Voting for Mr. Ryan’s bill probably did not help many of the 60 or so Republican representatives whose districts were carried by Barack Obama in 2008. Still, if the public regarded the vote as more or less the usual partisan posturing on the budget — Democrats vote one way, Republicans the other — the down side of backing the Ryan plan might have been limited.

Once some Republicans start to defect, though, the public may come to view the bill in a different way. Instead of seeing it as a division between Republicans and Democrats — neither of whom are trusted much on budget issues — voters may instead start to see it as a division between moderate Republicans and extremely conservative ones. Voters who are not steeped in the bill’s particulars may well take that as a signal that it is too extreme, and that the “reasonable” majoritarian position is to oppose the plan.

The bigger problem for the Republicans, though, is a snowball effect: each Republican lawmaker who comes out against the bill makes it a bit less popular — and that in turn increases the incentive for other Republicans to break ranks too. Some Republican House members might be willing to stomach voting for a bill that has the support of 45 percent of the voters in their districts, but if popular support is just 40 percent, or 35 percent, they may throw in the towel. So a feedback loop develops, and one defection begets another.