This lesson shows how backward induction can lead to some strange (and irrational?) problems.
Backward induction might not reflect how players actually play. (Alternatively, we may just be specifying the game incorrectly, as players might not have an understanding of what is going on. That would change things to a game of incomplete information, which backward induction does not solve.)
People who "misplay" the centipede game score higher utilities than those who play it according to backward induction. Game theory is the study of maximizing payoffs, but here it fails us.
We may want to induce our opponent to believe that we are crazy by making moves off the equilibrium path of play. If this is true, then backward induction cannot help us.
It is hard to pin down what exactly "rational" behavior is here.