Which "Extra time per move" methods are there?

For more than 50 years, it was standard in chess to play a serious game in two periods separated by a break. This had two disadvantages:

Games could not always be decided after two periods.

The increasing strength of chess computers and end-game databases added what some considered an unfair advantage in the ability to analyze adjourned games.

Various proposals were made for ways to finish games in one session, without having to resort to the Guillotine method. The Fischer, Bronstein, and FIDE methods provided a solution by giving players a predetermined amount of extra thinking time after making each move.

The earliest proposal (1969) came from IGM David Bronstein. His method applies from the first move. Principal thinking time is reduced by delay. Before the principal thinking time is reduced, the player has a fixed amount of time to complete a move. Unlike the FIDE and Fischer methods, it is not possible to increase the thinking time by playing more quickly.

This method applies from the first move. A basic time is given to each player. Each time a player makes a move and presses the clock button, several seconds are added to that player's basic time. By completing moves in less time than originally allotted for him, the player's overall time is increased.

The Fischer-tournament method is the most complex one in the way thinking time is regulated. In addition to the extra time available per move, the player is also given an extra amount of principal thinking time after a predetermined number of moves has been made. Since the introduction of the DGT Chess Clock, this method has rapidly gained popularity. The Fischer-Tournament method is now standard in many top-level events like the World Championship and Chess Olympiads.

The FIDE-Rapid method begins a game with a traditional period during which a predetermined number of moves must be completed. When this period ends, each player gets extra time for each subsequent move. The thinking time that is unused at the end of a move is carried over to the next move. By making moves in less time than the extra time given per move, a player can increase the thinking time available for subsequent moves.

In this method, two periods of time are allotted before extra time per move is available.

Go with byo-yomi
By its very nature, the game of go lends itself to allowing players extra time to complete a game. The traditional byo-yomi method is used for this. Byo-yomi gives the player who has used up his thinking time a fixed amount of time for each subsequent move.
In normal games of go, the principal thinking time is 1½ to 2 hours, usually combined with a byo-yomi of 20 to 30 seconds. After the principal thinking time has been used, the clock jumps to byo-yomi time. Each time a player completes a move, the clock jumps back. If the player has not completed the move before the clock reaches zero, a flag appears in the display.
For top-level go matches, the principal thinking time is 9 hours, followed by 5 byo-yomi periods of 1 minute each. At the end of the 9-hour period the clock jumps to 5 minutes. If the player completes a move before a period of four minutes is reached, the clock jumps back to 5 minutes. If he completes a move after the 4-minute period has ended, the clock reverts to 4 minutes. Thus, the clock reverts each time to the beginning of the current byo-yomi period.

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