5. A box containing 2 silver buttons is up for sale. Emma and Peili are the only two interested
parties. Emma attaches a value of 3 to each button (so her value for the box is 6), whereas
Peili thinks each button is worth 5 (so his value for the box is 10). The (quirky) owner of
the button wants to use the following protocol for selling: he asks Emma and Peili to each
write down 0, 2 or 4 in a piece of paper and submit it to him.
❼ If they write down the same number, they each get a button and each pay exactly one
half of what they write down: so if Emma wrote 2 and Peili wrote 2, Emma’s profit
will be 3-1 = 2 and Peili’s profit is 5-1 = 4.
❼ Otherwise, whoever wrote down the larger number gets both buttons and pay what
they wrote down; the other person gets nothing and pays nothing.
(a) Write down the matrix representation of this game.
(b) Does Emma have a strictly dominated strategy? What about Peili?
(c) If you iteratively eliminate strictly dominated strategies, which strategies survive?
(d) Going back to the original game in part (a), write down all strategies that form a Nash
equilibrium. Justify your answer.
6. On page 55 of the Tadelis textbook, he talks about “self-enforcement” as a natural requirement of a solution concept. The general view is that Nash equilibria are “self-enforcing.”
To assess whether it is correct, consider the following game:
Bob
Alice
L
R
U
10, 10
-100, 9
D
9, −100
8, 8
(a) Verify that (U, L) is a Nash equilibrium, as is (D, R).
(b) If Alice does not know Bob, can you envision a scenario in which she would play D
even though (U, L) is a “better” equilibrium than (D, R)?
(c) Suppose Alice and Bob talk it out and they “agree” to play U and L respectively. Each
then has to make a choice privately. Could you envision a scenario in which Alice still
might play D?
(Note that if indeed Nash equilibria are self-enforcing, then if Alice and Bob communicate
before playing and agree to play U and L respectively, then that is the only outcome to be
expected. Recall the Battle of the sexes game in which Alice and Bob are choosing between
going to the opera and going to the fight. In that game, a communication between Alice
and Bob that results in an agreement to (for e.g.) play O is self-enforcing.)
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