’s record closely enough to adjust for these round effects, and it simply wouldn’t be worth it.

To come up with parameters for my match model that will make the tournament simulator yield results that look like they might have come from a backgammon tournament, I simply played with my parameters until the top player in the simulation won a 16SE about 15% of the time.  I shaded this down because, after all, not every tournament has Mochy in it – his very presence, I suspect, tends to increase the advantage the best player has over everyone else. Besides, I got an acceptable number, 14.855%, at a satisfyingly round number for the luck factor: 3.0.

In my match model, skill is fixed at 1.0, so setting luck to 3.0 means that I’m assuming that backgammon is 25% skill and 75% luck. That’s an awful lot of luck, and in my next post I’ll ruminate a little about what this means for how the game is played and who plays it.

One other parameter adjustment to note. My sense is that the skill levels entries in some tournaments aren’t really normally distributed. The distribution of skill across the population as a whole may be normal, but the entries in the tournament don’t include most of the less skillful players, either because they don’t come, or because they’re not invited. The major tennis championships, for example, don’t start with 128 players drawn randomly from the world’s tennis players, or even from the world’s tennis professionals. There are various exceptions, like wild cards to home-country players, and qualifiers who win their way into the main draw by doing well in a preliminary tournament. But, as a first approximation, the 128 players in the main draw at a tennis major are more or less the best 128 players in the world.

Some backgammon tournaments are more plausibly elite than others. The typical stop on the American Backgammon Tour has three divisions for the main event: an open (or “championship”) division, an intermediate (or “advanced”) division, and a small section for beginners. There’s a certain amount of fudging the line between open players and intermediates – more on this in the next post – but in general I think it’s reasonable to consider the top division to be an elite tournament. To make my simulator track better to real open-level backgammon tournaments, I’ve applied an elite threshold of zero, which means that the entries are drawn only from the top half of a skill distribution that’s assumed to be normal.