**5.1** This problem makes clear the connection between a pair of numbers (, ) that maximizes both payoff functions and a Nash equilibrium. Suppose that we have a two-person game with payoff functions *u*_{i}(*q*_{1}, *q*_{2}), *i* = 1, 2. Suppose there is a pure strategy pair (, ) that maximizes both *u*_{1} and *u*_{2} as functions of the pair (*q*_{1}, *q*_{2}).

**(a)** Verify that (, ) is a Nash equilibrium.

**5.1.a Answer:** Since (, ) maximizes both *u*_{1} and *u*_{2}, we have

and

Thus,

and

for every , . Thus, (, ) automatically satisfies the definition ...

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