Ok, there’s the foundations-significant Kirby/Paris hydra battle, where we can show that Hercules always wins, by a transfinite induction up to ε_{0}.

Now, there must be lots of other hydra games — maybe some of them even a bit interesting! — that terminate (or at least terminate on best play) with a win for Hercules, and can be shown to do so by inductions along smaller ordinals.

Does anyone happen to know if there’s a discussion out there of examples of such shorter games?

[Full disclosure: I’m writing a chapter in which I want to work up to the Kirby/Paris example through just a few warm-exercises on simpler hydra games. I don’t want to be re-inventing the wheel, and it would be nice to have some examples which are better than my so-far feeble efforts!]