In game theory, the Nash equilibrium is a concept that describes a stable state in which no player has an incentive to change their strategy. It is named after mathematician John Nash, who won the Nobel Prize in Economics for his work on the concept.
In a game with multiple players, each player has a set of strategies that they can choose from, and each strategy has a corresponding payoff. The Nash equilibrium is reached when each player is playing the best strategy for themselves, given the strategies of the other players. In other words, no player can improve their payoff by switching to a different strategy, because doing so would not change the strategies of the other players.
For example, consider a game in which two players, Alice and Bob, can choose to either cooperate or defect. If both players cooperate, they both receive a payoff of 2. If one player defects and the other cooperates, the defector receives a payoff of 3 and the cooperator receives a payoff of 1. If both players defect, they both receive a payoff of 0. In this game, the Nash equilibrium occurs when both players defect, because neither player has an incentive to change their strategy. If Alice defects and Bob cooperates, Alice receives a higher payoff than if they both cooperate. If Bob defects and Alice cooperates, Bob receives a higher payoff than if they both cooperate. Therefore, the only stable state is for both players to defect.
Nash Equilibrium Game theory examples
Here are a few examples of how the Nash equilibrium can be applied in game theory:
The prisoner’s dilemma
In the prisoner’s dilemma game, two prisoners must decide whether to confess or remain silent. If both prisoners remain silent, they both receive a sentence of one year in jail. If one prisoner confesses and the other remains silent, the confessor goes free and the other prisoner receives a sentence of 10 years in jail. If both prisoners confess, they both receive a sentence of five years in jail.
The Nash equilibrium in this game occurs when both prisoners confess, because neither player has an incentive to change their strategy. If one player confesses and the other remains silent, the confessor receives a better payoff than if they both remain silent. Therefore, the only stable state is for both players to confess.
This game is often used to illustrate the concept of cooperation, as it shows that both prisoners are better off if they cooperate and remain silent, even though they could potentially receive a higher individual payoff by defecting and confessing. However, if one prisoner defects and confesses, it can lead to a suboptimal outcome for both prisoners.
The chicken game
In the chicken game, two drivers must decide whether to swerve or stay on course. If both drivers stay on course, they both receive a payoff of 0. If one driver swerves and the other stays on course, the swerving driver receives a payoff of 1 and the other driver receives a payoff of 3. If both drivers swerve, they both receive a payoff of 2.
The Nash equilibrium in this game occurs when both drivers stay on course, because neither player has an incentive to change their strategy. If one player swerves and the other stays on course, the swerving player receives a lower payoff than if they both stay on course. Therefore, the only stable state is for both players to stay on course.
This game is often used to illustrate the concept of brinkmanship, as it shows that both players are better off if they cooperate and stay on course, even though they could potentially receive a higher individual payoff by defecting and swerving. However, if one player defects and swerves, it can lead to a suboptimal outcome for both players.
The stag hunt
In the stag hunt game, two hunters must decide whether to hunt a stag or a rabbit. If both hunters hunt a stag, they both receive a payoff of 3. If one hunter hunts a stag and the other hunts a rabbit, the stag hunter receives a payoff of 0 and the rabbit hunter receives a payoff of 1. If both hunters hunt a rabbit, they both receive a payoff of 2.
The Nash equilibrium in this game occurs when both hunters hunt a stag, because neither player has an incentive to change their strategy. If one player hunts a stag and the other hunts a rabbit, the stag hunter receives a lower payoff than if they both hunt a stag. Therefore, the only stable state is for both players to hunt a stag.
This game is often used to illustrate the concept of cooperation, as it shows that both players are better off if they cooperate and hunt a stag, even though they could potentially receive a higher individual payoff by defecting and hunting a rabbit. However, if one player defects and hunts a rabbit, it can lead to a suboptimal outcome for both player.
Imagine that you and a friend are trying to decide where to go for dinner. You both have a list of restaurants that you would prefer to go to, ranked in order of preference. You agree to choose the restaurant that both of you prefer the most, as long as it is on both of your lists. If there is no restaurant that is on both lists, you will choose the restaurant that each of you prefers the most, even if the other person doesn’t like it as much.
In this situation, the Nash equilibrium occurs when both of you choose the restaurant that is on both of your lists and is the highest on both lists. Neither of you has an incentive to change your strategy, because going to a different restaurant would result in a lower payoff.
For example, if your list is:
- Italian restaurant
- Sushi restaurant
- Mexican restaurant
And your friend’s list is:
- Italian restaurant
- Steakhouse
- Sushi restaurant
The Nash equilibrium would be for both of you to go to the Italian restaurant, because it is the highest on both lists and is on both lists. If you both choose the Italian restaurant, you will both receive the highest payoff. If one of you chooses a different restaurant, you will both receive a lower payoff.
There are a few ways to determine if a game has a Nash equilibrium:
Solve for the best response of each player: For each player in the game, determine what their best response would be to each of the possible strategies of the other players. If there is a strategy that is a best response to every other strategy, then that strategy is a Nash equilibrium.
Look for a pure strategy equilibrium: A pure strategy equilibrium occurs when each player is playing a single strategy, rather than a combination of strategies. If a game has a pure strategy equilibrium, then it also has a Nash equilibrium.
Look for a mixed strategy equilibrium: A mixed strategy equilibrium occurs when each player is playing a combination of strategies, rather than a single strategy. To find a mixed strategy equilibrium, you can use a mathematical technique called linear programming.
It is important to note that not all games have a Nash equilibrium. Some games may have multiple Nash equilibria, while others may not have any. It is also possible for a game to have a Nash equilibrium that is not a Pareto optimal outcome, meaning that it is not the best outcome for all players.
Can a game have no Nash equilibrium?
Yes, it is possible for a game to have no Nash equilibrium. This can occur if there is no strategy that is a best response to every other strategy, or if a game has multiple pure strategy equilibria or mixed strategy equilibria.
Can you have 3 Nash equilibriums?
It is possible for a game to have multiple Nash equilibria. For example, a game with three players could have three different Nash equilibria, one for each player.
What is the opposite of a Nash equilibrium?
The opposite of a Nash equilibrium is a state in which at least one player has an incentive to change their strategy. In other words, a state in which one or more players can improve their payoff by switching to a different strategy. This state is called an unstable equilibrium, because it is not a stable state and players will have an incentive to change their strategies.
