Name:___________________ For ease of grading, please write your answer in the space provided below each question. 1. Consider the following two person game. Player 1 begins the game by choosing A or B. If Player 1 chooses A the game ends and Player 1 receives $100 and Player 2 receives $100. If Player 1 chooses B then Player 2 must choose C or D. If Player 2 chooses C then Player 1 receives $150 and Player 2 receives $250. If Player 2 chooses D then Player 1 receives $0 and Player 2 receives $400. a. Draw the complete game tree for this situation. Be sure to accurately label the tree and include the payoffs. (10 points) b. Rational Prediction: Using backwards induction (look forward and reason backwards) determine the rational outcome to this game. (5 points) c. Behavioral Prediction: Given how experimental subjects have behaved in the Ultimatum game and the Centipede game, provide a behavioral explanation for why an \"average\" Player 1 and 2 might deviate from the rational prediction. (5 points) 2. \"To be or not to be, that is the question.\" Imagine that in answering this question Hamlet had the following data to consider. Hamlet must choose either \"To Be\" or \"Not to Be.\" If Hamlet chooses \"To Be\" then his enemy the King will make a decision to either \"Kill\" Hamlet or let him \"Live.\" If Hamlet is killed by the King his payoff in this game is -25 since he will have been killed by the King who is his uncle and who also killed Hamlet's father, the King's brother, who was the prior king. If the King decides to let Hamlet live then Hamlet will be able to avenge his father's death and his payoff will be +45. Hamlet also considers suicide an option but given the uncertainty of the afterlife if he chooses not to be then his payoff is -5 since his doesn't what the after life holds for one who kills himself. a. Draw the complete game tree for this situation. Be sure to accurately label the tree and include the payoffs. b. Assume Hamlet will choose the course of action that offers the highest expected value. Under what circumstances will Hamlet choose \"To Be?\" Complete the following sentence: Hamlet will choose \"To Be\" if.... 3. In each of the games shown below, players 1 and 2 must move at the same time without knowledge of the other player's move. The first payoff is for the row player (Player1) and the second payoff is for the column player (Player 2). Solve each game using game theoretic logic. a. If either player has a dominant strategy, circle what it is (i.e. circle up or down, left or right). Do not circle an action if is not a dominant strategy. b. Circle the predicted outcome(s) to the game (i.e. circle (8, 3) or (4, 5) or (7, 2) or (1, 3 Player 2 Player 1 Left Right Up 8, 5 6, 9 Down 9, 6 7, 5 Player 2 Player 1 Left Middle Right Up 11, 4 3, 6 4, 7 Middle 13, 5 6, 10 5, 11 Down 12, 8 5, 2 2, 3 Player 2 E F G H A 6, 3 4, 4 3, 4 5, 5 B 5, 7 5, 8 4, 9 1, 2 C 6, 6 6, 8 5, 7 4, 3 D 0, 10 2, 13 0, 15 4, 3 Player 1 4. Consider the following 3 person color game between Hillary, Laura, and Michelle. Each player can choose either Red or Blue. The payoffs for these choices are shown below. The first payoff in each cell is for Hillary, the second payoff is for Laura, and the third payoff is for Michelle. Hillary Red Hillary Blue Michelle Red Laura Red 1, 1, 10 2, 1, 2 Laura Blue 1, 2, 2 2, 2, 2 Hillary Red Hillary Blue Michelle Blue Laura Red 6, 6, 6 10, 1, 1 Laura Blue 1, 10, 1 2, 2, 1 a. Assume this game is only played one time. Using dominant strategy logic, what outcome would game theory predict? If a player has a dominant strategy, circle the dominant strategy. b. Assume this game is repeated and played 200 times. What strategy would you advise the players to use assuming that the goal is to maximize total payoff after 200 trials? Explain your reasoning. 5. In the game shown below, Players 1 and 2 are competing over how to divide up $100. Each player must move at the same time without knowledge of the other player's move. The table shows the payoff for Player1 (Player 2's payoff is $100 - Player 1's Payoff). Find the Nash equilibrium profile of strategies for this game. That is, what strategy should Player 1 and Player 2 use for this game? If the Players use their optimal strategy, what will be the payoff for each player in this game? You will need to use a combination of techniques to solve this problem. Player 2 Player 1 Left Middle Right Up 60 40 30 Middle 50 30 65 Down 70 50 40 Homework Assignment Name:___________________ For ease of grading, please write your answer in the space provided below each question. 1. Consider the following two person game. Player 1 begins the game by choosing A or B. If Player 1 chooses A the game ends and Player 1 receives $100 and Player 2 receives $100. If Player 1 chooses B then Player 2 must choose C or D. If Player 2 chooses C then Player 1 receives $150 and Player 2 receives $250. If Player 2 chooses D then Player 1 receives $0 and Player 2 receives $400. a. Draw the complete game tree for this situation. Be sure to accurately label the tree and include the payoffs. (10 points) (P 1) 1 $100 (P 2) (P 1) A $100 0 (P 1) B (P 1) 2 (P 2) C 3 $150 $250 (P 1) (P 2) D (P 2) $0 4 (P 2) $400 b. Rational Prediction: Using backwards induction (look forward and reason backwards) determine the rational outcome to this game. (5 points) c. Behavioral Prediction: Given how experimental subjects have behaved in the Ultimatum game and the Centipede game, provide a behavioral explanation for why an \"average\" Player 1 and 2 might deviate from the rational prediction. (5 points) 2. \"To be or not to be, that is the question.\" Imagine that in answering this question Hamlet had the following data to consider. Hamlet must choose either \"To Be\" or \"Not to Be.\" If Hamlet chooses \"To Be\" then his enemy the King will make a decision to either \"Kill\" Hamlet or let him \"Live.\" If Hamlet is killed by the King his payoff in this game is -25 since he will have been killed by the King who is his uncle and who also killed Hamlet's father, the King's brother, who was the prior king. If the King decides to let Hamlet live then Hamlet will be able to avenge his father's death and his payoff will be +45. Hamlet also considers suicide an option but given the uncertainty of the afterlife if he chooses not to be then his payoff is -5 since his doesn't what the after life holds for one who kills himself. a. Draw the complete game tree for this situation. Be sure to accurately label the tree and include the payoffs. b. Assume Hamlet will choose the course of action that offers the highest expected value. Under what circumstances will Hamlet choose \"To Be?\" Complete the following sentence: Hamlet will choose \"To Be\" if.... 3. In each of the games shown below, players 1 and 2 must move at the same time without knowledge of the other player's move. The first payoff is for the row player (Player1) and the second payoff is for the column player (Player 2). Solve each game using game theoretic logic. a. If either player has a dominant strategy, circle what it is (i.e. circle up or down, left or right). Do not circle an action if is not a dominant strategy. b. Circle the predicted outcome(s) to the game (i.e. circle (8, 3) or (4, 5) or (7, 2) or (1, 3 Player 2 Player 1 Left Right Up 8, 5 6, 9 Down 9, 6 7, 5 Player 2 Player 1 Left Middle Right Up 11, 4 3, 6 4, 7 Middle 13, 5 6, 10 5, 11 Down 12, 8 5, 2 2, 3 Player 2 E F G H A 6, 3 4, 4 3, 4 5, 5 B 5, 7 5, 8 4, 9 1, 2 C 6, 6 6, 8 5, 7 4, 3 D 0, 10 2, 13 0, 15 4, 3 Player 1 4. Consider the following 3 person color game between Hillary, Laura, and Michelle. Each player can choose either Red or Blue. The payoffs for these choices are shown below. The first payoff in each cell is for Hillary, the second payoff is for Laura, and the third payoff is for Michelle. Hillary Red Hillary Blue Michelle Red Laura Red 1, 1, 10 2, 1, 2 Laura Blue 1, 2, 2 2, 2, 2 Hillary Red Hillary Blue Michelle Blue Laura Red 6, 6, 6 10, 1, 1 Laura Blue 1, 10, 1 2, 2, 1 a. Assume this game is only played one time. Using dominant strategy logic, what outcome would game theory predict? If a player has a dominant strategy, circle the dominant strategy. b. Assume this game is repeated and played 200 times. What strategy would you advise the players to use assuming that the goal is to maximize total payoff after 200 trials? Explain your reasoning. 5. In the game shown below, Players 1 and 2 are competing over how to divide up $100. Each player must move at the same time without knowledge of the other player's move. The table shows the payoff for Player1 (Player 2's payoff is $100 - Player 1's Payoff). Find the Nash equilibrium profile of strategies for this game. That is, what strategy should Player 1 and Player 2 use for this game? If the Players use their optimal strategy, what will be the payoff for each player in this game? You will need to use a combination of techniques to solve this problem. Player 2 Player 1 Left Middle Right Up 60 40 30 Middle 50 30 65 Down 70 50 40