theorem, security levels of matrix game players

Theorem

In every matrix game $A = \{a_{ij}\}$,

1. the security level of each player is unique
2. there exists at least one security strategy for each player
3. The security level of P1 (the minimizer) never falls below the security level of P2 (the maximizer), i.e.$$\min_i a_{ij^*} = \underline{V}(A) \leq \overline{V}(A) = \max_j a_{i^* j}$$ where $i^*$ and $j^*$ denote security strategies for P1 and P2 respectively

(note: P1 as minimizer, P2 as maximizer is a convention which is reversed in some textbooks)

References

1. T. Başar and G.J. Olsder, Dynamic Noncooperative Game Theory, 2nd edition, Classics in Applied Mathematics, SIAM, Philadelphia, 1999, p. 19.