b consider the ordinary di erential equation ode d y dx x dy dx x use the substitution of variables u x y x to

subgame perfect Nash equilibrium? Question 3: In which situations should we need the mixed extension of a game? Question 4: Find, if any, all Nash equilibria of the following famous matrix game: L R U (2,0) (3,3) D (3,4) (1,2) Question 5: What is the difference between a separating equilibrium and a pooling equilibrium in Bayesian games? Question 6: Give another name for, if it exists, the intersection of the players’ best-response « functions » in a game? Question 7: assuming we only deal with pure strategies, the Prisoner’s Dilemma is a situation with: No Nash equilibrium One sub-optimal Nash equilibrium One sub-optimal dominant profile No dominant profile Question 8: If it exists, a pure Nash equilibrium is always a profile of dominant strategies: True False Question 9: All games have at least one pure strategy Nash equilibrium: True False Question 10: If a tree game has a backward induction equilibrium then it must also be a Nash equilibrium of all of its subgames: Tr 2/2 Question 11: The mixed Nash equilibrium payoffs are always strictly smaller than the pure Nash equilibrium payoffs: True False Question 12: Which of the following statements about dominant/dominated strategies is/are true? I. A dominant strategy dominates a dominated strategy in 2x2 games. II. A dominated strategy must be dominated by a dominant strategy in all games. III. A profile of dominant strategies must be a pure strategy Nash equilibrium. IV. A dominated strategy must be dominated by a dominant strategy in 2x2 games. I, II and IV only I, II and III only II and III only I and IV only I, III and IV only I and II only Question 13: A pure strategy Nash equilibrium is a special case of a mixed strategy Nash equilibrium: True False Question 14: Consider the following 2x2 matrix game: L R U (3,2) (2,4) D (-1,4) (4,3) The number of pure and mixed Nash equilibria in the above game is: 0 1 2 3 Exercise (corresponding to questions 15 to 20 below): assume a medical doctor (M) prescribes either drug A or drug B to a patient (P), who complies (C) or not (NC) with each of this treatment. In case of compliance, controlled by an authority in charge of health services quality, the physician is rewarded at a level of 1 for drug A and 2 for drug B. In case of noncompliance, the physician is « punished » at -1 level for non-compliance of the patient with drug A and at -2 level for non-compliance with drug B. As for the compliant patient, drug A should give him back 2 years of life saved and drug B, only 1 year of life saved. When noncompliant with drug A, the same patient wins 3 years of life (due to avoiding unexpected allergic shock for instance), and when non-compliant with drug B, the patient loses 3 years of life. Question 15: You will draw the corresponding matrix of the simultaneous doctor-patient game. Question 16: Find, if any, the profile(s) of dominant strategies of this game. Question 17: Find, if any, the pure strategy Nash equilibrium/equilibria of this game. Question 18: Find, if any, the mixed strategy Nash equilibrium/equilibria of this game. Questions 19 and 20: Now the doctor prescribes first, then the patient complies or not: draw the corresponding extensive-form game (= question 19) AND find the subgame perfect Nash equilibrium/equilibria (=

, ACD®B, D®EF, BE®C, CF®BD, CE ®AF). (a) Find (BD)+ (b) Find (AB)+ (c) Find Candidate Keys for R Question 3 Given the scheme R= (ABC), and following set of functional dependencies: F = (A®BC, B® AC, C ® AB). (a) Find closure (BC, F) (b) Find at least one candidate key for R. (c) Find at least one super-key for R which is not the same as your answer in (b). (d) Find at least one minimal cover for a relational scheme (ABC). Show work. (e) Provide a 3NF decomposition for R. Question 4 Consider the following set of FDs: F = (A ® B, AB ® C, D ® AC, D ® E) G = (A ® BC, D ® AE) H = (A ® BC, B ® C, A ® B, AB ® C, AC ® D). (a) Is F ≡ G? Show your work. (b) Find the minimal cover for H. Show work.

f A is heated to 500 K, what is the concentration of B at equilibrium?

d are linked to a higher risk of developing coronary heart diseases. A researcher is investigating two types of interventions (e.g. A, B) to reduce the cholesterol level. Study participants were randomized to receive either intervention A or B. Some subjects dropped from the study due to reasons not related to the interventions. In order to determine which intervention was more effective, the mean cholesterol concentrations were compared between the two groups at the end of the study. The total cholesterol levels for the patients undergoing the intervention A were: 240, 238, 245, 256, 289, 190, 189, 205, 200, 214, 237, 247, 254, 209, 211, 200 mg/dl. The total cholesterol levels for the patients undergoing the intervention B were: 213, 188, 199, 247, 248, 239, 256, 244, 206, 223, 241, 261, 226, and 231 mg/dl. Which intervention was more effective? Consider alpha=0.05. Intervention B was more effective. Intervention A was more effective. Impossible to tell based on the information provided. The two interventions were not statistically significant different.

e in the Euclidean plane. In this exercise, we will show this is not true in the Poincar´e plane. As usual, let Γ be the unit circle in R2 , and let O be the origin. Let A = (3/4, 0) and B = (0, 3/4). Consider the P-triangle △OAB. Let γ be the E-circle through O, A, and B. Show that γ contains points outside of Γ and is therefore not a P-circle. Explain why this means that the P-triangle △OAB cannot have a circumscribed circle in the Poincar´e plane.