All PSP games are the same, there are no version specific games. No matter what model a PSP it is, as long as it has a UMD drive then it can play any UMD game.
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Something could be wrong with the psp.. or are you sure it isn't a PSP Go specific game? As a new hand-held console, I'm sure it would have better graphics than the old PSP, causing the game you have not to start up.
That's happening because you dropped your PSP or your battery is really old. Now to fix this just buy a brand new PSP battery after that it should be fine.
The old PS3 can only play the oldest PS3 games. A new PS3 can play both new and old games.
If you mean play xbox 360 games on the original xbox then no. If you mean play ORIGINAL xbox games on the 360 then yes. Some games work straight away when you put them in the 360 but some you have to download something off live so it does work i think.
Xbox games will not work on a Wii, and Wii games will not work on a Xbox. however, if you mod soft your wii, you can play games from old platforms.