Modal Logic – Problem Set 2

Game Theory – Problem Set 2
[due February 17 at 1:30pm]
Problems 1(c) and 5(b) are only required for those enrolled in the graduate level course (i.e., those taking 705 rather
than 405).
1. Consider a strategic form game G = (N, (Ai )i∈N , (ui )i∈N ).
(a) Give an example where N = {1, 2}, |A1 | = |A2 | = 2, and there is a (pure) Nash equilibrium in which
both players’ actions are weakly dominated.
(b) Given an example where N = {1, 2}, |A1 | ≤ 3, |A2 | ≤ 3, there is a unique (pure) Nash equilibrium, and
in that Nash equilibrium both players’ actions are weakly dominated.
(c) Prove that when N = {1, 2} and |A1 | = |A2 | = 2, if there is a (pure) Nash equilibrium in which
both players’ actions are weakly dominated, then it is not unique (i.e., there is a different (pure) Nash
equilibrium).
2. Recall that in class we defined
ui (α) =
X
αi (ai )ui (ai , α−i ).
ai ∈Ai
Show that
X
Y
ui (α) =
ui (a) ·
αj (aj ) .
a∈A
j∈N
3. Two players sit at a table, and each places a penny on that table, either heads up or heads down, covered by
their hand. They then remove their hands simultaneously; player 1 wins if the pennies match (either both
heads up or both tails up), and player 2 wins otherwise. This is summarized by the following game table:
heads
tails
heads
1, −1
−1, 1
tails
−1, 1
1, −1
(a) Show that this game has no (pure) Nash equilibrium.
(b) Construct each player’s best response function in the mixed extension of this game.
(c) Find a Nash equilibrium of the mixed extension of this game and prove that it is unique.
4. Two people are collaborating on a task that won’t get done unless they both work at it. Working incurs a
certain cost, but the benefit of completing the task outweighs this cost. However, if only one person chooses
to work, the work is wasted because the task is not completed. This is summarized by the following game
table, where 0 < c < 1:
work
lazy
work
1 − c, 1 − c
0, −c
lazy
−c, 0
0, 0
Find all the Nash equilibria of the mixed extension of this game. How do the equilibria change as c increases?
Explain.
5. Answers to the following questions need not be completely formal; however, they should be clear.
(a) Draw a picture of the subset of R2 defined by
A = {(x, y) : 1 ≤ x2 + y 2 ≤ 4};
explain why A is not convex, and describe a continuous function f : A → A that has no fixed point.
(b) Draw a picture of the subset of R2 defined by
B = {(x, y) : x2 + y 2 < 4}.
If f : B → B is continuous, must it have a fixed point? If so, explain why. If not, describe a continuous
function f : B → B that has no fixed point.
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