# Nash Equlibrium

The most important solution concept in Game Theory is that of the **Nash Equilibrium**, discovered in 1950 by the mathematician John Nash (1928 - 2015). A Nash Equilibrium in a game is a pair of strategies that, combined, yield an outcome with certain properties that make it natural to say that it is a solution for the game at hand.

The definition of a Nash Equilibrium relies on that of a **Best Response**. Player one's strategy X is a best response to player two's strategy A if, for all other strategies Y that player one could choose, the payoff for X vs. A is at least as great as the payoff for Y vs. A. In other words, player one cannot do better by unilaterally changing their strategy.

To see an example of Best Responses in action, consider this simple game based on the classic "Gift of the Magi" fable. Short on funds, Della decides to sell her hair in order to buy her lover Jim a chain for his watch. Unbeknownst to her, Jim decides to sell his watch in order to buy Della a collection of combs. The couple are left with gifts that neither can use (please, set aside the true moral of "Gift of the Magi" for just a moment).

Jim's best response to Della selling her hair is to keep his watch, and his best response to Della not selling her hair is to sell his watch to buy the combs.

Della's best response to Jim selling his watch is to keep her hair, and her best response to Jim not selling his watch is to sell her hair to buy the chain.

Two of these strategies are best responses to each other: Della sells her hair and Jim keeps his watch. Likewise, Della keeping her hair and Jim selling his watch are also best responses to each other. The outcomes that come from these strategy pairs are asymmetric: Jim prefers one, while Della prefers another. This can help explain how the miss-coordination in the story came about.

These pairs of strategies are also special in another regard: they are each a Nash Equilibrium of the Gift of the Magi game. **A Nash Equilibrium is a pair of strategies that are best responses to each other.**

Consider the Stag Hunt game. The best response to one player hunting Stag is for the other to hunt Stag as well, while the best response to one player hunting Hare is for the other to hunt Hare as well. Like the Gift of the Magi, the Stag Hunt has two Nash Equilibrium pairs: both hunt Stag, or both hunt Hare. And, also like Gift of the Magi, the differences between those two pairs explain the unique situation that the game represents. In the Stag Hunt, one Nash Equilibrium leads to a better outcome for all, but the other is risk dominant.

Consider the Prisoner's Dilemma. It has one Nash Equilibrium: both players Betray each other. Even though both players would each be better off if they both played Quiet, that pair of strategies is *not* a Nash Equilibrium, as the best response to Quiet is Betray. This helps explain why mutual cooperation is unattainable. Sure, the outcomes are better for each player, but the pair of strategies are not in equilibrium, as each player can do better by switching to Betray (with the unfortunate result that mutual Betrayal makes both players worse off).

Consider, finally, the true moral of the Gift of the Magi. By sacrificing their only possessions in order to buy each other something special, Della and Jim realize the true meaning of love, which is truly the best outcome for each of them. In Game Theory terms, this means that Jim's best response to Della selling her hair is to sell his watch to buy combs, and Della's best response to Jim selling his watch is to sell her hair to buy the chain. These strategies are best responses to each other, making the pair a Nash Equilibrium. Game Theory captures how priceless their love really is.

The reason these pairs of strategies are called an Equilibrium is because they are in balance with respect to the forces of utility maximization. No player can increase their expected utility by deviating, given the choice of the other player. If a player is *not* playing a strategy that is part of a Nash Equilibrium (if they are *not* playing a best response), then they could do better by switching strategies. The force of maximizing utility pulls them to a new strategy, and hence a new outcome.