Decision Making using Game Theory: An introduction for managers

188Appendix Aso a mixed strategy has therefore been found for player 1 such that:ξ 1 Σw 1j q j ξ 2 Σw 2j q j ··· ξ m Σw mj q j 0for every q, so that:min q Σw ij ξ i q j 0Since this holds for ξ, it must hold for the mixed-strategy p thatmaximisesmin q Σw ij p i q jTherefore:max p min q Σw ij p i q j 0So it has been shown that:if min q max p Σw ij p i q j 0, then max p min q Σw ij p i q j 0Proof: step 5Let k be any number and consider the pay-oV matrix which has w ij kin place of w ij everywhere. All pay-oVs are reduced by k in both pureand mixed strategies. So:(max p min q Σw ij p i q j ) is replaced by (max p min q Σw ij p i q j k)and(min q max p Σw ij p i q j ) is replaced by (min q max p Σw ij p i q j k)It was proved in step 4 that:if (min q max p Σw ij p i q j k) 0, then (max p min q Σw ij p i q j k) 0Soif min q max p Σw ij p i q j k, then max p min q Σw ij p i q j kSince k can be as close as necessary to min q max p Σw ij p i q j , it followsthat:max p min q Σw ij p i q j min q max p Σw ij p i q jBut we have already seen that: