“My opponent is a no-good, rotten cheater out to destroy America…”

Well, the primary campaign season sure has been heating up. As expected, we see that the candidates are really going after each other, making outlandish remarks, the whole shebang. This isn’t anything new: even back in June, former Minnesota governor Tim Pawlenty accused former Massachusetts governor Mitt Romney of being complicit in the formulation of the Democrats’ controversial health care reform plan, calling it “Obamneycare.” Clearly, with the jam-packed field and a necessity to beat the other contenders to get to the main event, there is a strong incentive for the candidates to go after each other’s throats.

Yet there is a major tradeoff in doing so. All of these candidates would very much prefer, even if they personally do not win the Republican nomination, that one of the other Republicans beat Obama in the general election. But by attacking their competitors, they decrease the chance of that happening.

Let’s assume that a candidate’s chance of winning is proportional to the amount of flak that is not directed at him or her. Thus, if each candidate i (of N total) generates f_{i,j} flak towards candidate j, then each candidate’s share of the flak is \frac{\sum_{j=1}^{N}f_{j,i}}{\sum_{k=1}^{N}\sum_{j=1}^{N}f_{j,k}}

However, by being the victim of more flak, the chances of beating Obama get smaller and smaller: voters are much less likely to vote for a candidate who has a terrible reputation, as bestowed upon him or her by his or her opponents. We can therefore set up a threshold at which the voters will not vote for candidate i: once he or she has taken more collective beatings than threshold F^{*}, Obama automatically wins (we can model Obama’s chances of winning as increasing in the amount of trashing his opponent has received; the main idea of the result will remain the same). Scary thought if you are a Republican, no?

Each candidate gets payoff R for being the nominee, as well as an additional payoff P if he or she wins the general election. The other candidates get payoff 0 if they lose the nomination and the presidency, while they get payoff V if a different Republican candidate wins the general election. Assume that R>V – candidates would rather be the nominee themselves, and get a shot at the presidency, no matter what the other candidates’ chances are.

This being the case, despite the preference to beat Obama, Republican primary candidates can always do better by hacking at each other as much as possible. Think about it – given any fixed amount of flak the other candidates are giving you, your chances of being the nominee go to one as the amount you attack them goes to infinity. This strategy (given finite amount of flak from others) will yield a payoff of approximately R, which we stipulated was greater than V. Hence the Democrats will automatically win.

Note that this isn’t quite a Nash equilibrium, since the set of possible options is not bounded. Yet some of the normal game theory ideas are still visible here: what will end up happening, based on the incentives, is that the Democratic candidate gets re-elected. Nevertheless, the result depends on several assumptions: that the Republicans can attack each other to an arbitrarily large extent, and that they would always rather be the nominee than let someone else win. Still, it is interesting to observe that the situation as modeled here leads to an automatic Republican loss in November 2012. Pretty ironic, given that one would think that the entire purpose of the campaign is to unseat Obama.

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