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Solutions Manual to Accompany Game Theory: An Introduction, 2nd Edition by E. N. Barron

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N-Person Nonzero Sum Games and Games with a Continuum of Strategies

5.1 The Basics

Problems

5.1 This problem makes clear the connection between a pair of numbers (images, images) that maximizes both payoff functions and a Nash equilibrium. Suppose that we have a two-person game with payoff functions ui(q1, q2), i = 1, 2. Suppose there is a pure strategy pair (images, images) that maximizes both u1 and u2 as functions of the pair (q1, q2).

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

5.1.a Answer: Since (images, images) maximizes both u1 and u2, we have

  and  

Thus,

  and  

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

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