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

Appendix E: N-Person Nonzero Sum Games and Games With a Continuum of Strategies

Definition E.1   A collection of strategies is a pure Nash equilibrium for the game with payoff functions {u_i(q₁, . . ., q_n)}, i = 1, . . ., N, if for each player i = 1, . . ., N, we have

A short hand way to write this is u_i for all q_i Q_i, where q_−i refers to all the players except the ith.

Definition E.2   Given payoff functions u_i(q₁, . . ., q_n), a best response of player k, written q_k = B R_k(q₁, . . ., q_k−1, q_k+1, . . ., q_n) is a value that satisfies

or . In other words, q_k provides the maximum payoff for player k, given the values of the other player’s .

In general, we say that q_k = B R_k(q_−k) arg max_x