security strategy

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 ($i^*$) whose largest entry is no bigger than the largest entry of any other row. The strategy "row $i$" that secures losses no greater than $\overline{V}$ is the loss ceiling of P1, which satisfies $$\overline{V}(A) = \max_j a_{i^* j} \leq \max_j a_{ij} \quad i=1,...,m$$ or equivalently, the security level for his losses.

P2 will similarly secure gains against P1, and will choose the column ($j^*$) whose smallest entry is no smaller than the smallest entry from any other column, the gain-floor of P2 $$\underline{V}(A) = \min_i a_{ij^*} \geq \min_i a_{ij}$$ or equivalently, the security level for his gains.

Thus, it may be shown that,

References

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