Nash Equilibrium

A **Nash Equilibrium** (NE) is a vector of strategies (pure or mixed), one per player, such that no player can improve her own payoff by unilaterally changing strategies. For an *N* player game , a NE is , such that,

- for all .

Note that it is equivalent to the following definition:

- for all .

Some nice examples of Nash Eq. are:

- Coordination game: it is a NE to all drive on the left side of roads, or all drive on right side of roads (there is one more... :)
- Prisoner's dilemma: it is a NE to not cooperate in prisoner's dilemma.

A game is finite if it has a finite number of players and each player has a finite number of actions (or pure strategies).

I watched the movie "A Beautiful Mind" last night where Nash's theory was explained as the best payoff for all participants is to pursue the brunettes who make up 4 out of the 5 available women, rather than disappoint and offend all the brunettes by all going for the blonde when only, theoretically, just one could win the blonde. It is also stated in the movie that the best payoff is to consider BOTH what is best for oneself AND what is best for the group. This would be the same as balancing Whole and Part, Female and Male, and Many and One.

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