In the last post, I showed how I used simulation to address the question of which of two brackets was better for my friend’s monthly backgammon tournaments. Before moving on with that analysis, let’s show how the question arose.
Say you’re running a tourney for nine to sixteen players, and need to achieve a result in a limited time. You can always run a simple four-round knockout tourney, but time is not that limited, and you’d like to offer people more play. So you want to run something along the lines of a double-elimination tourney.
Unfortunately, a full double-elimination tourney with all the bells and whistles can take as many as nine rounds. You don’t have that much time. What are your options?
The most common first step in paring down your event is to eliminate any possible recharge round. Without the recharge, the tourney that otherwise plays in eight or nine rounds always plays in eight. There’s a cost for this. Giving up the recharge round can be neutral or even positive in a larger tourney, but on a small bracket it sacrifices fairness (C) to some extent.
Let’s say that doesn’t save enough time. You can shift the lower bracket thus: 16lowershift. Depending on the skill level of the event, this may be positive or negative with respect to fairness (C) – for backgammon it’s always going to be positive. Good job, you’re now down to seven rounds.
Still not enough? Then you’ll have to do some major surgery on your bracket. Amputate the reconciliation round. Now that there’s no way back to the championship from the lower bracket, your event is no longer a double elimination. But it’s a lot like a double elimination because a player who loses only once wins something. It’s a consolation tourney, and you’re now down to six rounds.
Still not enough? Then don’t drop loser of the upper-bracket final into the lower bracket at all. Pay the person for second place, and save another round! Here, however, you run into a problem. If you don’t drop the D round, your lower bracket has shrunk to a size where it’s no longer possible to do a shift. If you were running a 32 bracket, there’s room for a shift even if you don’t drop the E round. But on a 16 you mess up the shift.
Now we’re at the specific problem my friend encountered. He wants to eliminate that seventh round, and he can accomplish that either by doing the shift or by not dropping the D round. But he can’t do both. So which one is better?
In the case of my friend’s backgammon tourney, not dropping the D round is better, or at least faster. To understand why, you need to look at another feature of his tourney.
Backgammon matches are nearly always played to an odd number of points. Five points is the minimum for serious play. Seven point matches are better, and Nine point matches are better still. Some very serious tourneys, like the finals of World Championship in Monte Carlo, are played to as many as 25 points.
Now, it’s common in backgammon tourneys to play longer rounds in the upper bracket than in the lower bracket, and this usually doesn’t matter a great deal. There are fewer rounds in the upper bracket, so they can take longer without extending the time of the tourney as a whole. My friend plays seven point matches in the upper bracket, except for a nine pointer in the final, and five point matches in the lower bracket, except for a seven pointer in the consolation final.
But that doesn’t work well with the shifted bracket. With the shift, the losing finalist drops into the semi-final of the consolation. So, even though both the shifted bracket and the bracket without the D drop play out in six rounds, the shifted bracket takes longer because the last two rounds of the lower bracket can’t begin until the upper bracket is finished. It doesn’t matter much that the lower bracket matches are shorter if the fifth and sixth rounds of the consolation are waiting on the completion of the fourth round of the championship.
As I’ll show, the tourney with the shifted drops is fairer than the one that omits the D drops. So it might be worthwhile to shorten the rounds a bit. Simply paring the upper final back from nine points to seven would save a good deal of time. To make a sensible judgment about whether it’s worthwhile to reduce the number of points in a round, it’s helpful to look more closely at the fairness differences between the two brackets. That’s what I’ll discuss in the next post.