ECOS3012 Practice mid-semester exam questions
September 21, 2020
• The formal mid-semester exam consists of 5 long questions. Below are sample questions
from exams in the past.
• Please attempt all of these questions on your own before checking the solutions.
Q1 Consider the game:
Player 2
L R
Player 1
U 10, 1 2, 2
M 2, 3 10, 3
D 5, 3 5, 5
(a) Is there any strictly dominated strategy? If so, please find all strictly dominated strategies.
If not, explain why.
(b) Find all Nash equilibria (pure and mixed).
Q2 Consider the following static game.
Player 2
A B C
Player 1
A 1, 1 9, 0 2, 0
B 0, 10 8, 8 1, 3
C 0, 2 4, 0 5, 5
(a) Does this game have strictly dominated strategies? If so, state all strictly dominated
strategies. If not, please explain.
(b) Find all Nash equilibria of this game.
(c) Suppose that the players act sequentially. Player 2 chooses her action after observing the
choice of player 1. Find the subgame perfect equilibrium(/ia) of this sequential game. State
each player’s equilibrium strategy.
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Q3 The products from firm 1 and firm 2 are imperfect substitutes. If the price of firm 1’s product
is p1 and the price of firm 2’s product is p2, then the market demand for firm 1 is q1 =
100− 2p1 + p2 and the market demand for firm 2 is q2 = 100− p2 + p1. The marginal
production cost is 10 for both firms.
(a) Express firm 1’s optimal price as a function of p2 and express firm 2’s optimal price as a
function of p1.
(b) What prices will these firms charge in equilibrium? Which firm earns a higher profit?
The owners of the two firms are considering a merge. If they merge, the firms become a
monopoly. The new monopoly will charge a single price pm, and the market demand is
qm = 200− pm. The two owners split the monopolistic profits evenly. The two firms will
merge if and only if both owners agree to merge.
(c) Will the firms merge to become a monopoly?
Q4 Three oligopolists operate in a market with inverse demand given by P(Q) = 50−Q, where
Q= q1+q2+q3 and qi is the quantity produced by firm i. Each firm has a constant marginal
cost of production, c = 5, and no fixed cost. First, firm 1 and 2 simultaneously chooses
q1 ≥ 0 and q2 ≥ 0. Then, after observing q1 and q2, firm 3 chooses q3.
(a) What quantities do firms choose in the subgame perfect equilibrium of this game?
(b) How much profit does firm 3 make?
(c) Suppose instead that firm 3 cannot observe q1 or q2 before choosing its own quantity.
Does it produce at a higher or lower quantity compared to your answer in (a)? Does it make
more or less profit compared to your answer in (b)? For full credit, please support your
answer with calculation.
Q5 Army 1 from country 1 must decide whether to attack army 2, of country 2, which is occupying
an island between the two countries. In the event of an attack, army 2 may stay on the island
and fight, or retreat over a bridge to its mainland. Each army prefers to occupy the island
than not to occupy it; a fight is the worst outcome for both armies.
(a) How many subgames does this game have? What are they?
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(b) Find the subgame perfect equilibrium. Remember, you should specify players’ complete
strategies following every possible history.
(c) Convert the game to its normal-form representation (i.e. draw the payoff matrix) and find
all Nash equilibria.
(d) For each Nash equilibrium that is not subgame perfect, identify the subgame in which
the test of subgame perfection fails.
(e) Show that army 2 can increase its subgame perfect equilibrium payoff (and reduce army
1’s payoff) by burning the bridge to its mainland (assume this act entails no cost), eliminating
its option to retreat if attacked.
Q6 Ann and Bob must choose among three movies, a, b, and c, for their movie night. Ann’s
payoff function isUA(a) = 3,UA(b) = 2, andUA(c) = 1. Bob’s payoff function isUB(a) = 1,
UB(b) = 2, and UB(c) = 3. The rules are that Ann moves first and can veto one of the three
alternatives. Then Bob chooses one of the remaining two alternatives.
(a) Draw an extensive-form game tree for this game.
(b) Find the subgame-perfect equilibrium.
(c) How many pure strategies does Ann have? How many pure strategies does Bob have?
(d) Find a pure-strategy Nash equilibrium that is not a subgame-perfect equilibrium.
Q7 The following game “5” is a simplified version of Blackjack (“21”).
Two players A and B take turns making choices. Player A starts off by choosing either 1
or 2. Player B observes this choice then increases the count by adding 1 or 2. Then player
A observes B’s choice and add 1 or 2. The game continues with the players taking turns,
incrementing the count by 1 or 2. The player who reaches 5 wins. If the current count is
already 4, the next player must choose 1 (and win the game).
Here is an example.
Round 1: A chooses 1
Round 2: B chooses 2
Round 3: A chooses 2 and wins the game
(a) Draw a game tree that describes this game.
(b) Find the subgame perfect equilibrium outcome(s).
(c) Explain whether this statement is correct: “The player that moves first always wins.”
(d) How many pure strategies does B have?
Q8 The Battle of the Exes
Suppose that Ann and Bob face the game below:
Bob
Frank Turner Sia
Ann
Frank Turner 1, 1 6, 2
Sia 2, 6 0, 0
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Ann and Bob met at a Frank Turner concert, and fell in, and then out of love (and had a
rough break-up). Both of them prefer Frank Turner to Sia, but really would prefer to not run
into each other. They first sought a love doctor (a sociologist) to help them with their woes,
and eventually turn to you, a practitioner of the dismal arts with the following questions:
a. If my Ex is still bitter about our break up and is doing whatever that minimizes my payoff,
what is the highest payoff I can get in the stage game?
b. If we repeat this game twice and our total payoff is the undiscounted sum of the stage
payoffs, please help us find a strategy profile so that (i) it is a subgame perfect equilibrium,
and (ii) in this equilibrium, we never go to the same concert.
c. If we repeat this game twice and our total payoff is the undiscounted sum of the stage
payoffs, please tell us: is it possible to have a subgame perfect equilibrium in which we both
go to the Frank Turner concert in the first period? Why? (Hint: for full credit, your answer
should contain the word “mixed”.)
Q9 Consider the following game.
Player 2
A B C
Player 1
A 1, 1 9, 0 2, 0
B 0, 10 8, 8 1, 3
C 0, 2 4, 0 5, 5
(a) Suppose that the players repeat this stage game twice. In each stage, they simultane-
ously choose their actions, and their total payoff from the twice-repeated game is the (undis-
counted) sum of the stage payoffs. Find a subgame perfect equilibrium in which the players
play (B, B) in the first stage. State the players’ strategies in this equilibrium and explain why
they constitute a subgame perfect equilibrium.
(b) Suppose that the players repeat this stage game infinitely many times. In each stage, they
simultaneously choose their actions, and their total payoff from the infinitely-repeated game
is the discounted sun of the stage payoffs with some discount factor δ ∈ (0,1). State the
lowest possible payoffs player 1 and player 2 can get in any subgame perfect equilibrium.
Explain your answer.
Q10 Suppose that the following prisoners’ dilemma is repeated infinitely with discount factor
δ ∈ (0,1).
Prisoner 2
Confess Not Confess
Prisoner 1
Confess 1, 1 5, 0
Not Confess 0, 5 4, 4
Consider a “tit-for-tat” strategy profile in which both players play NC in the first period. In
later periods, each player mimics the action that the opponent picked in the previous period.
(For example, if the outcome in period t is (C, NC) then players play (NC, C) in period t+1.)
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a) If both prisoners play the tit-for-tat strategy, what is the outcome path? What is each
prisoner’s average payoff?
b) Suppose that prisoner 1 plays C forever while prisoner 2 follows tit-for-tat. What is
prisoner 1’s average payoff from this infinite-shot deviation?
c) Consider the one-shot deviation from the equilibrium path: suppose that prisoner 1 chooses
to deviate only at the first period, but then follows tit-for-tat forever after. What is his average
payoff? Under what condition on δ is this not a profitable one-shot deviation?
d) Can tit-for-tat be a subgame perfect equilibrium? If so, find the condition on δ . If not,
please explain why.
Q11 Suppose the following game is repeated infinitely with a discount factor close to 1.
Player 2
L R
Player 1
U 0, 2 4, 1
D 1, 4 2, 0
Can the following payoff pairs be achieved as the average payoff in some subgame perfect
equilibrium of this repeated game? Please explain your answer in details.
(a) (2.5, 3)
(b) (1.5, 3)
(c) (1.5, 2)
(d) (0.5, 2)
Q12 Suppose that the following game is repeated infinitely with a discount factor δ .
Player 2
L R
Player 1
U 5, 0 1, 1
D 3, 3 1, 4
Can the payoff pair (1.5, 1.5) be achieved as the average payoff in some subgame perfect
equilibrium of this repeated game if δ is sufficiently close to 1? Please explain your answer
in details.
Q13 Consider the following “trust game”.
Player 1 first chooses whether to ask for the services of player 2. He can trust player 2 (T)
or not trust him (N).
If player 1 plays T, then player 2 can chooses to cooperate (C), which represents offering
player 1 some fair level of service, or he can defect (D), which is better for player 2 at the
expense of player 1.
The players prefer engaging in a cooperative exchange over not interacting at all (no trust).
This game captures many real-life exchanges in which one party must trust another in order
to achieve some gains from trade, but the second party can abuse that trust.
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(a) Suppose that this trust game is played only once. Find the subgame-perfect equilibrium
outcome.
(b) Suppose that this trust game is repeated infinitely with a discount factor of δ = 0.6. Can
you construct a subgame-perfect equilibrium in which player 1 always trusts player 2 and
player 2 always cooperates? Prove your answer.
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