Nine. 2) facts: the need half is fast from the definition of a Nash equilibrium. certainly, the stipulations (1. 1) carry real for arbitrary suggestions x and y (including natural strategies). The sufficiency of the stipulations (9. 2) could be proven as follows. Multiply the 1st inequality H1 (i, y∗ ) ≤ H1 (x∗ , y∗ ) through xi and practice summation over all i = 1, … , m. those operations yield the H1 (x, y∗ ) ≤ H1 (x∗ , y∗ ) for an arbitrary method x. Analogous reasoning applies to the second one inequality in (9. 2). The evidence of Theorem 1. three is done. Theorem 1. four (on complementary slackness) permit (x∗ , y∗ ) be a Nash equilibrium process profile in a bimatrix video game. If for a few i: xi∗ > zero, then the equality H1 (i, y∗ ) = H1 (x∗ , y∗ ) happens. equally, if for a few j: y∗j > zero, we have now H2 (x∗ , j) = H2 (x∗ , y∗ ). facts is through ex contrario. consider that for a definite index i′ such that xi∗′ > zero one obtains H1 (i′ , y∗ ) < H1 (x∗ , y∗ ). Theorem 1. three means that the inequality H1 (i, y∗ ) ≤ H1 (x∗ , y∗ ) is legitimate for the remaining indexes i ≠ i′ . accordingly, we arrive on the procedure of inequalities H1 (i, y∗ ) ≤ H1 (x∗ , y∗ ), i = 1, … , n, (9. 2′ ) the place inequality i′ seems strict. Multiply (9. 2′ ) by way of xi∗ and practice summation to get the contradiction H(x∗ , y∗ ) < H(x∗ , y∗ ). through analogy, one simply proves the second one a part of the theory. Theorem 1. four claims Nash equilibrium contains in basic terms these natural suggestions resulting in the optimum payoff of a participant. Such techniques are referred to as equalizing. STRATEGIC-FORM TWO-PLAYER video games 15 Theorem 1. five a technique profile (x∗ , y∗ ) represents a combined technique Nash equilibrium profile iff there exist natural approach subsets M0 ⊆ M, N0 ⊆ N and values H1 , H2 such that { { } } ∑ i ∈ M0 ≡ ∗ H1 (i, j)yj H1 , for (9. three) i ∉ M0 ≤ j∈N0 { { } } ∑ j ∈ N0 ≡ H2 (i, j)xi∗ H2 , for (9. four) i ∉ N0 ≤ i∈M0 and ∑ xi∗ = 1, i∈M0 ∑ y∗j = 1. (9. five) j∈N0 evidence (the necessity part). imagine that (x∗ , y∗ ) is an equilibrium in a bimatrix video game. Set H1 = H1 (x∗ , y∗ ), H2 = H2 (x∗ , y∗ ) and M0 = {i ∈ M : xi∗ > 0}, N0 = {j ∈ N : y∗j > 0}. Then the stipulations (9. 3)–(9. five) at once stick to from Theorems 1. three and 1. four. The sufficiency half. think that the stipulations (9. 3)–(9. five) carry actual for a definite technique profile (x∗ , y∗ ). formulation (9. five) signifies that (a) xi∗ = zero for i ∉ M0 and (b) y∗j = zero for j ∉ N0 . Multiply (9. three) by way of xi∗ and (9. four) by way of y∗j , in addition to practice summation over all i ∈ M and j ∈ N, respectively. Such operations convey us to the equalities H1 (x∗ , y∗ ) = H1 , H2 (x∗ , y∗ ) = H2 . This end result and Theorem 1. three convey that (x∗ , y∗ ) is an equilibrium. The evidence of Theorem 1. five is concluded. Theorem 1. five can be utilized to guage Nash equilibria in bimatrix video games. think that we all know the optimum method spectra M0 , N0 . it truly is attainable to hire equalities from the stipulations (9. 3)–(9. five) and locate the optimum combined concepts x∗ , y∗ and the optimum payoffs H1∗ , H2∗ from the procedure of linear equations. even though, the program can generate detrimental options (which contradicts the concept that of combined strategies).