[ExI] nash equilibrium again

[spike]

A few months ago we got talking about Nash equilibrium here.  Anders was up
to speed on it, so I am hoping he or one of the other math/philosophy
hipsters will be able to untangle this.

 

Scenario: the pirate ship is spotted by the navy, battle ensues, cannonball
thru the pirate ship's hull, blub blub blub, navy is hanging the survivors
from yard arms, but three pirates manage to escape in a lifeboat.  They row
like crazy in the moonless night.  When morning comes, it appears fortune is
with them, for they discover the navy ship is nowhere to be seen, the
lifeboat has drifted ashore near a town, and that the captain hid the loot
in the lifeboat!  Along with three loaded guns!  Each take one gun, the kind
they had in the old days, a huge lead ball driven by an absurd amount of
powder, so if one gets shot with one of those, they are ruined, game over
man, arrrr.

 

They sit on the beach and start to divide the gold doubloons in the usual
way, dealer starts the old one for you, one for you, one for me, repeat a
number of times, but the dealer was a logic professor before she gave it up
for the piracy business.  She knows about Nash equilibria.  The next round
goes one for you, one for you, two for me. Both the other pirates turn their
weapons on her, but she knows neither will fire, for as soon as either one
does, that guy is unarmed with one dead pirate (the dealer) one armed pirate
and a sack of gold coins.  The outcome of that doesn't require a math degree
to predict.  

 

She is right of course, both aim their weapons, neither fires.  So she
continues, one for you, one for you, three for me, and the next time, five
for me, and keeps Fibonnacci-ing for a few rounds, but then abruptly stands
up with the remaining gold, and bids them good day.  She walks away with
most of the gold, knowing neither will shoot, for the same reason as before:
he who shoots, dies by the other.  Nothing personal, it's just business,
it's what pirates do, along with saying arrrr much too often.

 

As she fades in the distance, she hears a shot.  This too is perfectly
logical: the remaining two pirates are not in a Nash equilibrium, not even a
metastable one that existed temporarily with the three of them.  She
quickens her pace, since there is now one living pirate with a few coins.
She and that guy aren't in a Nash equilibrium either.  She isn't motivated
to risk her many coins for a few more, but he is motivated to risk his few
for her many.

 

OK now could we argue that the three pirates were never in a Nash
equilibrium to start with?

 

spike

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