This question is an extension of the Centipede Game.
My prof. posed this to me in class and I can't figure out how to approach this problem.
Imagine in this game, there is an alternative possibility (with very low probability, 0.0001, 0.0000001, etc.) that the first player is irrational, and always goes right instead of down. If this were true, it would make sense for player 2 to always go right (except at the last node) to achieve the highest payoff.
The tricky part for me is: now assume that P1 and P2 are both rational, and that they know that each other are rational, but P1 does not know that P2 knows that P1 is rational — how does that affect the game (with regards to the equilibrium outcome)?
Backwards induction is pretty straightforward in the original form of the game, is there an alternative method that can be better used for this incomplete information version?
(I was thinking of assigning probabilities — starting from the second-to-last node — that would make the payouts equivalent for P2's response to an irrational/rational version of P1, but that seems quite tedious for the ~100 nodes, and I feel there must be come generalized approach to this.)