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A "properly mixed strategy" means a mixed strategy that does not assign all the probability to one pure strategy. In other words, it is not a pure strategy. Consider a simultaneous move game with the following payoff matrix:
Player 2
L
R
Player 1
A
4, 0
0, 3
B
2.5, 0
C
1, 1
a) Is pure strategy C strictly dominated by any other pure strategy of player 1? If so, write one out. If not, explain.
b) Is pure strategy C strictly dominated by any proper mixed strategy for player 1? If so, write one out. If not, explain.
c) Is pure strategy C a best response to some mixed strategy of player 2? If so, write one out. If not, explain.
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TC = Q3 – 8Q2 + 68Q + 4
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