Definition (bimatrix games)

Two (m×nm \times n) bimatrix games (A,B)(A,B) and (C,D)(C,D) are said to be strategically equivalent if there exists positive constants α1,α2\alpha_1, \alpha_2 and scalars β1,β2\beta_1, \beta_2 such that

aij=α1cij+β1bij=α2dij+β2 \begin{aligned} a_{ij} = \alpha_1 c_{ij} + \beta_1 \\ b_{ij} = \alpha_2 d_{ij} + \beta_2 \end{aligned}

for all i=1,...,m;j=1,...,ni=1,...,m; j = 1,...,n.

notes

This is an equivalence relation as it is symmetric, reflexive, and transitive.

Proposition (same Nash equilibria)

All strategically equivalent bimatrix games have the same Nash equilibria.

(NN-person) All strategically equivalent nonzero-sum finite static games in normal form have the same set of Nash equilibria.

Proposition (interchangeable Nash equilibria)

Multiple Nash equilibrium of a bimatrix game (A,B)(A,B) are interchangeable if (A,B)(A,B) are strategically equivalent to (A,−A)(A, -A).

References

  1. T. Başar and G.J. Olsder, Dynamic Noncooperative Game Theory, 2nd edition, Classics in Applied Mathematics, SIAM, Philadelphia, 1999, pp. 81, 90.