payoff functions

payoff functions

This problem makes clear the connection between a pair of numbers (q ∗ 1 , q ∗ 2 ) 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 (q ∗ 1 , q ∗ 2 ) that maximizes both u1 and u2 as functions of the pair (q1, q2). (a) Verify that (q ∗ 1 , q ∗ 2 ) is a Nash equilibrium. (b) Construct an example in which a Nash equilibrium (q ∗ 1 , q ∗ 2 ) does not maximize both u1 and u2 as functions of (q1, q ).

payoff functions