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 \(f: S \to 2^S\).
2) the graph of \(f\) is closed.
Hence it comes naturally there is at least one fixed point by Kakutani's fixed point theorem.