... entire Bangalore is out in cars today! why can't folks car pool or take public transport?
- A facebook friend's status message
Has the same thought crossed your mind? Why does everyone take a car when everyone is better off taking the bus?
As it happens, there is a "truly elegant" explanation for this, but it was too small to fit in a facebook comment box (with apologies to Pierre Fermat). So, blog post it will have to be.
Remember the movie "A Beautiful Mind"? Remember the mad mathematician John Nash? His eponymous contribution to Game Theory - the Nash Equilibrium - shows up in the said answer. And what may that be, you ask? The Mad Hatter, being qualified by way of similar mental states to interpret John Nash, will gladly answer.
Remember the Prisoner's dilemma?
Tanya and Cinque have been arrested for robbing the Hibernia Savings Bank and placed in separate isolation cells. Both care much more about their personal freedom than about the welfare of their accomplice. A clever prosecutor makes the following offer to each. “You may choose to confess or remain silent. If you confess and your accomplice remains silent I will drop all charges against you and use your testimony to ensure that your accomplice does serious time. Likewise, if your accomplice confesses while you remain silent, they will go free while you do the time. If you both confess I get two convictions, but I'll see to it that you both get early parole. If you both remain silent, I'll have to settle for token sentences on firearms possession charges. If you wish to confess, you must leave a note with the jailer before my return tomorrow morning.”
The “dilemma” faced by the prisoners here is that, whatever the other does, each is better off confessing than remaining silent. But the outcome obtained when both confess is worse for each than the outcome they would have obtained had both remained silent.
So, both prisoners confessing is what usually happens. That is a worse outcome for them than both prisoners remaining silent. Why does this happen?
John Nash generalized this idea into the concept of Nash Equilibrium. Assume you are a player in a game. You know everyone's strategy, and they know yours. Given knowledge of everyone's choices, if you could then change your strategy and come out ahead, the game is not in Nash Equilibrium. In a Nash Equilibrium, therefore, no individual player can incrementally change strategy and come out ahead. The globally optimal strategy of both players being silent in a prisoners' dilemma situation is not a Nash Equilibrium. A prisoner, knowing that the other is going to be silent, can improve his outcome by confessing. A globally optimal state that is not a Nash Equilibrium is unlikely to be reached. On the other hand, both prisoners confessing *is* a Nash Equilibrium, though not globally optimal. Knowing that the other would confess, a prisoner would rather confess than be silent.
For the curious reader, here are more examples of Nash Equilibria in games. And here are more.
All well and good, what does this have to do with people driving cars?
As the astute reader (and, as I never tire of saying, all the Mad Hatter's readers are by definition astute) would have guessed, the Bangalore traffic game is a generalized prisoner's dilemma situation. And the global optimum of everyone taking public transport is not a Nash Equilibrium. Knowing that everyone takes the bus, I can travel by car and travel in more comfort and save a bit of time. Thus, we end up in a non-optimal Nash Equilibrium where everyone takes their car.
Of course, this is simplistic, the situation involves a lot more variables (like my not actually wanting the stress of driving) and a lot of people still end up taking the bus. But, the non-optimal solution being a Nash Equilibrium captures the essence of the problem. Short sighted utility maximizers that we are, we get stuck in a globally non-optimal situation.
And that also is why Indians tend to drive very well in the US, and not in India. Given knowledge of everyone's strategy in the US (follow the rules, drive fast), Indian drivers pick the same strategy. It is not incrementally optimal in that situation to break rules, for reasons of safety. In India, given the expectation of slower and less rule-bound driving, the same drivers tend to pick rule-breaking as the right strategy. Both situations are Nash Equilibria. One is distinctly more optimal than the other.
So, is there no hope? Are we destined to be stuck in mad equilibria?
It turns out, there may just be a glimmer.
Douglas Hofstader came up with the concept of Superrationality in one of his columns. Simply put, Superrationality is going a step beyond simple utility maximizing rationality. If one assumes other actors in a symmetrical game are also logical thinkers, one must assume that they will come up with the same answer as one does. Therefore, one must find one's utility maximizing strategy assuming that all others will pick exactly the same strategy. That is, I cannot pick a strategy assuming everyone else will do something else - whatever I do, I have to assume everyone else will do the same. This gives superrational players a way to rationally "cooperate" to reach non Nash-Equilibrium solutions.
So clearly, if one knows that all other players in the game are superrational, one could end up in a good situation. But what does one do when one is not sure if other players are superrational or merely rational? This and other questions have been discussed but not definitively answered. If this sort of question interests you, perhaps this (slightly dated) survey of the field would as well.
Back to practical matters now. If we were all superrational, clearly we wouldn't be bound by Nash Equilibria in matters of public concern. Knowing that everyone would do the same as I, I would take the bus. Knowing that everyone would drive exactly as I, I would follow the rules of the road.
How do we transform our public from a rational Nash-Equilibrium-seeking herd to a superrational group? Should we teach game theory in school?
What say you, gentle reader?
ps: Interestingly, Emmanuel Kant came up with something resembling superrationality with his "categorical imperative" of moral philosophy
Act only according to that maxim whereby you can at the same time will that it should become a universal law.
pps: The "prisoners' dilemma" description above is quoted from the Stanford Encyclopedia of Philosophy
ppps: Thanks Ms. A. R. for inspiring this post! Dunno if you like to see your name on the blogosphere, so I've left you relatively anonymous :)
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