## Sunday, July 5, 2015

### Nash Equilibrium in Layman's Terms

NE is a game theory concept, so it applies to games.  Economists use the word game a bit differently than other people, so you can think of a game as a scenario where more than one player interact with each other and the payoffs (perhaps who wins and who loses and by how much) are determined by the decisions of all the players.  A simple game is played on The Price is Right.  The winner is the player who guesses the number closest to the value of the item without going over.  All four players make a guess and therefore whether or not one player is a winner depends on the value of the item and the guesses of the other three players.  Likewise, paper-rock-scissors is a game.  If you throw rock, your payoff depends on what your opponent throws.

In a NE, no player can increase his own payoff by deviating if everyone else keeps their move the same.  Think about letting just one player (at a time) have a do-over...if they change their move, then we say they have deviated.  Another aspect of NE that hurts brains is that there can be multiple NE in one game.  I am going to describe a two player game for simplicity.

THE GAME:
You and I get to pick an integer between 1 and 9 inclusive.  We do not get to communicate with each other about what we will pick.  Say that you pick a number by writing it down and giving the paper to a person organizing the game.  The payoffs are as follows:  if we pick the same number, then we each get paid that number of dollars (by the person who is organizing the game).  If we pick different numbers then we each have to pay the same number of dollars as the number we picked (to the person organizing the game).

Example 1 payoff:
I pick 5 and you pick 7.  Now I have to pay 5 and you have to pay 7.

Example 2 payoff:
We both pick 4.  Now we both get paid \$4.