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 \(f: S \to 2^S\).

2. the graph of \(f\) is closed.

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