FOM: Johnson's Irreproducible Result
JoeShipman
JoeShipman at aol.com
Sat Jan 17 21:17:02 EST 1998
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
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