ah right, when i responded I had forgotten we were talking specifically about strikes.
I guess I’m not clear on the TGP calculation for strikes tournaments. this calculator: https://strikestgp.slapsave.com/ says 40 players, head to head, 3 strikes, is ~12 games towards tgp (4% * 12 = 48% TGP), ~12 rounds
if you change it to 40 players, 3 strikes, 4 player fair strikes, it gives ~11.6 games towards TGP, ~6 rounds. Is that 11.68% or 68% for the TGP calculation?
I think the TGP/duration deficiency of Head-to-Head Strikes fundamentally comes from increased variance (or rather coefficient of variation) in game length.
As a thought experiment, consider two extremes:
Imagine we had 1 player per game in each round. The game length (using my crude heuristic) would be precisely the lognormal distribution I assumed for game score, with its long right tail. The round duration would be driven by a single player’s performance on one game.
Now imagine we had 20 players per game in each round. One player may get a very high score, but this will usually be balanced out by several other players getting low scores on that game. The distribution’s long right tail is severely shortened, and the distribution overall becomes narrow relative to its mean value (thanks to the central limit theorem).
It’s the same reason you’d expect better time efficiency when using multiple games in a round. If you assign a 4-player group to 3 games in a round, you get the sum of 12 player performances, and this has lower variance (that is, lower coefficient of variation) than 4 player performances on a game.
After some experimentation, I think I am mostly correct about coefficient of variation being the source of the inefficiency of 2-player games.
I made another script which generates 100 players of randomized skill in no particular order, partitions them into either 4- or 2-player groups, and estimates their game duration as their sum of game scores, then returns the maximum game duration across all groups. This is repeated 20,000 times for both 4- and 2-player groups.
Results
The average round duration for 25 groups of 4 players each is 22.141. The average round duration for 50 groups of 2 players each is 15.129.
The histograms for round duration with each type of group partitioning is shown below:
I believe the Extreme Value Distribution would describe this data very well.
Conclusions
The 4-player rounds are of course longer than the 2-player rounds. But they are not twice as long. Two rounds of 2-player games has an average duration of 30.26. A single round of 4-player games is, on average, only 73% as long as two rounds of 2-player games! This is the ~25% deficiency we saw in the bulk tournament analysis.
I would have to guess that most of the problems of 2-player rounds go away if you assign 2 games per round.
This ignores the fact that 2 player games are often shortened by player 2 winning and only needing 2 balls. It seems significant.
I analyzed that in an earlier post. It ended up reducing head-to-head round times by about 3.8% (relative to always playing every ball), which is not enough to counteract the coefficient of variation effect. But it should not be ignored!
Great stuff!
Would love to see some sims for other common formats as well (and playoff styles)! 
Two things to mention.
The skill information from an individual player, for a certain tournament duration is typically higher with H2H games vs 4-player games. You play more per hour basically.
However, for a TD, 2-player games also require more machines. This could limit the amount of participants in the tournament. So with that and TGP I guess 4-player games are often preferred, especially for larger tournaments.
Edit:
Also, let’s see the sim comparison of regular matchplay vs frenzy. I wanna see the Kendall tau…
The larger a tournament the lower effect of 2 player games being shortened. A round is as long as its longest match, so walk off wins don’t necessarily have an impact.
This is a good insight about the low impact of reduced game times due to walk-offs. It’s common for a walk-off to include a very long ball 1 or ball 2, where the player put in all of the flip-time that would more generally occur over the course of 3 balls in the typical game. I think that your simulation was also accounting for that.
And a minor quibble from an earlier post:
I believe you mean 23 Meaningful Games Played.
This makes me think that all of this analysis of the walk-off effect for 2P vs 4P would have a different impact for a tournament with Swiss rather than Random/Balanced pairings. In that situation I’d think the 2P format would gain some time efficiency since a 4P Swiss game would have less of a balancing effect than two 2P Swiss games among those same 4 players. Though maybe that’d be a pretty small difference.
Time for another simulation ‘experiment’!
Recipe
Note: In a previous post, I looked at the top group of (2/3/4) from a total of 100 players. This experiment is slightly different - I generate game durations for all groups and take only maximum game duration, as you would in a real tournament.
Results
| Groupings | Players Per Game | Avg Duration (No Skipping) | Avg Duration (Skipping) | Avg Skip Reduction |
|---|---|---|---|---|
| Sorted | 2 | 44.145 | 42.436 | 4.0% |
| Sorted | 3 | 60.670 | 59.315 | 2.3% |
| Sorted | 4 | 76.373 | 75.192 | 1.6% |
| Random | 2 | 37.638 | 35.783 | 5.2% |
| Random | 3 | 48.205 | 47.273 | 2.0% |
| Random | 4 | 58.426 | 57.898 | 0.9% |
Conclusions
Random groupings finish their round in significantly less time than sorted groupings. But a previous experiment showed that (at least for Fair Strikes with 4 players per game), random groupings lengthen the tournament by ~10%, because they increase the number of rounds more than they reduce the duration of each round. I did not include ‘skipping the last ball’ in the main tournament simulation code. I still think that Swiss groupings are preferred over random groupings.
In comparison, the effects of skipping the last ball are smaller. Two-player groups benefit the most from skipping the last player’s third ball if they’ve already won. For 4-player groups, the effect of skipping is quite small, regardless of which grouping method is used.
