UNIVERSITY OF SOUTHERN DENMARK FACULTY OF BUSINESS AND SOCIAL SCIENCES Strategy and markets 1. 2. 3. 4. 5. Assigner: Rafael Treibich Hand-out: Hand-in: All pages, incl. the front page, should include the following: Exam no. All pages must be numbered. Course description: Duration: 3 hours. Place: The test takes place at the university. The test is carried out using a personal computer to be able to access the university's wireless network. Internet access: Forbidden. Handing out the assignment: Physical in the examination room. Submission of answer: Through SDUassignment in the course page in Blackboard. Scope: No limitation. Aids: Aids allowed but internet access forbidden. It is not allowed to communicate with others. Exercise 1: Consider the following sequential-move games (the payoff of Peter is on the left, the payoff of Ida is on the right). (4,1) Left (-2,-3) East Ida Up Right Peter Peter (2,4) (-3,5) Soft Down West Ida (-1,1) Quick Hard Peter Slow (4,2) a) How many decision nodes does the game consist of? How many terminal nodes does the game consist of? Decision nodes: 5 Terminal nodes: 5 b) Describe and list the strategies (complete plan of actions) of Peter and Ida. Peter: UEQ, UES, UWQ, UWS, DEQ, DES, DWQ, DWS Ida: LS, LH ,RS, RH c) Find the Rollback equilibrium (equilibrium path, strategies and payoffs). d) Write the game in strategic form (game table). e) Find the Nash equilibria of the game and identify the subgame perfect Nash equilibria. Consider now the following game in extensive form: (2,1) Left Ida Up Right (5,-1) Peter Left (1,3) Down Ida Right (2,2) f) What is the rectangle around Ida’s two decision nodes? How do you interpret it? This shows us that this game is actually playes simultaniously. Ida makes a choice without knowing if Peter chose Up or Right. Hence, Ida’s nodes (information set) in the rectangle tells us that we have imperfect information and it illustrate Ida’s strategy uncertainty. g) Is this game simultaneous or sequential? Write the game in strategic form (game table) and find all the Nash equilibria in pure strategies. Exercise 2: Ida considers buying a lottery ticket with two possible outcomes: - with probability1/3, she wins 100. - with probability 2/3, she wins 400. Assume Ida’s utility function is given by u_I(x)=ln(1+2x). a) b) c) d) e) Compute the expected payoff of this lottery. Compute Ida’s expected utility for the lottery. Is Ida risk averse, risk neutral or risk lover? What is Ida’s certainty equivalent for the lottery? What is Ida’s risk premium? Peter also considers buying a ticket for the same lottery but his utility function is u_P(x)= x. f) Are either Peter or Ida willing to buy the lottery ticket for a price of 260? g) Who is more risk averse: Peter or Ida? Exercise 3: In (american) football the offense can either run the ball or pass the ball, whereas the defense can either anticipate (and prepare for) a run or anticipate (and prepare for) a pass. The objective for the offense team is to move forward so as to bring the ball in the end zone, in which case it scores a touchdown. The objective for the defense team is to prevent the offense team from moving forward. Assume that the expected gains (in meters) for the two teams on any given play are as follows: Defense Offense Run Pass Anticipate Run -3,3 7,-7 Anticipate Pass 4,-4 -3,3 a) Show that this simultaneous game has no pure-strategy Nash equilibrium. b) Find the unique mixed-strategy Nash equilibrium to this game. c) Explain why the mixture used by the offense is different from the mixture used by the defense. d) How many meters is the offense expected to gain per down in equilibrium? Exercise 4: Assume there are two type of mechanics, Competent and Incompetent. The population of mechanics consists of 40% Competent and 60% Incompetent. Mechanics can either work in licensed garages or as free lance. Competent mechanics generate a revenue of 100 in a licensed garage, while incompetent mechanics only generate a revenue of 40. Competent mechanics generate a revenue of 65 in free lance, while incompetent mechanics only generate a revenue of 30 in free-lance. There is enough demand for mechanics such that licensed garages must pay them what they expect the mechanics to produce. However, licensed garages must hire each mechanic without observing his type and pay him before knowing his actual output. a) At equilibrium, which type of mechanics end up working for the licensed garages and at which salary? Is this equilibrium efficient or not? How do you call this particular phenomena? Mechanics can now signal their qualification by getting a certification, but for the incompetent types certification takes extra time and effort. Competent mechanics have to spend C months preparing for the certification exam; incompetent ones take three times as long, 3C. Assume the payoff of each mechanic is equal to (x) – t, where x is his yearly salary and t is the number of months spent working for the certification. The licensed garages assumes anyone with a certification is competent and pays them 100. b) What minimum level of C will achieve separation between the two types? c) Who gains and who loses compared to the initial situation?
