What is Nash equilibrium?

In a 2-player static gameA static game is any interaction between 2 entities in which they both simultaneously make a decision about the outcome of their interaction., there can be many possible outcomes depending on the choices available to each player. For example, consider the famous game theory problem - the battle of the sexes.

There are four possible outcomes here, with each cell in the above table corresponding to the payoff for each player associated with one of the outcomes.

Before we jump into what Nash equilibrium is and how it can be found, we must first understand strategic uncertainty and the notion of rationalizability.

Rationalizability and strategic uncertainty

The concept of rationalizability encompasses the assumptions about what the players know about the game, what beliefs they have about each other, and how they respond to these beliefs. In other words, when talking about rationalizability, we assume that if a player’s action determines several outcomes that can occur in a game, then this player will select the action that leads to the outcome he most prefers.

This assumption that players behave according to these beliefs and their preferences lays the foundation for the game-theoretic analysis of interactions. However, it isn't enough to determine which specific outcome will occur. For example, in the battle of the sexes game whose standard form is depicted above, player 1 may rationally believe that player two will select Movie and thus select Movie in response. However, player two might rationally deduce that player 1 will select Opera and thus select Opera in response.

This leads to the strategy profileA strategy profile (sometimes called a strategy combination) is a set of strategies for all players which fully specifies all actions in a game. (Movie, Opera) being played, which has a payoff of 0 for both players, and this is not the best response for either player. This happens because rationalizability merely requires that the players' beliefs and behavior be consistent with common knowledge of rationality. It does not require these beliefs to be correct. This is called strategic uncertainty and can lead to very uncoordinated outcomes.

The concept of Nash equilibrium was introduced to solve strategic uncertainty and determine the best response behavior.

Nash equilibrium

Strategic uncertainty can be reduced by allowing communication between the players. For instance, if the players can communicate before a game starts, they can discuss how the game will be played out and what strategies each player will choose. By harmonizing the players' beliefs with their actual behavior, this strategic uncertainty can be resolved because players now choose their best response to the strategy chosen by the other player. In other words, the players' strategies are mutual best responses.

For a strategy profile to be a Nash equilibrium, neither player should gain any benefit by deviating from that outcome. For example, consider a setting in which the players meet before playing a game and agree on a strategy profile $s$ that should be played. The players each have an individual incentive to abide by the agreement only if each player’s prescribed strategy is the best response to the strategy of the other player. If, however, a player has some incentive to not abide by the agreement, they will choose a different strategy, and this won't be a Nash equilibrium.

Example

The example below demonstrates how we can find a Nash equilibrium using the definition introduced above. The best response of each player to each strategy of the other player is underlined.