Muth Rationality and the Pirate's Game

Rationality has to do with making optimal choices by means of a effectively computable calculus. There are other ways to make optimal choices- e.g. 'expert cognition' which is apophatic- and situations where some wholly intuitionistic or oracular process can be shown to yield a better result than any effectively computable method.

What happens when you set up a gedanken where you specifically prohibit the rational choice, yet demand that a rational methods be applied?
The answer is- you get nonsense. Take the following example-

The Pirate's Game. (from Wikipedia)

There are 5 rational pirates, A, B, C, D and E. They find 100 gold coins. They must decide how to distribute them.

The pirates have a strict order of seniority: A is superior to B, who is superior to C, who is superior to D, who is superior to E.

The pirate world's rules of distribution are thus: that the most senior pirate should propose a distribution of coins. The pirates, including the proposer, then vote on whether to accept this distribution. In case of a tie vote the proposer has the casting vote. If the distribution is accepted, the coins are disbursed and the game ends. If not, the proposer is thrown overboard from the pirate ship and dies, and the next most senior pirate makes a new proposal to begin the system again.^[1]

Pirates base their decisions on three factors. First of all, each pirate wants to survive. Second, given survival, each pirate wants to maximize the number of gold coins each receives. Third, each pirate would prefer to throw another overboard, if all other results would otherwise be equal.^[2] The pirates do not trust each other, and will neither make nor honor any promises between pirates apart from a proposed distribution plan that gives a whole number of gold coins to each pirate.

The result

It might be expected intuitively that Pirate A will have to allocate little if any to A for fear of being voted off so that there are fewer pirates to share between. However, this is quite far from the theoretical result.

This is apparent if we work backwards: if all except D and E have been thrown overboard, D proposes 100 for D and 0 for E. D has the casting vote, and so this is the allocation.

If there are three left (C, D and E) C knows that D will offer E 0 in the next round; therefore, C has to offer E 1 coin in this round to win E's vote, and get C's allocation through. Therefore, when only three are left the allocation is C:99, D:0, E:1.

If B, C, D and E remain, B considers being thrown overboard when deciding. To avoid being thrown overboard, B can simply offer 1 to D. Because B has the casting vote, the support only by D is sufficient. Thus B proposes B:99, C:0, D:1, E:0. One might consider proposing B:99, C:0, D:0, E:1, as E knows it won't be possible to get more coins, if any, if E throws B overboard. But, as each pirate is eager to throw each other overboard, E would prefer to kill B, to get the same amount of gold from C.

Assuming A knows all these things, A can count on C and E's support for the following allocation, which is the final solution:

• A: 98 coins
• B: 0 coins
• C: 1 coin
• D: 0 coins
• E: 1 coin^[2]

Also, A:98, B:0, C:0, D:1, E:1 or other variants are not good enough, as D would rather throw A overboard to get the same amount of gold from B.

Is this solution 'Muth rational'? Does it conform to the prediction of the correct economic theory- viz that of Shapley such that the booty will be divided according to fighting ability- i.e. Expected Value of their marginal contribution to any possible victorious coalition?

Clearly not.

This is a contrived paradox demonstrating something everyone already knows.

Induction is useless unless it also applies to the base case.

Here, voting on dividing the booty is irrational- pirates will always gang up to rob and kill any one with a gold coin- so no one is safe if they own gold. Thus Pirate A should offer 0,0,0,0,0 - unless Pirate A's seniority arises by reason of his superior ability to objectively and truthfully estimate each player's marginal product and this is common knowledge. However, in that case, the other rules stipulated are redundant. 

The fact that the Pirates are rankable at all means they must have a Expected Marginal Product based on their contribution to a victorious coalition. This may not be exactly known but it is something the Pirates can thoroughly discuss and, after a few iterations, you have a robust solution because it can incorporate newcomers & deal with deaths or defections.

Not so with the 'official' solution given above. A's proposal is voted down and he is thrown overboard because his solution concept is not robust at all. 

Consider the following scenario-  E is paralyzed completely. Should he vote for the 98/0/1/0/1 solution? No. Because he will be thrown overboard immediately for the sake of his gold.  Suppose D lacks arms and legs but can still roll around biting ankles and causing a minor nuisance . Should he vote for the proposed solution? No. Whoever gets the gold coin won't be him and anyway pirates like throwing each other overboard and there's little resistance he can put up. Suppose C has one leg and thus can deliver one potentially disabling kick but after which he just rolls around uselessly because he has no teeth. Should he vote for the conventional solution? No. He'll be killed for his gold. He votes with D and E and gets the pleasure of seeing A killed. Suppose B has one leg and one arm and can hop around waving a sword but will eventually be bested by A who has all his limbs- though there is a small chance that if D bites A's leg at just the right time and he trips over E, then B can deliver the killing blow.He votes yes, just in case C, D and E defect because only A is strong enough to actually throw anyone off the boat and they like contemplating that spectacle.

Of course, one can change the rules and say 'pirates can only throw people overboard after a vote of this type and are forbidden to steal from each other'. But this is equivalent of saying rationality means people can guess what I want them to do, just from the way I set up a voting game,  without my having to explicitly tell them because I have infinite power over them.' This is silly. Rationality means sometimes rebelling against a tyrant because there is no rule that says you have to follow his rules.

The Pirates' game is not wholly silly. It may have reference to poorly designed computer systems or stupid Indian politicians. But it isn't part of Economics which is about robust ergodic systems based on Muth Rational Expectations- i.e. ordinary people making situational, not trait based, hysteresis led decisions.

One other point, the question of dividing the booty would never be mooted unless it was common knowledge that no hegemonic coalition of a Barbarik type obtained.