Should "arbitrary" suffice on its own, or does it make sense to include "any?"
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If you leave out the any in that sentence, you could be talking about a known set of criteria that happen to be arbitrarily selected. Add it, and you are broadening the scope to include criteria that are not specific, germane, or qualified in some way.
My Webster's defines arbitrary as
I suspect you may be worried that arbitrary may mean the same thing as any, but it does not. There is no sense of randomness to the term any; it only expresses a degree of inclusiveness.
It makes very little difference, but I would remove the 'any' as it sounds awkward. Or if you choose to keep the any, remove the arbitrary as 'any' includes the meaning of arbitrary to an extent to which arbitrary can be reasonably removed.