**3.1** Show that (*X*^{*}, *Y*^{*}) is a saddle point of the game with matrix *A* if and only if (*X*^{*}, *Y*^{*}) is a Nash equilibrium of the bimatrix game (*A*, −*A*).

**3.1 Answer:** Use the definitions. For example, if *X*^{*}, *Y*^{*} is a Nash equilibrium, then *X*^{*} *AY*^{*}^{T} ≥ *X AY*^{*}^{T}, ∀*X*, and *X*^{*} (−*A*)*Y*^{*}^{T} ≥ *X*^{*} (−*A*)*Y ^{T}*, ∀

**3.2** Suppose that a married couple, both of whom have just finished medical school, now have choices regarding their residencies. One of the new doctors has three choices of programs, while the other has two choices. They value their prospects numerically on the basis of the program itself, the city, staying together, and other factors, and arrive at the bimatrix

**(a)** Find all the pure Nash equilibria. Which one should be played?

**3.2.a Answer:** There are two pure Nash equilibria at *X* = (0, 1), *Y* = (0, 1, 0), and *X* = (1, 0), *Y* = (1, 0, 0).

Note that column 3 is dominated and may be dropped. There is also a mixed Nash at . We will see how to calculate mixed ...

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