Introduction to Game Theory, Equilibrium, and John Nash

Game theory is a fascinating subject.  It is the mathematical study of decision making.  The name is a little mislead as it is not just the games we think of.  A ‘game’ is any situation in which two or more players make choices which result in gains or losses.  It is the study of making decisions logically.

                There are two types of games.  Zero-sum games non-zero sum games.  A zero-sum game is one in which a gain by any player has to come from a loss by another player.  This is why it is zero-sum, if you add up every players gains and losses you will get 0.  This is not true for a non-zero sum game.  In this game there are choices that can be made that result in gains for both players involved.   This makes cooperation possible.  However, it is still cut throat.  All players are still trying to optimize their gains.  In games like this alliances might form and players may get punished.  It is obvious that non-zero sum games are the ones that model reality much better.  However the zero-sum games are much simpler.  In fact, the field of finite zero-sum games for 2 players is completely solved.  Meaning for any situation each player can find their optimal decision by using math. 

                The solution to the problem is also called an equilibrium, because this is the spot in which both players lay.  If a player strays from that equilibrium at all, they lower their net-pay off and so thus both players stay.  The method for finding this equilibrium was proven by John von Neumann (the founder of Game Theory as a discipline!).

                This is where John Nash comes in.  His breakthrough was finding equilibriums in non-zero sum games.  This is gigantic because non-zero sum games are much more complicated.  It was his most famous contribution and he won the Nobel Prize in Economics for it, (because economic situations could be represented in this way.  There is a funny story (only to me) where a young mathematician is walking by John von Neumanna and asks John if he thought his zero-sum game equilibrium could be generalized to non-zero sum games.  John just kind of waves it off dismissing it and mutters “yes, I’m sure of it”.  Implying that he thought it could be done but that it wasn’t very important.  I guess this was fortunate for John Nash.  Sometimes even the smartest mathematicians can’t always predict which of their ideas will be the biggest, even many years after their death.

This introduction to game theory took most of this post length so I will have to make succinct the things that I would like to learn more about.  I will include a link below them where I can learn more about it:

a)      I want to learn more about non-zero sum games involving more than two people

http://students.cs.byu.edu/~cs670ta/Lectures/Noncoop.html

(a webpage/article)

b)      I want to learn the ways in which game theory has been put into practice in real life

http://blog.ted.com/2013/03/28/further-readings-in-game-theory-how-it-applies-to-marriage-kidney-donation-chains-and-government-gridlock

(a TED talk)

c)       I want to learn which activities I can use to introduce my students to Game Theory

http://faculty.lebow.drexel.edu/McCainR/top/eco/game/dilemma.html

(A simple introductory game called “Prisoners Dilemma”)

d)      I want to learn more about John Nash and his struggle with mental illness. http://www.amazon.com/The-Essential-John-nash/dp/0691096104/ref=zg_bs_917108_7

(a Biography)