JOHN NASH, R. I. P.

My old student, Dr. Andrew Blais, asks in an e-mail whether I might write a tutorial on the work of Nobel Laureate John Nash, who died recently with his wife in a tragic automobile accident.  I replied that I did not know enough to write such a tutorial, but there might be one or two folks out there who would like to know something about Nash’s contributions to the mathematical discipline known as Game Theory.  Herewith a brief explanation.  In order to keep this short, I shall have to refer interested readers to my second blog, entitled Formal Methods in Political Philosophy, for which I wrote a book length primer on a number of branches of formal analysis, including Game Theory.

Game Theory was invented by the great Hungarian-American mathematician John von Neumann, who set forth the theory in a classic book, The Theory of Games and Economic Behavior co-authored by Oskar Morgenstern and published in 1944.  The central theorem of the book asserts that every two person zero sum game with mixed strategies has a solution.  Each term in this sentence has a precise mathematical definition.  In particular, the term “zero sum game” is given an exact definition by von Neumann and Morgenstern.  By the way, almost nobody who uses the phrase knows what it means, not even professional economists who should know better.

Assuming that in a two person game, the first person has n pure strategies [a finite positive integer] and the second player has m pure strategies, and understanding that a mixed strategy is a probability distribution over a player’s pure strategies, the triads formed by the first player’s mixed strategy, the second player’s mixed strategy, and the payoff to the first player can be represented by points in an n+m-1 dimensional simplex.  A solution is a point in that space – called a saddle point – which has the property that if either player holds firm to his or her mixed strategy choice, the other player can only do worse by moving away from that point to an adjacent mixed strategy.

Von Neumann, using L. E. J. Brouwer’s famous “fixed point theorem,” proved that every such game has a solution.  What is more [this is the point of all of this hand waving], if there are multiple solutions, they all have the same payoff to each player.  Hence, they are interchangeable.

Powerful as this theorem is, its application is extremely limited, because the games that meet the stringent definition of being zero sum are rare indeed.  What John Nash did was to extend von Neumann’s proof by demonstrating that a very much wider set of games have solutions in the von Neumann sense.  However, in this wider set of games, if there are multiple solutions [multiple saddle points], it is not in general the case that they all result in the same payoffs to the players.  Hence these solution points are local but not global solutions.

O. K.  That is about as much as I know about John Nash, except that he was played by Russell Crowe in the movie A Beautiful Mind.