Getting the Most from an Afternoon

A common desire for many tournaments is to organize the competition to make the best use of limited time.

A friend runs a monthly backgammon tournament that, in the past year, has drawn from nine to 15 entries. It starts at noon on a Sunday, and needs to be finished by six. For some years, this tourney has been played with a rather unusual lower bracket. The upper bracket is a standard 16. The lower bracket is like a standard (unshifted) 16 double elimination bracket, except that takes only the first three rounds of drops, and has no provision for the winners of the two brackets to play for the championship.

My friend want to know if he would be better off by using a shifted 16 consolation bracket. We speculated on the effects of the change for a while, but I’ve been looking at brackets long enough to know that the best way to get a bracket to reveal its secrets is to run it through my simulator.  And, sure enough, comparative simulations help to show some features of his non-standard bracket that were not initally apparent to either of us.

In this post, I’ll report the results of the simulation with respect to the bracket architecture, and in the next I’ll explore some other questions, including the question of what payout schedule ought to be used.

I decided to run the experiment with 13 players. My friend has not yet had enough entries to run a full 16, so I thought it might be interesting to see how it affects the results to have three byes in each bracket.

Here are the analyzed brackets: 13UpperRevN, 13LowerUnshiftedN, and 16LowerShiftedN. (Note: In an earlier version of this post, I linked to incorrect analyzed brackets, which did not properly account for byes in the flow calculations.)

Here’s a picture of a bit of a bracket to help explain how to interpret this sort of analyzed bracket. This is match H1 from the unshifted consolation bracket – I’ve omitted the match number to reduce clutter on the diagram.   

The top line shows that this player drops into the consolation as the loser of match C1 in the upper bracket. The green $4.99 is the expectation of the player on that line, assuming that the prize fund is $100. The blue 0.259 represents the average skill of this player, expressed as a Z score. The red 16 shows that this player will have to wait an average of 16 minutes before the match can be played.

3:15 is the average starting time for the match, assuming that the event as a whole started at noon. In 13% of simulations, this match is between two players who already played each other in the upper bracket. Repeat matches can happens only in the last two rounds of the consolation, and the information is not given for other matches.

The lower line shows that that player, who reaches the line as the winner of G1, is a slightly better player, with a skill score of 0.286. For this reason, that player has a slightly better expectation. And while C1 usually finishes before G1, so that this player would have no wait, sometimes G2 does finish first, so that this player has a average wait of five minutes.

The winner of this match has an expectation that’s the sum of the two. And the winner, on average, will be a much stronger player, at 0.479.

Because the upper bracket plays the same no matter what lower bracket is in use, I’ve provided only one analyzed upper bracket. In that bracket, there are two expectations in green on the top line: the first represents the expectation when the unshifted lower bracket is in use, and the second the expectation with the shifted lower. The other numbers can be slightly different because even a million trials is not enough to remove all random variation. To simplify the presentation, I just used the values from the run with the shifted bracket.

So what does the analysis show with respect to the question of whether my friend should start using the shifted bracket? There’s really just one pair of numbers that I think clinches the decision. On average, the last match of the tourney goes off at 3:57 when the unshifted bracket is in use. But where a shifted bracket is in play, the last match goes off at 4:56! That last match may well take an hour to play. 4:56 is only an average start time for the consolation final – some of the time it will be later than that. If my friend adopts the shifted bracket, he’s going to find that sometimes he won’t have time to finish the event before six o’clock, which is something he really needs to do. So he’ll probably be wise to stay with the bracket that he’s been using.

This is a shame, because the shifted bracket is, in other respects, better than the unshifted one. I’ll discuss some of these factors in a subsequent post.