The Median Voter Model under normal distribution
The median voter model states that the candidate who is closest to the average voter will end up winning the election. Or if it a series of bills in the legislature, then the one closest to the median member will be the one that ends up passing. Consider the chart above is a normal bell curve of the population. There are people on the left (L), people on the right (R), but most people are in the middle near the median (M). Candidate A will win all votes to the left of him (red) while candidate B will win all votes to the right of him (blue). They will split the votes in the middle (grey), but most likely candidate A will win as he is closer to the median voter. It's an argument for centrism.
The problem is that the body politic is usually not a bell curve. The political preferences of the population usually follow the two major parties. Therefore moving away from a bell curve and to one that is bi-modal; that is as long as you remain on a single left-ring axis. Political strategy then becomes to capture all of your voters by getting them out to the polls, while being moderate enough to still capture enough of the votes in the middle.
If both candidates win their party's primary by being closest to the median voter of their party, then they have positioned themselves atop the peak of their distribution. The votes in the middle (the grey area) would be evenly split between the two parties. The end result is that Team Blue would win the general election as their party is slightly bigger than Team Red. (Note the height of the distribution).
Team Red and Blue both have to remain cautious that winning their respective primaries by being moderate does not lead to the more extreme factions of their party breaking off prior to the general election. Third parties usually succeed when there is dissatisfaction with one of the two major parties and/or their candidates. Third parties usually do not, at least initially, succeed as centrist parties, but rather are an outgrowth of a more extreme wing of one of the two major parties.
The Nolan Chart.
There is a problem still where most voters do not fall on a simple left-right axis. A better way to go about it is to separate social (personal) issues from economic issues. By doing so, you add another dimension to political preferences. David Nolan gave birth to his chart in 1970 as a way to express the views of libertarians as neither right nor left nor something in the middle, but rather a completely different attitude towards society.
There are many libertarians out there that believe in small government and free enterprise like traditional conservatives do, but also believe in equality for gays, decriminalization of drugs, and (maybe) affirmative action for minorities if an educational institutional chooses to do so (not state-mandated affirmative action) - more traditional positions for liberals. Basically, Libertarians believe that the government should leave everyone alone.
People will fall all over the place - as few purely subscribe to one ideology or another. You can even do polling over this by adapting The World's Smallest Political Quiz and then adapting the questions to meet qualitative research methodology standards. According to Wikipedia:
In August 2011, the libertarian Reason Magazine worked with the Rupe organization to survey 1,200 Americans by telephone and place their views within Nolan chart based categories. The Reason-Rupe poll found that "Americans cannot easily be bundled into either the 'liberal' or 'conservative' groups". Specifically, 28% expressed conservative views, 24% expressed libertarian views, 20% expressed communitarian views, and 28% expressed liberal views.
A similar Gallup survey in 2011 included possible centrist/moderate responses. That poll reported that 17% expressed conservative views, 22% expressed libertarian views, 20% expressed communitarian views, 17% expressed centrist views, and 24% expressed liberal views.Let's take this one step further.
Instead of just doing a simple summary of what quadrant people fell in like Gallup and Reason did, you can plot the results on the Nolan Chart into a 3rd dimension for population. It would end up looking like 3D topographic map of political beliefs. By doing a stratified sample by state, then adjusting the raw data for race, gender, voting history, and political affiliation - you would have the most detailed rational voter model for elections yet created.
Using some complicated math, you could figure out where the true median voter lies state by state. It may work in predicting the winner of the electoral college. It might explain why certain candidates win their primaries in landslide, but go on to lose the general election by just as much. It could be used for Senate and gubernatorial races as well to help future candidates stake out their future positions.
Just imagine how complicated and interesting this would be for multiparty electoral systems, like that in Canada, Europe, or the rest of the democratic world for that matter.
I see PhD thesis over this....