Suppose that we have two players, A and B. Player A can choose to either go Up or Down; B can choose to either go Left or Right.

• If A chooses Up and B chooses Left, each player receives 30 utils (a satisfaction level measure).
• If A chooses Down and B chooses Right, each player receives 50 utils.
• If A chooses Up and B chooses Right, A gets 100 utils and B gets 10 utils; the payoff values are reversed if A chooses Down and B chooses Left.

1. Write the normal form of this simultaneous game and find the pure strategy Nash equilibrium or equilibria (PSNE) using the underlining method. Then, if there is/are (a) PSNE(s), write it/them using the notation we learned in class.

1. Does any of the players have a dominant strategy?Define "dominant strategy" and explain your answer using the 4 appropriate if-then statements.

Next, suppose that A and B have the same strategies: to go Up or Down:

• If A and B both choose "Up" A receives 30 utils (utility units) and B receives 45 utils.
• If A and B both choose "Down", A yields 30 utils and B yields 40 utils.
• If A chooses "Up" and B chooses "Down," A gets 50 utils and B gets 75 utils
• If A chooses "Down" and B chooses "Up", A gets 80 utils and B gets 50 utils

1. Represent this simultaneous game in normal form and find the pure strategy Nash equilibrium or equilibria using the underlining method. Then, if there is/are (a) PSNE(s), write it/them using the notation we learned in class.

1. does any of the players have a dominant strategy?Define "dominant strategy" and explain your answer using the 4 appropriate if-then statements.

1. Now suppose that A and B can also choose "Straight" as a third action or strategy, that the above payoffs remain the same for the possible (Up, Down) combinations, and that the new payoffs associated with this third action are:

• If A and B choose Straight, they each get 60 utils
• If A chooses Up and B Straight, A gets 55 utils and B gets 85 utils
• If A chooses Down and B chooses Straight, A gets 50 utils and B gets 60 utils
• If A chooses Straight and B chooses Up, A gets 80 utils and B gets 45 utils
• If A chooses Straight and B chooses Down, A gets 30 utils and B gets 80 utils

e1) Represent this game in normal form.

e2) Find the pure strategy Nash equilibrium or equilibria of this game using the underlining method.Then, write the pure strategy Nash equilibrium or equilibria solution using the notation we learned in class.

e3) Does any player have a dominant strategy?

1. Suppose that player C joins A and B, now becoming a 3x2 game with possible actions being Up or Down.If C chooses Up, C's payoffs become the average of A and B's payoffs in each cell of the 2x2 normal form from part 'b' PLUS 10 utils; if C selects Down, C's payoffs are the sum of A and B's payoffs in each cell of the other 2X2 normal form from part 'c' MINUS 15 utils.Assuming that A and B's payoffs are the same ones used in part c), represent this 3X2 game in normal form and find the pure strategy Nash equilibrium or equilibria by using the underlining method and writing the solution of this game. Does any player have a dominant strategy? Explain.