Existence theorems of equilibrium points

Abstract

Let a game Γ be given in normal form

$$\Gamma = \left\{ {{{\sum }_{1}}, \ldots ,{{\sum }_{n}};{{K}_{1}}, \ldots ,{{K}_{n}};{{\Phi }_{1}}, \ldots ,{{\Phi }_{n}}} \right\} $$

.

As we have already seen an equilibrium point of the game Γ is an n-tuple of strategies $(\sigma _1^*,...,\sigma _n^*),(\sigma _i^* \in {\sum _1},i = 1,...,n)$ for which $\begin{array}{l}(a)\sigma _i^* \in {_1}(\sigma _i^*,...,\sigma _n^*)(i = 1,...,n),\\(b){K_i}(\sigma _i^*,...,\sigma _{i - 1}^*,{\sigma _{i + 1}},...,\sigma _n^*) \mathbin{\lower.3ex\hbox{$\buildrel<\over{\smash{\scriptstyle=}\vphantom{_x}}$}}{K_i}(\sigma _1^*,...,\sigma _n^*) \end{array}$for each for each $\sigma _i^* \in {_1}(\sigma _i^*,...,\sigma _n^*)(i = 1,...,n).$

The equilibrium point thus defined is generally referred to as a (generalized) Nash-equilibrium point.