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Microeconomic and GamesMay - June 2013
G. Fabbri
Home test - 1
The exercises of the test do not require a complete anddetailed
response. You are asked only to give the synthetic re-sult.You can
directly write the answers on this sheet once you haveprinted them
or you can simply send me an email with the an-swers if you
prefer
Name and surname of the student :
Exercise 1
Consider the following normal form of a certain game.
Player 2l m r
Player 1U (5,7) (0,7) (4,6)M (6,2) (7,2) (5,3)D (6,5) (0,4)
(3,7)
(i) What is the best response of Player 2 to the strategy M of
player 1 ?
(ii) One of the strategies of Player 1 is strongly dominated by
anotherstrategy of Player 1. Find them.
(iii) Find the Nash Equilibriums of the game (if they exist)
Exercise 2
Are the following statements True or False ?
(i) Consider a game (in normal form) N with two players : i {1,
2}.Let us call Si the set of the strategies of player i. Assume
that thestrategy s1 weakly dominates all the strategies s1 S1 and
that thestrategy s2 weakly dominates all the strategies s2 S2. Then
(s1, s2)is a Nash Equilibrium.
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(ii) Consider a game (in normal form) N with two players : i {1,
2}.Let us call Si the set of the strategies of player i. Assume
that (s
1, s
2)
S1S2 is a Nash Equilibrium. Then the strategy s1 weakly
dominatesall the strategies s1 S1 and the strategy s2 weakly
dominates all thestrategies s2 S2.
(iii) Consider a game (in normal form) N with two players : i
{1, 2}.Let us call Si the set of the strategies of player i. Assume
that thestrategy s1 S1 is the best response to the strategy s2 S2
and s2 isthe best response to the strategy s1. Then (s1, s2) is a
Nash Equilib-rium.
(iv) Consider a game (in normal form) N with two players : i {1,
2}.Let us call Si the set of the strategies of player i. Assume
that (s
1, s
2)
S1S2 is a Nash Equilibrium. Then the strategy s1 S1 is in the
setof best responses to the strategy s2 S2 and s2 is in the set of
thebest responses to the strategy s1.
Exercise 3
The people of a certain city have to finance the construction of
a newhospital. We assume for simplicity that the number of
inhabitants of the cityis two. Each of them can choose how much pay
for the hospital given thebenefit he will get (the benefit will
depend on the total amount employed inthe project).
More formally :Citizen 1 choses her contribution x1 > 0.
Citizen 2 chooses her contribu-
tion x2 > 0. The choice is simultaneous (and any player
chooses her numberwithout knowing the choice of the other
player).
At the end they will get an utility that we can estimate (in
monetaryterms) as follows
- The payoff of Citizen 1 isx1 + x2 x1
- The payoff of Citizen 1 isx1 + x2 x2
(i) Find the best response of player 2 if player 1 choose x1
=16
(ii) Find all possible Nash Equilibrium of the game (hint : they
aremany)
(iii) Find the unique symmetric Nash Equilibrium (i.e. the
unique NashEquilibrium for which x1 = x2)
Exercise 4
Assume that
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- In a market there are two producers.- Any producer has a cost
function of the form c(q) = 2q.- The inverse demand function of the
market is p(q) = 10 qFind : the quantity produced by each producer,
the price, the profit of
each producer if they compete a` la Cournot
Exercise 5
Assume that- In a market there are two producers.- Any producer
has a cost function of the form c(q) = 2q.- The inverse demand
function of the market is p(q) = 10 qFind : the quantity produced
by each producer, the price, the profit of
each producer if they compete a` la Bertrand
Exercise 6
Assume that- In a market there are two producers.- Any producer
has a cost function of the form c(q) = 2q.- The inverse demand
function of the market is p(q) = 10 qFind : the quantity produced
by each producer, the price, the profit of
each producer if they compete a` la Stakelberg, producer 1 is
the leaderand producer 2 is the follower.
Exercise 7
The following tree represents a two-player game. At each node
the playercan choose Left or Right (the actions are not written in
the tree tolighten the notation). In the tree the payoff (a, b)
represents an utility a forplayer 1 and an utility a for player
2.
Find a Nash Equilibrium using the backward induction.
Player1
Player2
(1,1) (2,2)
Player2
(1,2) Player1
(3,1) Player2
(2,5) Player1
(3,4) (4,4)
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Write the corresponding payoffs for Player1 and Player2. (Write
only towrite the final payoff, not the whole strategy)
Exercise 8
Consider the game represented in Exercise 7 :(i) How many are
the strategies of Player1 ?(ii) How many are the strategies of
Player2 ?
Exercise 9
The meeting in HCMC game :The structure of the game is the
following- There are two players, Player 1 and player 2 (that would
like to meet
in a bar in HCMC).- Each player can decide if going to the bar A
or to the bar B (we
abbreviate the two actions with A or B).- Player 1 moves first
and choose A or B.- Player 2 moves after player 1 and chooses A or
B knowing the choice
of Player 1.- If both players have chosen A then both win 1
dollar, if both choose
B then both win 2 dollars. If the two decisions are different
they getnothing.
(i) Describe the game in its extensive form drawing a tree.(ii)
Write the possible strategies of Player 2.(iii) Are some of the
strategies of Player 2 weakly dominated by others ?
(you can just answer yes or no, you dont need to justify
theanswer)
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