FOM: Johnson's Irreproducible Result

[JoeShipman]

Sorry, Fred, you picked a bad example.
A should intentionally miss (fire into the air).  The key point is that B and
C 
will fire at each other because each is more dangerous than A.

Start by looking at the simpler case of an A vs. B duel.  Let P be the
probability that A wins if A gets the first shot.  P = 1/3 + (2/3)*((1/3)*P).
Solving we get P=3/7.
If B gets the first shot the probability A wins is (2/3)*0 + (1/3)*P =1/7.

So if A hits B he is dead for sure because C shoots him.  If A hits C he is in
a duel with B in which B shoots first and his chance of surviving is 1/7.  If
A misses there is a 2/3 chance B will hit C, in which case A survives with
P=3/7, and a 1/3 chance B misses, in which case C shoots B and A has one
chance to shoot C (P=1/3) before C shoots him.  So A's overall chance of
survival is (2/3)*(3/7) + 1/9 = 25/63 if he chooses to miss, but only
(1/3)(1/7)+(2/3)(25/63) = 59/189 if he aims at C.  The cost of the wrong
decision is a 16/189 greater probability of dying.

It's really obvious in the version where B and C are perfect shots, then A
doesn't dare kill one of them while the other one is still alive.  There is
quite a large range 
of probabilities in which it makes sense for a player to shoot into the air.

-- Joe Shipman