H. Consider the game known as Rock, Paper, Scissors. Most of you are familiar with this game. There are two players, player 1 and player 2. Simultaneously each player chooses either rock, paper or scissors. When player 1 and player 2 play the same strategy each wins zero. When player 1 plays paper and 2 plays rock, paper wraps up rock so player one wins 1 and player 2 wins -1. If player 1 plays scissors and player 2 plays paper, scissors cuts paper, hence, player 1 wins 1 and player two wins -1. If player one plays rock and player 2 plays scissors, rock breaks scissors and player one wins 1 and two wins -1. a. Is this a constant sum game? Is it is zero sum game? b. Show that R/P/S has no pure strategy Nash equilibrium by drawing arrows on the bimatrix. Do you think that the arrow method is useful for larger games (i.e.games with more players and more strategies)? c. Show R/P/S has no pure strategy Nash equilibrium using the underlining method. d. Show that the strategy profiles (R, R) and (S,P) are not Nash using the definition method. 2. Consider the 3-player game matching pennies. You can find the description of this game in the Coordination Game Lecture. a. Represent the 3-player game matching pennies in normal form. (Hint: you will need two matrices. One matrix for player 3 plays Heads and one matrix for player 3 plays Tails). Make sure that the top left hand cell in matrix 1 is labeled H for player 1 and H for player 2. This is so we are all looking at the same matrix. b. Show that 3-player matching pennies has 6 pure strategy Nash equilibria using the underling method. c. Show that (T, T, H) is Nash and (T, T, T) is not Nash using the definition method. 1 3. Three tennis players want to enter a tennis tournament. There is room left in the tournament for one doubles team and one singles player. Each player may apply for only one slot-a singles slot or a doubles slot. Each player knows there are two other applicants, but doesn't know who they are so there is no chance for communication. If one player applies for the singles slot, she gets it. If two players apply for the singles slot they cut cards and each has a 50% chance of getting it. if all three apply for the singles slot, each has a 33 1 3% chance of getting it. If one player applies for the doubles team, she doesn't play in the tournament. If two players apply for the doubles team, they are in. If three apply for the doubles team, they cut cards and each of the three has a 66 2 3% chance of getting into the tournament. a. Put the game in normal form. Make sure that the top left cell of matrix one where Player 3 plays Singles, is labled S for player 1 and S for player 2. This is so we are all all looking at the same picture. b. Find the pure strategy Nash equilibria using the underlining method. c. Show (S,S,S) is Nash using the definition method. Show (S, S, D) is not Nash using the definition method. d. What would you do if you were one of the players
