Strategic Decision Theory
Problem Set 1
Due 5:00 PM on Thursday, September 14th.
Please submit a PDF, hand-written or typed, to the Brightspace page.
This is to get you started. You can work on the first two problems right away. The third might
be easier after the next lecture.
You may collaborate with a small study group. If so, please indicate who you collaborated with
and write up your answers individually so that it is clear that you understand the solution and
it is not simply a copy of someone else's solution
Problem 1 - Dominant Strategies
1. Give an example of a game in which no strategy is strictly dominant, but some strategy
is strictly dominated.
2. Recall our definition of "iterative elimination".
In that context prove that
, . . .
and that the procedure terminates in a finite number of steps.
confine attention to the simpler pure strategy definition of dominance.
3. Consider a game with a Nash Equilibrium strategy profile (one strategy for each player)
. Now consider Γ
, a modified version of Γ, in which a strictly dominated strategy has
been deleted for each player. Is
a feasible strategy profile of Γ
? Is it a Nash Equilibrium
? What about the converse: is any Nash Equilibrium strategy of the game Γ
Nash Equilibrium of Γ?
4. A strategy
does at least as well as
always, and strictly
better sometimes. Write down this casual definition formally.
5. It is well known that the final result of iterative elimination of
on the order of the elimination. Please show this by constructing a simple
The simplest example is a 3
2 game in which a player with three
strategies has two weakly dominated strategies.)
Problem 2 - Common Knowledge
(Please justify your answer as clearly as you can.)
Many centuries ago in a land far far away there was a village of 100 married couples, who
were all perfect logicians but had somewhat peculiar social customs. Every evening the men
of the village would have a meeting, in a great circle around a campfire, and each would talk
about his wife. If when the meeting began a man had any reason to hope that his wife had
been always faithful to him, then he would praise her virtue to all of the assembled men. On
the other hand, if at any time before the current meeting he had ever gotten proof that his wife
had been unfaithful, then he would moan and wail. Furthermore, if a wife had been unfaithful,
then she and the lover would immediately inform all of the other men in the village except her
husband. All of these traditions were
among the people of this village.