>In ROUND100, Alice knows Bob will take whatever non-zero offer she makes.

Wait, how does she know that? Can you explain why you consider that the most rational choice for Bob to make? As experiment shows, this is not the optimal strategy, so I don't see why you would call it the "rational" one.

Also, keep in mind that you're describing an iterative game (where the same people play the same game with the same circumstances repeatedly) and the pirate game is non-iterative, since the players and circumstances change between rounds. As you note, iterative games are much more complicated and the sort of induction used in the pirate game doesn't really work.