An increment is bonus time, usually a few seconds, added to your clock after every move you make.
In a control written 3+2 or 15+10, the second number is the increment: two or ten seconds credited after each of your moves. Bobby Fischer patented a digital clock built around the idea in 1989, which is why it is often called Fischer time. Move faster than the increment and your clock actually grows.
Increment differs from delay. With a delay, the clock simply waits a few seconds before counting down and unused delay is lost; with an increment the seconds are added to your total, so they bank and accumulate. Both prevent the pure mad scramble of sudden death finishes.
For endgame play the difference is huge. With even a two second increment, a technical win like the Lucena position can always be converted calmly; without one, plenty of won endgames are lost or drawn on the clock alone.
Fifteen minutes on the clock for each player, plus ten seconds added after every move you make. The added seconds are yours to keep, so quick moves build your time back up.
Most players prefer increment because saved seconds accumulate, which rewards fast play and makes long technical endgames finishable. Delay gives a similar safety net, but nothing you save is banked.
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