Consider the ultimatum game that we discussed in class except now there are only four dollars. Here is the game. Player 1 begins by proposing a split of the four players. Player 1 can propose a split of (x,4x), which means that player 1 gets to keep x dollars and the remainder goes to player 2. The splits must be whole numbers so that player 1 can offer splits of (0,4),(1,3),(2,2),(3,1),(4,0). Player 2 then observes this proposed split and then chooses whether or not to accept it. If player 2 accepts then player 1 obtains the dollar amount specified in the proposed split and player 2 gets the remaining money. Otherwise, they don't reach an agreement and both get zero dollars.

Part a: Suppose that each player's utility is exactly equal to the amount of money that the player himself receives at the end of the game. Find all of the backward induction solutions of this game.

Part b: Now suppose instead that the players actually care about the money that the other receives since the players care about each other. To model this, suppose that if the ultimate monetary outcomes are x dollars for player 1 and y dollars for player 2, then player 1 's utility is given by u1(x, y) =x|xy|. Similarly, player 2's utility is given by u2(x, y) =y|xy|. In this situation, find all of the backward induction solutions.