Some months ago, there were a series of posts (here, here, and here) about the allocation of byes – whether it was better to spread them evenly though the field, or to group them together so that they could begin play immediately by starting the second round with the byes playing each other.
One of the arguments made in favor of grouping the byes together is that it improved the flow of the tourney. Players would have to spend less time waiting for their opponents to become available, and the tourney as a whole would play our more quickly. Another observation was that the grouped-byes example I chose was intended specifically for a consolation backgammon tournament, and that my simulation was not for a full double elimination rather than a consolation, and that I used a luck parameter unsuitable for backgammon.
Armed with the new simulator, I can revisit the issue, addressing in particular tournament flow.
Here are new analyzed brackets, for a 24 consolation tourney with grouped byes: 24groupedupper and 24groupedlower, and with byes spread: 24spreadupper and 24spreadlower. In each case, the tournament was modeled to approximate a backgammon tournament, in which the upper bracket matches are played to seven points, which averages about an hour or virtual time, and the consolation bracket matches are played to five points, which average about 40 minutes. The tourney notionally begins at noon.
These simulations yield results very similar to the earlier ones on matters of fairness. Fairness (C) shows a small difference, 4.36 to 4.21 for spread byes and grouped byes, respectively, and fairness (B) shows a large one, 2.70 to 1.15. It appears that neither the high luck factor appropriate for backgammon nor the consolation format affects the basic conclusion.
The argument in favor of grouping byes is based chiefly on considerations of flow. There are two related advantages claims: that the tournament as a whole will run more quickly, and that there will be less waiting time for participants.
Were this a single-elimination tourney, it seems this is a pretty good argument. Looking only at the upper bracket of the simulation, it appears that grouping the byes causes one long wait, but avoids eight others. The player on the upper line of match E1 waits an average of 65 minutes for the rest of the bracket to catch up, but all other waits are on the order of ten minutes or so. In contrast, with spread byes eight players wait nearly an hour before playing at all, though once the initial waits are done, the rest of the delays are also on the order of ten minutes.
It’s important to recognize, however, that where the bracket with the grouped byes feeds a lower bracket for a double elimination or a consolation, the situation is different. The delays haven’t gone away – there are still players who are running ahead of the rest of the bracket. The delays have been pushed down into the lower bracket, where they reappear.
In the spread bracket, the matches receiving the four C drops have an average wait of 33 minutes, and the two matches with D drops wait 53 minutes. In contrast, the grouped-bye bracket has delays of 34, 84, 34, and 84 minutes – two of the drops delay proceedings nearly an hour. And the two D drops cause delays of 19 and 109 minutes! The match with the D2 drop is the source of the severely unbalanced match used as an example in the previous post.
For the A and B drops, the grouped byes show an advantage, with four delays of 61 minutes, as compared to eight delays of 54 minutes for the spread bracket.
The figures for “mean match latency” show the total wait time for the bracket divided by 44, the total number of matches. And the mean latency for the grouped bracket is seven and a half minutes less than it is for the spread bracket. So not all of the delays that are avoided by grouping the byes reappear in the consolation.
Nonetheless, I think that the format with the spread byes had better flow. That’s because I think that some delays are worse than others. Players rarely complain about wait times that are related to their own byes. True, there are eight players waiting for a match at the start of tourney, but these players can easily understand the wait, and are not unhappy. They’ve been lucky enough to draw byes, and the time they spend waiting is simply the time needed for the match that they don’t have to play.
The long delays that crop up in the consolation bracket are more difficult to stomach. The relationship between those delays and a bye that happened earlier is more tenuous. And the worst of the delays are worse than anything that happens in the spread bracket.
Perhaps the preference of many directors for grouped byes is related, in a perverse way, to these bad bottlenecks. Directors know, from experience, that long delays like the 109-minute wait for the D2 match can crop up in their brackets. Not having the benefit of the close analysis of these simulations, these bad delays seem mysterious. And the more directors encounter bad delays of mysterious origin, perhaps the more likely they are to want to group the byes – they imagine that they’re doing what they can to avoid delays by starting as many matches as possible early.
The fact of the matter, however, is that grouping the byes actually causes the bad delays deeper in the bracket. If you spread the byes, you’ll have some delays, but for the most part they’ll come early on, when the waiting players are mollified by the knowledge that they’re enjoying the advantage of a bye. Once you’ve taken your medicine at the beginning of the tourney, the rest of it should flow much more smoothly.
The evidence from these simulations is not entirely unambiguous, but I think that on balance it militates against grouping byes. We knew that grouping byes impaired fairness, but may have been inclined to do it anyway to get better flow. But the simulation shows that the flow is not clearly better – for my money, it’s worse.
The previous result has been, at least largely, confirmed. Grouped byes are bad byes.