Nash Equilibrium: Definition and Examples of Nash Equilibrium

Nash equilibrium is a concept in game theory in which every participant in a noncooperative game can optimize their outcome based on the other players’ decisions. Nash equilibrium is achieved in a game when no player has any incentive for deviating from their own strategy, even if they know the other players’ strategies.

In economic theory, the Nash equilibrium is used to illustrate that decision-making is a system of strategic interactions based on the actions of other players. It can be used to model economic behavior to predict the best response to any given situation.

The Nash equilibrium concept is named after the American mathematician John Nash who defined the theory and won the Nobel Prize for his work.

The theory of the pure-strategy Nash equilibrium, or finding the optimal strategy for each player in a game, was developed as a concept before John Nash fully defined it. It would also go on to be iterated on and developed to be more accurate and more widely applicable.

• The Cournot equilibrium: Antione Augustin Cournot created a similar theorem in 1838 called the theory of oligopoly, or the Cournot equilibrium. This theorem used business firms to explore how the ideal output for each firm to maximize their profit is dependent on the output of the other.
• Mixed-strategy equilibrium: In 1944, game theorists John von Neumann and Oskar Morgenstern introduced the concept of a mixed-strategy equilibrium, which proposes that a Nash equilibrium exists for a finite game with a specific set of actions with players choosing probability distributions over pure strategies or specific strategy profiles. Their theorem was largely constrained to simple games, particularly two-player games with rational players and zero-sum games.
• John Nash’s equilibrium: In 1951, John Nash published multiple articles while he was studying at Princeton University that outlined his Nash equilibria theory of games, including “Equilibrium Points in N-person Games” in National Academy of Sciences and “Non-Cooperative Games” in Annals of Mathematics. In these publications, he was able to prove that any game with finite actions must have at least one mixed-strategy Nash equilibrium, or multiple mixed-strategy Nash equilibria. Unlike previous theorems that were restricted to specific types of games or two-person games, Nash’s theorem applied to a wider class of games and larger number of players (called n-player games) using the Kakutani fixed-point theorem.
• Later development: The Nash equilibrium as defined by John Nash was a major development in game theoretics and would later be refined by other theorists. In 1967, John Harsanyi developed Bayesian game models, in which players had incomplete information about the other players. Then, in 1974, Robert Aumann introduced correlated equilibrium, in which communication between players or a mediator was modeled into the normal form game model, which depicts payoff functions as a payoff matrix with different strategies and expected payoffs. In 1975, Reinhard Selten showed that the normal-form model generated too many irrational equilibria as solution concepts, but David M. Kreps and Robert Wilson solved this problem in 1982 with their definition of sequential equilibrium, which models both strategies and beliefs for the set of players.

These classic examples from the fields of social sciences and economics illustrate how optimal game strategies are dependent on other players’ actions.

• The prisoner’s dilemma: In this hypothetical game, two criminals are taken to separate rooms and—without communicating with each other—must make the decision to testify against their partner and convict them or remain silent. If they both betray each other, they each serve two years in prison. If prisoner A testifies against prisoner B but prisoner B remains silent, prisoner A will be released and prisoner B stays in prison for ten years. If both stay silent, they both serve one year in prison. The Nash equilibrium or dominant strategy in this special case is for both players to testify against the other to get only two years in prison, even if mutual cooperation (staying silent) yields a better result, since, if one prisoner stays silent and the other betrays their partner, one player will have a much worse result. This is an example of a pure-strategy Nash equilibrium in which there is one strategy profile for the higher payoff or optimal result.
• Battle of the sexes: A man and a woman are hoping to attend an event together, but they must choose between a prizefight or a ballet. They cannot communicate where they will go beforehand, and the man in this story would prefer to go to the prizefight while the woman in this scenario would like to go to the ballet. This game has two unique Nash equilibrium strategies: one in which they both go to the prizefight and one in which they both go to the ballet.
• Matching pennies: This game is played with two players who each must secretly turn a penny to heads or tails and then reveal their penny. If the pennies are matching heads or tails, then player A keeps both pennies. If the pennies don’t match, player B keeps both pennies. This is an example of a game that has no Nash equilibrium, as the loss or gain of each player is directly correlated to the loss or gain of the other.

To find Nash Equilibrium, share each player’s strategy with every other player. If the players do not change their strategies, you have found the Nash equilibrium. Alternatively, use tables to map each set of strategies.

For example, in the battle of the sexes scenario, draw a table with two rows and two columns. The vertical axis represents the woman’s options: ballet or prizefight. The horizontal axis represents the man’s choices: also ballet or prizefight. In the top left quadrant, they both choose ballet (Nash equilibrium). In the top-right quadrant, the woman chooses ballet, but the man chooses prizefight. In the bottom left quadrant, the woman chooses prizefight and the man chooses ballet. In the bottom right quadrant, they both choose prizefight (Nash equilibrium, again). This table illustrates there are two possible Nash equilibria: Both players choose ballet, or both players choose prizefight.

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