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