Game Theory 101 (#5): What Is a Nash Equilibrium?

Introduction to Nash Equilibrium

In this section, William Spaniel introduces the concept of Nash equilibrium and its importance in game theory.

What is a Nash Equilibrium?

• A Nash equilibrium is a set of strategies, one for each player, where no player has an incentive to change their strategy given what the other players are doing.

• Pure strategy Nash equilibrium occurs when players do not randomize between multiple strategies.

• Mixed strategy Nash equilibrium will be discussed in future videos.

Interpreting Nash Equilibrium

• One interpretation of a Nash equilibrium is that it represents a law that no one would want to break even in the absence of an effective police force.

• For example, following traffic stoplights can be seen as a Nash equilibrium because drivers would still follow the rules even if there were no police to enforce them.

Example of Stag Hunt Game

In this section, William Spaniel provides an example of a stag hunt game and analyzes its payoff matrix to determine if it is a pure strategy Nash equilibrium.

Payoff Matrix Analysis

• The stag hunt game involves two players who can choose between hunting a stag or hunting a hare.

• If both players choose to hunt a stag, they receive a payoff of -5 (negative 5).

• If both players choose to hunt a hare, they receive a payoff of -1 (negative 1).

• If one player hunts a stag while the other hunts a hare, the player who hunts the stag receives a payoff of 1 and the other player receives 0.

• Analyzing the payoffs, it can be determined that both players choosing different strategies (one hunting stag and one hunting hare) is not beneficial for either player. Thus, pure strategy Nash equilibria occur when both players choose the same strategy.

Identifying Nash Equilibria

In this section, William Spaniel identifies the pure strategy Nash equilibria in the stag hunt game and explains why they are Nash equilibria.

Pure Strategy Nash Equilibria

• In the stag hunt game, there are two pure strategy Nash equilibria:

• Player 1 stops and Player 2 goes.

• Player 1 goes and Player 2 stops.

• These equilibria occur because neither player has an incentive to deviate from their chosen strategy given what the other player is doing.

• Deviating from these strategies would result in a worse payoff for each player.

Conclusion

Nash equilibrium is a fundamental concept in game theory that represents a set of strategies where no player has an incentive to change their strategy. In the example of the stag hunt game, pure strategy Nash equilibria occur when both players choose either to stop or go. These equilibria are stable because any deviation from them would result in a worse outcome for at least one player. Understanding Nash equilibrium is crucial for analyzing strategic interactions in various fields such as economics, politics, and biology.

New Section Nash Equilibrium and Self-Interest

In this section, the concept of Nash equilibrium is explained in the context of self-interest and following laws.

Understanding Nash Equilibrium and Self-Interest

• Nash equilibrium is reached when individuals make decisions based on their own self-interest.

• The example of a stoplight game is used to illustrate this concept.

• Players at a stoplight choose whether to go or stop based on their own self-interest, not because of police enforcement.

• The goal is to avoid car accidents and save time by making decisions that align with one's self-interest.

Importance of Self-Interest in Decision Making

• Players do not want to switch strategies at the stoplight because it may lead to car accidents or waste time waiting for the other player.

• Following laws and making decisions based on self-interest becomes a rational choice for individuals.

Nash Equilibrium as a Law

• Nash equilibrium can be seen as a "law" that individuals would not want to break even without an effective police force.

• It provides stability in decision-making by aligning individual interests.

Next Steps: Finding Best Responses

In the next video, the focus will be on finding best responses, which is a useful method for identifying pure strategy Nash equilibria in more complex games.