Suppose again, as in exercise 24.5, that two players have $100 to split between them.
A: But now, instead of one player proposing a division and the other accepting or rejecting it, suppose that player 1 divides the $100 into two piles and player 2 then selects his preferred pile.
(a) What is the sub game perfect equilibrium of this game?
(b) Can you think of a Nash equilibrium (with an outcome different than the sub game perfect outcome) that is not sub game perfect?
(c) In exercise 24.5, we considered the possibility of restricting offers to be in integer amounts, to be in pennies, etc. Would our prediction differ here if we made different such assumptions?
(d) Suppose that the pot was $99 and player 1 can only create piles in integer (i.e. full dollar) amounts. Who would you prefer to be: player 1 or 2?
(e) Suppose that player 2 has three possible actions: Pick up the smaller pile, pick up the large pile, and set all of it on fire. Can you now think of Nash equilibria that are not sub game perfect?
B: In exercise 24.5, we next considered an incomplete information game in which you split the $100 and I was a self-righteous moralist with some probability ρ. Assuming that the opposing player is some strange type with some probability can sometimes allow us to reconcile experimental data that differs from game theory predictions.
(a) Why might this be something we introduce into the game from exercise 24.5 but not here?
(b) If we were to introduce the possibility that player 2 plays a strategy other than the “rational” strategy with probability ρ, is there any way that this will result in player 1 getting less than $50 in this game?