GAME THEORY: SOME EXAMPLES:

Here are a few examples of what the author Siegfried uses to illustrate Von Neumann's game theory in his book 'A Beautiful Math'.
The first is from real life and has a trivial outcome ie a single best strategy fort each player no matter what the other player does.
During WW2 General George Kenney was in charge of the allied air forces in the southwest Pacific. During the battle of the Bismark Sea he knew that the Japanese would be sending a convoy of supply ships to New Guinea. The Allies wanted to bomb this convoy and could get in three days of possible bombing if the convoy too either of two possible routes, either north or south of New Britain. The complication was that the northern route would be stormy for one of these days, leaving only two clear days for bombing. Kenney could send his reconnaissance planes either north or south but not both (too few planes to cover both routes I guess). If they sighted the convoy the attack planes would follow. If not the attack force would head in the opposite direction the next day by a process of elimination.
I think you can see where the complications in this scenario lead, complications not mentioned by Siegfried and apparently not by the many textbooks that use this example. This first is why not send the attack planes in the opposite direction if the convoy is not sighted in due time ? The second is that one has to assume that reconnaissance is 100% effective. If the planes say that the Japanese are not there they are really and truly not there. this becomes particularly acute of day 1 is the rainy day. Can reconnaissance be 100% effective at the same time as bombing is 100% ineffective ?
But anyways Kenney made a decision by brute logic without the aid of game theory. The basis of his decision is best illustrated by the typical matrix used in game theory, outlined below(please excuse the limitations of the blogger format):
Japanese
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North South

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Allies North 2 2

South 1 3
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The game is "zero sum". Whatever the Allies gain the Japanese lose. The above matrix represents the days of bombing available given the choices of both the Japanese to go either north or south and the choice of Kenney to sent his reconnaissance planes either north or south. From the allied point of view sending the recon planes north would give two days of bombing in either case, and sending them south would give either one or three days depending upon what the Japanese did. Hard to choose from these possibilities. From the Japanese point of view, however, the choice is obvious. Going south results in a loss of either -2 or -3 (the Japanese "gains" are the obverse of the allied gains tabled above). Going north results in a loss of either -2 or -1. The best strategy is to go north no matter what the allies do. This indeed what happened in the real world. Both the convoy and the recon planes went north.
The secret of Von Neumann's proof is that it applied only to those cases where "the opponent plays as well as possible". It doesn't apply to an erratic opponent. Because of this there is a "minimax" strategy for the allies. Send the planes to the north because the Japanese are not fools. And maybe they aren't good at "bluff" either. Both "bluff" and "stupidity" are complications that often occur in real life.
Where Von Neumann's concept of "minimax" really becomes interesting is where the game is not, like the example cited above, a "one shot deal" but is repeated. Siegried uses an example modified from one given in Martin D Davis' book 'Game Theory:A Non-Technical Introduction' (Dover, Mineola NY 1983/1997). It goes as follows:
Two players, call them Alice and Bob are each dealt a single card. Black always beats red. Each player antes up "5" on each round so there is always "10" in the pot. Alice plays first, and she can either call or bet an additional "3". If she calls both players show their cards, and the black card wins the pot. If they both have black or they both have red they split the pot 50/50. Bob, on the other hand can, if Alice bets, either fold or match her "3". If he folds Alice takes the "13" in the pot. If he matches and calls the one with the black card wins the pot, and if both players have the same colour they once more split the pot 50/50.
The matrix for this game is considerably more complex than that given above. I'll spare you the attempt, but try it out if you are interested. Instead of the 4 possible entries in the matrix above there are 128 different possibilities because both Alice and Bob can follow 4 different strategies. From Alice's point of view she can either 1)always bet, 2)always call, 3)bet with red and pass with black or 4)bet with black and pass with red. The result with Bob's options is a 4X4 matrix with 8 entries in each cell because the different winnings have to be listed beside each other. This sort of game introduces the idea of "bluff", and it turns out that the best minimax strategy is a "mixed" one for both players. Alice's best strategy is to bet 60% of the time no matter what card she has (ie to "bluff" if she has black) and 40% of the time to bet only if she has black. Bob, on the other hand, should call Alice's bet 40% of the time no matter what card he has and 60% of the time call if he has black and fold if he has red. The choice of what to do should be totally random so that the percentages equal the above.
This game, by the way, is stacked in favour of Alice. She'll come away with a gain of about 30% if both players play their optimum strategies. So Von Neumann showed that there are minimax strategies that are "mixed strategies" where a given course of action is chosen randomly a certain percentage of the time.
Molly

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