It’s easy to look at the final fiscal-cliff vote, which 150-odd Republicans opposed, and assume that only the moderates came around. In fact, the reason Boehner was able to bring it up for a vote at all—in violation of the Hastert Rule—was that the pragmatic conservatives were with him too. That’s not just literally true, in that Boehner gave his troops the option of effectively killing the compromise, and he appeared to get few takers. It’s also true more figuratively, in that Boehner apparently wasn’t worried about pragmatic conservatives turning on him when he stood for re-election as Speaker two days later. “[I]n the end, most of our members wanted this to pass, but they didn’t want to vote for it,” he later told The Wall Street Journal. And the reason they wanted it to pass is that the public would have beat the party senseless for rejecting a deal, having largely accepted Obama’s case that taxes should rise on the wealthy.

Even more encouraging, the cliff deal wasn’t the only example of this dynamic recently. On Tuesday night, Boehner broke the Hastert Rule all over again, bringing up a $50 billion Hurricane Sandy relief bill that only 49 Republicans voted for. As before, pragmatic conservatives were almost certainly behind his decision to do this, even if they didn’t end up supporting the bill. Also as before, it’s almost certainly the case that their views on this were driven by anxieties over public opinion, which was shaped in this instance by high-profile Republicans like Chris Christie.

The lesson in all this for Obama is simple: Don’t bother engaging Republican leaders behind closed doors. The only way to move the leadership is to move pragmatic conservatives. And the only way to move pragmatic conservatives is to arrange it so that the political consequences of siding with the pure conservatives are brutal.