GAME THEORY 2021-2022
Week 4
Extensive form games
Exercise 1. Give an example of a game in extensive form with perfect information in which player I has
2 strategies {A, B} and player II has 4 strategies {X, Y, W, Z}.
Write the pay-offs of the game in order to have has a unique optimal strategies A, Y .
Can you find another example of such a game?
Exercise 2. Describe a game in extensive form with two players, where the first one has 8 strategies
and the second one has 7 strategies. Provide different examples.
Exercise 3. Consider the extensive form games represented in Figure 1a and in Figure 1b. Enumerate
the strategies of players and apply backward induction to find the rational outcome.
I
I
a
a
b
II
e
d
(1,2)
N
II
c
I
g
f
I
h
N
1/3
2/3
II
(2,3)
i
b
l
g
(2,1) (0,0) (0,0) (1,2)
1/2
II
h
i
(3,2)
l
(3,1) (2,2)(1,2) (3,0)
(a)
(b)
Figure 1: Exercise 3
1
1/2
(1,3)
I
A
B
II
Exercise 4. Given the following game, find its solution when a
and b range over the real numbers. Represent the
game in strategic form.
c
II
d
e
f g
(1,4) (2,2) (a,2) (2,b) (4,1)
Exercise 5. Consider the zero sum game in extensive form in Figure 2. Write all the strategies of the
two players, solve the game using backward induction and say which strategies are played.
I
a
b
II
II
c
e
d
I
N
f
N
3
g
h
1/2
1/2
1/3
2/3
2
1
2
-1
-1
-2
Figure 2: Exercise 5
Exercise 6. Given the game:
A=
(3, 2) (0, 1)
(2, 3) (2, 3)
,
find the pure Nash equilibria of the game. Then, find a game in extensive form with perfect information
such that A represent its strategic form and solve the game by using backward induction. Compare the
results.
Exercise 7. Consider the following game in strategic form:
(1, 2) (1, 3)
(0, 2) (a, b)
What are the conditions on a and b for the game to be representable as a game in extensive form with
perfect information?
2
Exercise 8. Consider the game in extensive form in Figure 3. Apply backward induction to solve the
game. Then, write the game in strategic form and find the Nash equilibria.
I
A
B
II
C
D
(3, 2)
(1, 2)
(2, 0)
Figure 3: Exercise 8
Exercise 9. Suppose that player I chooses at the beginning x1 ∈ [0, +∞); then after observing the
choice of the first player, player II chooses publicly y ∈ [0, +∞). Finally player I chooses x2 ∈ [0, +∞).
The utility functions of the two players are respectively
f (x1 , x2 , y) = −x21 − x22 + 2x2 y + 4x1 ,
g(x1 , x2 , y) = 2x1 x2 y − y 4 + 4x1 .
Find the rational outcome of the game.
Combinatorial games
Exercise 10. Two players. A pile of 15 chips. A move consists in removing 1, 3, or 4 chips. The player
removing the last one wins. Who wins?
Same game as before, with 79 chips. Who wins?
Exercise 11. Consider the Nim game with initial position (13, 12, 8). Is this a P-position? If not, what
is a winning move for the first player?
Exercise 12. Two players. A pile of 38 chips. A move consists in removing 1, 3 or 6 chips. The player
removing the last one wins. Who wins?
Exercise 13. We consider a Nim game with 3 piles. We sort the number of chips in each pile between
0 and 7. Evaluate the probability of obtaining a P-position (in this situation player II is going to win).
Generalize to an n chips situation.
Exercise 14. Consider the nim game with starting position (4,3,1). How long is the longest rational
play?
Exercise 15. Who wins in a 2 × ∞ Chomp? And in a 3 × ∞?
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