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# ps2 - Networks Fall 2011 David Easley and Eva Tardos...

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Networks: Fall 2011 Homework 2 David Easley and Eva Tardos Due in Class September 16, 2011 As noted on the course home page, homework solutions must be submitted by upload to the CMS site, at https://cms.csuglab.cornell.edu/ . This means that you should write these up in an electronic format (Word files, PDF files, and most other formats can be uploaded to CMS). Homework will be due at the start of class on the due date, and the CMS site will stop accepting homework uploads after this point. We cannot accept late homework except for University- approved excuses (which include illness, a family emergency, or travel as part of a University sports team or other University activity). Reading: The questions below are primarily based on the material in Chapters 6 and 8 of the book. (1) In this question we will consider several two-player games. In each payoff matrix below the rows correspond to player A’s strategies and the columns correspond to player B’s strategies. The first entry in each box is player A’s payoff and the second entry is player B’s payoff. Both players prefer higher payoffs to lower payoffs. (a) Find all pure (non-randomized) strategy Nash equilibria for the game described by the payoff matrix in Figure 1. Player A Player B L R U 3 , 2 7 , 0 D 1 , 8 3 , 5 Figure 1: The payoff matrix for Question (1a). (b) Find all pure (non-randomized) strategy Nash equilibria for the game described by the payoff matrix in Figure 2. Player A Player B L R U 3 , 1 1 , 7 D 5 , 4 6 , 2 Figure 2: The payoff matrix for Question (1b). (c) Find all Nash equilibria for the game described by the payoff matrix in Figure 3. [Hint: This game has two pure strategy equilibria and a mixed strategy equilibrium. To find the mixed strategy equilibrium let the probability that player A uses strategy U be p and 1

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Player A Player B L R U 4 , 4 2 , 2 D 2 , 4 6 , 6 Figure 3: The payoff matrix for Question (1c). the probability that player B uses strategy L be q . As we learned in our analysis of matching pennies, if a player uses a mixed strategy (one that is not really just some pure strategy played with probability one) then the player must be indifferent between two pure strategies. That is the strategies must have equal expected payoffs. So, for example, if p is not 0 or 1 then it must be the case that 4 q + 2(1 - q ) = 2 q + 6(1 - q ) as these are the expected payoffs to player A
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