I'm not sure why that should apply. The unexpected hanging worked by exploiting the fact that days that were "ruled out" were especially good candidates for being "unexpected". Other readings employ similar linguistic tricks.

The reasoning in the first case does not work in practice because in a tournament premise (1) is false; tit-for-tat agents, for example, will cooperate in every round against a cooperative opponent.

But that is not even relevant to the fact that the mathematical induction does not work for unknown numbers of rounds.