Just one problem from this test, more would be appreciated but just that one.

Economics 4340/7340: Midterm Exam 1 Spring 2014 Answer each of the seven questions below. The weight given to each question appears in parentheses following the question number. Show your work in order to receive partial credit. You have 75 minutes to complete the exam. Good luck! For instructor's use only: q1 q2 q3 q4 q5 q6 q7 total Question 1 (18 points) Consider the following sequential-move game between Ann and Bob. External uncertainty is handled by an outside player called \"Nature\". Nature chooses \"Left\" with probability 50% , \"Right\" with probability 50%. A. The inclusion of a random event (a choice made by Nature) means that the players need to determine their expected payos when Nature moves. Calculate the players' expected payos in the situation where Ann chooses \"South\" and Bob chooses \"Risky,\" giving rise to Nature's choice of \"Left\" or \"Right\". B. Use backward induction to nd all rollback equilibria. C. How many strategies does each player have? For each player, list all his/her strategies. 1 Question 2 (15 points) The Saga of Big Panda and Little Panda. Big Panda (BP) and Little Panda (LP) live on a deserted island in the South Pacic. Their food is the warifruit which grows only from the highest branches of the waritree. To get the fruit, one of the pandas must climb all the way up the tree, creep out on the precarious branch and shake vigorously until the fruit drops. Measurements show that a warifruit is worth 10 kilocalories (Kc) of energy. The cost to BP of climbing the tree is 2 Kc but is negligible (meaning 0 Kc) for LP. If both pandas climb the tree to get the fruit, BP takes most of the fruit and gets 7 Kc leaving LP only 3 Kc of fruit. If only LP climbs the tree, BP has eaten 9 Kc of the fruit by the time LP gets back down leaving him only 1 Kc. If BP climbs the tree, LP gets to eat 4 Kc of fruit before BP can climb down and get the other 6 Kc of fruit. If no pandas climb the tree, then they do not get the fruit, and both get 0 Kc. A. Assume the pandas decide simultaneously whether to climb the tree. Write down the payo matrix for this game. B. Find all Nash equilibria in pure strategies. 2 Question 3 (15 points) Consider the same setting as in Question 2. Now assume that Big Panda gets to move rst and that Little Panda observes this decision and then gets to choose. A. Draw the game tree for this game. B. Find the rollback equilibrium. What is the equilibrium outcome? 3 Question 4 (7 points) Which of the following quotes could have been spoken by a player after a game in which every player acted in a way that was consistent with the denition of a Nash equilibrium? Explain. You can give an example to justify your answer. 1. \"I am really annoyed. I could have been much better o. If only I'd known that you'd do 'that', I would have done 'this' instead of what I did.\" 2. \"I am really annoyed. All of us players could have been much better o if only we'd cooperated with each other.\" (a) quote 1 (b) quote 2 (c) both (d) neither Answer (explain): 4 Question 5 (5 points) When a player has a dominant strategy in a simultaneous-move game, we can say that if he uses that strategy, then (given whatever actions his opponent has taken) his payo at the end of the game is certainly higher than . (a) his opponent's payo (b) the payo he would have earned if he used a dierent strategy (c) any other payo that is possibly available to him in the game (d) both (b) and (c) are correct (e) all of (a), (b) and (c) are correct Answer (no explanation needed): 5 Question 6 (15 points) Consider the following zero-sum game: Row Up Straight Down Column Left Middle 1 2 2 4 1 3 Right 5 3 3 A. Does Row player have a strictly dominant strategy? A strictly dominated strategy? B. Does Column player have a strictly dominant strategy? A strictly dominated strategy? C. Use the method of iterated elimination of strictly dominated strategies to nd the Nash equilibrium/ equilibria. Give the order in which the elimination occur. D. Use the minimax method to nd the Nash equilibrium/ equilibria. Did you get the same answer as in C? 6 Question 7 (25 points) Firms 1 and 2 are in the pizza business. Each rm's sales depends on both its own price and the other rm's price and can be expressed as q1 = 10 2p1 + p2 and q2 = 10 2p2 + p1 , where p1 is the price charged by rm 1, q1 are the sales of rm 1, and p2 and q2 are dened similarly for rm 2. The cost of making one pizza equals 4 (i.e., c1 = c2 = 4). The rms set their prices simultaneously. A. Write down the prot functions for the two rms. Find the best-response rules. B. Use the best-response rules to nd the Nash equilibrium prices. (Hint: Noting that the solution to the system of equations must be symmetric will greatly simplify your algebra.) Calculate the equilibrium prots. C. Suppose that the two rms collude and set prices jointly to maximize the sum of their prots. Find the joint prot-maximizing prices. Calculate the resulting prots. 7