[ExI] who should white shoot?

[spike]

> Avant, let me clarify the question.  Each player has 
> arbitrarily many bullets.  Any hit anywhere instantly slays 
> the unfortunate gunslinger.  They start with Mr. White, then 
> they take turns with Brown following White and Black to 
> follow Brown, continuing the murderous game until a single 
> prole walks away with two cooling corpses behind... spike 


Given below is what I think is the answer, but I would like if some calculus
hipster would check my numbers, as my own frail skills in McLaurin and
Taylor series have grown tragically feeble with the passing of time.  So
much time that my high school calculus book is written in Sumerian cuneiform
with Roman numerals.  But I think I nailed this one.  Think more if you
wish, then scroll down.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Mr. White maximizes his chances of survival by shooting at no one,
intentionally firing into the air.  Reasoning: if White shoots Brown, then
only one target remains for Mr. Black, who never misses.  So White will not
shoot at Brown.  White will not shoot at Black either, for if he slays
Black, that sets up a duel between Brown and White in which Brown gets the
first shot.  The probability of White winning such a duel is one in 8.
(Calculus hipsters?  Lee?  Alex?)  If on the other hand White gets the first
shot in a White/Brown  duel, White's chances of survival are 3 in 7.  So by
firing into the air, White sets up a Black/Brown duel.  If Black wins that
match, then White has 1 in 3 chance of surviving to old age.  If Brown wins
that duel then White has the previously calculated 3 in 7 chance.  So White
should fire his first shot into the air.

spike