Nash equilibrium theorems:

There are few theorems, which guarantee the existence of Nash equilibrium  in a game. We will only state these theorems without proof.

Theorem 1:  (Nash's  theorem) Every finite  strategy form game has mixed strategy Nash equilibrium.

By  finite  strategy  form  game,  we  mean  a  normal form  game  where every player has a finite number of strategies at her disposal. For example, the game of "battle  of sexes" and "prisoners' dilemma"  are  finite  strategy  form  games whereas "Cournot's  duopoly"  is an  infinite strategy form game.

Theorem  2: Consider a strategy form game where strategy  spaces Si are non-empty,  compact, convex subset of  a Euclidean space.  If  the  payoff function  Π_i  are continuous in  s= (s₁,  s₂,  s₃  ,... s_n) and quasi-concave  in  si, then there exist a pure strategy Nash equilibrium.

Here we consider a general  case where there are n players  in a game. S,, the strategy space of the  ith  player,  is the set containing all the strategies available to the  ith player. S, is non-empty means there  is at least one element  in  the set S, When we say S,  is compact closed, we mean it is closed as well as bounded set. In a closed  set the end points of the set are inside the set and a bounded set can always be put  inside a bigger set. When  the linear combination of any two elements of a set lies inside of that set, it the set is called convex set. The payoff function of the  ith  player  is Π_i(s₁,  s₂,  s₃  ,... s_n), where  (s₁, s₂,  s₃  .... s_n,)  is the vector of strategies chosen by all the players.

If along with all  the above stated conditions on  the strategy  space the payoff function  is continuous and quasi-concave  in si, then there always exists a pure strategy Nash equilibrium of  the game.