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

Appendix C: Two-Person Nonzero Sum Games

Definition C.1    A pair of mixed strategies (X* S_n, Y* S_m) is a Nash equilibrium if E_I(X, Y*) ≤ E_I(X*, Y*) for every mixed X S_n and E_II(X*, Y) ≤ E_II(X*, Y*) for every mixed Y S_m. If (X*, Y*) is a Nash equilibrium we denote by v_A = E_I(X*, Y*) and v_B = E_II(X*, Y*) as the optimal payoff to each player. Written out with the matrices, (X*, Y*) is a Nash equilibrium if

,    for every X S_n,

,    for every Y S_m.

Definition C.2    A strategy X⁰ S_n is a best response strategy to a given strategy Y⁰ S_m for player II, if

Similarly, a strategy Y⁰ S_m is a