Before we learn about pure strategy Nash Equilibrium, it’s necessary to understand the “Game Theory” and “Dominant Strategy.”

## What is a Game Theory?

Game theory is the study of strategic decision-making. It is often used in economics, political science, and psychology, as well as in other fields. Game theory attempts to identify the optimal decisions that people make in various situations.

Game theory is based on the assumption that people are rational and will always try to make the best choices for themselves. This means that they will take into account all of the available information and make the decision that will give them the most benefit.

Game theory can be used to analyze situations where there is competition, such as in business or politics. It can also be used to analyze situations where there is cooperation, such as in relationships or team sports.

Game theory is a powerful tool that can help us to understand the complex decisions that people make. It can help us to find the optimal solutions to various problems and to make better decisions in our own lives.

## What is Dominant Strategy?

A dominant strategy is a decision rule that is optimal for each player, given the other player’s decision. In-game theory, a dominant strategy is a strategy that is best for a player, regardless of what the other player does.

For example, suppose you are trying to decide whether to go to the movie or to stay at home. If you prefer the movie, then it doesn’t matter what the other person does – you are better off going to the movie. In this case, going to the movie is a dominant strategy.

There can be more than one dominant strategy in a game. For example, in the game of rock-paper-scissors, all three strategies (rock, paper, and scissors) are dominant. This is because no matter what the other person does, you can always do something better for yourself.

A dominant strategy is I am doing the best I can, no matter what you do, or you are doing the best you can no matter what I do.

The concept of a dominant strategy is important in game theory because it can be used to find the Nash equilibrium of a game. The Nash equilibrium is the point at which no player has an incentive to change their strategy, given the other player’s strategy.

## What is Nash Equilibrium in Economics?

In-game theory, the Nash equilibrium is a set of strategies for a game where no player has an incentive to change their strategy given the strategies of the other players. The Nash equilibrium is named after mathematician John Nash, who first described the concept in his 1950 paper “Equilibrium Points in n-Person Games”.

In-game theory, the Nash equilibrium is often used to analyze competition between different players or groups.

The Nash equilibrium is important because it provides a way to predict the outcome of a game without having to know the details of the players’ strategies. If all the players in a game are in Nash equilibrium, then the game will have a predictable outcome.

There are many different types of Nash equilibria, but the most common is the pure Nash equilibrium. In a pure Nash equilibrium, each player has a strategy that is best for them given the strategies of the other players.

There are also mixed Nash equilibria, where each player has a mix of strategies, and there is no single best strategy for any player. Mixed Nash equilibria are less common than pure Nash equilibria, but they can still be useful in predicting the outcome of a game.

## Nash Equilibrium definition

Nash equilibrium is a concept in game theory that suggests that in any given situation, each player will make the best possible decision for themselves, given the decisions of the other players. This means that no one player can improve their situation by making a different decision.

A Nash equilibrium is a situation in which opposing players in a game reach a mutual understanding and agreement about the optimal way to play the game. In other words, each player knows what the other is doing and they both know that they cannot improve their position by deviating from their current strategy.

Nash equilibrium is I am doing the best I can, given what you are doing or you are doing the best I can, given what I am doing now.

The Nash equilibrium is named after John Nash, a mathematician who developed the concept in the 1950s. It is widely used in game theory and economics to analyze situations in which there is competition or conflict.

The Nash equilibrium is often used to explain why people might cooperate in a Prisoner’s Dilemma game. In this game, two players are each given the choice to either cooperate or defect. If both players cooperate, they each receive a small payoff.

If one player defects while the other cooperates, the defector receives a large payoff while the co-operator receives nothing. If both players defect, they each receive a small payoff.

The Nash equilibrium occurs when both players know that the other is going to cooperate. In this case, it is in both players’ best interests to cooperate because they will receive a small payoff. However, if one player defects while the other cooperates, the defector will receive a large payoff while the co-operator will receive nothing. Therefore, it is not in either player’s best interests to defect.

The Nash equilibrium is a powerful tool for analyzing situations in which there is competition or conflict. It can be used to explain why people might cooperate in a Prisoner’s Dilemma game, and it can also be used to analyze other games and economic situations.

## Nash equilibrium with example

Let’s understand this concept with the help of an example. Look at the table given below. In the table we have two Firms; Firm A and Firm B with an option to advertise their product or not to advertise.

TABLE

So, If Firm A advertises and Firm B also advertises the payoffs are 10 and 5. Now, what is a payoff? The payoff is any value associated with a possible outcome.

If Firm A chooses to advertise and Firm B also chooses to advertise the payoff to Firm A will get a reward or will get a profit of 10 while Firm B will get a profit of 5. Similarly, if Firm A advertises and Firm B does not then the payoffs are 15 and 0.

We can also see that if Firm A chooses not to advertise and Firm B chooses to advertise the payoff to Firm A is 6 and the payoff to Firm B is 8. If both of them choose not to advertise their product then 10 is the reward that goes to Firms A and 2 is the reward that goes to Firm B.

Now the question is which of these Firms has a Dominant Strategy? For that, we need to consider their payoffs. Now, if we consider Firm A the payoffs that belong to Firm A is 10, 15, 6, and 10 all before the comma. All values after the comma represent the payoffs belonging to Firm B i.e., 5,0,8,2.

### Does Firm A have a Dominant Strategy?

So here while checking whether Firm A has a dominant strategy or not. We need to consider its payoffs so here, 10 and 15 are the payoffs when the Firm chooses to advertise. When Firm A chooses to advertise and not to advertise the payoffs are 6 and 10.

We can see that 10 and 15 are greater than 6 and 10 respectively which means the payoffs that Firm A gets when it chooses to advertise are greater than when it chooses not to advertise.

It also means Firm A has a dominant strategy in advertising because both 10 and 15 are greater than 6 and 10 respectively and no matter whether Firm B chooses Firm A will always choose to advertise because Firm A has a dominant strategy.

### Does Firm B have a Dominant Strategy?

Now let’s check whether Firm B has a dominant strategy or not? So, considering the payoffs, first of all, we have to consider the payoffs of Firm B that is 5, 0, 8, and 2.

When Firm A chooses to advertise and Firm B also chooses to advertise then 5 is the payoff or 5 is the reward that goes to Firm B and similarly to all the other cases.

5 and 8 are the payoffs that the Firm gets when the Firm chooses to advertise. When the Firm chooses not to advertise, it’s 5 and 2. We can see that 5 is greater than 0 and 8 are greater than 2 which means Firm B also has a Dominant Strategy in advertising.

### Outcome

The result will be that both of these Firms will choose to advertise and Firm A will get a payoff of 10 and Firm B will get a payoff of 5. Now the logical outcome of this game is that both of these Firms will advertise because Firm A is doing the best it can, given what Firm B is doing and Firm B is also doing its best given what Firm A is doing

## Pure Strategy Nash Equilibrium

The pure strategy Nash equilibrium is a game theory concept that describes a situation in which each player in a game has chosen the best possible strategy for themselves, given the strategies of the other players. In other words, each player has no incentive to change their strategy, because doing so would only make them worse off.

The pure strategy Nash equilibrium is an important concept in game theory because it provides a way to analyze games in which there are multiple possible outcomes. It can be used to determine which strategies are likely to lead to a particular outcome and to predict how players will respond to changes in the game.

## Mixed Strategy Nash Equilibrium

In-game theory, the mixed strategy Nash equilibrium is a concept of a game where players randomize their strategies and no player has an incentive to change their strategy. The randomization of strategies means that each player has a probability distribution over the set of possible strategies. The mixed strategy Nash equilibrium is considered to be the most general and robust solution concept in game theory.

The mixed strategy Nash equilibrium has several important properties. First, it is always Pareto efficient. This means that there is no other game situation that would make all players better off. Second, the mixed strategy Nash equilibrium is always self-enforcing. This means that no player has an incentive to deviate from their equilibrium strategy, because doing so would only make them worse off.

The mixed strategy Nash equilibrium has several applications in economics and game theory. In particular, it is often used to model situations where players have incomplete information about each other’s preferences or strategies.

## Subgame perfect Nash Equilibrium

In-game theory, the subgame perfect Nash equilibrium is a refinement of the Nash equilibrium. It is a Nash equilibrium in which, at each information set, each player has chosen the best possible action for the rest of the game, given the other player’s action at the previous information set.

The subgame perfect Nash equilibrium is often considered to be the more realistic of the two concepts, as it takes into account the fact that players often have to make decisions without complete information. It is also more likely to occur in practice, as it is usually easier to achieve than a Nash equilibrium.

## Nash equilibrium calculator

In the below online combo game theory calculator enter the details for Player 1 and Player 2 then submit to know the results of the game theory. Overall combo calculates the equilibrium strategies for simultaneous games and sequential games (and more).

Nash equilibrium calculator

## Prisoner’s dilemma Nash equilibrium

The prisoner’s dilemma is a well-known problem in game theory that illustrates the difficulty of achieving cooperation between two or more parties. The dilemma is usually stated as follows: two prisoners are each in separate cells and cannot communicate with each other.

Each prisoner must decide whether to confess to a crime or to remain silent. If both prisoners confess, then each will receive a reduced sentence. However, if one prisoner confesses and the other remains silent, then the confessor will receive a reduced sentence while the other prisoner will receive a harsher sentence. If both prisoners remain silent, then each will receive a milder sentence.

The Nash equilibrium for this game is for both prisoners to confess, as this is the strategy that results in the best possible outcome for each player. However, this is not always the most desirable outcome for the prisoners as a whole, as it results in both of them receiving reduced sentences.

## How to find the Nash equilibrium?

There are a few different ways to find the Nash equilibrium of a game. One way is to use a payoff matrix, which shows the payoffs for each player for every possible combination of moves. To find the Nash equilibrium, you need to find the combination of moves that results in the highest payoffs for both players.

Another way to find the Nash equilibrium is to use a graph. To do this, you need to plot the payoffs for each player for every possible combination of moves. The Nash equilibrium is the point on the graph where the payoffs for both players are the highest.

## How to determine Nash equilibrium?

There are a few different ways to determine Nash equilibrium. The most common method is to use game theory. Game theory is a way of mathematically analyzing how two or more people interact with each other. This can be used to figure out what each person’s best move is, given what they know about the other person’s move.

Another way to determine Nash equilibrium is to use the best response function. This is a function that takes in all of the possible moves that each person could make, and then outputs the move that will give the person the best possible outcome.

Finally, you can also use a payoff matrix to determine Nash equilibrium. A payoff matrix is a table that shows what each person’s payoff would be for every possible combination of moves. To find the Nash equilibrium, you simply need to find the combination of moves that results in the highest payoffs for both people.

## Pure strategy Nash equilibrium 3×3

In a pure strategy Nash equilibrium, each player is playing their best possible strategy, given the strategies of the other players. In a 3×3 game, there are 9 possible strategies for each player.

A pure strategy Nash equilibrium is when each player has chosen one of these 9 strategies, and no player has an incentive to change their strategy. This is because they are already playing their best possible strategy, given the other player’s strategies.

## Criticism of Nash equilibrium

The Nash equilibrium is a powerful tool for game theorists, but it is not perfect. One criticism of the Nash equilibrium is that it assumes that all players are rational and will always make the best decision for themselves. This is not always the case in real life, where people often make irrational decisions.

Another criticism is that the Nash equilibrium can give rise to sub-optimal outcomes, where the overall outcome of the game is not as good as it could be. This is because the Nash equilibrium only considers the strategies of the individual players, and not the overall welfare of the game.

Despite these criticisms, the Nash equilibrium is still a widely used concept in game theory and has helped to predict the outcomes of many games.