Econ 650 PROBLEM SET 2- Differentiated Bertrand Competition

PROBLEM SET 2Econ 6501. WATSON Chapter 7, Exercise 2.2. Differentiated Bertrand Competition. There are two firms i = 1, 2 simultaneouslychoosing prices pi ? [0, 1]. The demand of firm i is Di (pi , p?i ) = 1 ? 3pi + p?i andit 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 youranswer.3. Consider the following Guessing Game. There are n = 10 players simultaneouslychoosing a number in {1, 2, 3}. The winners are those closest to 1/2 the averageguess (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 20for each player, so the winner would receive 20 and the loser 0. Given the norms ofthe country they are in, it is common to bribe the judge of a case. Each player cansecretly offer a bribe of 0, 9 or 20, and the one whose bribe is the largest is awardedthe patent. If both choose not to bribe, or if the bribes are the same amount, theneach has an equal chance of being awarded the patent. (If a player decides to bribethen 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 survivingIDSDS. 1