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In professional chess, I often see oddly arcane time controls. For instance, in the 2021 WCC

each game was 120 minutes per side for the first 40 moves, 60 minutes for the next 20 moves, and 15 minutes for the rest of the game, with a 30-second increment per move starting with move 61. — Wikipedia

I have the impression that, in some of these games, the players even play rather pointless moves when close to the next time bonus in order to get it more quickly. Are there arguments against using just an initial amount of time with a fixed increment after each move (as it's often the case online)?

I really do not want to incite a religious crusade about time controls, I'm just a beginner wanting to understand the advantages of these complicated rules.

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Why are such complicated time controls popular? ... For instance, in the 2021 WCC

First point, such time controls are neither popular nor unpopular with the vast majority of chess players for the simple reason that we never experience them. "WCC" stands for "World Chess Championship". Only two players ever play that once every two years. In general the time controls you talk about are only experienced at the very top of the game by the small number of players who can actually make a full time living at the game.

Secondly, the time control you quote has a total playing time for the game of 6 hours 30 minutes plus 1 minute per move after move 60. That is a very long time to be playing one game of chess with no breaks. Yes, the players are allowed to go to the bathroom during that time and they are usually allowed to also drink while playing but eating at the board is not allowed.

The intermediate time controls provide natural break points during the game which allow the players to have a break, go to the bathroom, splash some water on their face to freshen up, grab a sandwich, etc. It is very clear why they are popular at the elite levels where players can be playing one game for 7 hours or more.

At the more mundane level that the vast majority of us play at the time control is 90+30, 90 minutes for the game plus a 30 second increment from move 1, so for most games 4 hours or less. We normal players do very occasionally go beyond 60 moves but it is rare compared to the top players. We can get by with no intermediate time controls.

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Part of it will be tradition. Note that for the example that you give, adding a time per move is only one done after move 60. Traditionally, it was not done at all for a very simple reason: It would not have been possible with the chess clocks at the time. Only the advent of digital chess clocks brought these new options.

In the example that you give, for the first section there are 3 minutes per move, which is a common choice for "serious" games. The second section is again 3 minutes per game. Also, it is exactly one hour, which means that with an analog clock there would have been no adjustment necessary after move 40 (as it would have been for example if it would have been 30 minutes).

Nevertheless, the time control for World Chess Championships varied. I just browsed Wikipedia a bit. For 1990 I found an even longer 2.5h for 40 moves (after which the game would be adjourned) and then 1h for every 16 moves. You will note that the additions are again full hours, but also that the time was even more than 3 minutes per move, which is fitting because after all a World Championship is more than serious ;)

Note: Someone with actual historical knowledge will certainly be able to correct some things above or give a more complete picture. I am no expert, but I played a bit of tournament chess before the adoption of digital chess clocks.

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Old chess clocks didn't allow for increment to be added after every move, so extra time would be awarded at certain fixed points (normally moves 40 and 60).

With the invention of digital chess clocks time controls with increment per move have appeared and become more popular, but they're still not the only option being used.

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To take only the "simple" time control as 1 hour after 40 moves, it is in my opinion to move the game to the "technical" phase. Middle game is over usually due to that part and so a player has gained the advantage.

It would be rather disappointing to be a piece up and yet lose because you have only 2 minutes. Of course, the argument can be "bad time management", but at least the modern FIDE rules have state that if you have decisive advantage or clear draw, then you can call the arbiter for a draw.

If the game is still unclear after the 40 moves, then the player who saved the time will still keep their time advantage.

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On one hand, you have to have time controls for a number of reasons. There are logistical issues. Like you can't rent a venue out indefinitely. Also players might drag games out to get an unfair advantage like to wear an opponent down or to get to an adjournment where they can cheat. Also, people get bored and multi-day games aren't going to bring new fans into chess.

But on the other hand, you want to have good quality games and games that people can still appreciate centuries from now. You don't want the time controls to be too restrictive. It would be terrible if two players are playing an incredible game and the game ends because one player blunders in time trouble.

So, I think the goal is to keep the game moving and finish it, but still give the players enough time to produce a quality game.

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