# Introduction to game theory: Nash equilibrium

John Nash is the most famous game theorist, his life story was popularized in the book and movie A Beautiful Mind. The concept of a Nash equilibrium is also named after him.

A Nash equilibrium is a situation where each player has chosen a strategy which has given them the best possible outcome given what the other players have chosen to to.

If the game was played again then no player could improve by being the only person to change their strategy.

As an example, imagine there are two women in a bar, one is clearly more attractive than the other. Two guys walk in and have to figure out which of the women they will try to chat up. It is easy for the two guys to reach a Nash equilibrium as long as they both know which of them is the most attractive to women.

The more attractive guy knows that he can get his best outcome by hitting on the most attractive woman. The other guy knows that if he also goes for the most attractive woman then he will lose out, so he goes for the other woman.

They have reached a Nash equilibrium because both of them has played the best strategy they could given what they know about the other player’s strategy.

If the two guys both thought that they were the most attractive to women then they wouldn’t reach a Nash solution as they would both go for the more attractive woman. If the situation was repeated then maybe one of them would start to realize that they are regularly losing out and change their strategy. Once they had realized this then they would move into an equilibrium.

Today’s takeaway: Strategies can be stable if no-one can do better by unilaterally changing their strategy.

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