Imagine you and your best friend decide to commit a crime. The police catch you and put you into two separate rooms. They give you the option to confess, and if your friend doesn't confess, you go free. If your friend confesses also, you will get four years in prison. And if none of you confess, you will spend only one year in jail. But if you don't confess, and your friend does confess, you will be locked away for ten years. What will you decide? What about your friend? Most likely, you will both confess, as confessing is a Nash equilibrium.
This fascinating concept in game theory offers insight into how individuals make choices that affect both themselves and their opponents. From the definition of Nash equilibrium to understanding how to find it, and exploring its applications in game theory, this article will provide a comprehensive and engaging look at this intriguing topic. So get ready to dive into strategic decision-making and learn all about Nash Equilibrium!
Nash Equilibrium Definition
Nash equilibrium is a concept in game theory where each player in a game makes the best decision for themselves based on the choices of the other players. In other words, it is a situation where no player can improve their outcome by changing their strategy while the other players keep their strategies constant.
Nash equilibrium definition refers to a situation in which every player in a competitive game may maximize their result depending on the choices made by the other players. The Nash equilibrium is a part of the game theory, which aims to model economic behaviors that maximize outcomes for each affected economic agent.
Nash equilibrium is a concept in game theory that describes a situation where every player in a game chooses the best strategy for themselves based on the strategies chosen by the other players, and no player can improve their outcome by unilaterally changing their strategy.
Nash equilibrium is achieved when no player is incentivized to deviate from their strategy, as this strategy maximizes their outcome. The player does not differ from their own strategy regardless of whether or not that player is aware of the strategies employed by the other players.
However, in decision-making, each player considers the other player's move to reach the Nash Equilibrium.
Imagine that two companies are in the market, selling their products for a specific price. Decreasing the price would mean that the company will be able to gain more market share. Each company will want to do the best that it can given the actions that are being taken by its rivals.
If one firm decreases the price, the other will also decrease the price. Assuming that both firms have the same cost they face, they will decrease the price to the point where they can no longer afford it.
At the point where the price can't decrease any longer, firms can't decrease the price further due to cost. Both companies will sell at the lowest price, as they don't have the incentive to sell at a higher price. They know that selling at a higher price means losing market share to the other company.
The Nash equilibrium is a concept used in economic theory to represent the idea that decision-making is a system of strategic interactions that depend on the actions taken by other participants.
The Nash equilibrium is vital as it does not have applications only in economics but across various social sciences such as psychology, sociology, law, and more. Nash equilibrium helps explain how people make decisions and interact with one another.
Nash Equilibrium vs Dominant Strategy
The main difference between Nash equilibrium vs. dominant strategy is that in Nash equilibrium, the players do not have the incentive to change their positions as changing their position would mean the player can create a worse outcome. On the other hand, the dominant strategy occurs when the player has one choice that produces better outcomes regardless of what the other player is doing.
A dominant strategy is a strategy that provides the highest outcomes for the player regardless of what the other player chooses to do.
- In the dominant strategy, the first player does the best he can, regardless of what the second player is doing. And the second player is doing the best he can regardless of what the first player is doing.
- In a Nash equilibrium, the first player is doing the best he can, given what the second player is doing. And the second player is doing the best he can given what the first player is doing.
While the Nash equilibrium leads to a dominant strategy, a dominant strategy doesn't always lead to a Nash equilibrium. That's because only in some games do all players have a dominant strategy.
To find out more about dominant strategies and how games with no dominant strategies for all players work. Click here:
- Dominant Strategy.
How to Find Nash Equilibrium?
To find the Nash equilibrium in a game, you need to determine the best strategy for each player given the strategies chosen by the other players. There are different methods to find Nash equilibrium depending on the type of game you are analyzing.
For example, in a game with a finite number of strategies and players, you can use a mathematical approach called the elimination method. You eliminate any strategy that is always worse than another strategy and iteratively eliminate it until you reach a point where no strategy can be eliminated. The remaining strategies represent the Nash equilibrium.
Alternatively, you can use a graphical method called the payoff matrix, where you create a table representing the possible outcomes of the game for each combination of strategies. Then, you can identify any dominant strategies where one strategy is always better than another and check if any strategy profiles have no dominant strategies. If so, that strategy profile represents the Nash equilibrium. Keep reading to find a step-by-step example of finding Nash equilibrium using a payoff matrix.
However, for more complex games with infinite strategies or imperfect information, finding Nash equilibrium can be more challenging and may require more advanced mathematical techniques.
Nash Equilibrium Example
One of the most common examples of Nash Equilibrium is the prisoner's dilemma. The prisoner's dilemma is a game theory example where two prisoners have been arrested and are presented with the opportunity to confess to the crime in separate rooms.
Let's assume that Bob and John went on to commit a crime. They stole some jewelry from a rich neighborhood. The police quickly saw them due to the surveillance camera in the area.
After being caught, both Bob and John are brought to the police station. They are set in separate rooms and have no chance of cooperating. The investigator gives them the following opportunities to both of them:
If one of them confesses while the other doesn't, the one who confesses gets 1 year in prison, while the other gets 8 years in prison.
If both of them confess, they get 4 years in prison.
If none of them confesses, they get 2 years in prison.
Fig. 1 - Nash equilibrium
Figure 1 is known as the payoff matrix, showing the outcomes for each player, in this case, Bob and John.
The outcomes of Bob at each matrix are shown on the right, whereas the outcomes of John are shown on the left.
So what decision will both make? When weighing options, it is helpful to compare outcomes while assuming the other player's choice, do this for each option the other player can make.
Assuming Bob will confess, John is choosing between confess (4 years) and not confess (8 years). In this scenario, John will choose to confess, as it is the better outcome for him.Assuming Bob will not confess, John is choosing between confess (1 year) and not confess (2 years). In this scenario, John will choose to confess, as it is the better outcome for him.If they collaborate and decide that they are on this together, they will not confess, and both will get only two years in prison.
However, they are both in separate rooms, unaware of what the other one will choose. So the best choice for both individuals would be to confess. That's because if Bob doesn't confess, he risks being taken advantage of by John and gets 10 years. The same problem is present for John.
Both of them decide to confess, and both get 4 years in prison each.
After all, it's better to confess and risk 4 years in prison than not confess and risk 8 years in prison, which is double the time.
Nash Equilibrium Payoff Matrix Example
The Nash equilibrium can be found by using the payoff matrix, which illustrates each company's outcomes for the game they play.
Fig. 2 - Nash equilibrium payoff matrix
Figure 2 shows the payoff matrix of two companies that choose to advertise or not.
Company 1 knows that company 2's best choice is to advertise as it gives the highest earnings to company 2.
Company 2 also knows that company 1 will choose to advertise as it gives the highest earnings to company 1.
Both of these companies will choose to advertise. The Nash equilibrium occurs at the underlined outcome.
The overall Nash equilibrium of this game is Advertise - Advertise
Nash Equilibrium Game Theory
Nash equilibrium game theory is a concept that illustrates how players in a non-cooperative game do not have any incentive to deviate from their chosen strategy.
In game theory, when the Nash equilibrium occurs, no players have any incentive, meaning they don't gain any additional benefit if they deviate from their strategy.
There may be more than one Nash equilibrium in a game, or there may be none.
The idea of Nash equilibrium is one of the cornerstones around which game theory is built.
- It analyzes the players' behaviors and how they interact with one another to reach the greatest possible results.
- It is also possible to forecast the choices the players will make if they are making decisions simultaneously and if the choices one makes consider the other players.
Nash Equilibrium is a concept developed by the prominent mathematician John Nash after which the concept is named. John Nash received the Nobel prize for his Nash Equilibrium theory, the application of which did not just benefit economics but other social sciences as well.
John Nash suffered from a mental illness known as Schizophrenia. However, this did not prevent John Nash from coming up with the theorem that led to the foundation of game theory.
There is a movie about John Nash called "A Beautiful Mind," and we suggest you watch it.
Cournot and Nash Equilibrium
The concept of Nash Equilibrium is not new, however. The first introduction of such a concept was made in the early 19th century when Cournot aimed to explain how oligopolistic firms choose their output to maximize their profit.
An oligopoly refers to a market where a few firms dominate the market. Check out our explanation of oligopoly to refresh your knowledge of it. It covers all aspects of oligopoly!
According to Cournot's model, firms compete against one another to determine how much output to generate so that each can maximize its profits. The most optimal output for any given company is contingent on the outputs of the other companies.
A Cournot equilibrium, which is also a Nash equilibrium, takes place when each company's output is such that it maximizes its profits given the output of the other companies in the market.
However, Cournot did not apply the concept in other contexts or attempt to define it comprehensively.
Cournot Model helps explain how a dominant firm competes in a market when only a few other dominant companies exist. It will help you learn a great deal about relevant markets and how competition occurs. Don't miss it!
Nash Equilibrium - Key takeaways
- Nash equilibrium occurs when each player is doing their best based on the other competitive party's actions.
- In game theory, when the Nash equilibrium occurs, no players have any incentive; that is, they don't gain any additional benefit if they deviate from their strategy.
- A Cournot equilibrium, which is also a Nash equilibrium, takes place when each company's output is such that it maximizes its profits given the output of the other companies in the market.
- A dominant strategy is a strategy that provides the highest outcomes for the player regardless of what the other player chooses to do.
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