Should we draft with an odd number of players?

 I heard the latest episode of the podcast from Lucky Paper Radio today. First, the episode was, as always, great and I agree with most points.

What I found really interesting was the comment from Andy at the end, namely "playing with an odd number of people is better than with an even number". And whereas some people might find this to be heretical, I sort of agree on initial thought. Namely, we draft very often with odd numbers and it always felt much more dynamical. 

So why should an odd number of players be better? Namely, as soon as two people are finished with their game, another match can immediately start, whereas in an even numbered pod, you usually wait until everybody is finished and you enforce a timeout. So, from a pure mathematical point of view, what is the better option? In the end, we will see that it is rather a question of feelings than pure math.

Right away, computing the total time of a person not playing a match is rather trivial in the odd numbered case. Namely, let's say we want to Magic for 3 rounds @60min in the even numbered case. That's a total duration of T = 180min. For this whole time, one person won't be playing. Meaning that there is a downtime per player of t = 180/7min = 25.7min. That doesn't sound too bad! Of course, I ignored a lot of aspects here. Namely, what if player A and B finished their match, so player C joins in to play player A. Then, player A and C also quickly finish their match whereas the other players are still playing. This would take a long time to address numerically in a proper way. However, I think that the scenario is not that likely, but let's be pessimistic and add another 5min to the average, so ~30min.

To address the even numbered case, it is more difficult. Namely, we have to do an assumption of the probability distribution of an average Magic game. I have no idea. In principle, I have played games of Magic that lasted 5min to 40+min. The problem is that the average down time of players heavily depends on how the distribution looks like, which I will demonstrate in the following. Namely, we model the average time of one game distribution by setting a minimal time of 5min (this was a first guess of mine) and a maximum time of 20,30,40 min as seen in the following picture:

I tried to keep the average as well as median kinda close to each other. These are beta distributions with maximal and minimal average time per game. A maximal time certainly doesn't exist and whereas there is in principle a minimal time I am extremely unsure about specifics. I might take some time measurements at our next cube night...

Anyway, I think that a median time of like 10-15min is reasonable and we have a nice selection of probability distributions here. What I do is that I perform a quick Monte-Carlo simulation to see how long each pair of people have to wait during each round. To do this, for each round, you draw from the above probability distribution four times to determine the time of one game t1, t2, t3, t4. This is meant as the average game time in a certain matchup. Now, a game finishes if a person wins two times, so the time is either 2*t1 or 3*t1 both with 50% probability (we assume that you win or lose with 50%). However, we also add that after 60min, the round ends (it turns out that this heavily reduces the downtime per player). From these probability distributions, I obtain average downtime per player of t = 26min, 32.5min and 36min. Note, that we do not weight these times vs the total amount of time played. Namely, it could be that a round ends really earlier. Furthermore, we haven't adressed the possibility of two players playing a "redundant" match to bridge time, because two other players take longer (this is kinda awful to implement, because usually, you only do it if the match ends really quickly). 

Hence, first considerations show two things:

1) The average waiting time for players in an 8 player pod heavily depends on the probability distribution. Further data is needed to get a grasp on this distribution. Namely, I believe that thinking about how this distribution looks like is more interesting than the question of drafting with 7 or 8 players is more optimal.l

2) From very crude first considerations, it seems like playing with 7 players instead of 8 certainly doesn't waste people's times more on average. What I take away here is that the waiting time for 7 and 8 player pods is on a similar time scale at least. I would personally expect this downtime to depend a lot on nuances.

What also wasn't considered here is of course the type of downtime. There can be a big difference between one person having to wait "alone" vs two players waiting together. Sometimes, a person waiting alone might feel excluded. Furthermore, there can be complications in that people won't play the same number of rounds. Then the average waiting time can heavily differ between people and it might feel unfair.

Thus, my final verdict is the following: If you are playing in groups where people don't know each other too well and you might have a lot of new players, playing with an even pod is probably better as nobody will feel excluded. However, not wanting to play with an odd number of people because "it is not efficient" doesn't make sense in my opinion.

If you are playing in established groups where people are vibing, then playing with an odd number of people might actually increase the numbers of games you play in a night. Furthermore, this odd number structure can be beneficial for a potential host as they could in principle set up snacks in the first round. For example, when I host with odd numbers of people, I do exactly that. It sometimes felt a bit bad because I felt like I am missing out, but looking at these considerations, I might revisit my actual downtime in even pods. Let me know if you measure your time at the next cube night!

So what's the point of this? First, I had time on Sunday night. As I wouldn't finish this little project anyway without writing up something small already, I figured that I write it quickly to further ignite this discussion. Feel free to proof me wrong, I am of course not fully confident in my calculations, which can be done much more nicely and also with more attention to detail.

The main message is probably that playing with odd number of players is fine. Don't overthink it. I did the overthinking now, so just play cube.