Exercise 2: Nash equilibrium Consider the following two-player game in which each player is given a set of cards and each card has a number on it. The players are Antonia (Player 1) and Bryson (Player2). Antonia's cards have the following numbers (one number on each card): 2, 4 and 6. Bryson's cards are marked 0,1 and 2. Each player simultaneously chooses one of their cards. The payoffs depend on the sum of the points of the chosen cards. - If the sum of the points on the two chosen cards is greater than or equal to 5 , Antonia gets $10 minus that sum; - otherwise (that is, if the sum is less than 5) she gets nothing; - if the sum of points is an odd number, Bryson gets as many dollars as that sum; - if the sum of points is an even number and is less than or equal to 6 , Bryson gets $2; otherwise he gets nothing. 1. Write down the payoff table/matrix of this game. 2. Identify and eliminate any dominated strategies. 3. In the reduced game, find the players' best responses. 4. Find all pure-strategy Nash equilibria. Are any of the Nash equilibria Pareto efficient? 5. Suppose the players choose their cards sequentially rather than simultaneously. Antonia chooses one of her cards first. Then Bryson chooses one of his cards after observing Antonia's choice. (a) Draw the game tree. (b) Use Backward Induction to find all subgame-perfect Nash equilibria. Exercise 2: Nash equilibrium Consider the following two-player game in which each player is given a set of cards and each card has a number on it. The players are Antonia (Player 1) and Bryson (Player2). Antonia's cards have the following numbers (one number on each card): 2, 4 and 6. Bryson's cards are marked 0,1 and 2. Each player simultaneously chooses one of their cards. The payoffs depend on the sum of the points of the chosen cards. - If the sum of the points on the two chosen cards is greater than or equal to 5 , Antonia gets $10 minus that sum; - otherwise (that is, if the sum is less than 5) she gets nothing; - if the sum of points is an odd number, Bryson gets as many dollars as that sum; - if the sum of points is an even number and is less than or equal to 6 , Bryson gets $2; otherwise he gets nothing. 1. Write down the payoff table/matrix of this game. 2. Identify and eliminate any dominated strategies. 3. In the reduced game, find the players' best responses. 4. Find all pure-strategy Nash equilibria. Are any of the Nash equilibria Pareto efficient? 5. Suppose the players choose their cards sequentially rather than simultaneously. Antonia chooses one of her cards first. Then Bryson chooses one of his cards after observing Antonia's choice. (a) Draw the game tree. (b) Use Backward Induction to find all subgame-perfect Nash equilibria