1
E
CONOMICS
S1050
H
ARVARD
U
NIVERSITY
S
UMMER
2009
S
UMMER
S
CHOOL
P
ROBLEM
S
ET
3
A
NSWERS
due
July 23
, in class
Show all calculations, explain all answers (up to 5 lines), and fully label graphs.
Also:
(a) staple sheets, (b) acknowledge assistance, (c)
highlight
final answers, and (d) write neatly!
1.
Consider the following game:

1.5

L
C
R
U
2, 2
6, 1
1, 1
M
1, 6
5, 5
1, 1
D
1, 1
1, 1
4, 4
a)
If the game is played once, what are the pure strategy Nash Equilibria?
NE = { (U,L), (D,R) } 
0.5

b)
If the game is played twice, and the results of the first stage is known before the second
stage begins, how many strategies does each player have?
In StageI, the first player has 1 node, with 3 actions.
Though the second player appears
to have 3 nodes, one each for each of the 3 actions of the first player, because the game is
simultaneous within each stage (as shown by the normal form above), all the 3 nodes are
part of an information set.
Hence, the second player has only 1 node, as well.
In StageII, first player has 9 possible nodes, one for each of the 3
×
3 outcomes of StageI,
giving her a total of 1 + 9 = 10 nodes in the game.
But for each node of the first player,
the second player still has the information set (as explained above).
Hence, the second
player in this stage has 9 possible nodes, as well, for a total of 10 nodes in the game.
(Note that the either the row or the column player can be the “first”player.)
Each player has 3 actions (at each node) so
each player has 3
10
strategies 
0.5

.