Problem Set 4
ECON 2112
Deadline: 14 April (Sunday), 11.59 pm
(No late submissions, please)
x Please write your name and student ID on the top of assignment
x Please submit a pdf document
x Please show your work for all parts unless a part is worth 1 point or less, or the
question explicitly states that no explanation is needed.
Exercise 1
There are two players A and B. Player B can be of two types t א {0,1} with Pr (t=1) = p א
[0,1]. The actions and payoffs of the game are given by:
L R
U 4, t 0, 4(1-t)
D 0, (1-t) 1, 4t
where the row player is player A. We will use the following notation:
x ıa: probability that player A plays U
x ıbt: probability that player B plays L if she is of type t.
Part I (2 marks): Complete information (No explanation needed)
(i) (1 mark) Suppose p = 1. That is, player B’s type is t = 1 for sure. State all Nash
equilibria in pure and mixed strategies in the following format:
(ıa, ıb1) = ( . , . )
(ii) (1 mark) Suppose p = 0. That is, player B’s type is t = 0 for sure. State all Nash
equilibria in pure and mixed strategies in the following format:
(ıa, ıb0) = ( . , . )
Part II (5 marks): Incomplete information
Suppose p = 0.5. Check whether the following strategy profiles constitute a BNE:
(i) ıaıb1ıb0) = (1,1,1)
(ii) ıa, ıb1ıb0) = (0,0,0)
(iii) (ıa, ıb1ıb0) = (1,1,0)
(iv) (ıa, ıb1ıb0) = (0,0,1)
(v) (ıa, ıb1ıb0) = (0.8,0.2,0.2)
x To prove something is not a BNE, you need to show that just one of the three – type-0
player B, type-1 player B, or player A has an incentive to deviate.
x To prove something is a BNE you have to check that no player-type has an incentive
to deviate from the proposed strategy profile.
Exercise 2 [This game has some similarity with one of the unmarked BNE exercises posted
under week 9]
Boeing is the sole supplier of aircrafts to all Asian airlines. Airbus is deciding whether to
enter the Asian market and compete with Boeing.
x Airbus can take an (a) aggressive entry strategy which we refer to as E1. Airbus can also
take a (b) soft entry strategy which we refer to as E2. Finally, Airbus can completely (c)
stay out of the market which we refer to as O(ut).
x Boeing can decide to engage in price war (P) or share (S) the market with Airbus.
x Airbus chooses first between E1, E2, and O, after which Boeing chooses between P and S.
(i) If Airbus stays out of the market, Boeing gets 4, Airbus gets 0.
(ii) If Airbus chooses E1 or E2 but Boeing chooses P, each gets (- 1)
(iii) If Airbus chooses E1 and Boeing chooses S, Airbus gets 3, Boeing gets 1.
(iv) If Airbus chooses E2 and Boeing chooses S, Airbus gets 2, Boeing gets 2.
x Moving second, Boeing knows whether Airbus has chosen to stay out or enter, but it does
not know whether Airbus has chosen E1 or E2. [Hint: think of information set]
(a) (1 mark) Draw the relevant game tree associated the sequential game described above.
Clearly label nodes/information sets, who moves at each node/information sets, actions,
payoffs (at terminal nodes) [No explanation needed]
(b) (1 mark) Draw the payoff matrix for the normal form game associated with the
sequential move game described above. State the two Nash equilibria in pure strategies.
[No explanation needed]
(c) (3 marks) Consider the two Nash equilibria found in 2b. Is any one of them a Perfect
Bayesian Equilibrium (PBE)? Explain. In particular, consider each NE and argue why
they are or are not part of a PBE. [Note: A complete description of PBE must specify
beliefs as a part of description of the equilibrium.]