Explain the concept of Nash equilibrium in game theory.

Nash equilibrium is a solution concept in game theory where each player's strategy is optimal given the other player's strategy.

In game theory, a Nash equilibrium is a situation where each player's strategy is optimal given the other player's strategy. In other words, no player can improve their outcome by changing their strategy, assuming the other player's strategy remains the same. This concept is named after John Nash, who introduced it in his 1950 paper "Non-Cooperative Games".

To find a Nash equilibrium, we must first identify all possible strategies for each player and their corresponding payoffs. Then, we must determine if there is a combination of strategies where neither player has an incentive to deviate from their chosen strategy. If such a combination exists, it is a Nash equilibrium.

For example, consider the classic game of "Prisoner's Dilemma". In this game, two suspects are arrested and held in separate cells. Each suspect is given the option to confess or remain silent. If both confess, they each receive a 5-year sentence. If one confesses and the other remains silent, the one who confesses receives a 1-year sentence while the other receives a 10-year sentence. If both remain silent, they each receive a 2-year sentence.

The payoffs for each player can be represented in a matrix:

| | Confess | Remain Silent |
|-------|---------|---------------|
| Confess| -5,-5 | -1,-10 |
| Silent| -10,-1 | -2,-2 |

In this game, there is a unique Nash equilibrium where both players confess. If one player were to switch to remaining silent, the other player would have an incentive to confess and receive a shorter sentence. Therefore, both players confess and receive a 5-year sentence.

Overall, Nash equilibrium is a useful concept in game theory for predicting the outcome of strategic interactions between rational players.