The Price of Anarchy

I've always been fascinated by the spectacle of people evacuating.  I've had a first-hand look at the evacuation ahead of Hurricane Ike, and I think it's time to say something about what is called the price of anarchy.  Before the price of anarchy can make sense, however, we need to discuss the idea of a Nash equilibrium.  Remember the guy from A Beautiful Mind?  Yes, I'm talking about that Nash.

A Nash equilibrium is a situation in which two or more parties are deciding what to do, and none of the parties can do anything to improve their own situation unless other parties change their mind.  This is what happens when individuals go through life looking out for their own interests.  Perhaps an example would help.  There is a famous problem called the Prisoners' Dilemma.  The situation is one in which two suspects have been detained in connection with a crime.  They did commit the crime, but the evidence is circumstantial, and wouldn't stand up in court without a confession.  The suspects have agreed beforehand that they will not squeal, but when they are taken downtown they are interrogated separately.  The interrogator confronts each prisoner with roughly this information:  

"If you squeal on your buddy and he doesn't confess, you'll both do time, but you will do a month while he will do a year.  If you both squeal, you'll both do about four months time. "

What the interrogator doesn't mention (but both prisoners understand) is that if neither of them confesses they'll both get off on the main charges, but they'll be charged with some lesser charge and probably end up doing two months.

The socially optimal (from the prisoners' point of view) solution is for them to both withhold a confession.  The total time served would be four months, after which both could return to their life of crime.  This, however, is not a Nash equilibrium.  Either prisoner (acting alone) could improve his or her own situation by confessing.   

If both confess, however, then the total amount of time served jumps to eight months.  This is, however, a Nash equilibrium, since neither prisoner can shorten his sentence by acting alone and not confessing.  Without a central plan, society generally moves toward Nash equilibria.

In practice, Nash equilibria are often well below the socially optimal solution.  The ratio of these two, (in this case 2 (8 months vs. 4 months)) is what is called the price of anarchy.  In this example the lack of a strong uniting force costs twice what the optimal solution would cost them.

 The reason I bring all of this up is that it highlights one of the reasons we, as a society, need the GNU Public Dictatorship.  We can't afford to live sub-optimally, and therefore we need a central authority that can remind us all to act in a socially responsible fashion that will benefit us all.  We have the resources and connections at our disposal to know what the right choices are.  Some may complain that we are trying to limit your choices or make you behave as we want you to.  Some even suggest that we optimizing things for our own benefit and not that of society.  Nothing could be further from the truth.  We, your dictators, are committed to helping each of you avoid the price of anarchy and reap the benefits of cooperation.  We are nothing if not dedicated to optimality!