Definition

A pair of strategies {row i∗,column j∗}\{\text{row } i^*, \text{column } j^*\} is said to constitute a noncooperative (Nash) equilibrium solution to a bimatrix game if the following pair of inequalities is satisfied for all i=1,...,mi=1,...,m and all j=1,...,nj=1,...,n:

ai∗j∗≤aij∗bi∗j∗≤bi∗j \begin{aligned} a_{i^* j^*} \leq a_{ij^*} \\ b_{i^* j^*} \leq b_{i^* j} \end{aligned}

The pair (ai∗j∗,bi∗j∗)(a_{i^* j^*}, b_{i^* j^*}) is known as a noncooperative (Nash) equilibrium outcome of the bimatrix game.

Definition (mixed strategies)

A pair {y∗∈Y,z∗∈Z}\{y^* \in Y, z^* \in Z\} is said to constitute a noncooperative (Nash) equilibrium solution in mixed strategies if the following inequalities are satisfied for all y∈Yy \in Y and z∈Zz \in Z:

y∗′Az∗≤y′Az∗,y∈Yy∗′Bz∗≤y∗′Bz,z∈Z\begin{aligned} {y^*}' A z^* \leq y' A z^*, \quad y \in Y \\ {y^*}' B z^* \leq {y^*}' B z, \quad z \in Z \end{aligned}

with the pair (y∗′Az∗,y∗′Bz∗)({y^*}' A z^*, {y^*}' B z^*) known as a noncooperative (Nash) equilibrium outcome of the bimatrix game in mixed strategies.

Proposition

Let Y˚,Z˚\mathring{Y}, \mathring{Z} denote sets of inner points (interiors) of YY and ZZ, respectively. If bimatrix game (A,B)(A, B) admits a mixed strategy Nash equilibrium solution {y∗∈Y˚,z∗∈Z˚}\{y^* \in \mathring{Y}, z^* \in \mathring{Z} \}, then this also serves as a mixed-strategy solution for the bimatrix game (−A,−B)(-A, -B).

(Known also as completely mixed Nash equilibrium solution or inner mixed-strategy Nash equilibrium solution.)

Theorem

Every NN-person static finite game in normal form admits a non-cooperative (Nash) equilibrium solution in mixed strategies.

See also

References

  1. T. Başar and G.J. Olsder, Dynamic Noncooperative Game Theory, 2nd edition, Classics in Applied Mathematics, SIAM, Philadelphia, 1999, pp. 78-79, 85-86, 91-94.