Why does your psp cut off?

It depends on the conditions of the shut off, but it is generally related to the battery. If the battery is loose, it will shut off immediately (if the psp is hit or shaken too hard). If the battery has lost its ability to hold a charge, the screen will go grey and display a battery icon with a line through it before shutting off. You can test a loose battery by turning on the psp with the back cover removed, then wiggle the battery slightly and see if it stays on. A test to determine if the problem lies in the battery at all is to plug it in and see if it works longer than it did without. Good luck.