Social choice literature on multiwinner elections tends to focus on characterizing two distinct tasks: majoritarian representation and proportional representation. However, many real world systems can be viewed as operating between these two extremes, such as majority jackpot systems used in several countries throughout history. Towards the goal of better understanding these systems from a social choice perspective, we introduce two frameworks that encompass existing desiderata for majoritarian and proportional settings: coalitions, which encompasses properties such as fixed majority and solid coalitions; and stability, which encompasses various generalizations of Condorcet's criterion for single-winner elections, such as Gehrlein stability and local stability, among others. We develop theory and tools that characterize the mathematical and computational aspects of both frameworks across a wide array of preference structures (such as the unrestricted domain over candidates and the restricted domain of party lists). Building on this foundation, we shed new light on majority jackpot electoral systems, which to date have received limited mathematical analysis in the social choice literature.

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