Solutions Manual to Accompany Game Theory: An Introduction, 2nd Edition

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 (, ) that maximizes both payoff functions and a Nash equilibrium. Suppose that we have a two-person game with payoff functions u_i(q₁, q₂), i = 1, 2. Suppose there is a pure strategy pair (, ) that maximizes both u₁ and u₂ as functions of the pair (q₁, q₂).

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

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

  and  

Thus,

  and  

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