In an absolute sense, it is wise to say that survival is the most important thing to a human being. When it comes down to it, even wealth is second to health- realistically money is of no use when one can’t use it. As such it is not surprising that research has been done in the game-theoretic aspects of dueling, which even if it might not have practical applications in our civilized world, is still interesting in its economic aspect. Historically, duels as ways of settling differences (swords over words) have been very important, with the Alexander Hamilton and Aaron Burr duel being perhaps the most famous for its tragic outcome.
Duels are formalized forms of combat, where two combatants armed with deadly weapons fight each other, often in the presence of their “seconds”. Duels have been around for a long time, and have often been used to settle points of honor or (imagined or real) affronts. According to an article in the Smithsonian Magazine on dueling, duels arise in many forms, including balloon dueling (with each combatant boarding a balloon and attempting to shoot down his opponent’s balloon) and billiard dueling (attempting to beat opponents senseless with billiard balls).
While game theory cannot do much to help in classical two-person duels (rigorous training, skill, bravery and luck will win most duels), there are certain precautions one can take to avoid fatal casualties. One is choosing appropriate dueling rules and conditions, such as halting the proceedings upon first blood and choosing proper weapons. Another one (in the case of using pistols) is to shoot in the air or to delope, implying the opponent is not worth shooting (and possibly averting a fatal conflict).
In the case of duels involving more people, however, besides being more fun because of the richness of choices (whom to shoot, for one), are also mathematically more interesting. Truels (interesting paper from NYU), or duels involving three people, greatly change the rules of the game. Depending on the rules (order of shooting, time frame allowed per shot, number of shots…), individual marksmanship skill (usually mathematically represented with percentage accuracy) and strategy chosen, the results can be greatly differing.
In both An Introduction to Probability and Its Applications by Larsen & Marx and Fifty Challenging Problems in Probability by Mosteller there are problems involving truels. The rules are as follows: 3 duelists A, B and C with different firing accuracies (95%, 50% and 30%) face each other. To make it fair, there is sequential firing, with C firing first, and assuming they survive, then B and finally A. What is C’s best response? The standard textbook response is that combatant C should delope, as B will subsequently engage in a shooting contest with A (as A is the most dangerous one), and C can then fight the remaining weakened opponent, maximizing chances of survival.
The math does not end here, however. According to this article, “Optimal play can be very sensitive to slight changes in the rules, such as the number of rounds of play allowed. At the same time, some findings for truels are quite robust: the weakness of being the best marksman, the fragility of pacts, the possibility that unlimited supplies of ammunition may stabilize rather than undermine cooperation, and the deterrent effect of an indefinite number of rounds of play (which can prevent players from trying to get the last shot). Some of these findings are counterintuitive, even paradoxical.”
There are many more in-depth research papers regarding truels in their variations, as well as duels involving more than 3 combatants, called N-uels. While many of these do involve advanced concepts like Markov Chain Theory, the essential concept remains the same, with Nash Equilibria and probability distributions, and they do make for interesting reads. Who knows, they could save your life someday.