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.[1]  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.[2] 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.

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[1]. 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.

[2]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).

A truel is a three person duel.  There are lots of ways to play this type of thing, but the basic idea is this: three people must each choose which of the other two to try to kill.  They could shoot simultaneously or in sequence.  The details matter…a lot.  I won’t get into the weeds on this, but let’s think about the GOP race following last night’s Iowa caucus results.  By any reasonable accounting, there are three candidates truly standing: Ted Cruz, Marco Rubio, and Donald Trump.  The three of them took, in approximately equal shares, around 75% of the votes cast in the GOP caucus.

The next event is the New Hampshire primary, and the latest polls (all conducted before the Iowa caucus results) have Trump with a commanding lead and Rubio and Cruz essentially tied for (a distant) second.  So, the stage is set.  Who shoots first?  And at whom?

The truel is a useful thought experiment to worm one’s way into the vagaries of this kind of calculus.  A difference between truels and electoral politics is that the key factor in a standard truel is each combatant’s marksmanship, or the probability that he or she will kill an opponent he or she shoots at.  What we typically measure about a candidate is how many survey respondents support him or her.  For the purposes of this post, let’s equate the two.  Trump is the leader, and Rubio and Cruz are about equal.

A relatively robust finding about truels is that, when the shots are fired sequentially (i.e., the combatants take turns), each combatant should fire at the best marksman, regardless of what the other combatants are doing (this is known as a “dominant strategy” in game theory).  Thus, if we think that the campaigns are essentially taking turns (maybe as somewhat randomly awarded by the vagaries of the news cycle and external events), then both Rubio and Cruz should be “shooting at Trump.”  This is in line with Cruz’s post-caucus speech in Iowa last night.

An oddity of this formulation of the truel is that it is possible that the best marksman is the least likely to survive.  This is true even if the best marksman gets to shoot first.

Is it current, or future, popularity? An alternative measurement of marksmanship, however, is not the current support, but the perceived direction of change in support.  After all, marksmanship is about the ability to kill someone on the next shot.

On this front, Rubio is currently the better marksman: his support in Iowa vastly exceeded expectations, while by many accounts (though not necessarily my own), Trump is the worst marksman.  If one buys this alternative measure, then the smart strategy for both Trump and Cruz is to “aim their guns” at Rubio.  We have a week to see who they each aim at.

Of course, a truel is a simplistic picture of what’s going on in the GOP nomination process. In reality, it is probably better to think that each candidate’s marksmanship depends on his (or her) choice of target.  Evidence suggests that it is harder for Trump to “shoot down” Cruz than it was for him to shoot down Bush.  Maybe I’ll come to that later.  For now, I’m still making sense of Santorum’s strategy of heading to South Carolina. For that matter, I’m trying to make sense of him being called “a candidate for President.”

With that, I leave you with this.