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They don't change arbitrarily. They change one letter when you resign without exiting the game. Aside from that they are static unless you change your game's files. The reason it changes when you resign a game is to force you to restart the game before engaging in multiplayer to ensure all games are in the same state. The reason it changes when your files change is to make sure everyone is playing the same version and to prevent cheating.
 
Two things. First, I dispute the words "a little" near the end of the above post. It takes 5 mins for some of us to fire up EU III

Second, and more importantly, AFAIK checksums are somehow the additions of all the whatevers in the program. There can be up to, apx, half a million checksums. There, however, can be more than half a million states the game can be put in to. Thus my question is what happens if you've cheated in such a way to provide an identical checksum but with radically different files?

You would cheat the system but the game would go out of sync shortly after the MP game has started.
 
"The reason it changes when your files change is to make sure everyone is playing the same version and to prevent cheating."

The first part of what I said is also important, not playing the same version = super fast out of sync. It's not really a complex system either: Restart game before rehosting MP, if you have the wrong checksum after that your install is broken or modded so reinstall or remove your changes.