ONLY FOR PROF DAN.
Problem Set 3 (Solutions)
ECO 444
Konrad Grabiszewski
I will use the notation from Handout 6. Let αA = (p, 1 − p) be Ann’s mixed strategy where p
is the weight assigned to strategy T. Let αB = (q, 1 − q) be Bob’s mixed strategy where q is
the weight assigned to strategy L. Let (α∗A,α ∗ B) denote an MSNE where α
∗ A = (p
∗, 1 − p∗) and
α∗B = (q ∗, 1 − q∗) is Ann’s and Bob’s equilibrium strategy, respectively.
In the graphs below, solid line represents Ann’s best-response correspondence, and dashed line
represents Bob’s best-response correspondence.
Game 1
p
q
1
1 ½
½
There are three NE, two pure and one mixed.
• Pure NE: (T,L) and (B,R). Note that we can re-write (T,L) as ((1, 0), (1, 0)).
• Mixed NE: ((
1 2 , 1 2
) , ( 1 2 , 1 2
)) .
Formally, the set of Nash equilibria in Game 1.
{ ((1, 0), (1, 0)), ((0, 1), (0, 1)),
(( 1
2 , 1
2
) ,
( 1
2 , 1
2
))} (1)
1
Game 2
p
q
1
1 ⅓
⅔
There are three NE, two pure and one mixed.
• Pure NE: (T,R) and (B,L).
• Mixed NE: ((
2 3 , 1 3
) , ( 1 3 , 2 3
)) .
Formally, the set of Nash equilibria in Game 2.
{ ((1, 0), (0, 1)), ((0, 1), (1, 0)),
(( 2
3 , 1
3
) ,
( 1
3 , 2
3
))} (2)
2
Game 3
p
q
1
1 ½
¾
There is only one NE and it is an MSNE, ((
3 4 , 1 4
) , ( 1 2 , 1 2
))
3
Game 4
p
q
1
1 ⅓
In the graph above, Ann’s and Bob’s best-response correspondences intersect at (0, 1) and on the
interval [ 1 3 , 1 ]
on the horizontal line. We have two pure NE, (B,L) and (T,R), and infinitely many
MSNE. Each MSNE is such that p∗ = 0 and q∗ ∈ [ 1 3 , 1 ] .
Formally, the set of Nash equilibria in Game 2.
{((1, 0), (0, 1)), ((0, 1), (1, 0))}∪ {
(α∗A,α ∗ B) : p
∗ = 0,q∗ ∈ [
1
3 , 1
]} (3)
4
Game 5
p
q
1
1
0.4
There is only one NE and it is an equilibrium in pure strategies, (T,L).
5
Game 6
p
q
1
1 ⅓
½
There is only one NE and it is an MSNE, ((
1 2 , 1 2
) , ( 1 3 , 2 3
))
6