Tuesday, 20 August 2013

Barbarik & Backward Induction

Barbarik, son of Ghatotkacha, has 3 arrows which return to his quiver after completing their task. The first picks out and marks all the things he wants to destroy. The second picks out and marks all the things he wants to save and the third destroys everything marked by the first while sparing those picked out by the second.
Barbarik travels to Kurukshetra with the determination to join the weaker side. However, Krishna stops him and persuades him to offer his own head as a sacrifice to sanctify the battle ground.
Why does Barbarik agree to part with his own head? In ancient times it was the practice to offer 'dakshina' as a fee to the Guru who reveals a great truth.
What was this truth?
The argument Krishna uses with Barbarik is similar to what Mathematicians call 'backward induction'. This means first considering the last time a decision might be made and choosing what to do at that time. Using this information, one can then determine what to do at the second-to-last time of decision. This process continues backwards until one has determined the best action for every possible situation (i.e. for every possible information set) at every point in time.
Suppose Barbarik is the only combatant alive on the battlefield. If he does not kill himself he isn't on the weaker side because the weaker side must also be the losing side. So he should kill himself. However, suppose there is one other combatant left on the battlefield. If he is on the same side as Barbarik, then together they constitute the stronger side. If Barbarik kills the other, then Barbarik is still on the stronger side. However, if Barbarik kills himself, the other combatant is victorious and thus proven to be on the stronger side. Thus Barbarik should kill himself to ensure that he has fought on the weaker side. Suppose there are two other combatants other than Barbarik left on the battlefield. If Barbarik kills himself one or other is victorious or both are victorious (if allied)- in either case Barbarik is on the weaker, because losing, side. Suppose there are n combatants, Barbarik should kill himself because otherwise he ends up on the winning side. This is because his arrow kills all enemy combatants present at the time. So Backward induction says Barbarik should kill himself no matter what the number of combatants or relative strength of their respective sides.
However there is another way to look at this. Suppose Barbarik says to the first arrow- I want you mark for death everybody on the stronger side- and to the second arrow- I want you to mark 'safe' everybody on the weaker side- before unleashing the third arrow. What would happen?
Well, anyone marked for death by the first arrow can't be on the stronger side because they are bound to lose and thus will be marked 'safe' by the second arrow. So nobody dies when the third arrow is unleashed.
I'm assuming that only an infinitesimal span of time separates the flight of each arrow and that nothing else changes over the period.
What happens if all combatants are given the choice of switching sides after the flight of the first arrow? Then the third arrow can only have casualties from the weaker side. But if combatants are rational and given the chance to also switch sides before the first arrow then once again no casualties arise.
More generally, if all feasible adversarial coalitions.of combatants can be ranked and this information is publicly available then every combatant who wants to be on the winning side should have a 'nearest possible world' feasible coalition which is stronger than what obtains such that a strategic action on his own part, or that of his sub-coalition, can call it into being. This is a dynamic notion of allegiance which sounds quite realistic for medieval Indian wars where commanders frequently switched sides on the battlefield. 
Consider the set of combatants who want to be on the winning side and aren't particularly bothered whether this is the Pandavas or the Kauravas. Call those who want to be on the stronger side the Hard-heads. Those who want to be on the weaker side are Soft-hearts, .
Suppose the world is divided into Hard-heads and Soft-Hearts.
Can a given number of Hard-heads ever by themselves decide to have a war?
Yes, so long as there are two equally strong feasible adversarial coalitions assuming zero risk aversion. 
Indeed, if coalitions are unstable then there can still be wars between unequal coalitions because there is some chance that sufficient last minute desertions will pull off a big upset with a consequent big pay out for those who bet on the right side.
Similarly a population of Soft-hearts could go to war as could a mixed population of Soft-hearts and Hard-heads.
Call a warrior Hegemonic if he changes any coalition into a strong one by adhering to it.
We have seen that a Hegemonic Soft-heart like Barbarik either kills himself and lets the War proceed or kills all combatants and then himself. But this means only Soft hearts will be killed. Rational Hard-heads will abstain from Combat till Barbarik chops off his own head after which they can fight over the spoils of war.
A Hegemonic Hard-head also only kills Soft hearts but he does not kill himself and thus retains a countervailing power over other hard-heads.

In the Mahabhrarata, Barbarik's decapitated head witnessed the events of the War thanks to a boon from Lord Krishna. 
But this means Barbarik realizes that he could have prevented the great slaughter of the Kurukshetra war- including the killing of his father and uncles- by choosing
1) to mark for destruction, with his first arrow, all those implacably resolved on a war to the death
2) saving all those who would abide by a compromise settlement with his second arrow
3) unleashing his third arrow.

Was Krishna perhaps a tad thoughtless in the boon he granted Barbarik?
Perhaps. But Sacrifice, truly so called, should be a path to truth.
In this sense, Barbarik is a true Martyr, a true Shaheed- both of which words mean Witness. Yet, as the story of Barbarik shows, it is only by hanging around for a while after your death that you get to see the true stupidity of the fucked up principles for which you sacrificed your life and screwed the pooch of eusocial Consilience.


Rajiv said...

Krishna's argument isn't from backward induction.
Wikipedia says 'Krishna tells that whichever side he supports will only make the other side weak due to his power. Nobody will be able to defeat him. Hence, he is forced to support the other side that has become weaker due to his word to his mother. Thus, in an actual war, he will keep oscillating between the two sides, thereby destroying the entire army of both sides and eventually only he remains. Subsequently, none of the side is victorious as he will be the only lone survivor. Hence, Krishna avoids his participation from the war by seeking his head in Charity.'
I also read that Krishna's weak spot in his heel was caused by Barbarika's arrow- I don't recall hearing that before.
I don't really see that there was any real dilemma here of the sort Krishna mentions.
Weaker side means the one with fewer troops (okay, adjusted for quality). Barbarika helps them win. He keeps his vow to his Mom and helps his Dad's side win.
There is no law which says Barbarika has to include himself in the evaluation when deciding which side is weaker.
If a Judge sees that one side has top lawyers and the other side is self-represented, the Judge may give more leeway to the weaker party. The Judge does not add in his own strength to the weaker party because then the other side would become weaker. It's a fine line, but people walk it all the time.

I suppose, since Krishna is God, he knows everything. From his point of view 'weaker side' means something different from the ordinary human meaning. Still, since God himself has taught the human being something- even though it's not within our human ken and no use to us- still the human might well want to offer his own head as payment.
But, that doesn't have anything to do with backward induction or Game theory.

windwheel said...

I think there are a number of interesting Mathematical ways- backward induction being one- in which Krishna's argument could have been presented so as to have had the desired effect. Whether or not Krishna made them does not matter provided Barbarika could have grasped them as a rational human being. I say this because human beings don't necessarily communicate or grasp mathematical arguments by class-room methods.
I've focused on Backward induction because it gives a Muth rational solution. Otherwise we get a cobweb type oscillation which isn't credible because once Barbarika shows his powers no rational being will fight anymore.
In fact, since Barbarika is hegemonic, there can't be a battle unless Krishna meanly and out of partisanship enforces an information asymmetry such that the Kauravas turn up in battle array and are faced with just some decrepit mercenaries so that they rush to their destruction at the hands of Barbarika.
What is interesting is the notion that the head-sacrifice of the best i.e. hegemonic warrior is necessary to consecrate a battlefield. Initially, we would be tempted to see this as some sort of primitive hero-stone cultic practice whereas of course, without the death of the hegemonic (or nearest to hegemonic) warrior the War couldn't have Schelling salience as solving an underlying co-ordination problem. I think backward induction can show this.
Still, I feel sure there's something I'm missing in the story of the 3 arrows. I have a feeling it can link up with the axiom of determinacy as opposed to that of choice such that we get a Monist Banach Tarski type reading of Barbarika as gaining Advaitic Union in Moksha as opposed to the Visishtadvaitic ideal of Viyoga such that permanent servitor or defeated status is gloried in as the outcome of a 2 person game.
Your point is that Krishna treats 'weaker side' in a manner consistent with the axiom of choice whereas human beings don't do this. Instead the Judge, you refer to, is using something like the axiom of determinacy to examine a 2 person game and making a decision about it.
I dunno how the arrows come into it. I think it's got something to do with post Vedic trignometry or something. It couldn't be the continuum hypothesis could it? The Jains had these nondenumerable numbers... dunno.
I iz jus' iggnirint.

windwheel said...

Sorry, the last bit was unclear. I did not mention that I'm thinking of Krishna as having a type of Maths (because he is god) such that mathematical structures have what Godel called positive properties whereas humans are evolved and always have strategies with more complexity than is computable for them. This means what looks stochastic to humans looks like negative complexity to God.
No. I don't understand what I mean either. Nor am likely to do till my next scheduled bender.