Nash Equilibrium Study Guide

Introduction

In game theory, Nash Equilibrium (named after John Nash) is a solution concept for a two or more player game. It is a state in which no player can improve their payoff by changing their strategy, assuming that all other players in the game keep their strategy unchanged. Nash Equilibrium is widely used in various fields of study, including economics, political science, and psychology.

Understanding Nash Equilibrium

Definition

Nash Equilibrium is defined as a set of strategies such that no player has an incentive to change their strategy, given the strategies of all other players. In other words, each player's strategy is the best response to the strategies of the other players.

Example

Consider the following example of a Nash Equilibrium:

In this game, there are three possible strategies for Player A and three possible strategies for Player B. The Nash Equilibrium of this game is (X, B), as neither player can improve their payoff by changing their strategy, assuming the other player's strategy is fixed.

Finding Nash Equilibrium

Dominant Strategy

A dominant strategy refers to a strategy that is the best response, regardless of what the other player does. If both players have a dominant strategy, then the game has a unique Nash Equilibrium.

Iterated Deletion of Dominated Strategies

The iterative deletion of dominated strategies refers to the process of eliminating dominated strategies from consideration. This process is repeated until no dominated strategies remain. The remaining strategies constitute the Nash Equilibrium.

Mixed Strategies

A mixed strategy refers to a strategy in which a player randomly chooses from a set of possible strategies based on a probability distribution. A mixed strategy Nash Equilibrium is a state in which all players choose their mixed strategies in such a way that no player has an incentive to change their mix.

Conclusion

Nash Equilibrium is a solution concept widely used in game theory to analyze strategic interactions between multiple players. With a thorough understanding of this concept and its applications, you can gain an advantage in various fields of study, including economics, political science, and psychology.

Nash Equilibrium Quiz

Test your knowledge of Nash Equilibrium with the following questions:

Nash Equilibrium

Definition

Nash Equilibrium is a concept in game theory that refers to a situation in which each player in a game is making the best decision they can, given the decisions of the other players.

Key Concepts

• Players: individuals or entities involved in the game
• Strategies: courses of action available to each player
• Payoff: the reward or outcome for each player depending on their strategy and the strategies of the other players.

Example

Imagine there are two companies considering whether to advertise or not. If both companies advertise it will be costly, if only one advertises the unadvertised company will have a larger market, and if neither company advertises there could be a less competitive, shared market. - In this case, the Nash Equilibrium would occur if both firms choose to advertise, despite the cost, because this results in the most profitable outcome for both.

Identification

• To identify a Nash Equilibrium, look for a combination of strategies where no player can improve their own payoff by changing their strategy, assuming others keep their strategy the same.

Importance

• Nash Equilibrium enables players to make informed decisions by ensuring they fully consider the impacts and influences of other players in the game.
• It is a powerful tool for predicting outcomes in competitive scenarios, and can thus be applied in many fields of study, including economics, politics, and even evolutionary biology.

Key Takeaways

• Nash Equilibrium occurs when each player makes the best decision they can, given the decisions of the other players.
• Game theory concepts, such as payoffs and strategies, are important to understand when identifying and analyzing Nash Equilibrium.
• Nash Equilibrium can be applied in a variety of fields and is a valuable tool for prediction and strategy development.

Practice Sheet: Nash Equilibrium

1. Two rival companies, A and B, are considering reducing their prices. The payoff matrix for this interaction is shown below. What is the Nash equilibrium?

1. In a simultaneous move game, two players are trying to decide whether to cooperate or defect. If both cooperate, they each earn 3 utility units. If one cooperates and the other defects, the defector earns 5 utility units and the cooperator earns 0 utility units. If both defect, they each earn 1 utility unit. What is the Nash equilibrium?

2. Two players are trying to decide what movie to watch tonight. If both players want to watch the same movie, then they each receive 10 utility units. If they watch different movies, then they each receive 2 utility units. What is the Nash equilibrium in this game?

3. Two countries, X and Y, are considering whether or not to legalize marijuana. The payoff matrix for this decision is shown below. What is the Nash equilibrium?

1. In a market, two firms are deciding on their advertising budgets. If both firms spend the same amount on advertising, they each earn 8 utility units. If one firm spends more than the other, the one who spends more earns 10 utility units and the one who spends less earns 4 utility units. If both firms spend nothing on advertising, they each earn 2 utility units. What is the Nash equilibrium in this game?

2. Two players are trying to divide a pizza. If one player takes more than half of the pizza, the other player earns 0 utility units. If the players split the pizza evenly, then each earns 10 utility units. What is the Nash equilibrium in this game?

3. In a market, two firms are deciding on their production levels. If both firms produce the same amount, they each earn 10 utility units. If one firm produces more than the other, the one who produces more earns 12 utility units and the one who produces less earns 8 utility units. If both firms produce nothing, they each earn 2 utility units. What is the Nash equilibrium in this game?

4. Two players are trying to decide what restaurant to go to for dinner. If they go to the same restaurant, they each earn 5 utility units. If they go to different restaurants, they each earn 2 utility units. What is the Nash equilibrium in this game?

5. Two countries, P and Q, are considering whether or not to invest in renewable energy. The payoff matrix for this decision is shown below. What is the Nash equilibrium?

1. Two players are trying to decide which route to take to get to their destination. If they take the same route, they each earn 8 utility units. If they take different routes, they each earn 4 utility units. What is the Nash equilibrium in this game?