Definition

Consider a matrix game (a two-player zero-sum game) where P1 is a minimizer, P2 is a maximizer.

P1 secures his strategy against any behavior of P2, picking row (i∗i^*) whose largest entry is no bigger than the largest entry of any other row. The strategy "row ii" that secures losses no greater than V\overline{V} is the loss ceiling of P1, which satisfies

V(A)=maxj⁡ai∗j≤maxj⁡aiji=1,...,m\overline{V}(A) = \max_j a_{i^* j} \leq \max_j a_{ij} \quad i=1,...,m

or equivalently, the security level for his losses.

P2 will similarly secure gains against P1, and will choose the column (j∗j^*) whose smallest entry is no smaller than the smallest entry from any other column, the gain-floor of P2

V(A)=mini⁡aij∗≥mini⁡aij\underline{V}(A) = \min_i a_{ij^*} \geq \min_i a_{ij}

or equivalently, the security level for his gains.

Thus, it may be shown that,
theorem, security levels of matrix game players#Theorem

References

  1. T. Başar and G.J. Olsder, Dynamic Noncooperative Game Theory, 2nd edition, Classics in Applied Mathematics, SIAM, Philadelphia, 1999, p. 18.