saddle-point equilibrium (game theory)

Definition

For a given $m \times n$ Matrix game $A=\{a_{ij}\}$ let {row $i^*$, column $j^*$} be a pair of strategies adopted by the players. Then if the pair of equalities $$a_{i^* j} \leq a_{i^* j^*} \leq a_{ij^*}$$ is satisfied for all $i=1,...,m$ and all $j=1,...,n$, then the strategies are said to constitute a saddle-point equilibrium (or said to be saddle-point strategies). Corresponding outcome $a_{i^* j^*}$ is called the saddle-point value or simply the value of the matrix game, and is denoted $V(A)$.

Theorem

Let $A=\{a_{ij}\}$ denote a $m \times n$ Matrix game with $\overline{V}(A) = \underline{V}(A)$, then

1. $A$ has a saddle point in pure strategies,
2. an ordered pair of strategies provides a saddle pair for $A$ iff the first of these is a security strategy for P1, and the second one a security strategy for P2,
3. $V(A)$ is uniquely given by $V(A) = \overline{V}(A) = \underline{V}(A)$.

Corollary

In a matrix game $A$, let {row $i_1$, column $j_1$} and {row $i_2$, column $j_2$} be two saddle-point strategy pairs. Then {row $i_1$, column $j_2$}, {row $i_2$, column $j_1$} are also in saddle-point equilibrium. This feature of saddle-point strategies is known as their ordered interchangeability property.

References

1. T. Başar and G.J. Olsder, Dynamic Noncooperative Game Theory, 2nd edition, Classics in Applied Mathematics, SIAM, Philadelphia, 1999, p. 21.