Advanced Math 2010-2011 Homework Set 6
Due Monday 10/25/10 at the beginning of class
Game Theory!
Problems are taken from: Game Theory and Strategy, by Philip D. Straffin
1. Chapter 3 Exercise 4, p. 21
“Some games have more than one solution. The value of the game is fixed, but
the players may have several different strategies which ensure this value.
a)
Draw the graph for the following game. What happens?
C
b)
O
L
I
R
O
A B
A 2 0
C
2
S
E
B
1
3
1
N
Show that there are two different optimal strategies for Colin,
corresponding to the solutions for two different 2 x 2 subgames. The third
2 x 2 subgame does not yield a solution. In the graph, what is different
about that subgame?”
2. Chapter 3 Exercise 5c, p. 22
“Solve the following game:
A
B
C
D
4
4
3
4
2
4
4
2
C 0
D 5
1
2
3
7
2
2
E
2
2
2
A
B
3
”
3. Chapter 3 Exercise 7, p. 22
“In a simplified version of the Italian finger game Morra, each of two players
shows one finger or two fingers, and simultaneously guesses how many fingers
the other player will show. If both players guess correctly, or both players guess
incorrectly, there is no payoff. If just one player guesses correctly, that player
wins a payoff equal to the total number of fingers shown by both players.
a)
Each player has four strategies. Write the 4 x 4 matrix for this game. (Note
that it should have a kind of symmetry).
b)
Since the game is symmetric, its value should be zero. Show that the
strategy 5/12 show two-guess one, 7/12 show one-guess two, is optimal
for both players. (In actual Morra, players can show one, two or three
fingers. For its solution, see [Williams, 1986], pages 163-165).”
4. Game from class (also found on p. 20 called Game 3.4)
Analyze the following game to determine the optimal strategies for Rose and
Colin, as well as the value of the game. Compare your results with the
experimental probabilities we found in class.
A B C D E F
A 4 4 3 2 3 3
B 1 1 2 0 0 4
C 1 2 1 1 2 3
5. Chapter 3 Exercise 9, p. 22 (This is the hard one!)
“Suppose Rose and Colin play many rounds of a game by the following
procedure. In the first round, each player chooses a pure strategy arbitrarily, say
Rose A and Colin A. In the (n+1)st round, each player plays the pure strategy
which has the best expected value against the mixture of strategies used by the
other player in the first n rounds.
a)
Write a computer program to carry out this procedure for the game we
played from class (see Exercise 4 above). You should find tha the players’
mixtures converge to their optimal mixed strategies. (This is a celebrated
theorem of Julia Robinson (1951)) How soon do Colin’s non-active
strategies stop being played?
b)
(Suggested by Peter Ungar) It would be dangerous to play strictly
according to this procedure, because your opponent might realize what
you are doing. Suppose Colin plays the game in Exercise 4 by this
procedure, but Rose realizes he is doing it and in each round plays her best
response to what she knows Colin is going to do. Adapt your computer
program to investigate the result. Poor Colin will do badly, but will the
long-run mixtures still converge to the optimal mixed strategies?”
On Exercise 5, you may work with a partner, as I’m aware that the
programming expertise varies from person to person. Just let me know
who you worked with, and write up your own solution.