Equilibrium points in n-person games

The two-page paper by John Nash made the cornerstone to game theory.

Let's review what has been conveyed in this classic work.

Nash argued that in n-person games, there are equilibrium points. The arguments expand as follows:

Suppose there are (n) players, each with a pure strategy.

A point of a strategy profile is a vector of (n) player's strategies.

A countering strategy for player (i) is that given all other players' strategy fixed, the strategy along with best payoff for player (i).

A countering strategy profile (p_{c}) counters (p) if and only if every strategy in (p_c) counters (p).

Let (p) be a (n-tuple), denote the countering operation as (f), then (f(p) = p_c).

Nash argued there must be a fixed point: (f(t) = t), since

1. (f) is defined as (f: S o 2^S).

2. the graph of (f) is closed.

Hence it comes naturally there is at least one fixed point by Kakutani's fixed point theorem.