Two players try to split $100. In an outcome where player 1 receives $x and player 2 receives $y, player 1's utility u1(x, y) = x and player 2's utility u2(x, y) = y x. For example, suppose player 1 receives $60 and player 2 receives $40. Then player 1's utility (or payoff) is 60, while player 2's utility is 20. On the other hand, suppose player 1 receives $40 and player receives $60. Then player 1's utility is 40 and player 2's utility is 20.

(a) Interpret the utility functions.

Suppose they can simultaneously choose whether to "split" or "steal" the money. If they both choose "split" then the money is split evenly between them. If one of them chooses "split" and the other chooses "steal", then the person who chooses "steal" gets all the money. If they both choose to "steal" the money, then they both get none of the money.

(b) Model the above strategic situation with a normal form game and find all pure-strategy Nash equilibria of the game.

(c) Write the above game as a game tree. Now suppose both players simultaneously name how much money they would like to take, s1 and s2, which are real numbers between 0 and 100 (both ends included). If s1 + s2 100, then the players receive the amount they named; if s1 + s2 > 100, then both players receive no money.

(d) Write down the normal form game. (e) (5 points) Find the best responses for each player. (f) (5 points) Report all pure-strategy Nash equilibria of this game