# 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 X*^{0} *S*_{n} is a **best response strategy** *to a given strategy Y*^{0} *S*_{m} for player II, if

*Similarly, a strategy Y*^{0} *S*_{m} is a