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HW 2 solns

# HW 2 solns - Networks Spring 2008 David Easley and Jon...

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Networks: Spring 2008 Homework 2 Solutions David Easley and Jon Kleinberg February 13, 2008 Grading: Part A is out of 10, Part B is out of 14. 1.a [2 points] D is a dominant strategy for player A and R is a dominant strategy for player B . Thus the only Nash equilibrium is ( D,R ). 1.b [2 points] U is a dominant strategy for player A and L is player B ’s unique best response to U . Thus the only Nash equilibrium is ( U,L ). 1.c [2 points] This game has two pure strategy Nash equilibria: ( D,L ) and ( U,R ). It also has a Nash equilibrium in which both players use mixed strategies. Let Pr A ( U ) = p and Pr B ( L ) = q . To have a Nash equilibrium p,q , with both p and q greater than 0 and less than 1, A ’s expected payoffs from U and D must be equal; and B ’s expected payoffs from L and R must be equal. So we have q + 4(1 - q ) = 3 q + 2(1 - q ) and p + 3(1 - p ) = 2 p + 2(1 - p ) . Solving we see that p = q = 1 / 2. 2.a [ 2 points] This game has two pure strategy Nash equilibria ( U,L ) and ( D,R ). Strategy L is a weakly dominated strategy for player B . So the equilibrium ( U,L ) uses a weakly dominated strategy. 2.b [2 points] Here we are just looking for a discussion of why a player might or might not use a weakly dominated strategy. There are at least two equally good answers and an argument along either line received full credit for this part: One way to reason about the game is that B knows that if he has an opportunity to make a move that matters, then A must have chosen D . As R is the unique best response to D , player B should chose R . A second possibility is to note that, no matter what A does, B has nothing to lose from choosing R rather than L . So he should chose R .

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