Nash equilibrium Totally Explained

Everything about Nash Equilibrium totally explained

In game theory, the Nash equilibrium (named after John Forbes Nash, who proposed it) is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his or her own strategy (for example, by changing unilaterally). If each player has chosen a strategy and no player can benefit by changing his or her strategy while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitute a Nash equilibrium. In other words, to be a Nash equilibrium, each player must answer negative to the question: "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?" Stated simply, Amy and Bill are in Nash equilibrium if Amy is making the best decision she can, taking into account Bill's decision, and Bill is making the best decision he can, taking into account Amy's decision. Likewise, many players are in Nash equilibrium if each one is making the best decision that they can, taking into account the decisions of the others. However, Nash equilibrium doesn't necessarily mean the best cumulative payoff for all the players involved; in many cases all the players might improve their payoffs if they could somehow agree on strategies different from the Nash equilibrium (eg. competing businessmen forming a cartel in order to increase their profits).

History

The concept of the Nash equilibrium (NE) in pure strategies was first developed by Antoine Augustin Cournot in his theory of oligopoly (1838). Firms choose a quantity of output to maximize their own profit. However, the best output for one firm depends on the outputs of others. A Cournot equilibrium occurs when each firm's output maximizes its profits given the output of the other firms, which is a pure-strategy NE. However, the modern game-theoretic concept of NE is defined in terms of mixed-strategies, where players choose a probability distribution over possible actions. The concept of the mixed strategy NE was introduced by John von Neumann and Oskar Morgenstern in their 1944 book The Theory of Games and Economic Behavior. However, their analysis was restricted to the very special case of zero-sum games. They showed that a mixed-strategy NE will exist for any zero-sum game with a finite set of actions. The contribution of John Forbes Nash in his 1951 article Non-Cooperative Games was to define a mixed strategy NE for any game with a finite set of actions and prove that at least one (mixed strategy) NE must exist.

Definitions

Informal definition

Informally, a set of strategies is a Nash equilibrium if no player can do better by unilaterally changing his or her strategy. As a heuristic, one can imagine that each player is told the strategies of the other players. If any player would want to do something different after being informed about the others' strategies, then that set of strategies isn't a Nash equilibrium. If, however, the player doesn't want to switch (or is indifferent between switching and not) then the set of strategies is a Nash equilibrium.
   The original uses of Nash solutions were limited to situations where players had restricted strategy choices (such as particular monetary values) and generally led to intuitive, and in fact enlightening, results.
   But it was soon discovered that this solution concept can have counter-intuitive consequences. Since the Nash equilibrium focuses on an individual's preferences given that the others keep their choices fixed, there can be Nash equilibria where, if players could coordinate, they'd all want to switch.
   Another example of a counter-intuitive Nash equilibrium is the following situation. Two people enter a room that has a large bomb in it. They each play the following strategy: "I will set off the bomb unless the other player burns all his money, and I'll burn all my money." They are both behaving optimally, given the other player's strategy. This solution is Nash. Consider the optimization of one player: Given that the other guy is planning to kill me if I don't burn my money, it's clearly optimal for me to burn my money. And given that the other player is planning to burn all his money, I lose nothing by committing to set off the bomb if he does not, so that committment is an optimal strategy choice. This type of counter-intuitive situation with incredible threats led to development of the "subgame perfect" Nash equilibrium solution concept.

Formal definition

Let

(S, f)

be a game, where

S

is the set of strategy profiles and

f

is the set of payoff profiles. Let

x_
ight) sigma^*_i(a)^2 > 0

where the last inequality follows since

sigma^*_i

is a non-zero vector. But this is a clear contradiction, so all the gains must indeed be zero. Therefore

sigma^*

is a Nash Equilibrium for

G

as needed.

Further Information

Get more info on 'Nash Equilibrium'.

External Link Exchanges

Do you know how hard it is to get a link from a large encyclopaedia? Well we're different and will prove it. To get a link from us just add the following HTML to your site on a relevant page:

<a href="http://nash_equilibrium.totallyexplained.com">Nash equilibrium Totally Explained</a>

Then simply click through this link from your web page. Our crawlers will verify your link, extract the title of your web page and instantly add a link back to it. If you like you can remove the words Totally Explained and embed the link in article text.
As long as your link remains in place, we'll keep our link to you right here. Please play fair - our crawlers are watching. Your site must be closely related to this one's topic. Any kind of spamming, dubious practises or removing the link will result in your link from us being dropped and, potentially, your whole site being banned.