Appendix E of the Magic Tournament Rules discusses the number of rounds that are run for a normal Swiss tournament. Reading down the line, you’ll see the player counts increasing at a powers-of-2 rate: 32 players is five rounds, 128 is seven. The implication behind this is that, after all rounds of Swiss are completed, there will be no more than one undefeated (X-0) player. For completeness-sake, you can see this with the 32-player tournament:

- After 1 round, there will be no more than 16 undefeated players, and those 16 players will be paired against each other.
- After 2 rounds, those 16 players become, at most, 8.
- After 3 rounds, there will be up to 4 undefeated players.
- After 4 rounds, the max is 2.
- After 5 rounds, there’s no way for there to be more than one.

So, why is it that the eight round cutoff happens at 226 players, instead of 256? While it’s true that eight rounds with 256 will create only one undefeated player, it will also mean something new that hasn’t occurred at lower round counts: in addition to that undefeated player, up to eight players can finish with a 7-1 record.

Put in another way, it means that eight rounds cannot guarantee that an X-1 player makes Top 8. In order to do that, you need to drop the round threshold down.

### Navigating a Tournament at Round Thresholds

If you are working as an event as a Head Judge, you will likely have to deal with match result slip issues, either because a player marked their slip incorrectly, or because the scorekeeper entered in the result incorrectly. Different judges will have different policies for handling these situations, depending on when it was caught by a player, and the root cause. In general, though, it is more common for a result to get fixed, than it is to not be fixed.

However, when a tournament is dealing with round thresholds, there is an additional layer that may need to be considered. This is best described with an example:

In the PTQ for a 2017 European GP weekend, the PTQ was separated into eight 32-player pods. Each pod ran 5-rounds, with the winner of each pod making a Top 8 for the invite. At the end of Round 2, a player came up saying his points were incorrect, and the source of the error was an incorrectly-filled result slip in Round 1. The judge who initially worked with the player instructed the player to come up, and let me know that the judge (who I know reasonably well) had already OK’d fixing the slip.

In a normal tournament, this is just a minor issue. However, because we are at a round threshold, we introduce a new wrinkle into things, because Round 2 now had two 1-0 players playing against 0-1 players. Using our previous example:

- Round 2 now begins with fourteen 1-0 players playing against each other, and two 1-0 players playing 0-1 players. Assuming the 1-0’s win, you now have nine 2-0 players.
- Round 3 has one pair-down amongst 2-0 players, meaning you can have five 3-0’s.
- Round 4 can end with three 4-0 players.
- Round 5 can end with two 5-0 players.

Now, it’s important to emphasize something at this point:

**Nowhere in Magic policy does it state that any record guarantees a particular place in final standings.**

That being said, from a player perspective, it feels like there’s a reason that the round cutoff is at 226 players instead of 256, or that a GP PTQ is capped at 256 players and is run with eight flights of 32 players for five rounds. Such specific numbers imply guarantees, and those implied guarantees create expectations, and most Magic players are able to connect those dots. Subverting those expectations may not have a very visible effect on a tournament, but the risk you run is talking to that 5-0 player in a 32-player event with reasonably high stakes, and explaining to them why they’re not in contention due to tiebreakers.

## Byes & Round Thresholds

Byes represent an interesting challenge towards determining the number of rounds in a tournament, because each bye can have wildly-different effects on a tournament, because of the effect of pair-downs on an event. At a maximum, any given bye can have the effect stated in the MTR:

In events where awarded byes are used, each player with a 1-round bye should count as 2 players, each player with a 2-round bye should count as 4 players, and each player with a 3-round bye should count as 8 players when using the above chart.

However, not every bye will necessarily affect the tournament the same way. For example, if you had a 120-player tournament, where two players had a 2-round bye, by the above definition, any additional bye would require another round. Mathematically, however, two of the remaining 118 players could additionally have a 1-round bye, and the event would still be able to be run in seven rounds. (the math on these is beyond my capability, but if you want to check some simulations on your own, I have a Swiss Triangle simulator here, and there are other resources also available online)

Functionally, the MTR’s definition of byes here works fine, because it will always cover the worst-case scenario.

## Mathematical Issues

All that being said, there are still two occasions that could cause issues with those implied expectations, even in a perfectly-run tournament:

- 8-Round events that are 225-226 players. (or the equivalent count when counting byes)
- 9-Round events that are 385-410 players. (or the equivalent count when counting byes)

These two player ranges have a specific issue, in that it’s theoretically possible for their Swiss rounds to end with nine players at X-1 or better. When simulating this:

- Given a 410-player tournament where pair-downs lose 50% of the time, approximately 5.7% (5676 in 100k simulations) end Round 9 with more than eight players at X-1 or better.
- Given a 410-player tournament where pair-downs lose 30% of the time, approximately 47% (47164 in 100k simulations) end Round 9 with more than eight players at X-1 or better.
- Given a 226-player tournament where pair-downs lose 50% of the time, approximately 2.4% (2401 in 100k simulations end Round 8 with more than eight players at X-1 or better.

As a result of this quirk, these tournaments may need to be monitored more closely from a organizer-perspective, just in case there is a player with these expectations who finish 9th with a X-1 record.

## Additional Notes

- If anyone is curious, the round threshold for 10 rounds is 736 players. At that number, no more than eight players can finish X-1 or better.
- Magic Online’s round thresholds are different from paper Magic. (I don’t have any insight as to why the 8-round threshold is 212 players, instead of 224)

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