Nash Equilibrium & Game Theory Explained

Game theory

Game theory is a branch of mathematical economics first theorised by Jon Von Neumann in the 1930s. It is a mathematical framework for conceiving social situations among competing players. It is the science of strategy and serves as a model of an interactive situation among rational players.

The key to game theory is that one player’s payoff is contingent on the strategy implemented by the other player. The game identifies the players’ identities, preferences, and available strategies and how these strategies affect the outcome. The theory offers a wide number of applications in different fields, including economics, political science, finance, psychology, and biology, among others.

Any time we have a situation with two or more players that involve known payouts or quantifiable consequences, we can use game theory to help determine the most likely outcomes.

Nash Equilibrium 

Nash Equilibrium is a concept in game theory that determines the optimal solution in a non-cooperative game in which each player lacks any incentive to deviate from his/her initial strategy. In game theory, a non-cooperative game is a game with competition between individual players. More specifically, Nash equilibrium is an outcome where the optimal outcome of a game is one where no player has an incentive to deviate from his chosen strategy after considering an opponent’s choice.

Let us take an example to understand this. This example is popularly known as the Prisoner’s dilemma.

Two criminals are arrested and each is held in solitary confinement with no means of communicating with the other. The prosecutors do not have the evidence to convict the pair, so they offer each prisoner the opportunity to either betray the other by testifying that the other committed the crime or cooperate by remaining silent.

If both prisoners betray each other, each serves five years in prison. If A betrays B but B remains silent, prisoner A is set free and prisoner B serves 10 years in prison or vice versa. If each remains silent, then each serves just one year in prison. The Nash equilibrium in this example is for both players to betray each other. Even though mutual cooperation leads to a better outcome if one prisoner chooses mutual cooperation and the other does not, one prisoner’s outcome is worse. The concept is that; in a closed cell, a prisoner would rather seek to maximize his gains than to “cooperate”.

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History

Nash equilibrium was discovered by American mathematician, John Nash. He was awarded the Nobel Prize in Economics in 1994 for his contributions to the development of game theory. It is considered one of the most important concepts of game theory, which attempts to determine mathematically and logically the actions that participants of a game should take to secure the best outcomes for themselves.

Applications

Nash Equilibrium is applicable in a wide field of disciplines ranging from economics to social sciences. It has also had an important impact in fields as diverse as computer science, political science, sociology and biology.

Examples

Consider a driver approaching an intersection. She stops when she approaches a red light and she continues without concern when she approaches a green light. It is a Nash equilibrium when all drivers behave this way. When approaching a red light it is best to stop since the crossing traffic has a green light and will continue. When approaching a green light it is best to continue since the crossing traffic has a red light and will stop. Thus it is in each driver’s own interest to play her part in the equilibrium, given that everyone else does. No traffic cop is required.

To quickly find the Nash equilibrium or see if it even exists, reveal each player’s strategy to the other players. If no one changes his strategy, then the Nash equilibrium is proven.


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