Editing Nash equilibrium (section)

==Definitions==

===Nash equilibrium===
A strategy profile is a set of strategies, one for each player. Informally, a strategy profile is a Nash equilibrium if no player can do better by unilaterally changing their strategy. To see what this means, imagine that each player is told the strategies of the others. Suppose then that each player asks themselves: "Knowing the strategies of the other players, and treating the strategies of the other players as set in stone, can I benefit by changing my strategy?"

For instance if a player prefers "Yes", then that set of strategies is not a Nash equilibrium. But if every player prefers not to switch (or is indifferent between switching and not) then the strategy profile is a Nash equilibrium. Thus, each strategy in a Nash equilibrium is a [[best response]] to the other players' strategies in that equilibrium.<ref name="preliminaries">{{cite web |last=von Ahn |first=Luis |title=Preliminaries of Game Theory |url=http://www.scienceoftheweb.org/15-396/lectures_f11/lecture09.pdf |website=Science of the Web |url-status=dead |archive-url=https://web.archive.org/web/20111018035629/http://scienceoftheweb.org/15-396/lectures_f11/lecture09.pdf |archive-date=2011-10-18 |access-date=2008-11-07}}</ref>

Formally, let <math>S_i</math> be the set of all possible strategies for player <math>i</math>, where <math>i = 1, \ldots, N</math>. Let <math>s^* = (s_i^*, s_{-i}^*)</math> be a strategy profile, a set consisting of one strategy for each player, where <math>s_{-i}^*</math> denotes the <math>N - 1</math> strategies of all the players except <math>i</math>. Let <math>u_i(s_i, s_{-i}^*)</math> be player ''i''{{'}}s payoff as a function of the strategies. The strategy profile <math>s^*</math> is a Nash equilibrium if 
<math display="block">u_i(s_i^*, s_{-i}^*) \geq u_i(s_i, s_{-i}^*)\ \text{for all}\ s_i \in S_i.</math>

A game can have more than one Nash equilibrium. Even if the equilibrium is unique, it might be ''weak'': a player might be indifferent among several strategies given the other players' choices. It is unique and called a ''strict Nash equilibrium'' if the inequality is strict so one strategy is the unique best response: 
<math display="block">u_i(s_i^*, s_{-i}^*) > u_i(s_i, s_{-i}^*)\ \text{for all}\ s_i \in S_i, s_i \neq s_i^*.</math>
 
The strategy set <math>S_i</math> can be different for different players, and its elements can be a variety of mathematical objects. Most simply, a player might choose between two strategies, e.g. <math>S_i = \{\text{Yes}, \text{No}\}.</math> Or the strategy set might be a finite set of conditional strategies responding to other players, e.g. <math>S_i = \{\text{Yes} \mid p = \text{Low}, \text{No} \mid p = \text{High}\}.</math> Or it might be an infinite set, a continuum or unbounded, e.g. <math>S_i = \{\text{Price}\}</math> such that <math>\text{Price}</math> is a non-negative real number. Nash's existing proofs assume a finite strategy set, but the concept of Nash equilibrium does not require it.