[ExI] who should white shoot?

[Eric Messick]

Another related question: which character would you rather play in
this scenario?

Assumptions:

  Everyone acts in their own best interest.  As others have already
  pointed out, this means that each person shoots at the best shooter
  left standing at each opportunity.

  White and Brown have an infinite number of bullets, and never give
  up no matter how often they miss each other.  Black, on the other
  hand, has only two bullets, as he never needs any more anyway.

After the first round, there are four sequences which could have
played out:

  White hits Black, Brown hits White, Brown wins (measure 2/9).
  White hits Black, Brown misses White, White and Brown duel in
    round 2 (measure 1/9).
  White misses Black, Brown hits Black, White and Brown duel in
    round 2 (measure 4/9).
  White misses Black, Brown misses Black, Black hits Brown, White
    and Black duel in round 2 (measure 2/9).

The White/Black duel is simpler, so we look at it first:

  White hits Black, White wins (measure 2/27).
  White misses Black, Black hits White, Black wins (measure 4/27).

After round 1, the measure of White and Black both standing is 5/9.
In any given iteration of this (potentially repeated) duel, there are
three sequences possible:

  White hits and wins (measure 1/3).
  White misses, Brown hits and wins (measure 4/9).
  White and Brown both miss, and start again (measure 2/9).

It's the infinite repeat of that last 2/9 that turns the other
measures into sevenths, as spike has pointed out.  So White wins the
two way duel with a measure of 3/7, and Brown with a measure of 4/7.
Since they only got into this situation with a measure of 5/9, their
final measures for these outcomes are 15/63 and 20/63 respectively.

So, White walks away 2/27 + 15/63, or about 31.2% of the time.
    Brown walks away 2/9 + 20/63,  or about 54.0% of the time.
    Black walks away 4/27,         or about 14.8% of the time.

Sucks to be Black, with everyone shooting at you all of the time.
Best to be a pretty good shot, but not known as the best, if you're
going to be getting into these kinds of situations often.

-eric