This question is from \" Networks, Crowds, and Markets: Reasoning about a Highly

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This question is from " Networks, Crowds, and Markets: Reasoning about a Highly Connected World" textbook in chapter 6

In this question we will consider several two-player games. In each payo matrix below the rows correspond to player A’s strategies and the columns correspond to player B’s 202 CHAPTER 6. GAMES strategies. The rst entry in each box is player A’s payo and the second entry is player B’s payo. (a) Find all pure (non-randomized) strategy Nash equilibria for the game described by the payo matrix below. Player A Player B L R U 2,15 4,20 D 6,6 10,8 (b) Find all pure (non-randomized) strategy Nash equilibria for the game described by the payo matrix below. Player A Player B L R U 3,5 4,3 D 2,1 1,6 (c) Find all Nash equilibria for the game described by the payo matrix below. Player A Player B L R U 1,1 4,2 D 3,3 2,2 [Hint: This game has a both pure strategy equilibria and a mixed strategy equilibrium. To nd the mixed strategy equilibrium let the probability that player A uses strategy U be p and 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 indierent between two pure strategies. That is the strategies must have equal expected payos. So, for example, if p is not 0 or 1 then it must be the case that q+4(1q) = 3q+2(1q) as these are the expected payos to player A from U and D when player B uses probability q.]

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