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Unformatted text preview: 14.03 Fall 2000 Optional Problem Set 8 Solutions 1. a) The pure strategy Nash Equilibria are (Up, C) and (Down, A). b) There is only one mixed strategy Nash Equilibrium: Player 1 plays Up with probability 1/2 and Player 2 plays A with prob. 2/3, B with prob. 0, C with prob.1/3, and D with prob. 0. To find this equilibrium follow these steps. First, notice that D is strictly dominated by C for Player 2, so that Player 2 will never play D in a mixed strategy equilibrium, since he can do better by playing C instead of D. Next find the probability p of Player 1 playing Up that makes Player 2 indifferent between playing A and playing C. You will find that p =1/2. Then notice that for p =1/2 the expected payoff for Player 2 of playing either A or C is 3, which is bigger than 2.5 the expected payoff of playing B, so that B is left out of this mixed strategy equilibrium. Finally, find the probability q with which Player 2 has to randomize between A and C that makes Player 1 indifferent between playing Up and Down. You will find that q =2/3. Which establishes the result given at the beginning. You can convince yourself that there are no other mixed strategy equilibria by noticing that there is no randomization over Player 1’s actions that makes Player 2 happy to leave out either A or C, since the payoff of what is left out is always bigger than the payoff of what is included. 2. a) The normal form for this game is given below. The game has two pure strategy Nash Equilibria: (Out, Fight) and (In, Accommodate). Firm I Fight Accommodate Firm E Out 0, 2 0, 2 In -3, -1 2, 1 b) To find the SPNE of the game use the Extensive form and proceed by backward induction. If Firm E enters, Firm I decides to accommodate, because 1 is larger than –1. So, Firm E compares the payoff of entering, which is 2 (since Firm I would accommodate), with the payoff of staying out, which is 0, and it decides to enter. Therefore the SPNE of the game is (In, Accommodate). Notice that only one of the two Nash equilibria that you Accommodate)....
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