MANAGERIAL DECISION MAKING
Prof. Luigi MARENGO
MOCK EXAMPLE of 1st ASSESSMENT- SOLUTIONS
QUESTION 1 (2 points): Consider a game with two players where each player has two pure
strategies. Suppose one of the players has a strictly dominant strategy. Thus, there cannot be a
mixed-strategy equilibrium of this game in which this player gives strictly positive probability to both
of his pure strategies.
☒TRUE
☐FALSE
BRIEF MOTIVATION: a mixed strategy equilibrium cannot give positive probability to a dominated
strategy
PROBLEM 1 (4 points): In the following game, call pL player 1’s belief about the probability that
player 2 plays strategy L. Find Player 1’s Best Response for every possible value of pL:
Player 1
T
M
B
Player 2
L
4,2
0,5
1,1
R
0,3
4,4
3,2
Expected payoff of T: EπT=4pL
Expected payoff of M: EπM=4-4pL
Expected payoff of B: EπB=pL+3-3pL=3-2pL
The figure below draws the expected payoff of T (blue), M (red) and B (yellow).
For 0≤pL<0.5 M is the Best Response, for pL>0.5 T is the Best Response, for pL=0.5 all T,M and B are
all equally Best Responses.
PROBLEM 2 (5 points): Find all the Nash equilibria (both in pure and mixed strategies) of the
following game
T
B
L
-1,-1
3,1
R
1,3
-1,0
(B,L) and (T,R) are pure strategies Nash equilibria. There will be also a mixed strategy one. Let’s find
it:
For the raw player:
EπT= -1pL+1(1-pL)=1-2pL
and
EπB= 3pL-1(1-pL)=4pL-1
We put EπT= EπB which gives 1-2pL=4pL-1 and pL=1/3
For the column player:
EπL= -1pT+1(1-pT)=1-2pT
and
EπR= 3pT+0(1-pT)=3pT
We put EπL= EπR which gives 1-2pT=3pT and pT=1/5
Thus we also have the mixed-strategy equilibrium [(1/3,2/3),(1/5,4/5)]