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Increment is FIDE's default for their tournaments, although their rules permit delay as well. Most internet chess servers also use increment exclusively.
As I understand it, delay is used by the USCF (for historic reasons?), and, for some reason, by the 2017 Grand Chess Tour.
Why is increment preferred over delay by pretty much everyone but the USCF?
Very simple. Move quickly (faster than the increment) and you end up with more time. Over the course of a dozen or so quickly made moves you can build up enough time to allow for another serious think if required.
With delay you can never get more time. Once in zeitnot you are permanently in zeitnot.
Virtually "Delay" can be considered as a special case of "Increment". Both gain time in their own way, but "Delay" is limited to first few seconds, while "Increment" always guarantees the time gain.
e.g. 10 sec "Delay" will allow the player to gain time only if the player makes a move within their first 10 sec. Kind of enforces a quick play. After certain number of moves, it typically gets difficult to play quickly especially in a lesser time controls. This may cause unexpected blunder or weaker play eventually.
In case of "Increment" the time always gets added (unless the player has timed out). This allows the player to think their moves properly within their total time frame. Usually such moves generates a better game from both the sides.
Hence majority of the tournament opt for "Increment" over "Delay".
