There is a very simple location model in economics called Hotelling's Law, which explains that in a one-dimensional space, two firms can each produce the largest market share by locating in the center of the "linear city." I realize that to those of you with a background in economics, this is likely unnecessary, and to those of you without one, this may come off as gibberish. So I'll try to give a brief primer to the latter, and the former can skip down a bit.
Imagine a narrow stretch of beach running east and west. A gentleman — we'll call him Abel — decides to build a tiki bar on the beach. Where is the best place for him to build his tiki bar? In the direct center of the beach, of course. He is equidistant from patrons on the east and west ends of the beach, so he can equally draw patrons from along the beach and maximize the number of customers he serves.
But now a new gentleman — Bert, let's say — decides he is also going to build a tiki bar. Where does he put it? It only makes sense for him to locate it in the center, right next to Abel's, based on the same logic. If he were to move slightly west, he would essentially forfeit all of the customers on the east end of the beach to Abel. He would likely capture all of the customers on the west end of the beach, but that would be a smaller number than the east end as he had already moved away from center, and the patrons in the middle, between the two bars, would be left to choose which bar they wished to walk to. (For the sake of simplicity, we will assume that the bars are the same in quality and that the patrons are equally spread across the beach, and we will restrict our analysis to two bars. The theory gets much more difficult when we mess with those assumptions.)
Even when we add in the wrinkle that, say, Abel and Bert start to build their bars at the same time, this outcome holds true: They will both still locate in the middle of the beach because, though the socially optimal locations would be one a bit farther from center on either side, they essentially forfeit some of the market by locating toward the east or west. This shouldn't come as a surprise to anyone who has seen a Lowe's and Home Depot located across the street from each other or two gas stations side by side or any number of businesses located in what amounts to the same spot in a town. (The theory is complicated by the difference in products they're selling, the number of firms and a more complicated space than in our linear model from before, but clearly, there is still some carryover to the real world.)
This is a long preface to get here: Hotelling's law is often applied to politics to explain why the dominant parties in the American system are really not so different at all. (The smaller parties are so practically irrelevant that we can focus on the two "firms" like in our model from before.) Both parties are, in the scheme of things, located in almost the center of the political spectrum — if they moved toward either extreme (liberal or conservative), they would in effect be sacrificing the moderate voters to the other party. This theory has come up recently, though not in name, in several articles I've seen regarding the Republicans' choice of a presidential nominee; selecting a Tea Party candidate in the primary, the logic goes, would doom the party in the general election as the Democrats would collect all the moderate votes.
This application conforms to theory above in three important points but differs in perhaps the most important way. We satisfy our requirement of a linear space (here, the political spectrum, which we can map as a line with liberal at one end and conservative at the other), two firms (addressed already) and having relatively indistinguishable firms. (To the last point, Republican and Democrat as names carry some weight, but in the end, they're just amorphous vessels for a vote. If Democrats suddenly became the more conservative party, I doubt they'd have many straight-ticket voters carrying over.)
However, we do not have, as in our beach example, have patrons equally spread across our linear space. The majority of voters are moderate, with fewer voters located at either extreme end, more like a bell curve. That seems like it should reinforce the idea that the parties should locate in the center, but there is one more important divergence from theory: Not everyone votes, and it is not random who votes. In fact, we may often assume that the more politically radical citizens are the most engaged in politics and vote the most because they feel they have the most at stake. Indeed, there is a fair amount of evidence that "independents" vote much less frequently and are much less politically knowledgeable than their partisan compatriots.
So here we get to the takeaway point: The goal of politics can be described less as attracting undecided voters (as traditionally depicted) than as mobilizing your party base. With mandatory voting, we would expect parties to stay in the middle, where the most votes would be. But in the case of voluntary voting, it might be beneficial for a party to stray from the center, energizing the more politically engaged voters toward an extreme while giving up on the moderate independents in the middle who, chances are, won't vote anyway. The Republicans, if this were true, could potentially increase their vote share with a Tea Party candidate, contrary to the popular analysis out there. It's a simple twist on the popular logic, though I expect many will be unconvinced. Consider this: For the past several decades, far more Americans have self-identified as Democrats than as Republicans, yet in that time Republicans have won plenty of major national elections (Reagan, Bush I, Bush II). We could also use this analysis to help explain why the parties have become increasingly polarized in recent years.
I have a hard time envisioning a study that could confirm this hypothesis. The ideal experiment would be comparing the elected candidates before and after a country (or community) goes from a mandatory voting to a voluntary voting system (or vice versa). (There are plenty of ways to measure conservatism or liberalism, like ADA scores.) There are limitations to this, though. Perhaps most important, it is much more difficult to assess the liberalism/conservatism of an executive than a legislator, whose voting records provide the primary evidence, and, further, this would restrict the analysis to the relatively local. (Legislators are chosen on behalf of a community, state or city, compared with, say, a presidential election, which draws votes from across the nation.) I expect the analysis would still hold true for a more local election in some fashion, though the small sample size would be problematic, and there is also the complicating factor that communities are much more homogenous (see: Tiebout) than the nation as a whole, in effect compressing the available spectrum to a fraction of the scale of the spectrum of the country (and perhaps depressing the voting frequency among those supporting the minority party.)
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