How do you determine Nash equilibrium?

How do you determine Nash equilibrium?

Science

73 years ago today on the 15th of January 1950, John Nash’s paper ‘Equilibrium Points in n-Person Games’ was published by the National Academy of Sciences. You can read the full paper by Professor Nash here: Equilibrium Points in n-Person Games on JSTOR

Nash would share the 1994 prize in economic sciences “for their pioneering analysis of equilibria in the theory of non-cooperative games.”

Moreover to many the Nash Equilibrium is much more than a mathematical equation! It is a way of thinking, which applies to many situations in our world.

Nash equilibrium is a concept in game theory that describes a state in which all players in a game have chosen the best strategy for themselves given the strategies chosen by the other players. 

In other words, it is a state in which no player can improve their outcome by unilaterally changing their strategy.

How to find a Nash equilibrium: tutorial to calculate the Nash equilibrium

  1. Check each column to find the one where player 1 has maximum payout.
  2. Check if the choice of 1 tends to always be the same, whatever the choice of player 2 (dominant strategy)
  3. Repeat for the same player the same procedure.

The Nash equilibrium can be applied to a wide range of situations, from economic markets to political negotiations to sporting competitions. It is a fundamental concept in the study of strategic decision-making and has important implications for understanding how individuals and groups interact in strategic situations.

To illustrate the concept of the Nash equilibrium, consider a simple game of two players, A and B, who must simultaneously choose between two strategies, “cooperate” or “defect.”

Payoffs for each player are determined by the combination of strategies chosen, as shown in the following payoff matrix:

| Cooperate | Defect

–|————|——-

A | 3,3 | 0,5

B | 5,0 | 1,1

In this game, the Nash equilibrium is the strategy combination (Defect, Defect) because neither player can improve their outcome by changing their strategy while the other player keeps theirs. For example, if player A chooses to cooperate. Player B will choose to defect. Resulting in a payoff of 0 for player A and 5 for player B. On the other hand, if player A chooses to defect. Player B will also choose to defect, resulting in a payoff of 1 for both players.

It’s important to note that a Nash equilibrium is not always the “fair” or “socially optimal” solution. It is simply a state in which no player can improve their outcome by changing their strategy. In the example above, the Nash equilibrium results in the lowest possible payoffs for both players.

However, there are other solution concepts in game theory such as Pareto efficiency, which take into account the payoffs of all players and attempt to find a solution that is beneficial for all players.

In conclusion, the Nash equilibrium is a concept in game theory that describes a state in which all players in a game have chosen the best strategy for themselves given the strategies chosen by the other players.

Moreover, a fundamental concept in the study of strategic decision-making and has important implications for understanding how individuals and groups interact in strategic situations. However, it’s important to note that a Nash equilibrium is not always the “fair” or “socially optimal” solution and other solution concepts such as Pareto efficiency may also be used.

John Forbes Nash Jr., a renowned American mathematician born on June 13, 1928, in Bluefield, West Virginia. And was the eldest of three children.

Nash received his undergraduate degree in mathematics from Carnegie Mellon University in 1948. And later went on to pursue graduate studies at Princeton University. In 1950, he earned his Ph.D. in mathematics at the age of 21. With a thesis entitled “Non-cooperative Games,” which laid the foundation for the field of game theory.

In the early 1950s, Nash began to experience symptoms of schizophrenia.

As a result, ultimately led to his hospitalization in 1959. Despite his illness, Nash continued to work on mathematics and made significant contributions to the field of game theory and differential geometry. He also received the Nobel Memorial Prize in Economic Sciences in 1994 for his work on non-cooperative games.

During the latter part of his life, Nash struggled with his mental illness and spent time in and out of psychiatric hospitals. He died in a car accident on May 23, 2015, at the age of 86.

Despite the challenges he faced, Nash’s contributions to mathematics and game theory have had a significant impact on the field. And his story has been the subject of several books and films. Most notably the 2001 film A Beautiful Mind, which portrayed his life and struggles with schizophrenia. His work continues to be widely studied and respected in the mathematical community.

In conclusion, John Nash was a brilliant mathematician who made significant contributions to the field of game theory. Furthermore, differential geometry, and partial differential equations. Despite his struggles with schizophrenia!

Nash’s work continues to be widely studied and respected, and his story continues to inspire people all over the world.

Science

How do you determine Nash equilibrium?