What is the average amount of time needed to complete a game of chess for each of the time controls

Hey lads, CC has recently launched the leagues and I've been trying to prove that playing bullet and or blitz is the most efficient way to gain trophies. I've done the maths assuming all the time in any given game is all going to be used and my hypothesis was vindicated, a mate of mine disagreed with my maths though saying that people rarely use all their time, I've put a modifier to the total time needed to make it a bit more accurate but my mate is still in denial and says only using the actual averages from CC will get the correct figures. Does anyone know how to get those averages?

For anyone curios about my maths so far here it is btw.

Assume each game of chess takes the sum of both side's time, with half of one side's time subtracted from that sum (A 10+0 game will take (10+10)-(10/2) = 15 minutes)

Bullet is the baseline, scoring with 3 for a win, Blitz is 3 * the bullet score, and Rapid is 5 * the bullet score.

Bullet is fastest at the 1 minute time control which means 1.5 minutes per game.

Blitz is fastest at the 3 minute time control which means 4.5 minutes per game

Rapid is fastest at the 10 minute time control which means 15 minutes per game.

100 bullet games, takes 150 minutes. Assuming a 50% win rate, that's 150 trophies in 150 minutes which means one trophy per minute.

100 blitz games, at fastest, takes 450 minutes. Assuming a 50% win rate, that's 450 trophies in 450 minutes which means one trophy per minute.

100 rapid games, at fastest, take 1500 minutes. Assuming a 50% win rate, that's 750 trophies in 1500 minutes which means 0.5 trophies per minute.

You also get a points for drawing but I didn't count for that in my calculations since the draw percentage changes for the different elo brackets, I'll add it to my calculations if I get the relevant data so that I can make an even more accurate estimate of trophies earned per minute but I doubt it'll change anything by a significant margin.

In other words 75% of the total time available.

Yeah, under that assumption rapid is the slowest for getting trophies.

For what it's worth, when I was preparing an attempt to win a 24 hour tournament, I wanted to know about how many wins and total games I'd need (based on past winners). I found that a good estimate for the 3|0 time control is 6 minutes per game on average.

This seems counter intuitive, because it's impossible for both players to use all their time, but after a game finishes, there are a few seconds of looking at the position, looking at your rating, that sort of thing. Then you click the button, and it takes a few seconds to pair you... and since no one is a machine, after an hour or 2, we get something to drink, or we get up to use the bathroom, etc. So it turns out 6 minutes is a reasonable estimate for 3|0 games.

Also FWIW... in my experience weaker players, and people who become nervous because of the clock are the ones who play longer time controls. Frequently these people don't even use half their time. They only choose the rapid time control so they can ignore the clock completely. In other words they play rapid time controls at blitz speed... I'm surprised chess.com made it so that rapid games are worth 1/2 as much, but I don't think they're so incredibly incompetent to have done that accidentally. They have a lot of data. They probably calculated that this was the correct amount.

Anyway, when assuming all games will have the same % taken off (in your case 75%) whatever the ratio of minutes to trophies is will be preserved, so you don't have to recalculate anything.

However what I brought up would change things i.e. the time in between games. That's a constant value, unrelated to the time control of the game.

So in that case rapid would be the fastest, because you're adding this extra time less frequently (it's only added between games, so fewer games is ideal).

However, since chess.com already gave rapid a handicap, then my guess is 3|0 blitz is very slightly preferable to bullet or rapid.

Oh, and one last thing...

I also noticed that winners of arena tournaments consistently have a lower time per game average... in other words, not only are they strong players, but they also play more games.

I downloaded their games, and chessbase makes some visualization of data easy... I noticed on average their games were fewer moves. In particular their losses were fewer moves...

... meaning a good strategy is to resign early when your position is bad (that way you get to play more games). It's pretty obvious once you hear it... but this sort of optimization is probably unnecessary until you're going for the last league or two.

#4
Arena rewards playing too fast. If you use only 50% of the available time, you play more games, but you also put pressure on your opponent as you deny him time to think during your move.