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.

8 thoughts on “Still More Better Bad Byes”

1. Sean Garber says:

Hi Dan,

The answer to your question in the email is, “No”. My comments will be made with a Central Indiana Fall Trophy Event type tournament in mind.

With the grouped byes format, we have 1102 minutes of total waiting. We have 1406 minutes of waiting with the spread out byes. This is a substantial difference.

“That’s because I thing that some delays are worse than others. Players rarely complain about wait times that are related to their own byes.”
For the most part, I do not agree. I have heard players complaining when there are more than two players not playing immediately that “We should be playing someone.” When I get a bye in a tournament, I am usually happy about it. When I get a bye and still get to play a match immediately, I am happier.
When a player goes 0-2, that player should be able to go home as early as possible. When a disproportionate amount of the waiting is early in the tournament, that will have a significant impact on players who go 0-2 or 1-2. If you have 4 guys from Bloomington in the same vehicle who have to wait for each other to go home though…

“The evidence from these simulations is not entirely unambiguous, but I think that on balance it militates against grouping byes.”
I really think that you have to be looking through rose colored glasses to come to this conclusion. You do a good job of drawing attention to the longest waits in the grouped bye format in your article. The grouped bye format has 8 spots where a player will have to wait 45 minutes or longer. The spread out byes format has 18 spots where a player has to wait 45 minutes or longer. I actually don’t think the data from these simulations is ambiguous at all.

Just for the record, my first round of the consolation pairings in bracket order with a 24 bracket: B8 v A1, B5 v A4, B3 v A6, B2 v A7, B7 v A3, B1 v A8, B4 v A5, B6 v A2. If and only if exactly 21 players show up, A8 and B7 are switched.

Let us look at the numbers you have for the top 4 players in the tournament.
Grouped byes:
1     \$22.87 3.97
2 15.36 3.53
3 11.61 3.21
4 9.13 2.95

The byes spread out numbers are about the same, so this is good enough. The highest possible loss total for a player is 2. The top players cash a significant amount of the time, so they will lose less than 2 matches on average per tournament. The top seed wins 3.97 matches per tournament, on average. If you give the top player 2 losses (it really isn’t that many on average), that player will have a W-L record of 3.97-2   66.5%. Since the loss number (2) is overcooked, #1 seeds win% is higher than this.

Let us look at the bottom 4.
21 0.22 0.83
22 0.14 0.71
23 0.08 0.58
24 0.04 0.41

Since these players rarely cash, we can assume 2 losses here. #21 seems to have a win% of 29.3, #22 is at 26.2, #23 is at 22.5, and #24 is at 17.0 . Some estimates from myself concerning the players that show up to the Central Indiana Fall Trophy Event. The worst player has a well established long term win% in the low 20s. The 2nd, 3rd and 4th worst players have win%s around 40% or 41%..

If a player has a 60% win probability in every one of his matches (I know win probabilities will be lower in matches late in the tournament. I just want to get a close enough for horseshoes number), that player would have an expected return of \$8.29 . He would win 9.504% of the time, 2nd place 6.336%, first consolation 8.91648%, and 2nd consolation 5.94432%.

“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 form backgammon nor the consolation format affects the basic conclusion.”
If this is supposed to imply that you used win rates appropriate to backgammon in the simulation, the statement is simply off. Using win rates that are too high at the top and too low at the bottom highly exaggerate or perhaps in some cases reverse the difference in the fairness (B) number of the two types of tournaments. Throwing this out there, spaced byes give us a fairer zero loss bracket. A well worked grouped byes bracket gives us a fairer consolation.

“The previous result has been, at least largely, confirmed. Grouped byes are bad byes.”
To get to this result, you have set the parameters that cause win percentages to be completely inappropriate for backgammon in these published reports at least. You have also put the number of participants at exactly 24 which is good for the fairness numbers with a spread out byes format tournament.

I will copy and paste a comment of mine about the fairness (B) numbers in a 21 player coin flipping tournament from a previous article of Dan’s. The grouped bye format gets better fairness (B) numbers than the spaced bye format. I used a coin flipping  tournament because with that, I could calculate fairness (B) without a simulator. Of course, backgammon is not flipping coins, but I think that these numbers just might be at least as close to backgammon truth than the ones Dan uses in his simulations.
“I did some fairness (B) calculations assuming a coin flipping tournament. First, I recalculated the 24 player single elimination and got a fairness (B) number of .66014 this time. Sorry about the corrections. Then a 21 player coin flipping tournament. I compared 3 different formats. First,“the 11 byes grouped at the top, with the first round of the consolation as follows:B8 v A1, B5 v A4, B3 v A6, B2 v A7, B7 v A3, B1 v A8, B4 v A5, B6 v A2. Obviously A1, A2, and A3 do not exist, so the opponents get byes. The fairness (B) number here is .81270 . The second bracket I tested also had the 11 byes grouped together at the top, with a small switch in the first round of the consolation:B8 v A1, B5 v A4, B3 v A6, B2 v A7,  A8 v A3,  B1 v B7, B4 v A5, B6 v A2. A1, A2 and A3 still don’t exist, so they create byes in the consolation. The fairness (B) number for this one is .83753 . The third bracket tested was Dan’s byes evenly spaced out bracket. The fairness (B) number for this one is .76212 . As expected, the byes grouped together formats get better fairness (b) numbers in a coin flipping tournament. In contests where the favorite is usually a bigger favorite, the spread out bye format will get better fairness (B) numbers. At some point, there is a crossover where fairness (B) numbers are about the same with either type of bracket. I am not smart enough to figure out exactly where that point is.”

Let us think about a 21 player backgammon tournament, one with spaced byes, one with grouped byes. Let us add these parameters:
1.The top player in the tournament has to have a win rate of 60% or less.
2.The bottom player in the tournament has to have a win rate of 40% or more, with those in between players having win rates in between, split up fairly evenly.
3.The pairings in the first round of the consolation of the grouped byes format would be B8 v A1, B5 v A4, B3 v A6, B2 v A7, A8 v A3,  B1 v B7, B4 v A5, B6 v A2. A1, A2 and A3  don’t exist, so they create byes in the consolation.
My prediction is that the fairness (B) numbers for the 2 tournaments would be pretty indistinguishable. The grouped byes format’s fairness (B) just might be a tad better.

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2. Sean Garber says:

Sorry guys. Let me edit the consolation bracket order for a 21.
B8 v A1, B5 v A4, B3 v A6, B2 v A7, B7 v B6, B1 v A8, B4 v A5, A3 v A2

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3. Sean,

I tested the grouped byes with my own drops because I thought that they were likely to do better, and I wanted to give the grouped format the best possible chance. But I reran the test with your revised drops, which have the A’s and B’s as in your comment, and swap C2 and C3.

The results are quite similar. Fairness (C) is 4.192; Fairness (B) is 1.129 – both a little worse than with my drops. There were considerably more repeat pairings, 0.72 rather than 0.42. But there was a small improvement in wait time, 1,045 total minutes rather than 1,102, and of course that’s still much better than the 1427 for the spread byes. The ugly 109 minute wait in the H round is still there, but one of the long G round waits has gone from 84 to 23. There’s a new 40-minute wait in the J round.

I’m running the tests with an even higher luck factor, and I’ll also try some runs with different numbers of players. But I want to change things one element at a time so that we can see what makes the difference when there is one, and they take a while to run.

One other comment for now. It’s certainly true that the spread byes had 18 matches with waits of more than 45 minutes as opposed to 8 for the grouped byes. But you have chosen your threshold a bit artfully. If, instead, we counted matches with more than an hour’s wait, the score would be 0 for the spread byes to 8 for the grouped byes.

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4. Sean Garber says:

“But you have chosen your threshold a bit artfully.”

Of course!

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5. I’ve now rerun the experiment again, this time ramping up the luck factor. With more luck, fairness (C) decreases markedly, and fairness (B) a little. But the same pattern obtains: (C) is 2.868 for grouped byes, as against 2.919 for spread byes. Fairness (B) came in at 1.047 and 1.829, respectively. Total latency was about the same: 1,051 minutes and 1,442 minutes, also respectively.

I don’t understand the idea that the fairness disparity should decline further, or even be reversed, with more byes or fewer. 24 players is a worst case to my way of thinking – every player more (or less) means that there will be less waiting to play in the first round. If you were to remove skill entirely, so that the tourney is essentially a coin-flipping contest, the only thing that would matter for fairness (B) would be equalizing the number of rounds played, and that fairness (C) would decline to some minimum value that’s not affected by the bracket structure one way or another. I assume that your experiment with the coin-flipping contest used some unusual bracket that tended to equalize the number of rounds played, but I can’t tell what you used well enough to simulate it. If you send me a picture of the bracket, I’ll simulate it if my program can be made to handle it.

I think maybe our disagreement is essentially a fairness (A) problem. I don’t doubt that you are sincere in disliking round-one sit-outs – they seem perfectly normal to me. But waits of more that one round that crop up unexpectedly in later rounds drive me nuts, while you’re made your peace with them. Knowing that there’s some avoidable unfairness bothers me, even though it’s subtle enough it takes at least 1000 trials to prove the effect. You like the kind of tourneys you’re accustomed to running. I like lots of things about then, too, but grouped byes have always seemed a mistake to me, and my efforts to understand the effects of grouping byes only make it clearer, to me at least that they should generally be avoided.

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6. Sean Garber says:

Dan–I just emailed you a couple of brackets. I hope this clears things up a little.

“I’ve now rerun the experiment again, this time ramping up the luck factor. With more luck, fairness (C) decreases markedly, and fairness (B) a little. But the same pattern obtains: (C) is 2.868 for grouped byes, as against 2.919 for spread byes. Fairness (B) came in at 1.047 and 1.829″

I think I can tell right away that this was run on a 24 bracket, not a 21 bracket. You did say earlier, ” I want to change things one element at a time”. This is good scientific methodology, but not so good for impatient people like me. I am also interested in just how far the luck factor was ramped up. “I ramped up the luck factor to 4” means little to me. I am more interested in the overall win%s of the best and worst players in the simulations. If this number is not available, you posted a \$ number for the players in the previous simulation, which was the average dollar amount a player took home out of the \$100 prize pool. This number would be a good substitution.

“If you were to remove skill entirely, so that the tourney is essentially a coin-flipping contest, the only thing that would matter for fairness (B) would be equalizing the number of rounds played”

EXACTLY!!!!

” I assume that your experiment with the coin-flipping contest used some unusual bracket”

No. This is because byes spread out tournaments have a weakness when, say, 21 or 22 players show up.

I do think that we have fairness (A) differences. At our weekly Wednesday night club tournament, if we have exactly 11 players show up, players almost universally consider the nonplaying bye spot to be the worst draw of all in the tournament. We came to play backgammon. In an ABT event, things are different. Also, you value avoiding repeat matches more highly than I do.

Other random thoughts–If 24 to 32 players show up (or 48 to 64 players, or 12 to 16 players), fairness (B) should always be better for a spaced out byes tournament unless we are flipping coins. In that case it should be a tie. I think that with a “luck” number suited to backgammon, the difference would be small and wouldn’t make up for the better flow in a grouped byes tourney. I also think that my grouped byes bracket needs some work. I think that a separate consolation bracket is needed for tournaments of 22-24 players, a bracket if exactly 21 show up, and an 18-20 consolation bracket. I also think that putting the D1 and D2 losers in separate halves of the consolation and spreading out the C losers would also increase fairness (B) by a few millipoints.

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7. Sean Garber says:

I think that I should clarify my own statement for accuracy sake.

“If 24 to 32 players show up (or 48 to 64 players, or 12 to 16 players), fairness (B) should always be better for a spaced out byes tournament unless we are flipping coins.”

This refers to a single elimination with progressive consolation tournaments, not double elimination. The 32 bracket assumes 2 cash from the zero loss bracket and don’t drop, The 64 bracket assumes that 4 cash from the zero loss bracket and don’t drop. The 16 bracket assumes 1 cashes from the undefeated bracket and the rest drop. Also, if 15, 16, 31, 32, 63 or 64 players show up, there is no difference between a grouped byes bracket and a spaced byes bracket.

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