Chandler Gregg
4/22/13
Math 7 – 13S – Professor Winkler
The Truel
As of yet, my favorite problem in Martin Gardner’s “The Colossal Book of
Short Puzzles and Problems” is the Triangular Duel (Gardner, 240). The problem
goes as follows (paraphrased from Gardner):
Three men, Smith, Brown, and Jones, agree to participate in a
pistol “duel.” The three men decide who goes first by drawing straws
and each man is to wait their turn before firing upon either of the
other men. After the first round of shots fired, the men will proceed in
the same order until there is one man standing. All three men know
that Smith is 100% accurate, Brown is 80% accurate, and Jones is
50% accurate.
Assuming that all three men are intelligent, logical thinkers,
who has the best chance of winning? What is each man’s probability of
surviving?
Let us first consider the first question, as it is significantly easier to answer
than the second. Knowing that Smith is the most accurate and therefore most deadly
dueler, we can assume that Smith would be the first target for Brown and Jones.
Smith knows that Brown is more accurate and deadly than Jones, so Smith would
first take aim at Brown. Furthermore, Jones will not be targeted until either Smith or
Brown is dead, so it is in Jones’s best interest to not interfere with the duel between
Smith and Brown. In summary, Smith would aim at Brown and Brown at Smith;
Jones would refrain from firing until Smith or Brown is killed. Jones is not initially
targeted and therefore has the greatest chance of survival.
The harder problem to answer is each man’s probability of surviving,
although it can be explained with grade-school knowledge of probability.
Let’s start with Smith: Smith has a ½ chance of getting off the first shot in his
duel with Brown (based on who draws a better straw), in which case he will kill
Brown and survive that round. If Brown shoots before Smith (a ½ chance), he has a
1/5 chance of missing, so Smith has a (1/2) * (1/5) chance of surviving. Smith has a
(1/2) + (1/10) = 3/5 chance of surviving the first round of shots. Now Jones will
shoot at Smith, with a ½ chance. If Jones misses, Smith will shoot him and win. We
multiply Smith’s chances for the first round (3/5) with his chances for the second
round (1/2) to obtain 3/10, or 30%.
Brown’s case is more complicated because if he wins the first round, he has
the potential for an indefinitely long exchange of shots with Jones. Because Smith’s
survival chances against Brown are 3/5, Brown’s chances against Smith are 2/5 (1 –
3/5). The math gets hairy between Brown and Jones – I am going to simplify by
providing Gardner’s insight. The duel between Brown and Jones becomes a
repeating decimal of the series from n = 1 to infinity of 4/(10^n) (for those less
mathematically inclined, it looks like 4/10 + 4/100 + 4/1000 . . .). This is equivalent
to 4/9. Multiplying Brown’s first and second round chances, we obtain (2/5) * (4/9)
= 8/45.
To solve for Jones, we could use the same process or simply subtract the
other two probabilities from 1 and obtain 47/90.
Gardner also presents a solution via a graph of a tree, the branches of which
represent different occurrences. You can find this graph on page 268.
Gardner notes that the earliest mention of the Triangular Duel problem is
from “Question Time” by Hubert Phillips, written in 1938 (Gardner, 269) and goes
on to mention other early texts that feature the problem. After doing some research,
I found that the problem was further explored and elaborated upon by economist
Martin Shubik and political scientist/game theorist Steven Brams (Barash). Math
professor D. M. Kilgour has also co-authored papers on the problem with Brams.
Since then, the problem has spawned countless papers and most of the recent ones
use computational techniques to plot solutions when conditions, such as the
accuracy of each man, are changed. These papers use the problem as a jumping-off
point for exploration of game theory, the study of how competing individuals or
groups make intelligent decisions.
Game theory is especially intriguing because it has applications across a large
number of fields and has been studied by mathematicians, economists, politicians,
and scientists. John Nash, a famous game theorist, won the 1994 Nobel Prize in
Economics for his contributions to the field (his life is depicted in the biopic “A
Beautiful Mind”).
The Triangular Duel problem (or “truel” as I have heard it called) is a perfect
example of strategic decision-making, encompassing classic elements of a game
theory problem. The Triangular Duel features multiple competing “players,” and it
has an optimal solution for each player. It can be studied as a microcosm for conflict
between multiple armed parties – warfare and international security, for example.
If you enjoy the Triangular Duel problem, you would also enjoy the Prisoner’s
Dilemma. The Prisoner’s Dilemma is an even more common game theory problem
that is taught in economics and is necessary to understand to delve into oligopoly
(economic markets defined by few, large players). Like the Triangular Duel problem,
the Prisoner’s Dilemma is sure to make you question proper decision-making.
Works Cited
Barash, David. Duels, Truels, and Game Theory Gunslinger Rules. 13 March 2012. 16
April 2013 <http://chronicle.com/blogs/brainstorm/duels-truels-and-game
-theory-gunslinger-rules/44764>.
Gardner, Martin. The Colossal Book of Short Puzzles and Problems. New York: W. W.
Norton & Company, 2006.