I think you may be thinking of the argument that humans have free will, so they can't force future versions of themselves to do something that would be against that future version's given its information

That is not what I was thinking of. Here, let me re-quote the whole sentence:

The classical game theorist assumes you can't look into people's heads, so whatever you say or do before the cheating, you're always free to not punish during the punishment round

The funny implication here is that if someone did look into your head, you would no longer be "free." Like a lightswitch :P And then if they erased their memory of what they saw, you're free again. Freedom on, freedom off.

And though that is a fine idea to define, to mix it up with an algorithmic use of "freedom" seems to just be used to argue "by definition."