Pay Now or Pay Later

Not all tournament organizers have the power, or perhaps the inclination, to limit the number of entries in a bracketed tournament to a power of two. In such cases, the usual practice is to use the next highest power-of-two bracket, filling empty lines with as many byes as needed. On the printable brackets page, the model top-level brackets all have seeding numbers that double as guides for allocating byes.

But there is more than one way to allocate byes. In this post, I’ll illustrate the possibilities by working through a very simple case, comparing four different ways to prune a basic 32 single-elimination bracket so that it accommodates 24 entries. By artfully arranging the byes so that they fall in later rounds, there can be fewer of them. So, should you pay now by putting all the byes in as early as possible, or should you pay later by putting fewer byes into later rounds?

If you use the seeding lines on the regular 32upper, entering byes for seeds 25-32, you’ll end up with eight byes in the first round. But this is not the only way that this could be done. Instead of eight first-round byes, you can have four second-round byes, two third-round byes, or a single fourth-round bye. These thumbnails illustrate the possibilities, with the byes represented by the long, round-skipping lines:

The longer you wait to consolidate the bracket to a round with a power of two, the fewer byes you need. Byes are bad – they decrease fairness. So why not wait as long as possible?

The answer is that, while all byes compromise fairness, byes in later rounds have a greater effect than byes in earlier rounds. Here’s a table showing the fairness (b), (B), and (C) results for a full 32-team bracket and for the 24-team brackets illustrated above:

full 32,
no byes
8 byes, round A 4 byes, round B 2 byes, round C

1 bye, round D

 f(C) 1.005 0.930 0.939 0.944 0.948
f(B)[=f(b:A)] 0.73 3.62 6.81 11.51 17.55
f(b:B) 0.58 29.98 26.94 22.49 16.80
f(b:C) 0.44 0.28 26.94 22.49 16.80
f(b:D) 0.28 0.19 0.17 22.49 16.80
f(b:E) 0.20 0.07 0.12 0.08 16.80
 finals start 4:09 4:06 4:04 4:03 4:01
 average wait 0:06 0:23 0:13 0:09 0:06

The fairness numbers are for the new versions of those statistics, in which lower numbers represent greater fairness. The simulations each had 500,000 trials, set the luck factor at one, and had no elite threshold.

The full bracket has the worst f(C), but that’s not because it’s structurally unfair. With more entrants, there’s simply more opportunity for the prizes to go to players other than the most skillful ones.

Despite there being fewer of them, moving the byes to later rounds eroded fairness (C) a little bit, and had a huge bad effect on fairness (B).

There are no structural biases in the full bracket, so the small f(b) statistics for it reflect, essentially, statistical noise. There are more lines in earlier rounds than in later rounds, so the noise is greater at the beginning. But all values are well below 1.00 – that’s what a fair bracket looks like in terms of f(b).

For the 24-team brackets, f(B) represents the accumulated unfairness of later rounds, together with a small noise factor comparable with the full bracket. Later rounds have a greater effect on overall f(B) than earlier ones. The serious local unfairness effects begin in round B, and stays at the same level until the balance of the bracket is restored. The eight round A byes show a bigger hit to fairness in round B than those where the byes come later because there are more of them – 8 byes are more unbalancing than 4, or 2, or 1. But the round 1 byes get resolved in round B, and the rest of the bracket does quite well in terms of f(b).

Though the individual effect is smaller in round B, where the byes come later so does the resolution. The overall f(B) figure absorbs the accumulating damage of later rounds. This is easiest to see when the single bye comes in round D, and is not resolved until the last round. The 16.80 damage has become characteristic of the entire bracket. Together with another 0.75 noise, comparable to that of the full bracket, the composite f(B) is 17.55. The other 24 brackets take a bigger hit, but they have more opportunity for the unfairness to dissipate in later rounds.

This is, generally, why despite the greater number of byes, it’s generally best to take your medicine early, so that the imbalance introduced by the byes has the greatest chance to be diminished by subsequent balanced rounds.

So, why would anyone consider postponing the rectification of the bracket? It’s sometimes done to improve flow. The single, round D bye is functionally the single-elimination equivalent of the grouped byes bracket explored in great detail in the series of posts beginning with Bad Byes, and the four round B byes represents a compromise proposal suggested in the comments to that thread.

The last two lines in the table above represent flow considerations. The simulations assumed matches that average a little less than an hour in length, but are quite variable. This first of these lines represents the average start time of the final match (assuming that the tourney began at noon). The effect of later byes on the overall run time for the tourney is modest – placing one bye in round D saves only five minutes, on average, over putting eight of them in round A.

But the last line in the table shows the average number of minutes that each player was idle waiting for their next opponent to become available, and there the difference is substantial. The greater part of the delay is suffered by players who were lucky enough to draw byes as they wait out their bye rounds, which takes about an hour. Thus, the eight players who suffer this hour-long wait when the byes come in round A cause the average wait time to grow to 23 minutes. When a single bye comes in round D, the avoidance of any wait at all in half of round B compensates for the single long wait, so that the average player waits no longer than when the bracket is full.

As discussed in the bad byes post, the fairness effects of the late-round byes are smaller where skill progression is lower, so that late-round byes are more likely to be tolerated when high levels of luck lead to low levels of skill progression, notably in backgammon tournaments.


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