Game Theory
Chapter 8
1.
Marge and Lydia are two grandmothers who supplement their retirement pensions by selling
muffins and cookies at a local swap meet.
They each have limited baking facilities, so they
must bake either all cookies or all muffins on a given week.
Each woman has profits that are
a function of what she bakes and what the other grandmother bakes.
The payoff matrix (in
terms of hundreds of dollars in profit per month) for the two women is as follows:
a)
Does the game have any dominant strategies?
Is there a Nash equilibrium?
b)
Suppose that Lydia threatens to always bake cookies.
Explain whether this threat is
credible and how Marge is likely to respond.
c)
Find the mixed strategies solution to the game, if the probability of
baking muffins is
P
M
and P
L
for Marge and Lydia, respectively.
d)
Find the expected payoff from the mixed strategy solution?
Answer:
a)
The game does not have any dominant strategies or Nash equilibria.
b)
The threat is not credible, because it is not rational in this game for Lydia to always
bake cookies.
If Lydia always baked cookies, then Marge would earn more profits by
always baking muffins.
But if Marge always baked muffins, then Lydia would make
more profits by switching to muffins.
Hence, the threat is not credible, and Marge will
ignore it.
c)
The equilibrium occurs where P
M
=.5, P
L
=.25.
d)
Marge will earn expected profits of 3.5 and Lydia will earn expected profits of 3.
2.
Consider the following payoff matrix between two players:
a)
If the players act simultaneously, what is the equilibrium outcome of the game?
Explain.
b)
Suppose that the game is played sequentially, and player 2 moves first.
Explain how
this would affect the outcome of the game.
a)
Discuss whether player 2 would prefer to move first instead of allowing player 1 to
move first.
b)
Find the mixed strategies solution to the game.
What is the expected payoff of the
mixed strategy solution?
Answer:
a)
There is no dominant strategy for either player.
There is no Nash equilibrium in the
game, so we cannot predict the outcome in pure strategies.
b)
If Player 2 moved first, he or she would pick left, then player 1 would pick top and we
would have the outcome (TL) as equilibrium (player 2 earns 3).
If Player 2 had picked
right, then player 1 would have picked bottom with player 2 earning only 1.
c)
In this game, there is a second mover advantage for player 2, i.e., player 2 is better off
if player 1 moves first.
If player 1 goes first, he or she will pick bottom and player 2
will pick left (player 1 earns 4), so BL is solution.
If player 1 had picked top, then
Lydia
Muffins
Cookies
Marge
Muffins
2,3
4,2
Cookies
5,3
3,4
Player 2
Left
Right
Player 1
Top
7,3
3,6
Bottom
4,6
5,1