1. Consider the following bargaining game. Two players are trying to divide 100 gold between each other. Each player announces a numbersimultaneously.The announced number reflects how much gold that player wants. The number does not have to be an integer so players can announce numbers involving fractions. Denote the number announced by player 1 ass1and the number announced by player 2 ass2. Each player gets payoff from the amount of gold they get. For example, if a player gets 30 gold, her payoff will be 30. The game has the following additional rule. If the sum of the two numbers announced by the two players are strictly greater than 100 (s1+s2>100), then both players get 0. If the two numbers announced by the two players are less than or equal to 100 (s1+s2100) then each of them gets the number they announce. For example, ifs1= 23 ands2= 45, the sum of the two numbers are less than 100, so player 1 will get 23 and player 2 will get 45. If s1=90 and s2=30, the sum of the two numbers are greater than 100 so both players get 0.

a) (15 points) Find all pure strategy Nash equilibria of this game. Arguewhythey are Nash equilibria. (Hint: There might be too many Nash equilibria, but actually some of them are very similar to each other. Think about those similarities.)

Now suppose that we are changing the payoff function of player 1, while keeping everything else the same as part a. Suppose that player 1 is a jealous person, she gets utility from the amount of gold she gets but she also gets a disutility (negative utility) from the amount of gold player 2 gets. So player 1's payoff is given by "amount of gold player 1 gets""amount of gold player 2 gets". Player 2's payoff is the same as part a, it is just the amount of gold she gets. For example, ifs1= 30 ands2= 20, each player gets the amount of gold they announce becauses1+s2100, but player 1 gets payoff 3020 = 10. Player 2's payoff is 20.

b) (5 points) Show thats1= 40 ands2= 60 is not a Nash equilibrium.

c) (5 points) Show thats1= 140 ands2= 60 is a Nash equilibrium.

Defend your answers carefully and show every step of your calculations.