Duels, Truels, and Game Theory Gunslinger Rules

Game theory is a branch of mathematics that I both love and hate: I hate it for the license it has provided for a certain breed of Strangelovian “strategic analysts” to shed their humanity in deference to an extended, bloodless exercise in mental masturbation whose presuppositions are frequently far removed from reality and which all too often serve to justify a pre-existing pro-military ideology, and I love it for the opportunity it provides to clarify one’s thinking via a seductive array of logical constructs … that are often reassuringly far removed from reality!

It’s seductive, often wrong, but great fun - enough so that I once wrote a book about it: The Survival Game, how game theory explains the biology of competition and cooperation (Henry Holt).

Game theory is especially relevant to situations in which there are two or more players, where each player has a discrete array of possible options (“moves”) and—this is key—the payoff to each depends not just on what any given player does, but also what the others do. Of the many possible games, the ones most interesting to game theorists, including myself, are Prisoner’s Dilemma and the Game of Chicken. But there are lots of others, and the current Republican Presidential primary festivities bring several to mind. Of these, the simplest is one we might call the Gunslinger Game.

Two gunfighters walk slowly toward each other in the dusty street, á la High Noon (except that Gary Cooper had multiple opponents). The outcome for each clearly depends not only on what he does, but on the other’s action, too. Neither wants to get shot; each wants to shoot the other. There is a payoff to drawing your gun first and shooting your opponent, but since each seeks that same payoff, the situation is unstable, to put it mildly. A frequent outcome: Both draw and each shoots the other.

The game gets more interesting, and no less lethal, if we add this complication: Each still wants to shoot the other, but neither wants to be the one who draws first, because the initiator can legally be seen as the attacker. The problem, of course, is that the attacker is also more likely to be the victor—unless the responder is so fast that he can wait until the attacker attacks, then still fire first. In the real world, most gunfighters are not that confident. Neither are countries, or aspiring presidential candidates, which is why the latter are typically inclined to “go negative,” the equivalent of defecting in a Prisoner’s Dilemma. (Animals, interestingly, are often more restrained. Thus, conflict among the bitterest of rivals typically involves lots of bluff and bluster, but little lethality. Even male rattlesnakes, capable of killing each other by a single bite, fight by seeking to push each other over; they keep their revolvers holstered, although we now understand that Konrad Lorenz exaggerated the degree of nonhuman restraints on violence.)

But human beings have discovered how to arm themselves via technology, and as a result, they confront each other with weaponry that goes beyond the “merely biological.” And as a result, gunslinger-type situations, in which each side fears being pre-empted by the other, are dangerously unstable. Think of the United States and the former Soviet Union during the darkest days of the Cold War. A great fear was that either side (or both) might reason as follows: “I’m not planning to attack, but they might be (or, they evidently think—incorrectly—that we are planning to attack them). As a result, they might well be intending to pre-empt us by attacking first. So, we’d better pre-empt their pre-emption.” In the fateful days leading up to World War I, both the Allies and the Central Powers knew that whoever mobilized first would have an advantage. Germany feared that France and Russia would get the jump; France and Russia had similar anxieties. No wonder there was a war.

A duel, whether between people or countries, can readily become a Prisoner’s Dilemma, with the participants locked in a devilish paradox whose outcome is disadvantageous to both. Dueling gunslingers, or competing candidates, like countries playing brinkmanship, can thus find themselves pushed into shooting each other.

An interesting variant on the duel was first introduced by economist Martin Shubik, and then elaborated by NYU political scientist and game theorist Steven Brams. Call it a truel, since it is a duel for three. Let’s imagine three gunslingers, approaching each other equally, like three angles of an equilateral triangle. Assume further that they each have only one bullet (this isn’t strictly necessary, but makes it easier to analyze). If the trio—call them Moe, Larry, and Curly (or maybe Mitt, Newt and Rick)—are all equally good shots, we might assume that, for example, Moe might shoot at Larry, Larry at Curly, while Curly fires at Moe. (This phenomenon, also known derogatively as a circular firing squad, had long been characteristic of competing Democrats rather than Republicans, the latter being reliably disposed—at least in the past and as befits their ideology—to suck up to power and established hierarchy.)

Alternatively, let’s also suppose that one of the “players,” Curly, is particularly incompetent, the worst shot among the three truelling gunfighters. You might then think that he would be a dead duck, but in fact, the ideal strategy for Curly—and, ironically, for the others—suggests that he has a fighting chance … in fact, the best chance of all!

Curly’s best move is to wait. The two better shots should proceed to fire at each other, since each is the greatest threat to the other. Having refrained, Curly, the worst shot, can then fire with impunity at the survivor. In this way, the poorest shot is paradoxically assured the greatest probability of survival. Think once more, if you haven’t done so already, of the current Republican presidential gunfight, in which each successive front-runner has been consistently knocked out early, as he or she became the target of everyone else’s “best shot.” (More accurately, the victim of their own incompetent proclivity to shoot themselves in the foot, whether via revealed incompetence, a penchant for extramarital dalliance, or a notable indifference to simple facts.) In such cases, it may well be smartest to stay in the middle of the pack, and not make your move until the leaders have damaged each other.

A kind of truel has even been observed among animals, including certain fish. In these cases, large aggressive males—equivalent to hotshot gunslingers—fight it out with each other while small, unobtrusive males take advantage of their mutual preoccupation to sneak in and attempt to fertilize the females.

In an actual human truel with guns, each individual has an even better and more paradoxical option: Fire into the air. A player who did so would no longer be a threat to the others; they would have no reason to shoot at the disarmed individual, and would be more likely, instead, to aim for one another. In fact, given the odds for survival, a disarmament competition might even ensue, in which all parties strive to be the first to waste their bullet, thereby removing themselves as a threat to the others! Alternatively, each ought to refrain from shooting at all, although Reservoir Dogs, directed by Quentin Tarantino, suggests otherwise: In the movie’s climactic scene, three gangsters confront each other in a tense triangle of drawn weapons … and everyone shoots everyone. Maybe it’s just Hollywood. Or maybe more people should study game theory.