Editing Nash equilibrium (section)

== Occurrence ==
If a game has a [[unique (mathematics)|unique]] Nash equilibrium and is played among players under certain conditions, then the NE strategy set will be adopted. Sufficient conditions to guarantee that the Nash equilibrium is played are:
# The players all will do their utmost to maximize their expected payoff as described by the game.
# The players are flawless in execution.
# The players have sufficient intelligence to deduce the solution.
# The players know the planned equilibrium strategy of all of the other players.
# The players believe that a deviation in their own strategy will not cause deviations by any other players.
# There is [[common knowledge (logic)|common knowledge]] that all players meet these conditions, including this one. So, not only must each player know the other players meet the conditions, but also they must know that they all know that they meet them, and know that they know that they know that they meet them, and so on.

=== Where the conditions are not met ===
Examples of [[game theory]] problems in which these conditions are not met:
# The first condition is not met if the game does not correctly describe the quantities a player wishes to maximize. In this case there is no particular reason for that player to adopt an equilibrium strategy. For instance, the prisoner's dilemma is not a dilemma if either player is happy to be jailed indefinitely.
# Intentional or accidental imperfection in execution. For example, a computer capable of flawless logical play facing a second flawless computer will result in equilibrium. Introduction of imperfection will lead to its disruption either through loss to the player who makes the mistake, or through negation of the [[common knowledge (logic)|common knowledge]] criterion leading to possible victory for the player. (An example would be a player suddenly putting the car into reverse in the [[game of chicken]], ensuring a no-loss no-win scenario).
# In many cases, the third condition is not met because, even though the equilibrium must exist, it is unknown due to the complexity of the game, for instance in [[Chinese chess]].<ref>T. L. Turocy, B. Von Stengel, ''[http://www.cdam.lse.ac.uk/Reports/Files/cdam-2001-09.pdf Game Theory]'', copyright 2001, Texas A&M University,  London School of Economics, pages 141-144. {{Citation needed span|text=Nash proved that a perfect NE exists for this type of finite [[extensive form game]]|date=April 2010}} – it can be represented as a strategy complying with his original conditions for a game with a NE. Such games may not have unique NE, but at least one of the many equilibrium strategies would be played by hypothetical players having perfect knowledge of all {{Citation needed span|text=10<sup>150</sup> [[game-tree complexity|game trees]]|date=April 2010}}.</ref> Or, if known, it may not be known to all players, as when playing [[tic-tac-toe]] with a small child who desperately wants to win (meeting the other criteria).
# The criterion of common knowledge may not be met even if all players do, in fact, meet all the other criteria. Players wrongly distrusting each other's rationality may adopt counter-strategies to expected irrational play on their opponents’ behalf. This is a major consideration in "[[Game of chicken|chicken]]" or an [[arms race]], for example.

=== Where the conditions are met ===
In his Ph.D. dissertation, John Nash proposed two interpretations of his equilibrium concept, with the objective of showing how equilibrium points can be connected with observable phenomenon.

{{Blockquote|(...) ''One interpretation is rationalistic: if we assume that players are rational, know the full structure of the game, the game is played just once, and there is just one Nash equilibrium, then players will play according to that equilibrium''. }}

This idea was formalized by R. Aumann and A. Brandenburger, 1995, ''Epistemic Conditions for Nash Equilibrium'', Econometrica, 63, 1161-1180 who interpreted each player's mixed strategy as a conjecture about the behaviour of other players and have shown that if the game and the rationality of players is mutually known and these conjectures are commonly known, then the conjectures must be a Nash equilibrium (a common prior assumption is needed for this result in general, but not in the case of two players. In this case, the conjectures need only be mutually known).

A second interpretation, that Nash referred to by the mass action interpretation, is less demanding on players:

{{Blockquote|[i]t is unnecessary to assume that the participants have full knowledge of the total structure of the game, or the ability and inclination to go through any complex reasoning processes. ''What is assumed is that there is a population of participants for each position in the game, which will be played throughout time by participants drawn at random from the different populations. If there is a stable average frequency with which each pure strategy is employed by the ''average member'' of the appropriate population, then this stable average frequency constitutes a mixed strategy Nash equilibrium.''}}

For a formal result along these lines, see Kuhn, H. and et al., 1996, "The Work of John Nash in Game Theory", ''Journal of Economic Theory'', 69, 153–185.

Due to the limited conditions in which NE can actually be observed, they are rarely treated as a guide to day-to-day behaviour, or observed in practice in human negotiations. However, as a theoretical concept in [[economics]] and [[evolutionary biology]], the NE has explanatory power. The payoff in economics is utility (or sometimes money), and in evolutionary biology is gene transmission; both are the fundamental bottom line of survival. Researchers who apply games theory in these fields claim that strategies failing to maximize these for whatever reason will be competed out of the market or environment, which are ascribed the ability to test all strategies. This conclusion is drawn from the "[[Nash equilibrium#Stability|stability]]" theory above. In these situations the assumption that the strategy observed is actually a NE has often been borne out by research.<ref>J. C. Cox, M. Walker, ''[http://excen.gsu.edu/jccox/research/learnplay.pdf Learning to Play Cournot Duoploy Strategies] {{Webarchive|url=https://web.archive.org/web/20131211182058/http://excen.gsu.edu/jccox/research/learnplay.pdf |date=2013-12-11}}'', copyright 1997, Texas A&M University,  University of Arizona, pages 141-144</ref>