Let's say the US population breaks down to 50% Democrat and 50% Republican. If I am a D candidate, I want the Ds to show up and the Rs to stay at home and same if I were R. If I defend the Ds too much, the Rs take it offensively and vice versa. Assuming they both use equivalent tactics to harm the other while improving their own popularity, how does that not lead to a Nash equilibrium? BOTH populations are riled enough to support their chosen candidate (and prevent the other from winning) or be apathetic enough to stay at home (and let the other candidate win). With voting, candidates cannot be satisfied with just 50% of those who voted i.e. become a commodity and split it. They need to use the fact that belittling someone has a more visible reaction that just praising themselves. The balancing act that BOTH candidates need to play leads to the Nash equilibrium. But really the fact is that the population is not 50/50 and there are regional/demographic differences which just makes candidates' strategies that much more complicated.
This is incomplete. Negative advertising a) motivates the already-motivated, but more importantly it demotivates the less motivated.
Essentially, your goal as the Democrat is to make the Republican moderate say "a pox on both their houses!"
And yes, it's more complicated than a simple straight line and a fifty-fifty split. We're talking about median positions floating in hyperdimensional issue spaces (the most interesting times in politics are the catastrophic jumps from one local minima to another). But the simplified model has surprising explanatory power.
Edit:
I didn't properly address your argument which, if I read it correctly, is that perfectly symmetrical strategies will cancel out in a perfectly symmetrical race.
That's true, as far as it goes. But the implementation varies, the candidates vary, the electorates vary (especially in the USA where you have gerrymandering -- over here electorates are carved out by an independent commission).
Nevertheless, the beauty-and-bile strategy fits the circumstances better than beauty or bile by itself. It's a minimum viable strategy.
One interesting quirk of the US is that the two main parties have strongly differing strategies with respect to turnout at the state/national level.
The Republicans have a stronger 'base' that is more likely to turn out, while Democrats traditionally capture more undecided voters, but have a smaller base.
What this means is that in elections with lower overall turnout (Non-presidential year house races, for example) tend to favor Republicans, and their game-theoretic response is to generally 'Go after the Base' and swing more conservative (E.g. Reagan, Bush II, adding Palin to the McCain ticket, and the US House races in 2010 and 1994).
Democrats tend to have much more impetus to 'get out the vote' and get undecided voters to show up to the polls, even if they risk adding Republicans to the ranks, so you tend to comparatively more centrist Democrat Candidates (Clinton, Kerry, Obama). They also tend to do better nationally in Presidential Election years with high turnout (Ceteris Paribus).
If you have any interest in how this plays out in the actual nuts & bolts of voter targeting, Hal Malchow's "Political Targeting" (2nd Ed) is the Bible for beginners.
Completely agree. Just to point out the logic you are using, for each factor that you deem relevant (i.e. now bile in addition to beauty), an exhaustive combination of factors will be the minimum viable strategy since it will fit better. I could continue to list additional factors in addition to these two and those additional ones would also be part of the minimum viable strategy.
The point I guess I was making was that this is part of the minimum viable strategy because Hotelling's Law applies only weakly in the voluntary voting case.
The US is far more complex than that with some areas being significantly more Democratic or Republican. Also candidate also choses a point on the political landscape and often needs to beat people within their own party before going after the general election. So there are plenty of Democrats that are far more conservative than elected republicans and vice versa.
This suggests that each race may have a seperate Nash equilibriam based on the canidates and the electrate.
From what I heard gerrymandering means ~2/3 of general elections are fairly uninteresting. But, you still see fairly wide swings in public sentiment leading to a fair amount of churn in public office. Also, in extreme cases people can treat the primary as if it was the general election.
