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.