Game Theory [15 marks total] 4.1 [8 marks] We call a number low if it belongs to the set {1,2}. We call it high if it belongs to the set {3,4}. Consider the following matrix game. Player C chooses a number from {1,2,3,4} and player R simultaneously chooses either high" or "low. If player R guesses correctly what set player C picked from, then player C must pay x to player R, where x is the number that player C picked. If player R guesses incorrectly, then that amount must be paid from player R to player C. Determine the payoff matrix. Then write down the linear program to determine the optimal randomized strategy for player C. Do not solve the linear program! [7 marks] Consider the matrix game given by the following payoff matrix (as usual, the column player is the maximisation player and the row player is the minimisation player). 4.2 1 4 4 2 2 -1 -3 4 3 0 -2 -2 The column player plays the randomized strategy (1,4, 0, 2). We assume that the row player knows this. What is the corresponding best strategy for the row player? Game Theory [15 marks total] 4.1 [8 marks] We call a number low if it belongs to the set {1,2}. We call it high if it belongs to the set {3,4}. Consider the following matrix game. Player C chooses a number from {1,2,3,4} and player R simultaneously chooses either high" or "low. If player R guesses correctly what set player C picked from, then player C must pay x to player R, where x is the number that player C picked. If player R guesses incorrectly, then that amount must be paid from player R to player C. Determine the payoff matrix. Then write down the linear program to determine the optimal randomized strategy for player C. Do not solve the linear program! [7 marks] Consider the matrix game given by the following payoff matrix (as usual, the column player is the maximisation player and the row player is the minimisation player). 4.2 1 4 4 2 2 -1 -3 4 3 0 -2 -2 The column player plays the randomized strategy (1,4, 0, 2). We assume that the row player knows this. What is the corresponding best strategy for the row player