Sunday, March 04, 2007

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).


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.

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