1.In a simple reaction A ↔ A*, a molecule is interconvertible between two forms that differ in standard free energy
ard free energy G° by 18 kJ/mole, with A* having the higher G°.
Use the table below to find how many more molecules will be in state A* compared with state A at equilibrium.
If an enzyme lowered the activation energy of the reaction by 11.7 kJ/mole, how would the ratio of A to A* change?
Table: RELATIONSHIP BETWEEN THE STANDARD FREE- ENERGY CHANGE, ∆G°, AND THE EQUILIBRIUM CONSTANT
Hint: ∆G° represents the free-energy difference under standard conditions (where all components are present at a concentration of 1 mole/litter). From this table, we see that if there is a favourable free-energy change of –17.8 kJ/mole for the transition Y→ X, there will be 1000 times more molecules of X than of Y at equilibrium (K = 1000).
2.Question 1: What is a player’s « reaction function » in a Bertrand game ?
Question 2: What is a subgame
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:
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
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:
Question 9: All games have at least one pure strategy Nash equilibrium:
Question 10: If a tree game has a backward induction equilibrium then it must also be a Nash
equilibrium of all of its subgames:
Question 11: The mixed Nash equilibrium payoffs are always strictly smaller than the pure
Nash equilibrium payoffs:
Question 12: Which of the following statements about dominant/dominated strategies is/are
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
Question 14: Consider the following 2x2 matrix game:
U (3,2) (2,4)
D (-1,4) (4,3)
The number of pure and mixed Nash equilibria in the above game is:
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
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