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Members of Congress are commonly thought to be more ideologically extreme than even their party's median voter. Recent work notes that "issue delegates" who adopt the electorate's majority position on each issue can be more ideologically extreme than the median voter, but is silent on when they will be. Our answer to the latter question sheds new light on the substantive importance of this "paradox" (Ahler and Broockman 2018). We show that the relationship between the median voter and an issue delegate's ideology is conditional: issue delegates are much more extreme than the median voter in left- and right-leaning electorates, but are not at all more extreme in evenly divided electorates or in simulated electorates containing only ideological purists. We argue that consequently, issue delegate representation has the most potential to amplify polarization and extremism under precisely the conditions that have arisen in contemporary American politics: moderately "sorted" parties, noncompetitive Congressional districts, and an increased focus on appealing to the party base. Two replication studies demonstrate the substantive importance of these new insights: we find that (1) party issue delegates are just as extreme as their Senators in the 109th and 110th Congresses, and (2) party issue delegates polarized at nearly twice the rate of party median voters. We conclude that neglect of the possibility that responsive politicians would engage their constituents' preferences on an issue by issue basis has caused scholars of politics to under-estimate the consistency between elite polarization and electoral preferences.