Editing Nash equilibrium (section)

=== Proof using the Kakutani fixed-point theorem ===
Nash's original proof (in his thesis) used Brouwer's fixed-point theorem (e.g., see below for a variant). This section presents a simpler proof via the [[Kakutani fixed-point theorem]], following Nash's 1950 paper (he credits [[David Gale]] with the observation that such a simplification is possible).

To prove the existence of a Nash equilibrium, let <math>r_i(\sigma_{-i})</math> be the best response of player i to the strategies of all other players.
<math display="block"> r_i(\sigma_{-i}) = \mathop{\underset{\sigma_i}{\operatorname{arg\,max}}} u_i (\sigma_i,\sigma_{-i}) </math>

Here, <math>\sigma \in \Sigma</math>, where <math>\Sigma = \Sigma_i \times \Sigma_{-i}</math>, is a mixed-strategy profile in the set of all mixed strategies and <math> u_i </math> is the payoff function for player i. Define a [[set-valued function]] <math>r\colon \Sigma \rightarrow 2^\Sigma </math> such that <math>r = r_i(\sigma_{-i})\times r_{-i}(\sigma_{i}) </math>. The existence of a Nash equilibrium is equivalent to <math>r</math> having a fixed point.

Kakutani's fixed point theorem guarantees the existence of a fixed point if the following four conditions are satisfied.
# <math> \Sigma</math> is compact, convex, and nonempty.
# <math>r(\sigma)</math> is nonempty.
# <math>r(\sigma)</math> is [[Hemicontinuity|upper hemicontinuous]]
# <math>r(\sigma)</math> is convex.

Condition 1. is satisfied from the fact that <math>\Sigma</math> is a simplex and thus compact. Convexity follows from players' ability to mix strategies. <math>\Sigma</math> is nonempty as long as players have strategies.

Condition 2. and 3. are satisfied by way of Berge's [[maximum theorem]]. Because <math> u_i </math> is continuous and compact, <math> r(\sigma_i) </math> is non-empty and [[Hemicontinuity|upper hemicontinuous]].

Condition 4. is satisfied as a result of mixed strategies. Suppose <math> \sigma_i, \sigma'_i \in r(\sigma_{-i}) </math>, then <math> \lambda \sigma_i + (1-\lambda) \sigma'_i \in r(\sigma_{-i}) </math>. i.e. if two strategies maximize payoffs, then a mix between the two strategies will yield the same payoff.

Therefore, there exists a fixed point in <math> r </math> and a Nash equilibrium.<ref>{{cite book |last1=Fudenburg |first1=Drew |first2=Jean |last2=Tirole |title=Game Theory |publisher=MIT Press |year=1991 |isbn=978-0-262-06141-4 }}</ref>

When Nash made this point to [[John von Neumann]] in 1949, von Neumann famously dismissed it with the words, "That's trivial, you know. That's just a [[fixed-point theorem]]." (See Nasar, 1998, p.&nbsp;94.)