I posted earlier about truels and how the current GOP nomination approximates one. In that post, I laid out the basics of the simple truel (i.e., a three person duel), assuming that the three shooters shoot sequentially. Things can be different when the three shooters shoot simultaneously. Short version: Trump and Rubio aren’t allies, but game theory suggests they should both attack Cruz, in spite of this.
This is arguably a better model for debates than the sequential version, in which candidates prepare extensively prior to debate, largely in ignorance of the other debaters’ preparations. Leaving that interesting question aside, let’s work this out. I assume that the truel lasts until only one shooter is left, and that each shooter wants to live, and is otherwise indifferent. I’ll also assume that the best shooter hits with certainty. The probability that the second-best shooter hits his or her target is 0<p<1 and the probability that the worst shooter hit his or her target is 0<q<p.
When there are two shooters left, each will shoot at the other. Not interesting, but important, because this implies that the worst shooter wants to shoot at the best shooter in the first round. In the first round, both the second-best and worst shooters shoot at the best shooter. Either the first best or second best shooter will be dead after this (if the second-best and worst shooter each get to shoot before the first best shooter, but miss, then the second-best shooter will be killed with certainty). There is also a chance that the worst shooter will win in the first round: the best shooter kills the second-best shooter (probability 1/3), and the worst shooter kills the best shooter (probability q<1).
What does this say about the GOP race? Both Rubio and Trump should be shooting at Cruz. This is a simplistic model, and it ignores a lot of real-world factors. But that’s why it’s valuable, from a social science perspective: if (and when) the behaviors of the three campaigns deviate from this behavior, we know that we need to include those other factors. Until then, you see, in this world there’s two kinds of models, my friend: Those with just enough to capture the logic and those who need to dig for more things to include. We’ll see if this one needs to dig.
With that, I leave you with this.
. For simplicity, I will assume that, if two shooters shoot at each other, then one of them, randomly chosen, will “shoot first” and, if he or she kits, kill the other shooter before he or she fires his or her weapon. Note that, with this assumption, if shooter A knows that shooter B (and only shooter B) is going to shoot at shooter A, then shooter A should definitely shoot at shooter B.
This assumption isn’t as strong as it appears. This is because the truel is already assumed to continue until only one player is left (note that it is impossible for zero shooters to survive, given the tie-breaking assumption).