Fred and Ellie need to decide how to split $1 between them. Fred is designing the rules of the bargaining process that the pair will use to make the decision. Fred's per period discount factor is and Ellie's is . Both of their discount factors are in the range [0,1].

a) Fred considers a three-period alternating offers bargaining game, where he makes the first offer and Ellie then decides whether to accept the offer or reject it. If Ellie rejects, then the game moves forward one period and Ellie makes an offer, which Fred can either accept or reject. If Fred rejects, then he gets to make a second offer in the third period of the game. Ellie can either accept or reject Fred's final offer. If she rejects it, both players receive 0. Derive the subgame perfect Nash equilibrium of this game and write down the equilibrium payoffs of each player.

b) Are there any circumstances in which Fred is indifferent between making the first offer himself in the game described in part a) and playing a version of the same game where he and Ellie switch places? In this alternative version, Ellie would make the first offer, Fred would make the second offer if he decided to reject Ellie's first offer, and Ellie would make a third offer if she decided to reject Fred's offer, and the game still ends after three periods.

c) Ellie has another idea. She proposes that she takes the $1 and places it in either Room A or Room B without letting Fred know the room that she has chosen. Fred simultaneously decides to enter one of the two rooms via another door. If he enters the room with the $1, Fred and Ellie share the $1 evenly between them. Otherwise, Ellie keeps the entire $1 and Fred gets nothing. Represent this game in Normal Form. Find the Nash Equilibria or Equilibrium of this game. Would Fred ever agree to this proposed way to share the $1 rather than the sequential bargaining game that is described in part a) where Fred moves first?