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select a random person and move on

Most games (or other programs) will pick an arbitrary player from the candidate list.
Can be randomized, just the one with the lowest/highest player number, by whoever manages to join the game first etc.

Making this an unhandled error condition would be lazy, not user-friendly and overall pretty stupid.

It kinda depends on how the programmer has written his or her code.

These are some possibilities that could happen:

1. The programmer noticed the possibility and decided to apply some form of ruling. Perhaps he or she made it so that the amount of players each get a the same probability to get chosen. So with two tied players, there's a 50% for each.
2. An error occurs, as you've said.
3. It could also continue the code without interruption. Perhaps it'll pick the first or last least-special player, and ignore the others. This may sound terrible, but might not be that awful, since it could then add a +1 to the times the chosen person has been special and next time it could be the other player.

Depends on the implementation and compiler - if you show us that function, we will be able to tell.

format c:

It would probably just throw out unhandled exception error, click ok to attempt to continue...
I'm guessing that the special player will simply be not unselected (code is skipped) and game continues.
Game will throw out next unhandled error at the next moment where 'special player' has to do something according to the code, as there is no special player. Then at some point after clicking trough error messages you either crash game or return to main menu if code supplies reasons for that event.

The code will do what you tell it to do.
Nothings happens magically. unhandled stuff behavior is determined by your implementation and language\compiler