THE NEVER ENDING REVIEW: CHAPTER THREE OF 'A BEAUTIFUL MATH' : 'NASH'S EQUILIBRIUM':
Welcome back to this continuing review of Tom Siegried's 'A Beautiful Math', on the growth of game theory. I spend a goodly amount of time on this review because it is, in my opinion, important for a clear sighted view of social action- not because I agree with the author's often overinflated claims for the relevance of game theory. Even a sceptic of some of the grander claims can see just how important this matter is for a rational radicalism of the future. Anyways...
It's the third chapter of this book before the character of the title, John Nash, makes an appearance. Nash entered Princeton University as a graduate student in 1948. This was Von Neumann's stomping grounds. Morgenstein worked in the economics department and Von Neumann was at the Institute for Applied Studies a mile away.
To this point game theory had been restricted to the rather sparse world of "two player zero-sum" games. Nash rapidly broke into new fields of analysis, and his 1950 paper ('The Bargaining Problem, Econometrica 18(1950) pp 155-162) on which he was advised by both Morgenstein and Von Neumann, expanded the world of game theory into "cooperative games" in which the two sides work together to achieve a mutual benefit, and what he provided was a mathematical map for finding the optimal bargain that maximized the utilities of both players.
In the same year that he published the above paper he also presented his doctoral thesis- Non-Cooperative Games. This introduced the idea of an "equilibrium strategy" towards which a repeated round game will evolve. At this equilibrium Nash wrote in his thesis the situation is such that,
"...each player's mixed strategy maximizes his payoff if the strategies of the other players are held fixed."
What this idea did was to take game theory and make it possible to describe multi-player games, something that Von Neumann's ideas floundered on. Nash's proof depended upon something known as the fixed point theorem , an idea borrowed from topology. The ideas presented in this thesis were also published in the PNAC as 'Equilibrium Points in N-Person Games' in 1950 and in 1951 as 'Non-Cooperative Games' in the Annals of Mathematics.
As a side note "cooperative" and "non-cooperative" have a rather restricted meaning here. Cooperative refers to the coalition forming that Von Neumann and Morgenstein used to get around the fact that their theories couldn't deal with more than two players. Non-cooperative refers to Nash's expansion which can deal with any number of players who don't collaborate or communicate with each other. What Nash showed is that there is an "equilibrium strategy" that each player (at least one such but sometimes more than one) which maximizes their payoff no matter what the other players do assuming they also try to maximize their payoff.
This is, of course a simplified version of the real world where the "equilibrium states" of perfectly rational actors who try to maximize their self interest rarely exist. But armed with this general description the author goes on to describe specific "games" have analyzed, especially 'The Prisoner's Dilemma' (1), first described by Nash's Princeton professor Albert W. Tucker in 1950 (2) .
The game is set up as follows, two criminals, call them "Alice and Bob', are arrested. The police interrogate them separately. They have enough evidence to convict each of them on a minor charge, but they need confessions for convictions on more serious charges. If both refuse to confess they each get one year on the lesser charge. If one confesses and the other stays silent the squealer goes free and the other gets five years. If both confess they each get 3 years (two years off for "copping a plea". The payoff matrix is as below (once more excuse the limitations of blogger).

Alice

Keep Mum Rat

Bob: Keep Mum 1, 1 5,0

Rat 0,5 3,3

The above game is actually set up as something like a "routine procedure" by police interrogators, often with the predictable outcome (criminals are usually not heroes after all).

The 'Nash equilibrium' for the above is for both players to confess. from the point of view of either player the best choice is to rat no matter what the other player does. The outcome where both players squeal is "worse for the group" as 6 combined years is the maximum sentence, but is the best outcome for an individual acting in their own self interest. A real life example of this can be seen in continual news of "eco-terrorists" acting as squealers time after time in the USA. The ideology of those who promote such acts (while often remaining aloof from same) is insufficient to overcome the self interest of those who are caught in such acts (which they usually are), and their attempted "punishment" is as quite puny as compared to that of ordinary criminals. Hence the great incentive to confess on the part of people who are caught for such crimes. "Spite" may play a part in this as well, as those who have been caught may come to realize the self-interest of many who have "egged them on" while remaining out of danger themselves.

Another game mentioned, one closer to actual reality rather than anarchist cultism, is the 'Public Goods Game'. The question of this game revolves around the provision of public goods by voluntary donation- something closer to the heart of real anarchism rather than the posturing of certain American cults. In this case "defectors", otherwise known as "free riders" who don't voluntarily contribute can still reap the benefits of a "public good". It seems OK for the defectors, but if too many decide to "free ride" then the public good becomes unavailable and the defectors get no benefit.

One of the variants of the public goods game that Siegfried mentions is set up as follows. Four players are given monetary tokens and told that they could keep as many as they wanted or put them into a "public pot" where the amount would be doubled by the experimenter. There were a certain number of "rounds" in this game wherein each player would be told how much had been contributed to the pot, and they would be offered the chance to change their contribution; either decrease or increase it.

When the game was played repeatedly a stable pattern began to emerge. As Siegried says,

"Players fell into three identifiable groups:cooperation, defection (or free riders) and reciprocaters. Since all the players learned at some point how much had been contributed, they could adjust their behavior accordingly. Some players remained stingy (defectors), some continued to contribute generously (cooperators) and others contributed more if others in the group had donated significantly (reciprocaters).

Over time, the members of each group earned equal amounts of money, suggesting that something like a Nash Equilibrium had been achieved- they all won as much as they could, given the strategy of others. In other words,in this kind of game, the human race plays a mixed strategy- about 13% cooperators, 20% defectors (free riders) and 60 % reciprocaters in this particular experiment". (Molly Note: this emphasizes the importance of what is called "altruistic punishment" in evolutionary psychology. In a "game" where knowledge of an "opponent's" previous interactions with other players is given the percentage of "defectors" can be reduced by such punishment inflicted by players who were not part of the original rounds).

The author ends the chapter with an overview on 'Game Theory Today'. He notes that the field has been broadened considerably to cover "games where coalitions form, where information is incomplete, where players are less than perfectly rational". He also notes that there are arguments about whether game theory predicts behavior or "proscribes" what a rational person should do. He goes on to answer some of the criticisms of game theory's ability to "predict" in real world situations.

Siegried notes that game theory, like other scientific theories, is a model of reality, not reality itself. It makes reality comprehensible by simplifying it. As it is tested in experimental situations it is modified and grows just like any other scientific theory. The author quotes Colin Cameron in 'Behavioral Game Theory':

"The goal is not to disprove game theory...but it is to improve it"

Siegried goes on to describe the contributions of Thomas Schelling who won the 2005 Nobel Prize in economics. Schelling focused on games where there is more than one Nash equilibrium. he particularly analyzed conflict in international relations and the role of "bluff" in same. He also analyzed games where a "coordinated outcome" is better than any particular outcome. These are situations where, as Siegried says,

"...where it is better for everybody to be on the same page, regardless of what the page is."

The work of the other 2005 economics Nobel Prize winner, Robert Aumann, is also mentioned. Aumann analyzed the prisoner's dilemma game as a "repeated rounds" situation rather than a "one shot" affair and showed how cooperation could evolve in such situations (Much closer to everyday life:Molly Note). He identified situations where cooperation is less likely ie many players, limited communication or limited game time (fewer rounds). (Molly Note: The eventual "goal" of studying matters such as these is to identify what sort of conditions lead to "increased cooperation" in the presumed society that we want. Some things are obvious from the above. "Fewer players" means a decentralized society. "Full communication" means not just decentralization but also the elimination of "socialist managers" who mediate such communication and add "noise" to same).

Siegried finishes this chapter by naming multiple applications of game theory, not just in economics but also in medicine, politics,ecology and especially !!! evolutionary biology.

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
-------------------------------------------------------------------
North South

-------------------------------------------------------------------

Allies North 2 2

South 1 3
-------------------------------------------------------------------

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

A BEAUTIFUL MATH:

BY TOM SEIGFRIED

On the surface this book is about John Nash (1), the Princeton mathematician most famous for his portrayal in the movie 'A Beautiful Mind' (2)about a mathematical genius cursed with schizophrenia. Nash won the 1994 Nobel prize for his pioneering work in the branch of mathematics called "game theory", but his contributions to algebraic geometry were what actually won him the most fame within the mathematics community.

The subtitle of this book is 'John Nash, Game Theory and the Modern Quest for a Code of Nature', but the author ranges far and wide across a number of theorists, fields of inquiry and ideas. From Asimov to Lan Zhou, from 'The Age of Reason' to 'Zero Sum Games', it all plays out on these pages.

Siegfried is an award winning science journalist who has taken on a rather grandiose project in this book. While Nash is indeed important to the development of game theory and its applications as diverse as evolutionary biology, information theory and experimental economics one can't help but feel that the title and cover were designed more to capitalize on the success of the movie rather than to describe the contents of the book. In his introduction the author tries to give a brief overview of game theory and how it touches on such fields as those mentioned above and others such as neurophysiology, anthropology and even, according to the author, quantum physics. Nash himself is rather peripheral to the central thread of the book, that game theory may be the sort of "psychohistory" that Sci-Fi author imagined in his 'Foundation Trilogy', ie a mathematically precise theory that can describe the changes and stases in society in the same sort of statistical but testable way that statistical mechanics describes the behavior of such things as gases even if the behavior of each and every molecule is inaccessible to analysis. Asimov put it as "the science of human behavior reduced to mathematical equations", but Siegried makes much larger claims for the utility of this branch of mathematics, some of them already being played out and some of them quite frankly speculative.

Molly has to admit to a certain amount of scepticism regarding such claims. While there is little doubt of the utility of game theory in evolutionary biology and in experimental economics some of the other claims are the purview of the fringes of certain fields. But...I'm reviewing this book as I read it, so many my scepticism will be overcome by the time I reach the index. For now...

Chapter One: Smith's Hand: Searching for the Code of Nature:

There's an old libertarian book, written I believe by Jerome Tuccille (3), entitled 'It Usually Begins With Ayn Rand' . In this case it begins way before that. Siegfried goes back a lot further to a much more respected figure, the economist Adam Smith(4), to begin his story. Chapter one is all about the theories of Adam Smith, with special reference to his "invisible hand" and how it was a precursor to the sort mathematical science of society that he sees in game theory. Along the way he gives a brief biography of Smith, how he was influenced by the French physiocrats, especially Francois Quesnay, and of how he came to formulate his theories in reaction to theirs, especially in regards to the source of wealth which Smith believed was labour rather than land. Both Quesnay and Smith believed that most (certainly not all in the case of Smith) government interference with the economy disrupted a natural process of economic interaction that usually produced results much superior to those produced by government action. Both authors opposed the prevailing merchantilist theories of the day that encouraged ceaseless government action to produce a favourable balance of trade, and both held to a free-trade "laissez-faire" attitude towards most economic questions.

Where this connects with the 'Code of Nature' that Siegfried sees in game theory is that Smith's was the first systematic "inquiry" that tried to build a theory of society that examined how the efforts of individuals could produce "macro" effects such as the changes in price that result from a competitive economy. This was the "invisible hand", and it described in at least a partial way how an equilibrium resulted from actions taken by individuals that have no such goal in mind. Along the way the author corrects a lot of misconceptions about Smith. Adam Smith was not the dogmatic advocate of free markets in all things that moderns tend to think he was. He saw at least a limited role for government. He also wrote on "moral sentiments" and did not believe that "rational selfish calculation" explained more than a subset of human society and its developments. What Smith, however, pointed to was that there could be a "natural order" of society that could be investigated by scientific methods. The present day experimental economists carry out this tradition investigating the often messy and even non-rational ways that choices are made by real people in the real world.

The chapter concludes with a brief subheading on 'The Origin of Darwinism'. In 'The Structure of Evolutionary Theory' Stephen J. Gould (5)has traced innumerable influences on Darwin's intellectual development. Smith was one of them, but it was not 'The Wealth of Nations' with which Darwin was familiar but rather another of Smith's works, 'The Theory of Moral Sentiments' in which Smith argues in a manner quite different from what he is usually portrayed as today.

What Darwin took from the general culture, of which was Smith was an illustrious part, is not some fantastic turn of political opinion. What he took was the general idea that small actions could produce-statistically- large effects which were part of an 'emergent order' not immediately implied by the actions themselves. If individual competition can produce an economy with regular laws then natural selection can produce the origin of species.

More on this book later,

Molly

MOLLY NOTES:

1) See also John Nash's home page at http://math.princeton.edu/jfnj and his Nobel prize address at http://nobelprize.org/nobel_prizes/economics/laureates/1994/nash-auobio.html which gives the account of his life in his own terms.

2) The official site for the movie is at http://abeautifulmind.com . The PBS Network did a much better biography on Nash that corrects many of the misstatements and omissions of the movie. this can be accessed at http://www.pbs.org/wgbh/amex/nash . The PBS site includes many more items on Nash including something of a "Game Theory for Dummies" guide.

3)Ahhh....Jerome Tuccille. In addition to the above link you can see his home page at http://jerometuccille.com . Tuccille is a long time libertarian, and one of the most amusing authors I have ever read. I can remember his 'It Usually Begins With Ayn Rand' from decades ago. At the time I read it as a leftist anarchist who had never met a libertarian in his life. The times have changed. At the time libertarianism was purely !!!! an American ideology. I had no end of pleasure of reading his account of attempting to form an alliance between the anti-statist realms of both the left and right at that time (the early 70s). I probably agree with him now about as much as I did then ie "only in a very limited sense", but his writing style was entertaining beyond belief, and I was as much gratified then as I am now by his descriptions of the "nuts on the right" and how it consoled me in my "hope" that insanity and creepiness was not just confined to my own position as a leftist. I can recommend Tuccille beyond all other writers with whom i disagree merely because of the skill of his description.

4) While researching this blog I came upon the on-line library of the 'Library of Economics and Liberty'. This site contains the online writings of a great number of economists, including Adam Smith. So, if you want to read the classics of economics go there. Everything from 'The Wealth of Nations' to Marx's 'Capital' is available there. The site is conservative in political orientation, but it is still the best resource for economics "in the original" that I have yet to come upon.

5)See also the Stephen J. Gould Archive at http://www.stephenjaygould.org, a collection of many of his writings and a brief biography at http://www.annonline.com/interview/961009/biography.html

6)One thing that Molly became aware of during the writing of this blog is the general impression that great mathematicians have a much greater incidence of insanity than others, especially other scientists. The list of famous mathematicians who were demonstrably insane or at least unstable enough to commit suicide is long and impressive. It began with many Greek mathematicians who committed suicide when their theories were proven wrong and continues on to the modern day with people such as Cantor, Godel, Turing, Boltzmann and Eherenfest. Even that supreme example of a failed human being, the 'Unibomber' apparently studied mathematics during his academic days, though he can hardly be counted as a "great mathematician" as even an "adequate one" as he was as much a failure there as he was in the rest of his life. Is it the subject matter and the talent needed for same that leads to this connection between mathematics and insanity ? Is the connection real or merely a widely believed myth ? If it is not the subject matter is there perhaps something about the "culture of mathematics" that drives successes to insanity or suicide or failures(both professional and personal) such as the Unibomber to murder ? All that is the matter of another blog entirely.

Molly