Econ 650 PROBLEM SET 2- Differentiated Bertrand Competition

2. Differentiated Bertrand Competition. There are two firms i = 1, 2 simultaneously
choosing prices pi ? [0, 1]. The demand of firm i is Di (pi , p?i ) = 1 ? 3pi + p?i and
it has zero production costs. That is, firm i’s payoff is pi Di (pi , p?i ).
(a) Find the best response function of firm i.
(b) Find the set of 1-rationalizable, 2-rationalizable and 3-rationalizable strategies.
(c) Do you think there is a unique rationalizable strategy profile? Justify your
answer.

3. Consider the following Guessing Game. There are n = 10 players simultaneously
choosing a number in {1, 2, 3}. The winners are those closest to 1/2 the average
guess (they evenly split the prize between the winners if there is more than one).
Find the set of rationalizable strategy profiles. Justify your answer.

4. Two players find themselves in a legal battle over a patent. The patent is worth 20
for each player, so the winner would receive 20 and the loser 0. Given the norms of
the country they are in, it is common to bribe the judge of a case. Each player can
secretly offer a bribe of 0, 9 or 20, and the one whose bribe is the largest is awarded
the patent. If both choose not to bribe, or if the bribes are the same amount, then
each has an equal chance of being awarded the patent. (If a player decides to bribe
then the judge pockets it regardless of who gets the patent).
(a) Derive the game matrix.
(b) Is the game dominance solvable? If so, find the strategy profile surviving
IDSDS. 1