A BEAUTIFUL MATH: CHAPTER TWO: VON NEUMANN'S GAMES:

The author of 'A Beautiful Math' devotes his second chapter to the contributions of John von Neumann to game theory. This particular chapter is riddled with errors and omissions. It credits Von Neumann with the standard mathematical formulation of quantum mechanics for instance. In actual fact von Neumann's formulation which was supposed to unite and supersede the matrix algebra of Heisenberg and the wave mechanics of Schrodinger was pretty well universally rejected by physicists in favour of the unification proposed by Paul Dirac. Way back in the 60s when Molly studied quantum chemistry we were required to learn the approach of both Heisenberg (which Molly understood best) and Schrodinger (which was more popular), but we also had to understand how Dirac had unified the two. Von Neumann was a non-name . His ideas had long since been discarded.

This whole matter could go on and on, and perhaps it is basically the difference between tastes. Tom Seigfried actually "likes" and admires von Neumann. He even wrote a previous book about him-The Bit and the Pendulum. Molly finds little to like in the biography of the man, whether it is his fake nobility (his father bought his title from the Austro-Hungarian Empire), his attitude to women, his close to insane approach as a cold warrior, his reputation as an evil drunk,etc.,etc.,etc..

All that being said Von Neumann was indeed a genius and a polymath who contributed to many different fields. Siegfried opens his chapter with a brief biography that omits quite a few of the "juicy" parts of Von Neumann's detestable personality. He then, however, goes on in the subchapter titled 'Utility and Strategy' to point out Von neumann's unique contribution to the field of economics. The author does note that von Neumann had been preceded by the German mathematician Ernst Zermelo and the French mathematician Emile Borel., though he understandably downplays the contributions of these two men. What the author sees Von Neumann as doing is laying out a mathematically precise formulation of the idea of "utility" that economists before then had always talked about but never defined. In later years von Neumann collaborated with the German Oskar Morgenstein , who accepted an appointment to von Neumann's Princeton University in 1938 . Their collaboration produced the groundbreaking 'Theory of Games and Economic Behavior' in 1944.

The essential point of what von Neumann and Morgenstein did was to produce a simplified model such as those useful in physics that could lead to research and greater understanding by a gradual process of experiment rather than ideological argument. As such they made the simplifying assumption that utility=money, something that is not necessarily true in the real world- as will become apparent later. Siegfried calls this "taking society's temperature" in his attempt to compare the development of game theory to that of physics. as the author says,

"With the basis for utility established at the onset, von Neumann and Morgenstern could proceed simply by taking money as utility's measure".

With this simplification in hand the authors went on to analyse the sort of games described as "two-person, zero-sum" games where there are only "two sides" and where whatever the one player wins the other loses. they came up with the concept of the "minimax" which basically means a game strategy that "minimizes ones losses and maximizes ones gains" at the expense of the opponent. The essential points to note about this are:

1)These games are zero sum ie competitive. What one player gains the other loses.

2)The "payoff" depends upon what the other "player" does. Because of this complication the "best strategy" is often a "mixed strategy" that keeps the other player guessing in games that are something other than trivial. The concept is called "bluff" in poker. There is actually a mathematical formula that describes the "best strategy" that a player should adopt in such games, though the formula varies with the rules of the game.

Siegried does a masterful job of laying out the payoff matrices of such games with illustrations from "real life examples", and his point is basically this,

"By choosing the best mixed strategy you can guarantee the best possible outcome you can get- if your opponent plays as well as possible. If your opponent doesn't know game theory you might do even better."

All of this is, of course, very simplified. It presumes only two players and a zero sum game. It hardly applies to real life where there are usually many players and the "games" are usually not "zero-sum". But that is for later chapters in this book. So, as usual...

More later,

Molly