15
Backward Induction
and Subgame Perfection
1.
(a) (I, C, X)
(b) (AF, C)
(c) (BHJKN, CE)
2.
(a) The subgame perfect equilibria are (WY, AC) and (ZX, BC). The
Nash equilibria are (WY, AC), (ZX, BC), (WY, AC), (ZY, BC), and
(WX, BD).
(b) The subgame perfect equilibria are (UE, BD) and (DE, BC). The Nash
equilibria are (UE, BD), (DE, BC), (UF, BD), and (DE, AC).
3.
(a)
A
B
2
A
B
A
B
1
O
I
x
, 1
3, 1
0, 0
0, 0
1, 3
1
x
, 1
x
, 1
x
, 1
x
, 1
OA
OB
A
B
2
1
3, 1
0, 0
IA
IB
1, 3
0, 0
(b) If
x >
3, the equilibria are (OA,A), (OB,A), (OA,B), (OB,B). If
X
= 3, add (IA, A) to this list. If 1
< x <
3, the equilibria are (IA,A),
(OA,B), (OB,B). If
x
= 1, add (IB, B) to this list. If
x <
1, the equilibria
are (IA,A), (IB,B).
(c) If
x >
3 any mixture with positive probabilities over OA and OB for
player 1, and over A and B for player 2.
If 1
< x <
3, then IB is dominated. Any mixture (with positive proba-
bilities) over OA and OB will make player 2 indi
ff
erent. Player 2 plays A
with probability
x/
3, and plays B with probability 1
−
x/
3.
110