1. Consider the following game:
L R
U 2, 4 -1, 6
D 0, 10 5, 0
a) For what range of beliefs θ2 = (θ(L), 1 - θ2(L))) is U a best response for player 1? [Your answer should take the form: a ≤ θ2(L) ≤ b for specific numbers a and b, which may or may not be distinct.]
Show your derivation clearly.
b) For what range of beliefs θ2 = (θ(L), 1 - θ2(L))) is D a best response for player 1? Show your derivation clearly.
c) For what range of beliefs θ2 = (θ(L), 1 - θ2(L))) are both U and D best responses for player 1?
2 Consider the partnership game with synergy. Player 1 chooses 0 ≤ x ≤ b, and Player 2 chooses 0 ≤ y ≤ b, if the parameter b is a positive number. If b = ∞, x and y can be any nonnegative real numbers. The effort levels x, y produce the partnership revenue 4(x + y + cxy). The personal effort costs are x 2 for Player 1 and y 2 for Player 2. Each player receives half the partnership revenue minus her personal effort cost. The synergy parameter c is a positive constant.
a) Derive the players' best-response functions. Show your derivation clearly.
b) For b = 10 and c =2/3 find the rationalizable strategies for each player.
Do three iterations of deletion of dominated strategies to show that your answer is correct.
c) For b = ∞ and c =2/3 , find the rationalizable strategies for each player.
Do three iterations of deletion of dominated strategies to show that your answer is correct.
d) For b = ∞ and c = 2, are there any rationalizable strategies? Explain your answer.