Moreover, while the winning candidate may have to spend more money or campaign harder to win in a divisive primary, he or she can also benefit from the organizational efforts required to win a tough primary fight. President Obama, for instance, seemed to perform slightly better during the 2008 general election in states that were more competitive during his nomination fight against Mrs. Clinton.

Why, then, is belief in the theory of divisive primaries so pervasive? One factor is the seeming correlation between divisive presidential primaries and general election losses. But vulnerable incumbents tend to attract credible challengers, whereas strong incumbents do not. When researchers take the state of the economy and the approval ratings of the president into account, the relationship disappears.

A variant of this claim is that contested primaries force candidates to the extremes, hurting them in the general election. For instance, a former Clinton strategist, Maria Cardona, suggested to Politico that defeating opponents like Senator Ted Cruz of Texas “is going to pull the Republican Party much more to the right, and we know how that turns out,” adding, “Just ask Mitt Romney.” Democrats and media analysts have offered similar predictions about the effects of Mr. Cruz’s candidacy in recent days.

However, the evidence that divisive primary campaigns increase extremism is also overstated. First, front-runners who avoid divisive primaries often do so precisely because they are near the center of their party ideologically. If they were too moderate (or extreme) for their party, they would be more likely to face a quality opponent.