Wednesday, April 1, 2015

Rationality, Game Theory and Statism

There is too much misconception about rationality.

People argues whether human are rational or not, and those who does not think so tends to point out impulsive human action that contribute to goes against their own interest.
One of those arguments is summarized by the irrationality principle of capital flow analysis.
They list several sources of irrationality:

  • Imperfect Knowledge
  • Conflicting Interests
  • Manipulative Aims
  • False Information
  • Custom and Long-Held Belief
But rationality among players in game theory is an axiom.
It follows that game theory can't be used for people holding the irrational principle as granted.

However, all those "source of irrationality" can be explained by using the game theory axiom of rationality, which permit us to use a powerful mathematical framework to model capital flow analysis, without dismissing such model.

First let's take an example of the prisoner dilemna game.
In a nutshell, two players are in prison for collaborating on a murder and have the following choice : 

Now, let's say that you are Prisoner A.
Let's say you know B betrays you, you have two choices : betray or cooperate. In the first case you serve 2 years, and the second case 3 years. So your best choice is to betray.

Let's say you know B did not betray you, you have two choice : betray or cooperate. In the first case you are free, and the second case 1 year. So your best choice is to betray.

So two rational prisoner A and B will choose to betray in all cases, this is called the nash equilibrium of the game.

What if, in real life, you observe prisoner A and B to not betray, should you conclude they are irrational ?
Since you can't conclude that, given the rationality axiom of game theory, such behavior seems to render game theory useless at explaining human behavior.
But maybe the problem is not the irrationality of A, but your assumptions. If A and B are part of the mafia, betraying means that they will loose a hand. Knowing that, you would say that they acted rationally. But in this case, they are not playing a prisoner's dilemna as specified by the game theory. In mathematical terminology, you guessed wrong their payoff function.
Some people argues that in game theory's real prisoner's dilemna, each player would betray even if the best social response would be to collaborate. (in other words, if each player collaborate, there is no other outcome which would not hurt at least one of them)
Thus the statist argument that something like the mafia or government should enforce them to cooperate. This is missing the fact that it just change the payoff function (change the game), but do not solve the prisoner's dilemna. With external force, the best selfish response would be to collaborate.

So back to our question : Does a rational person would play the best social response in prisoner's dilemna (both collaborate), without the use of external force ? (which would change the game)
The response is yes.
Imagine that A and B are part of a gang which does not promote violence, and no additional consequence are taken for betrayal. Members ends up very often in prison, repeatedly, as part of their business.
Now here is this rule in the gang that everybody knows: "If you have betrayed someone, other members will betray you forever. Otherwise, they will collaborate".
In game theory terminology, they are playing a repeated game infinitely.
As an observer, you are seeing that one of this gang member is collaborating and thus think he is irrational. But again, you are wrong.
From the point of view of the gang member, he is playing a repeated game. If he expects to go 10 times in prison, then betraying for his first time, it will assure him a total 18 years in prison.
If he collaborates, he will go only 10 years in prison in his lifetime.

You can reject the irrationality principle of capital flow analysis, and replace it with the rationality principle of game theory and thus inherit its tools.
Then consider the old "sources of irrationality" enumerated earlier as variables which just change the rule of the game players are participating by changing their payoff functions. In doing that, you can leverage the game theory framework.

When you observe someone acting irrationally, do not dismiss game theory for explaining the behavior, you just guessed wrong the game he is playing.
As Francisco in Atlas Shrugged would say : 

Whenever you think that you are facing a contradiction, check your premises. You will find that one of them is wrong.