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EC202 – Course test
Microeconomics 2
March Examinations 2021/22
1. Ram is faced with a lottery which pays £100 in a good state and £0 in a bad state with each state occuring with equal probability. Ram says he prefers to receive £60 for certain rather than taking the lottery. Which statement is true? (6 marks)
A. Ram is risk loving.
B. Ram is risk averse.
C. Ram is risk neutral.
D. Ram is not risk loving.
E. We have insufficient information to determine whether Ram is risk averse, risk loving or risk neutral.
2. Consider Hugo whose preferences can be represented by the following utility function:
where x is wealth. Which statement is true? (6 marks)
A. Hugo’s coefficient of relative risk aversion is the same regardless of his level of wealth.
B. Hugo’s coefficient of absolute risk aversion is constant.
C. Hugo would choose £30 for certain over a lottery that would pay £100 or £0 with equal probability.
D. Hugo would choose £20 for certain over a lottery that would pay £100 or £0 with equal probability.
E. The utility function can represent both risk averse and risk loving preferences.
3. Consider a game with two players, L = {1, 2}, with actions S1 = S2 =
{Rock, Paper, Scissors, Well}. The Rock (R) beats Scissors (S) and is beaten by Paper (P) and Well (W); P beats R and W and is beaten by S; S beats P and is beaten by R and W ; W beats R and S and is beaten by P. Every action draws with itself. A win gives a player a payoff equal to 1, lose gives -1, and a tie gives a payoff equal to 0. The normal-form representation takes the following form:
Which of the following statements is true? (6 marks)
A. There is a unique Nash equilibrium in which each player mixes between all actions with equal probability.
B. There is no Nash equilibrium of this game.
C. There are infinitely many Nash equilibria.
D. We cannot say whether there is a Nash equilibrium or Nash equilibria.
E. There are no pure-strategy Nash equilibria of this game.
4. Consider the following game:
Which statement is true? (6 marks)
A. There are two pure-strategy Nash equilibria of this game – (U, L) and (M, M) – but neither are Pareto efficient.
B. There is only one pure-strategy Nash equilibrium – (M, M) – and this is Pareto efficient.
C. There are two pure-strategy Nash equilibria of this game – (U, L) and (M, M) – both of which are Pareto efficient.
D. The only Pareto efficient outcome of the game is (D, R).
E. There are two pure-strategy Nash equilibria of this game – (U, L) and (M, M) – but only the latter is Pareto efficient.
5. Consider the following game:
Which of the following statements is true? (6 marks)
A. R weakly dominates L.
B. There are only two Nash equilibria of the game.
C. There are two pure-strategy Nash equilibria of the game – (D, M) and (U, R) – both of which are Pareto efficient.
D. There is a mixed-strategy Nash equilibrium in which Player 2 plays R with probability ρ2 = 1 and Player 1 plays U with probability ρ 1 = 10/1 .
E. There is only one Pareto efficient outcome of the game.
6. Consider the following game:
Which statement is true? (8 marks)
A. There is one unique pure-strategy Nash equilibrium (B, X).
B. There are two Nash equilibria of the game – (B, X) and (B, Y).
C. There are infinitely many Nash equilibria of this game.
D. All outcomes of the game are Pareto efficient.
E. There is no pure-strategy Nash equilibrium of the game.
7. Consider the following game:
Suppose the game is played 5 times. How many subgames are there? (6 marks)
A. 406,901
B. 406,911
C. 250,000
D. 408,911
E. 251,000
8. Consider the following “war of attrition” game, which is played over discrete periods of time. Player 1 and Player 2 can play Stop (S) or Continue (C). We can represent the game in normal form as follows:
The length of the game depends on the players’ behaviour. Specifically, if one or both players select S in a period, then the game ends at the end of this period. Otherwise, the game continues into the next period. Suppose the players discount payoffs between periods according to the discount factor δ ∈ (0, 1). What is the probability that the game will be played exactly three times if the players play the mixed-strategy Nash equilibrium? (8 marks)
A. 4096/565.
B. 4096/566.
C. 4096/567.
D. 4096/568.
E. 4096/569.
9. Consider an individual facing the following gamble. With probability ρ = 0.2 they will end up in a “bad state” and consume xb =£0 and with probability 1 − ρ = 0.8 they will consume xg =£1, 000 in a “good state”. The individual’s utility can be represented as u(xi ) = √xi , where i = b,g. Suppose there is an insurance contract available to the individual offering K units of insurance for a premium of 5/2 K. How much insurance will the individual buy? (6 marks)
A. 7/1500
B. 1000
C. 1100
D. 1200
E. 0
10. Consider the following simultaneous-move three-player game:
The three players make their choices simultaneously and independently. The sets of actions available to Player 1, Player 2 and Player 3, respectively, are S1 = {U, D}, S2 = {L, R} and S3 = {A, B}. The payoffs are listed in the table above where the first entry refers to the payoff to Player 1, the second to Player 2 and the third to Player 3. A profile of actions is written (x,y, z) where the first entry is the action of Player 1, the second entry is the action of Player 2 and the third entry is the action of Player 3. Which statement is true? (8 marks)
A. There are only two Nash equilibria of the game – {(D, L, A) and (U, R, A)}.
B. There is no Nash equilibrium of the game.
C. There are infinitely many Nash equilibria of the game.
D. There are three Nash equilibria of the game.
E. There is one unique Nash equilibrium of the game – {(D, L, A)}.
11. Consider the following dynamic game of incomplete information:
There are two players in the game – Player E and Player P. The timing of the game is as follows. Nature draws a type ti , from the set of types, T = {t1 , t2 }. Each type, t1 and t2 are drawn with equal probability. Player E observes their type, ti , and chooses a message mi from a set of messages, M = {m1 , m2 }. Player P observes mi (but not ti ) and then chooses an action ak from a set of feasible actions A = {a1 , a2 }. The payoffs from each action are illustrated in the extension form representation of the game above as pairs (x, y) with the first entry denoting the payoff to Player E and the second the payoff to Player P. Which of the folloing statements is true? (8 marks)
A. There is a separating equilibrium in which a type t1 Player E selects m2 and in which a type t2 Player E selects m1 .
B. There is a separating equilibrium in which a type t1 Player E selects m1 and in which a type t2 Player E selects m2 .
C. There is a pooling equilibrium in which both types of Player E select m2 .
D. There is no perfect Baysian equilibrium.
E. There is a unique separating equilibrium in which a type t1 Player E selects m2 and in which a type t2 Player E selects m1 .
12. Consider the following prisoner’s dilemma:
where C is cooperate and D is defect. Suppose the two players alternate between actions D and C. In period t, Player 1 plays C and Player 2 plays D; in period t + 1, Player 1 plays D and Player 2 plays C. Assume that Player 1 and Player 2 discount the future by δ ∈ [0, 1).
For which values of the discount factor δ will the players be able to sustain cooperation if the game is repeated infinitely many times? Assume that any deviation from the alternating
equilibrium is followed by infinite defection (grim-trigger). (8 marks)
A. δ ∈ [3/1, 1)
B. δ ∈ [4/1, 1)
C. δ ∈ [8/1, 1)
D. δ ∈ [2/1, 1)
E. δ ∈ [10/1, 1)
13. Consider the following game of chicken:
Suppose an ‘umpire’ proposes the following mechanism: a random device selects one cell in the game matrix with the following probabilities:
When a cell is selected, each player is told by the ‘umpire’ to play corresponding to pure strategy. Each player is told what to play but not what the other player is told although the probability distribution is common knowledge. The payoffs to the payoffs is a pair (x, y),
where the first entry is the payoff to Player 1 and the second is the payoff to Player 2. Which statement is true? (6 marks)
A. The payoffs to the players in the correlated equilibrium is (3/2, -3/2), and in the mixed strategy it is (-3/2, -1).
B. The payoffs to the players in the correlated equilibrium is (3/2, 3/2), and in the mixed strategy it is (-8/7, -7/8).
C. The payoffs in the mixed-strategy Nash equilibrium exceed those in the correlated equilibrium for both players.
D. The payoff in the mixed-strategy Nash equilibrium exceed those in the correlated equilibrium for Player 1.
E. The mechanism proposed by the ‘umpire’ cannot be sustained.
14. Consider the following game:
Suppose this game is played T times. Which statement is true? (6 marks)
A. The total number of information sets belonging to Player 1, Player 2 and Player 3 exceed the number of subgames.
B. The total number of information sets is equal to the number of subgames
C. The total number of information sets is less than the number of subgames
D. The number of information sets belonging to Player 1 exceeds the number of subgames
E. There are no proper subgames
15. Consider a game in which, simultaneously, Player 1 selects x ∈ {1, 2, 3} and Player 2 selects y ∈ {1, 2}. The payoffs are given as:
Which of the following statements is true? (6 marks)
A. y = 1 is a strictly dominated strategy.
B. There is a pure-strategy Nash equilibrium in which Player 1 selects x = 3 and in which Player 2 selects y = 2 but this equilibrium is not unique.
C. There is a pure-strategy Nash equilibrium in which Player 1 selects x = 2 and in which Player 2 selects y = 1 and there is a mixed strategy Nash equilibrium in which each player selects each action with equal probability.
D. There is no pure strategy Nash equilibrium.
E. There is a unique mixed strategy Nash equilibrium in which each player selects each action with equal probability.