The Importance of A Beautiful Mind

If you have ever read anything about game theory, you have probably heard of John Nash. He was an influential mathematician that made major contributions to the fields of game theory, differential geometry, and differential equations. In 1994, he was awarded a Nobel Prize for his work. You would think that a mathematician’s life would be quite dull, but his definitely was not. There was even a movie made about his life called A Beautiful Mind. It actually won four Oscars, including Best Picture. I mention Nash because of his most famous addition to game theory, the Nash equilibrium.

One of the most important concepts related to a Nash equilibrium is an idea called a best response. A best response is exactly as it sounds. It is a player’s best available option in response to the specific choice of another player. A mutual best response, or a Nash equilibrium, is the convergence of two best responses. The Nash equilibrium creates a sort of common ground, in which no player benefits by deviating away from it and both players are rewarded for responding to it. That is, if the other players are also playing the same Nash equilibrium. This does not always provide players with the highest possible payoff, but it will lead to the most optimal payoff for both players in in a given scenario.

Let me give you a very basic example. Let’s take a simple game with two players involving their hands. Each player has a choice of whether to stick out their left hand or stick out their right hand. For the sake of this example, assume this choice is done simultaneously and that each player does not know what the other player is going to do. The game is set up such that the two players are not playing against each other. Instead, they each have the same goal, which is to accumulate the most points. Here’s how the game is scored: if both players extend their left hands, they will each get two points, and if they both extend their right hands, they will each get zero points. If one player sticks out their right hand, while the other sticks out their left, then the player sticking out their right hand would get one point, and the player sticking out their left would get three points.

To give you another way to visualize it, here is the payoff matrix:

 

Player 1, Player 2 Left Right
Left 2, 2 3, 1
Right 1, 3 0, 0

 

(For those who do not know how payoff matrices work, here is a short translation. The left column is Player 1’s choices, and the right is Player 2’s choices. The intersection of the choices is the payoffs. For example if Player 1 choose Left and Player 2 choose Right, the payoff with be the top right column or 3, 1. Player 1’s payoff is on the left, 3, and Player 2’s payoff is on the right, 1.)

 

So, if you want to choose the best option, you need to find the Nash equilibrium. An easy way to find the Nash equilibrium is to mark the best responses of each player, and find where they converge. Any time you are trying to find all of the best responses, go through each option available to each player one by one. First, let’s find the best response if Player 2 chooses to stick out their right hand. Player 1 has two possible choices when Player 2 chooses to stick out their right hand. Player 1 can either stick out their left hand, which has a payoff of three, or their right hand, which has a payoff of zero. Since three is the highest payoff, Player 1’s best response when Player 2 chooses right, is to stick out their left hand.

Player 1’s best response to Left is Left, since two is greater than one. Player 2’s best response to Right is Left, since three is greater than zero. Player 2’s best response to Left is Left, since two is greater than one. Now let me show a payoff matrix with each of the best responses highlighted.

 

Player 1, Player 2 Left Right
Left 2, 2 3, 1
Right 1, 3 0, 0

 

It is clear that the Nash equilibrium is at Left for both players, since that is the convergence of their best responses. Therefore, it is the optimal choice for both players to make. And while this example may be oversimplified, the Nash equilibrium can be used to analyze far more complex scenarios. One example of this in daily life would be convincing your roommate to clean your room together. In this scenario, there would be a Nash equilibrium if each person put in the same amount of work to fully clean the room. And while it may be a nerdy argument with your roommate, using a Nash equilibrium is logically sound.  
I hope you learned something and thanks for reading. Next time, I will go into the more complex factors that can cause changes in a Nash equilibrium.

 

For anyone who wants to follow their own interest in game theory, here are a few resources:

http://oyc.yale.edu/economics/econ-159#overview

http://mindyourdecisions.com/blog/category/game-theory/

Images:

http://www.newyorker.com/news/john-cassidy/the-triumph-and-failure-of-john-nashs-game-theory

 

5 thoughts on “The Importance of A Beautiful Mind

  1. Max D.

    Hey Tom,
    I’ve also heard about a term: pareto optimal / pareto efficiency
    does this have anything to do with Nash equilibrium?

    Reply
    1. tkbarnet Post author

      I have not yet learned anything about pareto optimal or pareto efficiency, but it sounds like an intriguing if simple concept from what I found online.

      Reply
  2. margaretjhaviland

    In the movie, Nash has his epiphany about equilibrium while contemplating how to pick up girls when he is out with his colleagues. What would At&T, Verizon and Sprint do with Nash’s Equilibrium?

    Reply
    1. tkbarnet Post author

      In any market where companies are offering identical products, there is a price point that exists which no company would technically have a reason to deviate from. If they go lower then that price point they would lose too much profit, and if they go higher then they lose too many customers. However due to other variables like cell coverage and advertising, the companies have other incentives to deviate from that price point.

      Reply
  3. anrowshan

    Ahhhhhh I love A Beautiful Mind! It’s one of my favorite movies. I can still remember the exact scene that shows Nash’s breakthrough. Was this Nash’s only contribution to game theory?

    Reply

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