The absence of a core means that a majority coalition can never choose a policy that will keep it safe from minority appeals to its pivotal members. In two dimensions, strategic minorities will always be able to offer pivotal voters attractive policy concessions. We argue that this instability of multidimensional politics explains why minorities raise wedge issues and how wedge issues result in partisan realignment in legislative politics. Applying agenda-constrained ideal point estimation techniques to immigration debates, we show that the Reagan coalition—pro-business and social conservatives—has been vulnerable on the wedge issue of immigration and that parties have switched their positions on immigration over the last three decades. We use the uncovered set as the best-fit theoretical solution concept in this legislative environment, to capture the limits of majority rule coalitional possibilities and policy change in the two-dimensional absence of a core.