Nash Equilibria Questions

1. Consider the following game in matrix form with two players. Payoffs for the row player Izzy are indicated first in each cell, and payoffs for the column player Jack are second.

X      Y       Z

S  5, 2 10, 6 25,10
T  10, 12 5, 6 0,0

(a) This game has two pure strategy Nash equilibria. What are they (justify your answer)? Of the two pure equilibria, which would Izzy prefer? Which would Jack prefer?

(b) Suppose Izzy plays a totally mixed strategy, where both S and T are chosen with positive probability. With what probability should Izzy choose S and T so that each of Jack’s three pure strategies is a best response to Izzy’s mixed strategy.

(c) Suppose Jack wants to play a mixed strategy in which he selects X with probability 0:7. With what probability should Jack plays actions Y and Z so both of Izzy’s pure strategies is a best response to Jack’s mixed strategy? Explain your answer.

(d) Based on your responses above, describe a mixed strategy equilibrium for this game in which both Jack and Izzy play each of their actions (pure strategies) with Positive probability. Explain why this is in fact a Nash equilibrium (you can rely on the quantities computed in the prior parts of this question).

(e) If we swap two of Izzy’s payoffs in this matrix—in other words, if we replace one of his payoffs r in the matrix with another of his payoffs t from the matrix, and replace t with r— we can make one of his strategies dominant. What swap should we make, which strategy becomes dominant, and why is it now dominant?

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