The incorrect submissions all made the assumption -- incorrect, it turns out -- that each man's best strategy is to shoot at the other man with the better aim. However, as R. McQuaid correctly observes, this is not the best strategy for Sir Soso; in fact, if after drawing lots, Soso is to fire first, his best strategy is to miss on purpose!
In every case, it is in Señor Casisiempre's
best interest to fire at Monsieur Toujours, should Toujours still be a
threat; similarly it is in Toujours best interest to shoot at
Casisiempre.
A straightforward check of probabilities gives the
following table.
Let's see how to compute the values in the table. For
example, in the third row, for Sir Soso to survive Casisiempre must either
kill Toujours, which happens with probability 8/10, then Soso must win his
duel with Casisiempre, which happens with probability 4/9; or Casisiempre
must fail to kill Toujours, which happens with probability 2/10, in which
case Toujours kills Casisiempre, thereby setting up a duel between Soso
and Toujours, which Soso wins with probability 1/2. This gives the total
probability
Firing Order
Survival Probability for
Toujours
Casisiempre
Soso
T - C - S
1/2
0
1/2
T - S - C
1/2
0
1/2
C - T - S
1/10
16/45
49/90
C - S - T
1/10
16/45
49/90
S - T - C
1/2
0
1/2
S - C - T
1/10
16/45
49/90
In order to compute these probabilities, you might start
out by figuring the likelihood of winning each possible two-man duel. The
only tricky duel, and the only one that you need to know for the
computation above, is the one between Soso and Casisiempre. With Soso
firing first, Casisiempre's chance of survival is 4/9, while Soso survives
with a probability of 5/9.
The other calculations are similar.
Since each of the firing orders is equally likely, you determine the final survival probabilities by taking the average of the individual probabilities. This gives the overall survival probabilities to be:
Note that the best shot has the worst chance of survival, while the worst shot has the best. This is due to the fact that being a better shot also makes you a greater threat and consequently a more likely target.
A paper has recently been published on such three-man duels. You can find the article, The Truel, by D. Marc Kilgour and Steven J. Brams, in Mathematics Magazine, vol. 70 (December 1997), pages 315 - 326. You may enjoy looking it up at the library. You might also want to look up an analysis of the situation in which three shooters, A, B, and C (how pedestrian!) shoot and hit with respective probabilities 30%, 100%, 50%; see especially the response by David Seal.
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